Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 916357 11 pageshttpdxdoiorg1011552013916357

Research ArticleBIBO Stabilization of Discrete-Time Stochastic Control Systemswith Mixed Delays and Nonlinear Perturbations

Xia Zhou1 Yong Ren2 and Shouming Zhong3

1 School of Mathematics and Computational Science Fuyang Teachers College Fuyang 236037 China2Department of Mathematics Anhui Normal University Wuhu 241000 China3 College of Applied Mathematics University of Electronic Science and Technology of China Chengdu Sichuan 611731 China

Correspondence should be addressed to Xia Zhou zhouxia44185163com

Received 19 March 2013 Revised 23 May 2013 Accepted 6 June 2013

Academic Editor Sakthivel Rathinasamy

Copyright copy 2013 Xia Zhou et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The problem of bounded-input bounded-output (BIBO) stabilization in mean square for a class of discrete-time stochasticcontrol systems with mixed time-varying delays and nonlinear perturbations is investigated Some novel delay-dependent stabilityconditions for the previously mentioned system are established by constructing a novel Lyapunov-Krasovskii function Theseconditions are expressed in the forms of linearmatrix inequalities (LMIs) whose feasibility can be easily checked by usingMATLABLMI Toolbox Finally a numerical example is given to illustrate the validity of the obtained results

1 Introduction

Many dynamical systems not only depend on the presentstates but also involve the past ones generally called the time-delay systems Generally as a source of poor or significantlydeteriorated performance and instability for the concernedclosed-loop system the time delays are unavoidable intechnology and nature Many works have been done on thestability of time-delay systems one can see [1ndash23] and thereferences therein The dynamics analysis of continuous-time systems with distributed delay has been well studiedin [9ndash12 20] The aspect of simulation and application incontrol systems whereas discrete-time control systems play amore important role than their continuous-time counterpartsin the practical digital world If one wants to simulateor compute the continuous-time systems it is essential toformulate the discrete-time analogue so as to investigate thedynamical characteristics It is necessary to take continuousdistributed delays into account formodeling realistic systemsfor example neural networks due to the presence of anamount of parallel pathways of a variety of axon sizes andlengths a neural network usually has a spatial nature Veryrecently Liu et al introduced the infinite distributed delayand distributed delay in the form of constant delay into thedelay neural networks See [17ndash19]

In order to track out the reference input signal in realworld the bounded-input bounded-output stabilization hasbeen investigated by many researchers one can see [20ndash32] and the references therein In [22 23] the sufficientconditions for BIBO stabilization of control systems with nodelays were proposed by the Bihari type inequality In [9 10]by employing the parameters technique and the Gronwallinequality the authors investigated the BIBO stability ofthe systems without distributed time delays In [20 27 29]based on Riccati equations and by constructing appropriateLyapunov functions someBIBO stabilization conditions for aclass of delayed control systems with nonlinear perturbationswere established In [30] the BIBO stabilization problemof a class of piecewise switched linear systems was furtherinvestigated It should be pointed out that almost all resultsconcerning the BIBO stability for control systems mainlyconcentrate on continuous-time models Seldom works havebeen done for discrete-time control systems one can see[21 28] In addition the previously mentioned works justconsidered the deterministic systems (see eg [31 32]) Thedeterministic systems often fluctuate due to noise which israndom or at least appears to be soTherefore we must movefrom deterministic problems to stochastic ones So the BIBOstabilization for stochastic control systems case is necessaryand interesting To the best of our knowledge there is no

2 Abstract and Applied Analysis

work reported on the mean square BIBO stabilization forthe discrete-time stochastic control systemswithmixed time-varying delays

It is well known that the classical technique applied in thestudy of stability is based on the Lyapunov direct methodHowever the Lyapunov direct method has some difficultieswith the theory and application to specific problems whilediscussing the stability of solutions in stochastic systems withtime delay In [33] the midpoint in the time delayrsquos variationinterval is introduced and the variation interval is dividedinto two subintervals with equal length by constructing theLyapunov functional which involved midpoint to reduce theconservatism of stability conditions This method was firstproposed to study the stability and stabilization problems forlinear continuous-time systems and then many successfulapplications were found in [13ndash15] In this paper we willreconsider this method by introducing a new piecewise-like delay method given that the point of the time delayrsquosvariation interval is arbitrary point rather than midpoint

Motivated by the aforementioned works in this paperwe investigate BIBO stabilization in mean square for a classof discrete-time stochastic control systems with mixed time-varying delays and nonlinear perturbations Some noveldelay-dependent stability conditions for the previously men-tioned system are derived by constructing a novel Lyapunov-Krasovskii function These conditions are expressed in theforms of linear matrix inequalities whose feasibility can beeasily checked by using MATLAB LMI Toolbox Finally anumerical example is given to illustrate the validity of theobtained results

The paper is organized as follows In Section 2 somenotations and the problem formulation are proposed Themain results are given in Section 3 In Section 4 a numericalexample is given to illustrate the validity of the obtainedtheory results The conclusion is proposed in Section 5

2 Notations and Problem Formulation

Firstly we propose some notations which will be needed inthe sequel The notations are quite standard Let 119877119899 and 119877119899times119898denote respectively the 119899-dimensioned Euclidean space andthe set of all 119899 times119898 real matrices The superscript ldquo119879rdquo denotesthe transpose and the notation 119883 ge 119884 (respective 119883 gt 119884)means that119883 and 119884 are symmetric matrices and that119883minus119884 ispositive semidefinitive (respective positive definite) Let sdot denote the Euclidean norm in 119877119899 let119873+ denote the positiveinteger set and let 119868 be the identity matrix with compatibledimension If119860 is a matrix denote by 119860 its operator normthat is 119860 = sup119860119909 119909 = 1 = radic120582max(119860

119879119860) where120582max(119860) (resp 120582min(119860)) means the largest (resp smallest)value of 119860 Moreover let (ΩF F

119905119905ge0 119875) be a complete

probability space with a filtration F119905119905ge0

satisfying theusual conditions (ie the filtration contains all 119875-null setsand is right continuous) 119864sdot stands for the mathematicalexpectation operator with respect to the given probabilitymeasure 119875 The asterisk lowast in a matrix is used to denote termthat is induced by its symmetry Matrices if not explicitlystate are assumed to have compatible dimensions Denote

119873[119886 119887] = 119886 119886 + 1 119887 Sometimes the arguments of thefunctions will be omitted in the analysis without confusions

In this paper we consider the discrete-time stochasticcontrol systemwithmixed time-varying delays and nonlinearperturbations with the following form

119909 (119896 + 1) = 119860119909 (119896) + 119861119909 (119896 minus 120591 (119896)) + 119862119906 (119896)

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))

+ (119866119909 (119896) + 119867119909 (119896 minus 119889 (119896))) 120596 (119896)

119910 (119896) = 119872119909 (119896)

(1)

where 119909(119896) = [1199091(119896) 1199092(119896) 119909

119899(119896)]119879

isin 119877119899 denotes the

state vector 119906(119896) = [1199061(119896) 1199062(119896) 119906

119898(119896)]119879

isin 119877119898 is the

control input vector 119910(119896) = [1199101(119896) 1199102(119896) 119910

119899(119896)]119879

isin 119877119899 is

the control output vector ℎ(119909(119896)) = [ℎ1(119909(119896)) ℎ

2(119909(119896))

ℎ119899(119909(119896))]

119879

119860 119861 119863 119866 119862 119867 and 119872 represent the weight-ing matrices with appropriate dimension and the positiveintegers 120591(119896) and 119889(119896) are the discrete-time-varying delaydistributed time-varying delay and respectively satisfyingthat

1205911le 120591 (119896) le 120591

2 1198891le 119889 (119896) le 119889

2 119896 isin 119873

+

(2)

with 1205911 1205912 1198891 and 119889

2being four known positive integers For

any given 120591lowast isin (1205911 1205912) 119889lowast isin (119889

1 1198892) The initial conditions of

the system (1) are given by

119909 (119896) = 120601 (119896) 119896 isin [minusmax 1205912 1198892 0] (3)

Thenonlinear vector-valued perturbation119891(119896 119909(119896)119909(119896minus120591(119896))) satisfies that

1003817100381710038171003817119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))1003817100381710038171003817

2

le 1205721119909(119896)

2

+ 1205722119909(119896 minus 120591(119896))

2

(4)

where 1205721and 120572

2are two positive constants 120596(119896) is a scalar

Wiener process defined on (ΩF F119905119905ge0 119875) with

119864 (120596 (119896)) = 0 119864 (120596(119896)2

) = 1

119864 (120596 (119894) 120596 (119895)) = 0 119894 = 119895

(5)

Remark 1 The 120591lowast divides the discrete-time delayrsquos variationinterval into two subintervals that is [120591

1 120591lowast

] and (120591lowast 1205912] and

119889lowast divides the distributed time delayrsquos variation interval into

two subintervals that is [1198891 119889lowast

] and (119889lowast 1198892] We will dis-

cuss the variation of differences of the Lyapunov-Krasovskiifunctional119881(119905 119909(119905)) for each subinterval Compared with theprevious results in the works of [27ndash32] the BIBO stabilityconditions are derived in this paper by checking the variationof119881(119905 119909(119905)) in subintervals rather than in the whole variationinterval of the delays

In what follows we describe the controller with the form

119906 (119896) = 119870119909 (119896) + 119903 (119896) (6)

Abstract and Applied Analysis 3

where119870 is the feedback gain matrix and 119903(119896) is the referenceinput

Assumption 2 For any 1205851 1205852isin 119877 120585

1= 1205852 let

120574minus

119894leℎ119894(1205851) minus ℎ119894(1205852)

1205851minus 1205852

le 120574+

119894 (7)

where 120574minus119894and 120574+119894are known constant scalars

Remark 3 The constants 120574minus119894 120574+119894in Assumption 2 are allowed

to be positive negative or zero Hence the function ℎ(119909(119896))could be nonmonotonic and is more general than the usualsigmoid functions and the recently commonly used Lipschitzconditions

At the end of this section let us introduce some importantdefinitions and lemmas as which will be used in the sequel

Definition 4 (see [28 31]) A vector function 119903(119896) = (1199031(119896)

1199032(119896) 119903

119899(119896))119879 is said to be an element of 119871119899

infin if 119903

infin=

sup119896isin119873[0infin)

119903(119896) lt +infin where sdot denotes the Euclid normin 119877119899 or the norm of a matrix

Definition 5 (see [28 31]) The nonlinear stochastic controlsystem (1) is said to beBIBO stability inmean square if we canconstruct a controller (6) such that the output 119910(119896) satisfiesthat

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198731+ 11987321199032

infin (8)

where1198731and119873

2are two positive constants

Lemma 6 (see [28]) For any given vectors V119894isin 119877119899 119894 = 1

2 119899 the following inequality holds

[

119899

sum

119894=1

V119894]

119879

[

119899

sum

119894=1

V119894] le 119899

119899

sum

119894=1

V119879119894V119894 (9)

Lemma 7 (see [28]) Let 119909 119910 isin 119877119899 and any 119899 times 119899 positive-

definite matrix 119876 gt 0 Then one has

2119909119879

119910 le 119909119879

119876minus1

119909 + 119910119879

119876119910 (10)

Lemma 8 (see [28]) Given the constant matricesΩ1Ω2 and

Ω3with appropriate dimensions where Ω

1= Ω119879

1and Ω

2=

Ω119879

2gt 0 then Ω

1+ Ω119879

3Ωminus1

2Ω3lt 0 if and only if

(Ω1Ω119879

3

lowast minusΩ2

) lt 0 119900119903 (minusΩ2Ω3

lowast Ω1

) lt 0 (11)

3 BIBO Stabilization for the System (1)In this section we aim to establish our main results based onthe LMI approach For the conveniences we denote

Γ1= diag 120574minus

1120574+

1 120574minus

2120574+

2 120574

minus

119899120574+

119899

Γ3= diag 120574+

1 120574+

2 120574

+

119899

Γ2= diag

120574minus

1+ 120574+

1

2120574minus

2+ 120574+

2

2

120574minus

119899+ 120574+

119899

2

119886 = 1205912minus 1205911+ 1

119887 =

(119889lowast

+ 1198891minus 1) (119889

lowast

minus 1198891) + 2119889

lowast

2 1198891le 119889 (119896) le 119889

lowast

(119889lowast

+ 1198892minus 1) (119889

2minus 119889lowast

) + 21198892

2 119889lowast

lt 119889 (119896) le 1198892

119888 = 119889lowast

1198891le 119889 (119896) le 119889

lowast

1198892 119889lowast

lt 119889 (119896) le 1198892

120579 (119896) = 119909 (119896 minus 120591

1) 1205911le 120591 (119896) le 120591

lowast

119909 (119896 minus 1205912) 120591

lowast

lt 120591 (119896) le 1205912

120591 = 120591lowast

minus 1205911 1205911le 120591 (119896) le 120591

lowast

1205912minus 120591lowast

120591lowast

lt 120591 (119896) le 1205912

120573 =

120591 (119896) minus 1205911

120591lowast minus 1205911

1205911le 120591 (119896) le 120591

lowast

1205912minus 120591 (119896)

1205912minus 120591lowast

120591lowast

lt 120591 (119896) le 1205912

(12)

Theorem9 For given positive integers 1205911 12059121198891 and119889

2 under

Assumption 2 the nonlinear discrete-time stochastic controlsystem (1) with the controller (6) is BIBO stabilization inmean square if there exist symmetric positive-definite matrix119875 119877 119876

119894 119894 = 1 2 5 119885

1 1198852 and 119883 with appropriate

dimensional positive-definite diagonal matrices Λ and somepositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868 (13)

Ξ

=

(((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))

)

le 0

(14)

4 Abstract and Applied Analysis

whereΞ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 119866

119879

120582lowast

119866 + 2120591lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860 minus 2119860119879

119883 minus 2119883119879

119860

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861 + 119866119879

120582lowast

119867

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ22= 119867119879

120582lowast

119867 + 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(15)Proof We construct the following Lyapunov-Krasovskiifunction for the system (1)

119881 (119896 119909 (119896)) = 1198811(119896 119909 (119896)) + 119881

2(119896) + 119881

3(119896)

+ 1198814(119896) + 119881

5(119896) + 119881

6(119896)

(16)

where

1198811(119896 119909 (119896)) = 119909

119879

(119896) 119875119909 (119896)

1198812(119896) =

119896minus1

sum

119894=119896minus120591lowast

119909119879

(119894) 1198761119909 (119894)

+

119896minus1

sum

119894=119896minus119889lowast

119909119879

(119894) 1198762119909 (119894)

1198813(119896) =

119896minus1

sum

119894=119896minus120591(119896)

119909119879

(119894) 1198763119909 (119894) +

1205912minus1

sum

119894=1205911

119896minus1

sum

119895=119896minus119894

119909119879

(119895) 1198763119909 (119895)

+

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198763119909 (119894)

+

minus1205911+1

sum

119894=minus1205912+1

119896minus1

sum

119895=119896minus1+119894

119909119879

(119895)1198764119909 (119895)

1198814(119896) = 120591

lowast

minus1

sum

119894=minus120591lowast

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198851120578 (119895)

120578 (119896) = 119909 (119896 + 1) minus 119909 (119896)

1198815(119896) =

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198765119909 (119894) + (120591

lowast

minus 1205911)

times

minus1205911minus1

sum

119894=minus120591lowast

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198852120578 (119895)

1205911le 120591 (119896) le 120591

lowast

119896minus1

sum

119894=119896minus1205912

119909119879

(119894) 1198765119909 (119894) + (120591

2minus 120591lowast

)

times

minus120591lowast

minus1

sum

119894=minus1205912

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198852120578 (119895)

120591lowast

lt 120591 (119896) le 1205912

1198816(119896) =

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

119896minus1

sum

119897=119896+119895

ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897))

1198891le 119889 (119896) le 119889

lowast

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus119889lowast

minus1

sum

119894=minus1198892

minus1

sum

119895=119894+1

119896minus1

sum

119897=119896+119895

ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897))

119889lowast

lt 119889 (119896) le 1198892

(17)

Calculating the difference of 119881(119896 119909(119896)) and taking the math-ematical expectation by Lemma 6 we have

119864Δ1198811(119896 119909 (119896))

= 119864 [119909119879

(119896 + 1) 119875119909 (119896 + 1) minus 119909119879

(119896) 119875119909 (119896)]

= 119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896)]

119864Δ1198812(119896)

= 119864 [ 119909119879

(119896) 1198761119909 (119896) minus 119909

119879

(119896 minus 120591lowast

) 1198761119909 (119896 minus 120591

lowast

)

+119909119879

(119896) 1198762119909 (119896) minus 119909

119879

(119896 minus 119889lowast

) 1198762119909 (119896 minus 119889

lowast

)]

119864Δ1198813(119896)

= 119864[(

119896

sum

119894=119896+1minus120591(119896+1)

minus

119896minus1

sum

119894=119896minus120591(119896)

)119909119879

(119894) 1198763119909 (119894)

+

1205912minus1

sum

119894=1205911

(

119896

sum

119895=119896minus119894+1

minus

119896minus1

sum

119895=119896minus119894

)119909119879

(119895)1198763119909 (119895)

+ (

119896

sum

119894=119896minus1205911+1

minus

119896minus1

sum

119894=119896minus1205911

)119909119879

(119894) 1198763119909 (119894)

+

minus1205911+1

sum

119894=minus1205912+1

(

119896

sum

119895=119896+119894

minus

119896minus1

sum

119895=119896+119895minus1

)

times 119909119879

(119895)1198764119909 (119895) ]

]

Abstract and Applied Analysis 5

le 119864[

[

(

119896

sum

119894=119896+1minus1205912

119909119879

(119894) 1198763119909 (119894) minus

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198763119909 (119894))

+

1205912minus1

sum

119894=1205911

(119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 119894) 1198763119909 (119896 minus 119894))

+ (119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1))

+

minus1205911+1

sum

119894=minus1205912+1

(119909119879

(119896) 1198764119909 (119896) minus 119909

119879

times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]

]

= 119864[

[

119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus 2119909119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1)

minus

119896minus1205911

sum

119894=119896minus1205912

119909119879

(119894) 1198764119909 (119894)]

]

le 119864 [119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus119909119879

(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]

119864Δ1198814(119896)

= 119864[

[

120591lowast

minus1

sum

119894=minus120591lowast

(

119896

sum

119895=119896+119894+1

minus

119896minus1

sum

119895=119896+119894

)120578119879

(119895) 1198851120578 (119895)]

]

= 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896) minus 120591

lowast

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851120578 (119894)]

le 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896)

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)]

(18)

Note that

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)

= (119909 (119896)

119909 (119896 minus 120591lowast

))

119879

(minus11988511198851

lowast minus1198851

)(119909 (119896)

119909 (119896 minus 120591lowast

))

(19)

119864Δ1198815(119896) = 119864 119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) minus 120591120595 (119896)

(20)

where

120595 (119896) =

119896minus1205911minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852120578 (119894) 120591

1le 120591 (119896) le 120591

lowast

119896minus120591lowast

minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894) 120591

lowast

lt 120591 (119896) le 1205912

(21)

When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that

minus 120591120595 (119896)

= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

)]

times

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852120578 (119894)

minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

))]

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894)

le minus120573

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894) minus (1 minus 120573)

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894)

(22)

When 1205911le 120591(119896) le 120591

lowast similarly we can have

minus120591120595 (119896) le minus (1 minus 120573)

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

minus 120573

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

(23)

6 Abstract and Applied Analysis

From (20) (22) and (23) we have

119864Δ1198815(119896) = 119864 [119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) + (

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

119879

times (

minus211988521198852

1198852

lowast minus1198852

0

lowast lowast minus1198852

)(

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

+ 120573(119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

))

119879

(minus11988521198852

lowast minus1198852

)

times (119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)) + (1 minus 120573)

times(119909 (119896 minus 120591 (119896))

120579 (119896))

119879

(minus11988521198852

lowast minus1198852

)

times(119909 (119896 minus 120591 (119896))

120579 (119896))]

(24)

When 1198891le 119889(119896) le 119889

lowast by Lemma 6 it is easy to get

119864Δ1198816(119896) = 119864[

[

(

minus1

sum

119894=minus119889(119896+1)

119896

sum

119895=119896+119894+1

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

)

times ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

(

119896

sum

119897=119896+119895+1

minus

119896minus1

sum

119897=119896+119895

)

times ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897)) ]

]

le 119864[

[

minus1

sum

119894=minus119889lowast

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1

sum

119894=minus119889lowast

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1

sum

119894=minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]

]

le 119864[(119889lowast

+ 1198891minus 1) (119889

lowast

minus 1198891) + 2119889

lowast

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119889lowast(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(25)

When 119889lowast lt 119889(119896) le 1198892 similarly we can have

119864Δ1198816(119896) le 119864[

(119889lowast

+ 1198892minus 1) (119889

1minus 119889lowast

) + 21198892

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

1198892

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(26)

From (25) and (26) we have

119864Δ1198816(119896) le 119864[119887ℎ

119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119888(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(27)

From (7) it follows that

(ℎ119894(119909 (119896)) minus 120574

+

119894119909119894(119896))

times (ℎ119894(119909 (119896)) minus 120574

minus

119894119909119894(119896)) le 0 119894 = 1 2 119899

(28)

which are equivalent to

(119909 (119896)

ℎ (119909 (119896)))

119879

(120574minus

119894120574+

119894119890119894119890119879

119894minus120574minus

119894+ 120574+

119894

2119890119894119890119879

119894

lowast 119890119894119890119879

119894

)

times(119909 (119896)

ℎ (119909 (119896))) le 0

(29)

where 119890119894denotes the unit column vector having one element

on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582

1 1205822

120582119899 gt 0 it follows that

(119909 (119896)

ℎ (119909 (119896)))

119879

(minusΓ1Λ Γ2Λ

lowast minusΛ)(

119909 (119896)

ℎ (119909 (119896))) ge 0 (30)

Abstract and Applied Analysis 7

Note that by Lemma 7 we get

119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896) + 120591lowast2

120578119879

(119896) 1198851120578 (119896)

+120591120578119879

(119896) 1198852120578 (119896)]

= 119864 [120578119879

(119896) (119875 + 120591lowast2

1198851+ 1205911198852) 120578 (119896) + 2120578

119879

(119896) 119875119909 (119896)]

= 119864 [119909119879

(119896 + 1) (119875 + 120591lowast2

1198851+ 1205911198852) 119909 (119896 + 1)

minus 2119909119879

(119896 + 1) (120591lowast2

1198851+ 1205911198852) 119909 (119896)

+ 119909119879

(119896) (120591lowast2

1198851+ 1205911198852minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) (119875 + 2120591lowast2

1198851+ 2120591119885

2) 119909 (119896 + 1)

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) 120582lowast

119868119909 (119896 + 1) + 119909119879

(119896)

times (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879

times 120582lowast

119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

times 120582lowast

119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]

119879

120582lowast

119868

times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

120582lowast

119868

times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

+ 120582lowast

119909119879

(119860 + 119862119870)119879

(119860 + 119862119870) 119909 (119896)

+ 120582lowast

119909119879

(119896 minus 120591 (119896)) 119861119879

119861119909 (119896 minus 120591 (119896))

+ 120582lowast

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

119863119879

119863

times (

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

+5120582lowast

1198622

1199032

infin

(31)

Then from (18) to (31) we have

119864Δ119881 (119896) le 119864120585119879

(119896) [Ξ1015840

+ (radic2119883 0 0 0 0 0 0 0)119879 1

120582lowast

times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)

+ 1205881199032

infin

(32)

where

Ξ1015840

119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582

lowast

1198632

120585119879

(119896) = [119909119879

(119896) 119909119879

(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast

) 119909 (119896 minus 119889lowast

)

120579119879

(119896) ℎ119879

(119909 (119896))

minus1

sum

minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119891119879

]

(33)

If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that

119864Δ119881 (119896) le minus120576119864119909 (119896)2

+ 1205881199032

infin (34)

It is easy to derive that

119864119881 (119896) le 1205831119864119909 (119896)

2

+ 1205832

119896minus1

sum

119894=119896minus1205912

119864119909 (119894)2

+ 1205833

119896minus1

sum

119894=119896minus1198892

119864119909 (119894)2

(35)

8 Abstract and Applied Analysis

with

1205831= 120582max (119875)

1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)

+ 2120582max (1198764) + 4120591lowast2

120582max (1198851)

+ 4(1205912minus 1205911)2

120582max (1198852)

1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)

times (1198892minus 1198891)]

times Γ2

120582max (119877)

(36)

For any 120579 gt 1 it follows from (34) and (35) that

119864 [120579119895+1

119881 (119895 + 1) minus 120579119895

119881 (119895)]

= 120579119895+1

119864Δ119881 (119895) + 120579119895

(120579 minus 1) 119864119881 (119895)

le 120579119895 [

[

(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ (120579 minus 1) 1205832

119895minus1

sum

119894=119895minus1205912

119864119909 (119894)2

+ 1205881205791199032

infin

+ (120579 minus 1) 1205833

119895minus1

sum

119894=119895minus1198892

119864119909 (119894)2]

]

(37)

Summing up both sides of (37) from 0 to 119896minus1 we can obtain

120579119896

119864119881 (119896) minus 119864119881 (0)

le (1205831(120579 minus 1) minus 120576120579)

119896minus1

sum

119895=0

120579119895

1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ 1205832(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

+ 1205833(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

(38)

Also it is easy to compute that

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1205912

119894+1205912

sum

119895=0

+

119896minus1minus1205912

sum

119894=0

119894+1205912

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1205912

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 12059121205791205912 sup119904isin[minus120591

20]

119864119909 (119904)2

+ 12059121205791205912

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1198892

119894+1198892

sum

119895=0

+

119896minus1minus1198892

sum

119894=0

119894+1198892

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1198892

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 11988921205791198892 sup119904isin[minus1198892 0]

119864119909 (119904)2

+ 11988921205791198892

times

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(39)Substituting (39) into (38) leads to

120579119896

119864119881 (119896) minus 119864119881 (0)

le 1205781(120579) sup119904isin[minus1205912 0]

119864119909 (119904)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

+ 1205782(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

+ 1205783(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(40)

where 1205781(120579) = 120583

2(120579 minus 1)120591

21205791205912 + 1205833(120579 minus 1)119889

21205791198892 1205782(120579) = 120583

2(120579 minus

1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578

3(120579) = 120583

3(120579minus1)119889

21205791198892 +1205831(120579minus1)minus120576120579

Since 1205782(1) lt 0 120578

3(1) lt 0 there must exist a positive

1205790gt 1 such that 120578

2(1205790) lt 0 120578

3(1205790) lt 0 Then we have

119864119881 (119896)

le 1205781(1205790) (

1

1205790

)

119896

sup119904isin[minus12059120]

119864119909 (119904)2

+ (1

1205790

)

119896

119864119881 (0) + 120588

119896minus1

sum

119895=0

1

120579119896minus119895minus1

0

1199032

infin

+ 1205782(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

+ 1205783(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+120588

1205790minus 1

1199032

infin

(41)On the other hand by (16) we can get

119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2

(42)Combining (41) with (42) we have

119864119909 (119896)2

le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892

120582min (119875)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+1

120582min (119875)

120588

1205790minus 1

1199032

infin

(43)

Abstract and Applied Analysis 9

Thus

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198722

119864119909 (119896)2

le 1198731+ 11987321199032

infin (44)

where 1198731= 119872

2

((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))

sup119904isin[minusmax120591

211988920]119864119909(119904)

2 1198732

= (1120582min(119875))(120588(1205790 minus

1))1198722 By Definition 5 the nonlinear discrete-time

stochastic control system (1) is BIBO stability in meansquare This completes the proof

If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained

Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889

2

under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876

119894 119894 = 1 2 5119885

11198852 and119883with appropriate

dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868

Ξ

=

(((((((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))))))

)

le 0

(45)

where

Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 120577119868 + 2120591

lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860

minus 2119860119879

119883 minus 2119883119879

119860

Ξ22= 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(46)

Proof The proof is straightforward and hence omitted

Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)

Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay

Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591

1= 1205912

Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves

Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval

4 An Example

In this section a numerical example will be presented to showthe validity of the main results derived in Section 3

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

work reported on the mean square BIBO stabilization forthe discrete-time stochastic control systemswithmixed time-varying delays

It is well known that the classical technique applied in thestudy of stability is based on the Lyapunov direct methodHowever the Lyapunov direct method has some difficultieswith the theory and application to specific problems whilediscussing the stability of solutions in stochastic systems withtime delay In [33] the midpoint in the time delayrsquos variationinterval is introduced and the variation interval is dividedinto two subintervals with equal length by constructing theLyapunov functional which involved midpoint to reduce theconservatism of stability conditions This method was firstproposed to study the stability and stabilization problems forlinear continuous-time systems and then many successfulapplications were found in [13ndash15] In this paper we willreconsider this method by introducing a new piecewise-like delay method given that the point of the time delayrsquosvariation interval is arbitrary point rather than midpoint

Motivated by the aforementioned works in this paperwe investigate BIBO stabilization in mean square for a classof discrete-time stochastic control systems with mixed time-varying delays and nonlinear perturbations Some noveldelay-dependent stability conditions for the previously men-tioned system are derived by constructing a novel Lyapunov-Krasovskii function These conditions are expressed in theforms of linear matrix inequalities whose feasibility can beeasily checked by using MATLAB LMI Toolbox Finally anumerical example is given to illustrate the validity of theobtained results

The paper is organized as follows In Section 2 somenotations and the problem formulation are proposed Themain results are given in Section 3 In Section 4 a numericalexample is given to illustrate the validity of the obtainedtheory results The conclusion is proposed in Section 5

2 Notations and Problem Formulation

Firstly we propose some notations which will be needed inthe sequel The notations are quite standard Let 119877119899 and 119877119899times119898denote respectively the 119899-dimensioned Euclidean space andthe set of all 119899 times119898 real matrices The superscript ldquo119879rdquo denotesthe transpose and the notation 119883 ge 119884 (respective 119883 gt 119884)means that119883 and 119884 are symmetric matrices and that119883minus119884 ispositive semidefinitive (respective positive definite) Let sdot denote the Euclidean norm in 119877119899 let119873+ denote the positiveinteger set and let 119868 be the identity matrix with compatibledimension If119860 is a matrix denote by 119860 its operator normthat is 119860 = sup119860119909 119909 = 1 = radic120582max(119860

119879119860) where120582max(119860) (resp 120582min(119860)) means the largest (resp smallest)value of 119860 Moreover let (ΩF F

119905119905ge0 119875) be a complete

probability space with a filtration F119905119905ge0

satisfying theusual conditions (ie the filtration contains all 119875-null setsand is right continuous) 119864sdot stands for the mathematicalexpectation operator with respect to the given probabilitymeasure 119875 The asterisk lowast in a matrix is used to denote termthat is induced by its symmetry Matrices if not explicitlystate are assumed to have compatible dimensions Denote

119873[119886 119887] = 119886 119886 + 1 119887 Sometimes the arguments of thefunctions will be omitted in the analysis without confusions

In this paper we consider the discrete-time stochasticcontrol systemwithmixed time-varying delays and nonlinearperturbations with the following form

119909 (119896 + 1) = 119860119909 (119896) + 119861119909 (119896 minus 120591 (119896)) + 119862119906 (119896)

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))

+ (119866119909 (119896) + 119867119909 (119896 minus 119889 (119896))) 120596 (119896)

119910 (119896) = 119872119909 (119896)

(1)

where 119909(119896) = [1199091(119896) 1199092(119896) 119909

119899(119896)]119879

isin 119877119899 denotes the

state vector 119906(119896) = [1199061(119896) 1199062(119896) 119906

119898(119896)]119879

isin 119877119898 is the

control input vector 119910(119896) = [1199101(119896) 1199102(119896) 119910

119899(119896)]119879

isin 119877119899 is

the control output vector ℎ(119909(119896)) = [ℎ1(119909(119896)) ℎ

2(119909(119896))

ℎ119899(119909(119896))]

119879

119860 119861 119863 119866 119862 119867 and 119872 represent the weight-ing matrices with appropriate dimension and the positiveintegers 120591(119896) and 119889(119896) are the discrete-time-varying delaydistributed time-varying delay and respectively satisfyingthat

1205911le 120591 (119896) le 120591

2 1198891le 119889 (119896) le 119889

2 119896 isin 119873

+

(2)

with 1205911 1205912 1198891 and 119889

2being four known positive integers For

any given 120591lowast isin (1205911 1205912) 119889lowast isin (119889

1 1198892) The initial conditions of

the system (1) are given by

119909 (119896) = 120601 (119896) 119896 isin [minusmax 1205912 1198892 0] (3)

Thenonlinear vector-valued perturbation119891(119896 119909(119896)119909(119896minus120591(119896))) satisfies that

1003817100381710038171003817119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))1003817100381710038171003817

2

le 1205721119909(119896)

2

+ 1205722119909(119896 minus 120591(119896))

2

(4)

where 1205721and 120572

2are two positive constants 120596(119896) is a scalar

Wiener process defined on (ΩF F119905119905ge0 119875) with

119864 (120596 (119896)) = 0 119864 (120596(119896)2

) = 1

119864 (120596 (119894) 120596 (119895)) = 0 119894 = 119895

(5)

Remark 1 The 120591lowast divides the discrete-time delayrsquos variationinterval into two subintervals that is [120591

1 120591lowast

] and (120591lowast 1205912] and

119889lowast divides the distributed time delayrsquos variation interval into

two subintervals that is [1198891 119889lowast

] and (119889lowast 1198892] We will dis-

cuss the variation of differences of the Lyapunov-Krasovskiifunctional119881(119905 119909(119905)) for each subinterval Compared with theprevious results in the works of [27ndash32] the BIBO stabilityconditions are derived in this paper by checking the variationof119881(119905 119909(119905)) in subintervals rather than in the whole variationinterval of the delays

In what follows we describe the controller with the form

119906 (119896) = 119870119909 (119896) + 119903 (119896) (6)

Abstract and Applied Analysis 3

where119870 is the feedback gain matrix and 119903(119896) is the referenceinput

Assumption 2 For any 1205851 1205852isin 119877 120585

1= 1205852 let

120574minus

119894leℎ119894(1205851) minus ℎ119894(1205852)

1205851minus 1205852

le 120574+

119894 (7)

where 120574minus119894and 120574+119894are known constant scalars

Remark 3 The constants 120574minus119894 120574+119894in Assumption 2 are allowed

to be positive negative or zero Hence the function ℎ(119909(119896))could be nonmonotonic and is more general than the usualsigmoid functions and the recently commonly used Lipschitzconditions

At the end of this section let us introduce some importantdefinitions and lemmas as which will be used in the sequel

Definition 4 (see [28 31]) A vector function 119903(119896) = (1199031(119896)

1199032(119896) 119903

119899(119896))119879 is said to be an element of 119871119899

infin if 119903

infin=

sup119896isin119873[0infin)

119903(119896) lt +infin where sdot denotes the Euclid normin 119877119899 or the norm of a matrix

Definition 5 (see [28 31]) The nonlinear stochastic controlsystem (1) is said to beBIBO stability inmean square if we canconstruct a controller (6) such that the output 119910(119896) satisfiesthat

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198731+ 11987321199032

infin (8)

where1198731and119873

2are two positive constants

Lemma 6 (see [28]) For any given vectors V119894isin 119877119899 119894 = 1

2 119899 the following inequality holds

[

119899

sum

119894=1

V119894]

119879

[

119899

sum

119894=1

V119894] le 119899

119899

sum

119894=1

V119879119894V119894 (9)

Lemma 7 (see [28]) Let 119909 119910 isin 119877119899 and any 119899 times 119899 positive-

definite matrix 119876 gt 0 Then one has

2119909119879

119910 le 119909119879

119876minus1

119909 + 119910119879

119876119910 (10)

Lemma 8 (see [28]) Given the constant matricesΩ1Ω2 and

Ω3with appropriate dimensions where Ω

1= Ω119879

1and Ω

2=

Ω119879

2gt 0 then Ω

1+ Ω119879

3Ωminus1

2Ω3lt 0 if and only if

(Ω1Ω119879

3

lowast minusΩ2

) lt 0 119900119903 (minusΩ2Ω3

lowast Ω1

) lt 0 (11)

3 BIBO Stabilization for the System (1)In this section we aim to establish our main results based onthe LMI approach For the conveniences we denote

Γ1= diag 120574minus

1120574+

1 120574minus

2120574+

2 120574

minus

119899120574+

119899

Γ3= diag 120574+

1 120574+

2 120574

+

119899

Γ2= diag

120574minus

1+ 120574+

1

2120574minus

2+ 120574+

2

2

120574minus

119899+ 120574+

119899

2

119886 = 1205912minus 1205911+ 1

119887 =

(119889lowast

+ 1198891minus 1) (119889

lowast

minus 1198891) + 2119889

lowast

2 1198891le 119889 (119896) le 119889

lowast

(119889lowast

+ 1198892minus 1) (119889

2minus 119889lowast

) + 21198892

2 119889lowast

lt 119889 (119896) le 1198892

119888 = 119889lowast

1198891le 119889 (119896) le 119889

lowast

1198892 119889lowast

lt 119889 (119896) le 1198892

120579 (119896) = 119909 (119896 minus 120591

1) 1205911le 120591 (119896) le 120591

lowast

119909 (119896 minus 1205912) 120591

lowast

lt 120591 (119896) le 1205912

120591 = 120591lowast

minus 1205911 1205911le 120591 (119896) le 120591

lowast

1205912minus 120591lowast

120591lowast

lt 120591 (119896) le 1205912

120573 =

120591 (119896) minus 1205911

120591lowast minus 1205911

1205911le 120591 (119896) le 120591

lowast

1205912minus 120591 (119896)

1205912minus 120591lowast

120591lowast

lt 120591 (119896) le 1205912

(12)

Theorem9 For given positive integers 1205911 12059121198891 and119889

2 under

Assumption 2 the nonlinear discrete-time stochastic controlsystem (1) with the controller (6) is BIBO stabilization inmean square if there exist symmetric positive-definite matrix119875 119877 119876

119894 119894 = 1 2 5 119885

1 1198852 and 119883 with appropriate

dimensional positive-definite diagonal matrices Λ and somepositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868 (13)

Ξ

=

(((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))

)

le 0

(14)

4 Abstract and Applied Analysis

whereΞ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 119866

119879

120582lowast

119866 + 2120591lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860 minus 2119860119879

119883 minus 2119883119879

119860

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861 + 119866119879

120582lowast

119867

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ22= 119867119879

120582lowast

119867 + 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(15)Proof We construct the following Lyapunov-Krasovskiifunction for the system (1)

119881 (119896 119909 (119896)) = 1198811(119896 119909 (119896)) + 119881

2(119896) + 119881

3(119896)

+ 1198814(119896) + 119881

5(119896) + 119881

6(119896)

(16)

where

1198811(119896 119909 (119896)) = 119909

119879

(119896) 119875119909 (119896)

1198812(119896) =

119896minus1

sum

119894=119896minus120591lowast

119909119879

(119894) 1198761119909 (119894)

+

119896minus1

sum

119894=119896minus119889lowast

119909119879

(119894) 1198762119909 (119894)

1198813(119896) =

119896minus1

sum

119894=119896minus120591(119896)

119909119879

(119894) 1198763119909 (119894) +

1205912minus1

sum

119894=1205911

119896minus1

sum

119895=119896minus119894

119909119879

(119895) 1198763119909 (119895)

+

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198763119909 (119894)

+

minus1205911+1

sum

119894=minus1205912+1

119896minus1

sum

119895=119896minus1+119894

119909119879

(119895)1198764119909 (119895)

1198814(119896) = 120591

lowast

minus1

sum

119894=minus120591lowast

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198851120578 (119895)

120578 (119896) = 119909 (119896 + 1) minus 119909 (119896)

1198815(119896) =

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198765119909 (119894) + (120591

lowast

minus 1205911)

times

minus1205911minus1

sum

119894=minus120591lowast

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198852120578 (119895)

1205911le 120591 (119896) le 120591

lowast

119896minus1

sum

119894=119896minus1205912

119909119879

(119894) 1198765119909 (119894) + (120591

2minus 120591lowast

)

times

minus120591lowast

minus1

sum

119894=minus1205912

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198852120578 (119895)

120591lowast

lt 120591 (119896) le 1205912

1198816(119896) =

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

119896minus1

sum

119897=119896+119895

ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897))

1198891le 119889 (119896) le 119889

lowast

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus119889lowast

minus1

sum

119894=minus1198892

minus1

sum

119895=119894+1

119896minus1

sum

119897=119896+119895

ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897))

119889lowast

lt 119889 (119896) le 1198892

(17)

Calculating the difference of 119881(119896 119909(119896)) and taking the math-ematical expectation by Lemma 6 we have

119864Δ1198811(119896 119909 (119896))

= 119864 [119909119879

(119896 + 1) 119875119909 (119896 + 1) minus 119909119879

(119896) 119875119909 (119896)]

= 119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896)]

119864Δ1198812(119896)

= 119864 [ 119909119879

(119896) 1198761119909 (119896) minus 119909

119879

(119896 minus 120591lowast

) 1198761119909 (119896 minus 120591

lowast

)

+119909119879

(119896) 1198762119909 (119896) minus 119909

119879

(119896 minus 119889lowast

) 1198762119909 (119896 minus 119889

lowast

)]

119864Δ1198813(119896)

= 119864[(

119896

sum

119894=119896+1minus120591(119896+1)

minus

119896minus1

sum

119894=119896minus120591(119896)

)119909119879

(119894) 1198763119909 (119894)

+

1205912minus1

sum

119894=1205911

(

119896

sum

119895=119896minus119894+1

minus

119896minus1

sum

119895=119896minus119894

)119909119879

(119895)1198763119909 (119895)

+ (

119896

sum

119894=119896minus1205911+1

minus

119896minus1

sum

119894=119896minus1205911

)119909119879

(119894) 1198763119909 (119894)

+

minus1205911+1

sum

119894=minus1205912+1

(

119896

sum

119895=119896+119894

minus

119896minus1

sum

119895=119896+119895minus1

)

times 119909119879

(119895)1198764119909 (119895) ]

]

Abstract and Applied Analysis 5

le 119864[

[

(

119896

sum

119894=119896+1minus1205912

119909119879

(119894) 1198763119909 (119894) minus

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198763119909 (119894))

+

1205912minus1

sum

119894=1205911

(119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 119894) 1198763119909 (119896 minus 119894))

+ (119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1))

+

minus1205911+1

sum

119894=minus1205912+1

(119909119879

(119896) 1198764119909 (119896) minus 119909

119879

times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]

]

= 119864[

[

119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus 2119909119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1)

minus

119896minus1205911

sum

119894=119896minus1205912

119909119879

(119894) 1198764119909 (119894)]

]

le 119864 [119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus119909119879

(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]

119864Δ1198814(119896)

= 119864[

[

120591lowast

minus1

sum

119894=minus120591lowast

(

119896

sum

119895=119896+119894+1

minus

119896minus1

sum

119895=119896+119894

)120578119879

(119895) 1198851120578 (119895)]

]

= 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896) minus 120591

lowast

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851120578 (119894)]

le 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896)

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)]

(18)

Note that

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)

= (119909 (119896)

119909 (119896 minus 120591lowast

))

119879

(minus11988511198851

lowast minus1198851

)(119909 (119896)

119909 (119896 minus 120591lowast

))

(19)

119864Δ1198815(119896) = 119864 119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) minus 120591120595 (119896)

(20)

where

120595 (119896) =

119896minus1205911minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852120578 (119894) 120591

1le 120591 (119896) le 120591

lowast

119896minus120591lowast

minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894) 120591

lowast

lt 120591 (119896) le 1205912

(21)

When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that

minus 120591120595 (119896)

= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

)]

times

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852120578 (119894)

minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

))]

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894)

le minus120573

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894) minus (1 minus 120573)

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894)

(22)

When 1205911le 120591(119896) le 120591

lowast similarly we can have

minus120591120595 (119896) le minus (1 minus 120573)

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

minus 120573

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

(23)

6 Abstract and Applied Analysis

From (20) (22) and (23) we have

119864Δ1198815(119896) = 119864 [119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) + (

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

119879

times (

minus211988521198852

1198852

lowast minus1198852

0

lowast lowast minus1198852

)(

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

+ 120573(119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

))

119879

(minus11988521198852

lowast minus1198852

)

times (119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)) + (1 minus 120573)

times(119909 (119896 minus 120591 (119896))

120579 (119896))

119879

(minus11988521198852

lowast minus1198852

)

times(119909 (119896 minus 120591 (119896))

120579 (119896))]

(24)

When 1198891le 119889(119896) le 119889

lowast by Lemma 6 it is easy to get

119864Δ1198816(119896) = 119864[

[

(

minus1

sum

119894=minus119889(119896+1)

119896

sum

119895=119896+119894+1

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

)

times ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

(

119896

sum

119897=119896+119895+1

minus

119896minus1

sum

119897=119896+119895

)

times ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897)) ]

]

le 119864[

[

minus1

sum

119894=minus119889lowast

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1

sum

119894=minus119889lowast

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1

sum

119894=minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]

]

le 119864[(119889lowast

+ 1198891minus 1) (119889

lowast

minus 1198891) + 2119889

lowast

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119889lowast(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(25)

When 119889lowast lt 119889(119896) le 1198892 similarly we can have

119864Δ1198816(119896) le 119864[

(119889lowast

+ 1198892minus 1) (119889

1minus 119889lowast

) + 21198892

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

1198892

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(26)

From (25) and (26) we have

119864Δ1198816(119896) le 119864[119887ℎ

119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119888(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(27)

From (7) it follows that

(ℎ119894(119909 (119896)) minus 120574

+

119894119909119894(119896))

times (ℎ119894(119909 (119896)) minus 120574

minus

119894119909119894(119896)) le 0 119894 = 1 2 119899

(28)

which are equivalent to

(119909 (119896)

ℎ (119909 (119896)))

119879

(120574minus

119894120574+

119894119890119894119890119879

119894minus120574minus

119894+ 120574+

119894

2119890119894119890119879

119894

lowast 119890119894119890119879

119894

)

times(119909 (119896)

ℎ (119909 (119896))) le 0

(29)

where 119890119894denotes the unit column vector having one element

on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582

1 1205822

120582119899 gt 0 it follows that

(119909 (119896)

ℎ (119909 (119896)))

119879

(minusΓ1Λ Γ2Λ

lowast minusΛ)(

119909 (119896)

ℎ (119909 (119896))) ge 0 (30)

Abstract and Applied Analysis 7

Note that by Lemma 7 we get

119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896) + 120591lowast2

120578119879

(119896) 1198851120578 (119896)

+120591120578119879

(119896) 1198852120578 (119896)]

= 119864 [120578119879

(119896) (119875 + 120591lowast2

1198851+ 1205911198852) 120578 (119896) + 2120578

119879

(119896) 119875119909 (119896)]

= 119864 [119909119879

(119896 + 1) (119875 + 120591lowast2

1198851+ 1205911198852) 119909 (119896 + 1)

minus 2119909119879

(119896 + 1) (120591lowast2

1198851+ 1205911198852) 119909 (119896)

+ 119909119879

(119896) (120591lowast2

1198851+ 1205911198852minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) (119875 + 2120591lowast2

1198851+ 2120591119885

2) 119909 (119896 + 1)

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) 120582lowast

119868119909 (119896 + 1) + 119909119879

(119896)

times (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879

times 120582lowast

119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

times 120582lowast

119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]

119879

120582lowast

119868

times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

120582lowast

119868

times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

+ 120582lowast

119909119879

(119860 + 119862119870)119879

(119860 + 119862119870) 119909 (119896)

+ 120582lowast

119909119879

(119896 minus 120591 (119896)) 119861119879

119861119909 (119896 minus 120591 (119896))

+ 120582lowast

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

119863119879

119863

times (

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

+5120582lowast

1198622

1199032

infin

(31)

Then from (18) to (31) we have

119864Δ119881 (119896) le 119864120585119879

(119896) [Ξ1015840

+ (radic2119883 0 0 0 0 0 0 0)119879 1

120582lowast

times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)

+ 1205881199032

infin

(32)

where

Ξ1015840

119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582

lowast

1198632

120585119879

(119896) = [119909119879

(119896) 119909119879

(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast

) 119909 (119896 minus 119889lowast

)

120579119879

(119896) ℎ119879

(119909 (119896))

minus1

sum

minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119891119879

]

(33)

If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that

119864Δ119881 (119896) le minus120576119864119909 (119896)2

+ 1205881199032

infin (34)

It is easy to derive that

119864119881 (119896) le 1205831119864119909 (119896)

2

+ 1205832

119896minus1

sum

119894=119896minus1205912

119864119909 (119894)2

+ 1205833

119896minus1

sum

119894=119896minus1198892

119864119909 (119894)2

(35)

8 Abstract and Applied Analysis

with

1205831= 120582max (119875)

1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)

+ 2120582max (1198764) + 4120591lowast2

120582max (1198851)

+ 4(1205912minus 1205911)2

120582max (1198852)

1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)

times (1198892minus 1198891)]

times Γ2

120582max (119877)

(36)

For any 120579 gt 1 it follows from (34) and (35) that

119864 [120579119895+1

119881 (119895 + 1) minus 120579119895

119881 (119895)]

= 120579119895+1

119864Δ119881 (119895) + 120579119895

(120579 minus 1) 119864119881 (119895)

le 120579119895 [

[

(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ (120579 minus 1) 1205832

119895minus1

sum

119894=119895minus1205912

119864119909 (119894)2

+ 1205881205791199032

infin

+ (120579 minus 1) 1205833

119895minus1

sum

119894=119895minus1198892

119864119909 (119894)2]

]

(37)

Summing up both sides of (37) from 0 to 119896minus1 we can obtain

120579119896

119864119881 (119896) minus 119864119881 (0)

le (1205831(120579 minus 1) minus 120576120579)

119896minus1

sum

119895=0

120579119895

1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ 1205832(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

+ 1205833(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

(38)

Also it is easy to compute that

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1205912

119894+1205912

sum

119895=0

+

119896minus1minus1205912

sum

119894=0

119894+1205912

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1205912

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 12059121205791205912 sup119904isin[minus120591

20]

119864119909 (119904)2

+ 12059121205791205912

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1198892

119894+1198892

sum

119895=0

+

119896minus1minus1198892

sum

119894=0

119894+1198892

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1198892

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 11988921205791198892 sup119904isin[minus1198892 0]

119864119909 (119904)2

+ 11988921205791198892

times

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(39)Substituting (39) into (38) leads to

120579119896

119864119881 (119896) minus 119864119881 (0)

le 1205781(120579) sup119904isin[minus1205912 0]

119864119909 (119904)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

+ 1205782(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

+ 1205783(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(40)

where 1205781(120579) = 120583

2(120579 minus 1)120591

21205791205912 + 1205833(120579 minus 1)119889

21205791198892 1205782(120579) = 120583

2(120579 minus

1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578

3(120579) = 120583

3(120579minus1)119889

21205791198892 +1205831(120579minus1)minus120576120579

Since 1205782(1) lt 0 120578

3(1) lt 0 there must exist a positive

1205790gt 1 such that 120578

2(1205790) lt 0 120578

3(1205790) lt 0 Then we have

119864119881 (119896)

le 1205781(1205790) (

1

1205790

)

119896

sup119904isin[minus12059120]

119864119909 (119904)2

+ (1

1205790

)

119896

119864119881 (0) + 120588

119896minus1

sum

119895=0

1

120579119896minus119895minus1

0

1199032

infin

+ 1205782(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

+ 1205783(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+120588

1205790minus 1

1199032

infin

(41)On the other hand by (16) we can get

119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2

(42)Combining (41) with (42) we have

119864119909 (119896)2

le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892

120582min (119875)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+1

120582min (119875)

120588

1205790minus 1

1199032

infin

(43)

Abstract and Applied Analysis 9

Thus

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198722

119864119909 (119896)2

le 1198731+ 11987321199032

infin (44)

where 1198731= 119872

2

((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))

sup119904isin[minusmax120591

211988920]119864119909(119904)

2 1198732

= (1120582min(119875))(120588(1205790 minus

1))1198722 By Definition 5 the nonlinear discrete-time

stochastic control system (1) is BIBO stability in meansquare This completes the proof

If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained

Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889

2

under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876

119894 119894 = 1 2 5119885

11198852 and119883with appropriate

dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868

Ξ

=

(((((((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))))))

)

le 0

(45)

where

Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 120577119868 + 2120591

lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860

minus 2119860119879

119883 minus 2119883119879

119860

Ξ22= 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(46)

Proof The proof is straightforward and hence omitted

Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)

Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay

Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591

1= 1205912

Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves

Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval

4 An Example

In this section a numerical example will be presented to showthe validity of the main results derived in Section 3

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

where119870 is the feedback gain matrix and 119903(119896) is the referenceinput

Assumption 2 For any 1205851 1205852isin 119877 120585

1= 1205852 let

120574minus

119894leℎ119894(1205851) minus ℎ119894(1205852)

1205851minus 1205852

le 120574+

119894 (7)

where 120574minus119894and 120574+119894are known constant scalars

Remark 3 The constants 120574minus119894 120574+119894in Assumption 2 are allowed

to be positive negative or zero Hence the function ℎ(119909(119896))could be nonmonotonic and is more general than the usualsigmoid functions and the recently commonly used Lipschitzconditions

At the end of this section let us introduce some importantdefinitions and lemmas as which will be used in the sequel

Definition 4 (see [28 31]) A vector function 119903(119896) = (1199031(119896)

1199032(119896) 119903

119899(119896))119879 is said to be an element of 119871119899

infin if 119903

infin=

sup119896isin119873[0infin)

119903(119896) lt +infin where sdot denotes the Euclid normin 119877119899 or the norm of a matrix

Definition 5 (see [28 31]) The nonlinear stochastic controlsystem (1) is said to beBIBO stability inmean square if we canconstruct a controller (6) such that the output 119910(119896) satisfiesthat

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198731+ 11987321199032

infin (8)

where1198731and119873

2are two positive constants

Lemma 6 (see [28]) For any given vectors V119894isin 119877119899 119894 = 1

2 119899 the following inequality holds

[

119899

sum

119894=1

V119894]

119879

[

119899

sum

119894=1

V119894] le 119899

119899

sum

119894=1

V119879119894V119894 (9)

Lemma 7 (see [28]) Let 119909 119910 isin 119877119899 and any 119899 times 119899 positive-

definite matrix 119876 gt 0 Then one has

2119909119879

119910 le 119909119879

119876minus1

119909 + 119910119879

119876119910 (10)

Lemma 8 (see [28]) Given the constant matricesΩ1Ω2 and

Ω3with appropriate dimensions where Ω

1= Ω119879

1and Ω

2=

Ω119879

2gt 0 then Ω

1+ Ω119879

3Ωminus1

2Ω3lt 0 if and only if

(Ω1Ω119879

3

lowast minusΩ2

) lt 0 119900119903 (minusΩ2Ω3

lowast Ω1

) lt 0 (11)

3 BIBO Stabilization for the System (1)In this section we aim to establish our main results based onthe LMI approach For the conveniences we denote

Γ1= diag 120574minus

1120574+

1 120574minus

2120574+

2 120574

minus

119899120574+

119899

Γ3= diag 120574+

1 120574+

2 120574

+

119899

Γ2= diag

120574minus

1+ 120574+

1

2120574minus

2+ 120574+

2

2

120574minus

119899+ 120574+

119899

2

119886 = 1205912minus 1205911+ 1

119887 =

(119889lowast

+ 1198891minus 1) (119889

lowast

minus 1198891) + 2119889

lowast

2 1198891le 119889 (119896) le 119889

lowast

(119889lowast

+ 1198892minus 1) (119889

2minus 119889lowast

) + 21198892

2 119889lowast

lt 119889 (119896) le 1198892

119888 = 119889lowast

1198891le 119889 (119896) le 119889

lowast

1198892 119889lowast

lt 119889 (119896) le 1198892

120579 (119896) = 119909 (119896 minus 120591

1) 1205911le 120591 (119896) le 120591

lowast

119909 (119896 minus 1205912) 120591

lowast

lt 120591 (119896) le 1205912

120591 = 120591lowast

minus 1205911 1205911le 120591 (119896) le 120591

lowast

1205912minus 120591lowast

120591lowast

lt 120591 (119896) le 1205912

120573 =

120591 (119896) minus 1205911

120591lowast minus 1205911

1205911le 120591 (119896) le 120591

lowast

1205912minus 120591 (119896)

1205912minus 120591lowast

120591lowast

lt 120591 (119896) le 1205912

(12)

Theorem9 For given positive integers 1205911 12059121198891 and119889

2 under

Assumption 2 the nonlinear discrete-time stochastic controlsystem (1) with the controller (6) is BIBO stabilization inmean square if there exist symmetric positive-definite matrix119875 119877 119876

119894 119894 = 1 2 5 119885

1 1198852 and 119883 with appropriate

dimensional positive-definite diagonal matrices Λ and somepositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868 (13)

Ξ

=

(((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))

)

le 0

(14)

4 Abstract and Applied Analysis

whereΞ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 119866

119879

120582lowast

119866 + 2120591lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860 minus 2119860119879

119883 minus 2119883119879

119860

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861 + 119866119879

120582lowast

119867

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ22= 119867119879

120582lowast

119867 + 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(15)Proof We construct the following Lyapunov-Krasovskiifunction for the system (1)

119881 (119896 119909 (119896)) = 1198811(119896 119909 (119896)) + 119881

2(119896) + 119881

3(119896)

+ 1198814(119896) + 119881

5(119896) + 119881

6(119896)

(16)

where

1198811(119896 119909 (119896)) = 119909

119879

(119896) 119875119909 (119896)

1198812(119896) =

119896minus1

sum

119894=119896minus120591lowast

119909119879

(119894) 1198761119909 (119894)

+

119896minus1

sum

119894=119896minus119889lowast

119909119879

(119894) 1198762119909 (119894)

1198813(119896) =

119896minus1

sum

119894=119896minus120591(119896)

119909119879

(119894) 1198763119909 (119894) +

1205912minus1

sum

119894=1205911

119896minus1

sum

119895=119896minus119894

119909119879

(119895) 1198763119909 (119895)

+

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198763119909 (119894)

+

minus1205911+1

sum

119894=minus1205912+1

119896minus1

sum

119895=119896minus1+119894

119909119879

(119895)1198764119909 (119895)

1198814(119896) = 120591

lowast

minus1

sum

119894=minus120591lowast

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198851120578 (119895)

120578 (119896) = 119909 (119896 + 1) minus 119909 (119896)

1198815(119896) =

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198765119909 (119894) + (120591

lowast

minus 1205911)

times

minus1205911minus1

sum

119894=minus120591lowast

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198852120578 (119895)

1205911le 120591 (119896) le 120591

lowast

119896minus1

sum

119894=119896minus1205912

119909119879

(119894) 1198765119909 (119894) + (120591

2minus 120591lowast

)

times

minus120591lowast

minus1

sum

119894=minus1205912

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198852120578 (119895)

120591lowast

lt 120591 (119896) le 1205912

1198816(119896) =

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

119896minus1

sum

119897=119896+119895

ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897))

1198891le 119889 (119896) le 119889

lowast

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus119889lowast

minus1

sum

119894=minus1198892

minus1

sum

119895=119894+1

119896minus1

sum

119897=119896+119895

ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897))

119889lowast

lt 119889 (119896) le 1198892

(17)

Calculating the difference of 119881(119896 119909(119896)) and taking the math-ematical expectation by Lemma 6 we have

119864Δ1198811(119896 119909 (119896))

= 119864 [119909119879

(119896 + 1) 119875119909 (119896 + 1) minus 119909119879

(119896) 119875119909 (119896)]

= 119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896)]

119864Δ1198812(119896)

= 119864 [ 119909119879

(119896) 1198761119909 (119896) minus 119909

119879

(119896 minus 120591lowast

) 1198761119909 (119896 minus 120591

lowast

)

+119909119879

(119896) 1198762119909 (119896) minus 119909

119879

(119896 minus 119889lowast

) 1198762119909 (119896 minus 119889

lowast

)]

119864Δ1198813(119896)

= 119864[(

119896

sum

119894=119896+1minus120591(119896+1)

minus

119896minus1

sum

119894=119896minus120591(119896)

)119909119879

(119894) 1198763119909 (119894)

+

1205912minus1

sum

119894=1205911

(

119896

sum

119895=119896minus119894+1

minus

119896minus1

sum

119895=119896minus119894

)119909119879

(119895)1198763119909 (119895)

+ (

119896

sum

119894=119896minus1205911+1

minus

119896minus1

sum

119894=119896minus1205911

)119909119879

(119894) 1198763119909 (119894)

+

minus1205911+1

sum

119894=minus1205912+1

(

119896

sum

119895=119896+119894

minus

119896minus1

sum

119895=119896+119895minus1

)

times 119909119879

(119895)1198764119909 (119895) ]

]

Abstract and Applied Analysis 5

le 119864[

[

(

119896

sum

119894=119896+1minus1205912

119909119879

(119894) 1198763119909 (119894) minus

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198763119909 (119894))

+

1205912minus1

sum

119894=1205911

(119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 119894) 1198763119909 (119896 minus 119894))

+ (119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1))

+

minus1205911+1

sum

119894=minus1205912+1

(119909119879

(119896) 1198764119909 (119896) minus 119909

119879

times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]

]

= 119864[

[

119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus 2119909119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1)

minus

119896minus1205911

sum

119894=119896minus1205912

119909119879

(119894) 1198764119909 (119894)]

]

le 119864 [119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus119909119879

(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]

119864Δ1198814(119896)

= 119864[

[

120591lowast

minus1

sum

119894=minus120591lowast

(

119896

sum

119895=119896+119894+1

minus

119896minus1

sum

119895=119896+119894

)120578119879

(119895) 1198851120578 (119895)]

]

= 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896) minus 120591

lowast

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851120578 (119894)]

le 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896)

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)]

(18)

Note that

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)

= (119909 (119896)

119909 (119896 minus 120591lowast

))

119879

(minus11988511198851

lowast minus1198851

)(119909 (119896)

119909 (119896 minus 120591lowast

))

(19)

119864Δ1198815(119896) = 119864 119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) minus 120591120595 (119896)

(20)

where

120595 (119896) =

119896minus1205911minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852120578 (119894) 120591

1le 120591 (119896) le 120591

lowast

119896minus120591lowast

minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894) 120591

lowast

lt 120591 (119896) le 1205912

(21)

When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that

minus 120591120595 (119896)

= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

)]

times

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852120578 (119894)

minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

))]

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894)

le minus120573

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894) minus (1 minus 120573)

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894)

(22)

When 1205911le 120591(119896) le 120591

lowast similarly we can have

minus120591120595 (119896) le minus (1 minus 120573)

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

minus 120573

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

(23)

6 Abstract and Applied Analysis

From (20) (22) and (23) we have

119864Δ1198815(119896) = 119864 [119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) + (

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

119879

times (

minus211988521198852

1198852

lowast minus1198852

0

lowast lowast minus1198852

)(

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

+ 120573(119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

))

119879

(minus11988521198852

lowast minus1198852

)

times (119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)) + (1 minus 120573)

times(119909 (119896 minus 120591 (119896))

120579 (119896))

119879

(minus11988521198852

lowast minus1198852

)

times(119909 (119896 minus 120591 (119896))

120579 (119896))]

(24)

When 1198891le 119889(119896) le 119889

lowast by Lemma 6 it is easy to get

119864Δ1198816(119896) = 119864[

[

(

minus1

sum

119894=minus119889(119896+1)

119896

sum

119895=119896+119894+1

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

)

times ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

(

119896

sum

119897=119896+119895+1

minus

119896minus1

sum

119897=119896+119895

)

times ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897)) ]

]

le 119864[

[

minus1

sum

119894=minus119889lowast

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1

sum

119894=minus119889lowast

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1

sum

119894=minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]

]

le 119864[(119889lowast

+ 1198891minus 1) (119889

lowast

minus 1198891) + 2119889

lowast

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119889lowast(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(25)

When 119889lowast lt 119889(119896) le 1198892 similarly we can have

119864Δ1198816(119896) le 119864[

(119889lowast

+ 1198892minus 1) (119889

1minus 119889lowast

) + 21198892

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

1198892

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(26)

From (25) and (26) we have

119864Δ1198816(119896) le 119864[119887ℎ

119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119888(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(27)

From (7) it follows that

(ℎ119894(119909 (119896)) minus 120574

+

119894119909119894(119896))

times (ℎ119894(119909 (119896)) minus 120574

minus

119894119909119894(119896)) le 0 119894 = 1 2 119899

(28)

which are equivalent to

(119909 (119896)

ℎ (119909 (119896)))

119879

(120574minus

119894120574+

119894119890119894119890119879

119894minus120574minus

119894+ 120574+

119894

2119890119894119890119879

119894

lowast 119890119894119890119879

119894

)

times(119909 (119896)

ℎ (119909 (119896))) le 0

(29)

where 119890119894denotes the unit column vector having one element

on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582

1 1205822

120582119899 gt 0 it follows that

(119909 (119896)

ℎ (119909 (119896)))

119879

(minusΓ1Λ Γ2Λ

lowast minusΛ)(

119909 (119896)

ℎ (119909 (119896))) ge 0 (30)

Abstract and Applied Analysis 7

Note that by Lemma 7 we get

119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896) + 120591lowast2

120578119879

(119896) 1198851120578 (119896)

+120591120578119879

(119896) 1198852120578 (119896)]

= 119864 [120578119879

(119896) (119875 + 120591lowast2

1198851+ 1205911198852) 120578 (119896) + 2120578

119879

(119896) 119875119909 (119896)]

= 119864 [119909119879

(119896 + 1) (119875 + 120591lowast2

1198851+ 1205911198852) 119909 (119896 + 1)

minus 2119909119879

(119896 + 1) (120591lowast2

1198851+ 1205911198852) 119909 (119896)

+ 119909119879

(119896) (120591lowast2

1198851+ 1205911198852minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) (119875 + 2120591lowast2

1198851+ 2120591119885

2) 119909 (119896 + 1)

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) 120582lowast

119868119909 (119896 + 1) + 119909119879

(119896)

times (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879

times 120582lowast

119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

times 120582lowast

119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]

119879

120582lowast

119868

times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

120582lowast

119868

times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

+ 120582lowast

119909119879

(119860 + 119862119870)119879

(119860 + 119862119870) 119909 (119896)

+ 120582lowast

119909119879

(119896 minus 120591 (119896)) 119861119879

119861119909 (119896 minus 120591 (119896))

+ 120582lowast

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

119863119879

119863

times (

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

+5120582lowast

1198622

1199032

infin

(31)

Then from (18) to (31) we have

119864Δ119881 (119896) le 119864120585119879

(119896) [Ξ1015840

+ (radic2119883 0 0 0 0 0 0 0)119879 1

120582lowast

times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)

+ 1205881199032

infin

(32)

where

Ξ1015840

119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582

lowast

1198632

120585119879

(119896) = [119909119879

(119896) 119909119879

(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast

) 119909 (119896 minus 119889lowast

)

120579119879

(119896) ℎ119879

(119909 (119896))

minus1

sum

minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119891119879

]

(33)

If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that

119864Δ119881 (119896) le minus120576119864119909 (119896)2

+ 1205881199032

infin (34)

It is easy to derive that

119864119881 (119896) le 1205831119864119909 (119896)

2

+ 1205832

119896minus1

sum

119894=119896minus1205912

119864119909 (119894)2

+ 1205833

119896minus1

sum

119894=119896minus1198892

119864119909 (119894)2

(35)

8 Abstract and Applied Analysis

with

1205831= 120582max (119875)

1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)

+ 2120582max (1198764) + 4120591lowast2

120582max (1198851)

+ 4(1205912minus 1205911)2

120582max (1198852)

1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)

times (1198892minus 1198891)]

times Γ2

120582max (119877)

(36)

For any 120579 gt 1 it follows from (34) and (35) that

119864 [120579119895+1

119881 (119895 + 1) minus 120579119895

119881 (119895)]

= 120579119895+1

119864Δ119881 (119895) + 120579119895

(120579 minus 1) 119864119881 (119895)

le 120579119895 [

[

(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ (120579 minus 1) 1205832

119895minus1

sum

119894=119895minus1205912

119864119909 (119894)2

+ 1205881205791199032

infin

+ (120579 minus 1) 1205833

119895minus1

sum

119894=119895minus1198892

119864119909 (119894)2]

]

(37)

Summing up both sides of (37) from 0 to 119896minus1 we can obtain

120579119896

119864119881 (119896) minus 119864119881 (0)

le (1205831(120579 minus 1) minus 120576120579)

119896minus1

sum

119895=0

120579119895

1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ 1205832(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

+ 1205833(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

(38)

Also it is easy to compute that

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1205912

119894+1205912

sum

119895=0

+

119896minus1minus1205912

sum

119894=0

119894+1205912

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1205912

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 12059121205791205912 sup119904isin[minus120591

20]

119864119909 (119904)2

+ 12059121205791205912

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1198892

119894+1198892

sum

119895=0

+

119896minus1minus1198892

sum

119894=0

119894+1198892

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1198892

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 11988921205791198892 sup119904isin[minus1198892 0]

119864119909 (119904)2

+ 11988921205791198892

times

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(39)Substituting (39) into (38) leads to

120579119896

119864119881 (119896) minus 119864119881 (0)

le 1205781(120579) sup119904isin[minus1205912 0]

119864119909 (119904)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

+ 1205782(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

+ 1205783(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(40)

where 1205781(120579) = 120583

2(120579 minus 1)120591

21205791205912 + 1205833(120579 minus 1)119889

21205791198892 1205782(120579) = 120583

2(120579 minus

1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578

3(120579) = 120583

3(120579minus1)119889

21205791198892 +1205831(120579minus1)minus120576120579

Since 1205782(1) lt 0 120578

3(1) lt 0 there must exist a positive

1205790gt 1 such that 120578

2(1205790) lt 0 120578

3(1205790) lt 0 Then we have

119864119881 (119896)

le 1205781(1205790) (

1

1205790

)

119896

sup119904isin[minus12059120]

119864119909 (119904)2

+ (1

1205790

)

119896

119864119881 (0) + 120588

119896minus1

sum

119895=0

1

120579119896minus119895minus1

0

1199032

infin

+ 1205782(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

+ 1205783(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+120588

1205790minus 1

1199032

infin

(41)On the other hand by (16) we can get

119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2

(42)Combining (41) with (42) we have

119864119909 (119896)2

le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892

120582min (119875)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+1

120582min (119875)

120588

1205790minus 1

1199032

infin

(43)

Abstract and Applied Analysis 9

Thus

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198722

119864119909 (119896)2

le 1198731+ 11987321199032

infin (44)

where 1198731= 119872

2

((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))

sup119904isin[minusmax120591

211988920]119864119909(119904)

2 1198732

= (1120582min(119875))(120588(1205790 minus

1))1198722 By Definition 5 the nonlinear discrete-time

stochastic control system (1) is BIBO stability in meansquare This completes the proof

If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained

Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889

2

under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876

119894 119894 = 1 2 5119885

11198852 and119883with appropriate

dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868

Ξ

=

(((((((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))))))

)

le 0

(45)

where

Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 120577119868 + 2120591

lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860

minus 2119860119879

119883 minus 2119883119879

119860

Ξ22= 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(46)

Proof The proof is straightforward and hence omitted

Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)

Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay

Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591

1= 1205912

Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves

Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval

4 An Example

In this section a numerical example will be presented to showthe validity of the main results derived in Section 3

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

whereΞ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 119866

119879

120582lowast

119866 + 2120591lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860 minus 2119860119879

119883 minus 2119883119879

119860

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861 + 119866119879

120582lowast

119867

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ22= 119867119879

120582lowast

119867 + 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(15)Proof We construct the following Lyapunov-Krasovskiifunction for the system (1)

119881 (119896 119909 (119896)) = 1198811(119896 119909 (119896)) + 119881

2(119896) + 119881

3(119896)

+ 1198814(119896) + 119881

5(119896) + 119881

6(119896)

(16)

where

1198811(119896 119909 (119896)) = 119909

119879

(119896) 119875119909 (119896)

1198812(119896) =

119896minus1

sum

119894=119896minus120591lowast

119909119879

(119894) 1198761119909 (119894)

+

119896minus1

sum

119894=119896minus119889lowast

119909119879

(119894) 1198762119909 (119894)

1198813(119896) =

119896minus1

sum

119894=119896minus120591(119896)

119909119879

(119894) 1198763119909 (119894) +

1205912minus1

sum

119894=1205911

119896minus1

sum

119895=119896minus119894

119909119879

(119895) 1198763119909 (119895)

+

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198763119909 (119894)

+

minus1205911+1

sum

119894=minus1205912+1

119896minus1

sum

119895=119896minus1+119894

119909119879

(119895)1198764119909 (119895)

1198814(119896) = 120591

lowast

minus1

sum

119894=minus120591lowast

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198851120578 (119895)

120578 (119896) = 119909 (119896 + 1) minus 119909 (119896)

1198815(119896) =

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198765119909 (119894) + (120591

lowast

minus 1205911)

times

minus1205911minus1

sum

119894=minus120591lowast

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198852120578 (119895)

1205911le 120591 (119896) le 120591

lowast

119896minus1

sum

119894=119896minus1205912

119909119879

(119894) 1198765119909 (119894) + (120591

2minus 120591lowast

)

times

minus120591lowast

minus1

sum

119894=minus1205912

119896minus1

sum

119895=119896+119894

120578119879

(119895) 1198852120578 (119895)

120591lowast

lt 120591 (119896) le 1205912

1198816(119896) =

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

119896minus1

sum

119897=119896+119895

ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897))

1198891le 119889 (119896) le 119889

lowast

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus119889lowast

minus1

sum

119894=minus1198892

minus1

sum

119895=119894+1

119896minus1

sum

119897=119896+119895

ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897))

119889lowast

lt 119889 (119896) le 1198892

(17)

Calculating the difference of 119881(119896 119909(119896)) and taking the math-ematical expectation by Lemma 6 we have

119864Δ1198811(119896 119909 (119896))

= 119864 [119909119879

(119896 + 1) 119875119909 (119896 + 1) minus 119909119879

(119896) 119875119909 (119896)]

= 119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896)]

119864Δ1198812(119896)

= 119864 [ 119909119879

(119896) 1198761119909 (119896) minus 119909

119879

(119896 minus 120591lowast

) 1198761119909 (119896 minus 120591

lowast

)

+119909119879

(119896) 1198762119909 (119896) minus 119909

119879

(119896 minus 119889lowast

) 1198762119909 (119896 minus 119889

lowast

)]

119864Δ1198813(119896)

= 119864[(

119896

sum

119894=119896+1minus120591(119896+1)

minus

119896minus1

sum

119894=119896minus120591(119896)

)119909119879

(119894) 1198763119909 (119894)

+

1205912minus1

sum

119894=1205911

(

119896

sum

119895=119896minus119894+1

minus

119896minus1

sum

119895=119896minus119894

)119909119879

(119895)1198763119909 (119895)

+ (

119896

sum

119894=119896minus1205911+1

minus

119896minus1

sum

119894=119896minus1205911

)119909119879

(119894) 1198763119909 (119894)

+

minus1205911+1

sum

119894=minus1205912+1

(

119896

sum

119895=119896+119894

minus

119896minus1

sum

119895=119896+119895minus1

)

times 119909119879

(119895)1198764119909 (119895) ]

]

Abstract and Applied Analysis 5

le 119864[

[

(

119896

sum

119894=119896+1minus1205912

119909119879

(119894) 1198763119909 (119894) minus

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198763119909 (119894))

+

1205912minus1

sum

119894=1205911

(119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 119894) 1198763119909 (119896 minus 119894))

+ (119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1))

+

minus1205911+1

sum

119894=minus1205912+1

(119909119879

(119896) 1198764119909 (119896) minus 119909

119879

times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]

]

= 119864[

[

119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus 2119909119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1)

minus

119896minus1205911

sum

119894=119896minus1205912

119909119879

(119894) 1198764119909 (119894)]

]

le 119864 [119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus119909119879

(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]

119864Δ1198814(119896)

= 119864[

[

120591lowast

minus1

sum

119894=minus120591lowast

(

119896

sum

119895=119896+119894+1

minus

119896minus1

sum

119895=119896+119894

)120578119879

(119895) 1198851120578 (119895)]

]

= 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896) minus 120591

lowast

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851120578 (119894)]

le 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896)

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)]

(18)

Note that

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)

= (119909 (119896)

119909 (119896 minus 120591lowast

))

119879

(minus11988511198851

lowast minus1198851

)(119909 (119896)

119909 (119896 minus 120591lowast

))

(19)

119864Δ1198815(119896) = 119864 119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) minus 120591120595 (119896)

(20)

where

120595 (119896) =

119896minus1205911minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852120578 (119894) 120591

1le 120591 (119896) le 120591

lowast

119896minus120591lowast

minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894) 120591

lowast

lt 120591 (119896) le 1205912

(21)

When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that

minus 120591120595 (119896)

= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

)]

times

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852120578 (119894)

minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

))]

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894)

le minus120573

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894) minus (1 minus 120573)

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894)

(22)

When 1205911le 120591(119896) le 120591

lowast similarly we can have

minus120591120595 (119896) le minus (1 minus 120573)

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

minus 120573

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

(23)

6 Abstract and Applied Analysis

From (20) (22) and (23) we have

119864Δ1198815(119896) = 119864 [119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) + (

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

119879

times (

minus211988521198852

1198852

lowast minus1198852

0

lowast lowast minus1198852

)(

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

+ 120573(119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

))

119879

(minus11988521198852

lowast minus1198852

)

times (119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)) + (1 minus 120573)

times(119909 (119896 minus 120591 (119896))

120579 (119896))

119879

(minus11988521198852

lowast minus1198852

)

times(119909 (119896 minus 120591 (119896))

120579 (119896))]

(24)

When 1198891le 119889(119896) le 119889

lowast by Lemma 6 it is easy to get

119864Δ1198816(119896) = 119864[

[

(

minus1

sum

119894=minus119889(119896+1)

119896

sum

119895=119896+119894+1

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

)

times ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

(

119896

sum

119897=119896+119895+1

minus

119896minus1

sum

119897=119896+119895

)

times ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897)) ]

]

le 119864[

[

minus1

sum

119894=minus119889lowast

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1

sum

119894=minus119889lowast

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1

sum

119894=minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]

]

le 119864[(119889lowast

+ 1198891minus 1) (119889

lowast

minus 1198891) + 2119889

lowast

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119889lowast(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(25)

When 119889lowast lt 119889(119896) le 1198892 similarly we can have

119864Δ1198816(119896) le 119864[

(119889lowast

+ 1198892minus 1) (119889

1minus 119889lowast

) + 21198892

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

1198892

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(26)

From (25) and (26) we have

119864Δ1198816(119896) le 119864[119887ℎ

119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119888(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(27)

From (7) it follows that

(ℎ119894(119909 (119896)) minus 120574

+

119894119909119894(119896))

times (ℎ119894(119909 (119896)) minus 120574

minus

119894119909119894(119896)) le 0 119894 = 1 2 119899

(28)

which are equivalent to

(119909 (119896)

ℎ (119909 (119896)))

119879

(120574minus

119894120574+

119894119890119894119890119879

119894minus120574minus

119894+ 120574+

119894

2119890119894119890119879

119894

lowast 119890119894119890119879

119894

)

times(119909 (119896)

ℎ (119909 (119896))) le 0

(29)

where 119890119894denotes the unit column vector having one element

on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582

1 1205822

120582119899 gt 0 it follows that

(119909 (119896)

ℎ (119909 (119896)))

119879

(minusΓ1Λ Γ2Λ

lowast minusΛ)(

119909 (119896)

ℎ (119909 (119896))) ge 0 (30)

Abstract and Applied Analysis 7

Note that by Lemma 7 we get

119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896) + 120591lowast2

120578119879

(119896) 1198851120578 (119896)

+120591120578119879

(119896) 1198852120578 (119896)]

= 119864 [120578119879

(119896) (119875 + 120591lowast2

1198851+ 1205911198852) 120578 (119896) + 2120578

119879

(119896) 119875119909 (119896)]

= 119864 [119909119879

(119896 + 1) (119875 + 120591lowast2

1198851+ 1205911198852) 119909 (119896 + 1)

minus 2119909119879

(119896 + 1) (120591lowast2

1198851+ 1205911198852) 119909 (119896)

+ 119909119879

(119896) (120591lowast2

1198851+ 1205911198852minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) (119875 + 2120591lowast2

1198851+ 2120591119885

2) 119909 (119896 + 1)

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) 120582lowast

119868119909 (119896 + 1) + 119909119879

(119896)

times (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879

times 120582lowast

119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

times 120582lowast

119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]

119879

120582lowast

119868

times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

120582lowast

119868

times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

+ 120582lowast

119909119879

(119860 + 119862119870)119879

(119860 + 119862119870) 119909 (119896)

+ 120582lowast

119909119879

(119896 minus 120591 (119896)) 119861119879

119861119909 (119896 minus 120591 (119896))

+ 120582lowast

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

119863119879

119863

times (

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

+5120582lowast

1198622

1199032

infin

(31)

Then from (18) to (31) we have

119864Δ119881 (119896) le 119864120585119879

(119896) [Ξ1015840

+ (radic2119883 0 0 0 0 0 0 0)119879 1

120582lowast

times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)

+ 1205881199032

infin

(32)

where

Ξ1015840

119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582

lowast

1198632

120585119879

(119896) = [119909119879

(119896) 119909119879

(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast

) 119909 (119896 minus 119889lowast

)

120579119879

(119896) ℎ119879

(119909 (119896))

minus1

sum

minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119891119879

]

(33)

If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that

119864Δ119881 (119896) le minus120576119864119909 (119896)2

+ 1205881199032

infin (34)

It is easy to derive that

119864119881 (119896) le 1205831119864119909 (119896)

2

+ 1205832

119896minus1

sum

119894=119896minus1205912

119864119909 (119894)2

+ 1205833

119896minus1

sum

119894=119896minus1198892

119864119909 (119894)2

(35)

8 Abstract and Applied Analysis

with

1205831= 120582max (119875)

1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)

+ 2120582max (1198764) + 4120591lowast2

120582max (1198851)

+ 4(1205912minus 1205911)2

120582max (1198852)

1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)

times (1198892minus 1198891)]

times Γ2

120582max (119877)

(36)

For any 120579 gt 1 it follows from (34) and (35) that

119864 [120579119895+1

119881 (119895 + 1) minus 120579119895

119881 (119895)]

= 120579119895+1

119864Δ119881 (119895) + 120579119895

(120579 minus 1) 119864119881 (119895)

le 120579119895 [

[

(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ (120579 minus 1) 1205832

119895minus1

sum

119894=119895minus1205912

119864119909 (119894)2

+ 1205881205791199032

infin

+ (120579 minus 1) 1205833

119895minus1

sum

119894=119895minus1198892

119864119909 (119894)2]

]

(37)

Summing up both sides of (37) from 0 to 119896minus1 we can obtain

120579119896

119864119881 (119896) minus 119864119881 (0)

le (1205831(120579 minus 1) minus 120576120579)

119896minus1

sum

119895=0

120579119895

1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ 1205832(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

+ 1205833(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

(38)

Also it is easy to compute that

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1205912

119894+1205912

sum

119895=0

+

119896minus1minus1205912

sum

119894=0

119894+1205912

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1205912

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 12059121205791205912 sup119904isin[minus120591

20]

119864119909 (119904)2

+ 12059121205791205912

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1198892

119894+1198892

sum

119895=0

+

119896minus1minus1198892

sum

119894=0

119894+1198892

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1198892

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 11988921205791198892 sup119904isin[minus1198892 0]

119864119909 (119904)2

+ 11988921205791198892

times

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(39)Substituting (39) into (38) leads to

120579119896

119864119881 (119896) minus 119864119881 (0)

le 1205781(120579) sup119904isin[minus1205912 0]

119864119909 (119904)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

+ 1205782(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

+ 1205783(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(40)

where 1205781(120579) = 120583

2(120579 minus 1)120591

21205791205912 + 1205833(120579 minus 1)119889

21205791198892 1205782(120579) = 120583

2(120579 minus

1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578

3(120579) = 120583

3(120579minus1)119889

21205791198892 +1205831(120579minus1)minus120576120579

Since 1205782(1) lt 0 120578

3(1) lt 0 there must exist a positive

1205790gt 1 such that 120578

2(1205790) lt 0 120578

3(1205790) lt 0 Then we have

119864119881 (119896)

le 1205781(1205790) (

1

1205790

)

119896

sup119904isin[minus12059120]

119864119909 (119904)2

+ (1

1205790

)

119896

119864119881 (0) + 120588

119896minus1

sum

119895=0

1

120579119896minus119895minus1

0

1199032

infin

+ 1205782(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

+ 1205783(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+120588

1205790minus 1

1199032

infin

(41)On the other hand by (16) we can get

119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2

(42)Combining (41) with (42) we have

119864119909 (119896)2

le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892

120582min (119875)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+1

120582min (119875)

120588

1205790minus 1

1199032

infin

(43)

Abstract and Applied Analysis 9

Thus

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198722

119864119909 (119896)2

le 1198731+ 11987321199032

infin (44)

where 1198731= 119872

2

((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))

sup119904isin[minusmax120591

211988920]119864119909(119904)

2 1198732

= (1120582min(119875))(120588(1205790 minus

1))1198722 By Definition 5 the nonlinear discrete-time

stochastic control system (1) is BIBO stability in meansquare This completes the proof

If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained

Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889

2

under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876

119894 119894 = 1 2 5119885

11198852 and119883with appropriate

dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868

Ξ

=

(((((((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))))))

)

le 0

(45)

where

Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 120577119868 + 2120591

lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860

minus 2119860119879

119883 minus 2119883119879

119860

Ξ22= 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(46)

Proof The proof is straightforward and hence omitted

Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)

Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay

Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591

1= 1205912

Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves

Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval

4 An Example

In this section a numerical example will be presented to showthe validity of the main results derived in Section 3

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

le 119864[

[

(

119896

sum

119894=119896+1minus1205912

119909119879

(119894) 1198763119909 (119894) minus

119896minus1

sum

119894=119896minus1205911

119909119879

(119894) 1198763119909 (119894))

+

1205912minus1

sum

119894=1205911

(119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 119894) 1198763119909 (119896 minus 119894))

+ (119909119879

(119896) 1198763119909 (119896) minus 119909

119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1))

+

minus1205911+1

sum

119894=minus1205912+1

(119909119879

(119896) 1198764119909 (119896) minus 119909

119879

times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]

]

= 119864[

[

119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus 2119909119879

(119896 minus 1205911) 1198763119909 (119896 minus 120591

1)

minus

119896minus1205911

sum

119894=119896minus1205912

119909119879

(119894) 1198764119909 (119894)]

]

le 119864 [119909119879

(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)

minus119909119879

(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]

119864Δ1198814(119896)

= 119864[

[

120591lowast

minus1

sum

119894=minus120591lowast

(

119896

sum

119895=119896+119894+1

minus

119896minus1

sum

119895=119896+119894

)120578119879

(119895) 1198851120578 (119895)]

]

= 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896) minus 120591

lowast

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851120578 (119894)]

le 119864[120591lowast2

120578119879

(119896) 1198851120578 (119896)

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)]

(18)

Note that

minus

119896minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198851

119896minus1

sum

119894=119896minus120591lowast

120578 (119894)

= (119909 (119896)

119909 (119896 minus 120591lowast

))

119879

(minus11988511198851

lowast minus1198851

)(119909 (119896)

119909 (119896 minus 120591lowast

))

(19)

119864Δ1198815(119896) = 119864 119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) minus 120591120595 (119896)

(20)

where

120595 (119896) =

119896minus1205911minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852120578 (119894) 120591

1le 120591 (119896) le 120591

lowast

119896minus120591lowast

minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894) 120591

lowast

lt 120591 (119896) le 1205912

(21)

When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that

minus 120591120595 (119896)

= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

)]

times

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852120578 (119894)

minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591

lowast

))]

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852120578 (119894)

le minus120573

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591lowast

minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894) minus (1 minus 120573)

times

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus1205912

120578 (119894)

(22)

When 1205911le 120591(119896) le 120591

lowast similarly we can have

minus120591120595 (119896) le minus (1 minus 120573)

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus1205911minus1

sum

119894=119896minus120591(119896)

120578119879

(119894) 1198852

119896minus120591minus1

sum

119894=119896minus120591(119896)

120578 (119894)

minus

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

minus 120573

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578119879

(119894) 1198852

119896minus120591(119896)minus1

sum

119894=119896minus120591lowast

120578 (119894)

(23)

6 Abstract and Applied Analysis

From (20) (22) and (23) we have

119864Δ1198815(119896) = 119864 [119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) + (

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

119879

times (

minus211988521198852

1198852

lowast minus1198852

0

lowast lowast minus1198852

)(

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

+ 120573(119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

))

119879

(minus11988521198852

lowast minus1198852

)

times (119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)) + (1 minus 120573)

times(119909 (119896 minus 120591 (119896))

120579 (119896))

119879

(minus11988521198852

lowast minus1198852

)

times(119909 (119896 minus 120591 (119896))

120579 (119896))]

(24)

When 1198891le 119889(119896) le 119889

lowast by Lemma 6 it is easy to get

119864Δ1198816(119896) = 119864[

[

(

minus1

sum

119894=minus119889(119896+1)

119896

sum

119895=119896+119894+1

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

)

times ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

(

119896

sum

119897=119896+119895+1

minus

119896minus1

sum

119897=119896+119895

)

times ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897)) ]

]

le 119864[

[

minus1

sum

119894=minus119889lowast

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1

sum

119894=minus119889lowast

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1

sum

119894=minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]

]

le 119864[(119889lowast

+ 1198891minus 1) (119889

lowast

minus 1198891) + 2119889

lowast

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119889lowast(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(25)

When 119889lowast lt 119889(119896) le 1198892 similarly we can have

119864Δ1198816(119896) le 119864[

(119889lowast

+ 1198892minus 1) (119889

1minus 119889lowast

) + 21198892

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

1198892

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(26)

From (25) and (26) we have

119864Δ1198816(119896) le 119864[119887ℎ

119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119888(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(27)

From (7) it follows that

(ℎ119894(119909 (119896)) minus 120574

+

119894119909119894(119896))

times (ℎ119894(119909 (119896)) minus 120574

minus

119894119909119894(119896)) le 0 119894 = 1 2 119899

(28)

which are equivalent to

(119909 (119896)

ℎ (119909 (119896)))

119879

(120574minus

119894120574+

119894119890119894119890119879

119894minus120574minus

119894+ 120574+

119894

2119890119894119890119879

119894

lowast 119890119894119890119879

119894

)

times(119909 (119896)

ℎ (119909 (119896))) le 0

(29)

where 119890119894denotes the unit column vector having one element

on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582

1 1205822

120582119899 gt 0 it follows that

(119909 (119896)

ℎ (119909 (119896)))

119879

(minusΓ1Λ Γ2Λ

lowast minusΛ)(

119909 (119896)

ℎ (119909 (119896))) ge 0 (30)

Abstract and Applied Analysis 7

Note that by Lemma 7 we get

119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896) + 120591lowast2

120578119879

(119896) 1198851120578 (119896)

+120591120578119879

(119896) 1198852120578 (119896)]

= 119864 [120578119879

(119896) (119875 + 120591lowast2

1198851+ 1205911198852) 120578 (119896) + 2120578

119879

(119896) 119875119909 (119896)]

= 119864 [119909119879

(119896 + 1) (119875 + 120591lowast2

1198851+ 1205911198852) 119909 (119896 + 1)

minus 2119909119879

(119896 + 1) (120591lowast2

1198851+ 1205911198852) 119909 (119896)

+ 119909119879

(119896) (120591lowast2

1198851+ 1205911198852minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) (119875 + 2120591lowast2

1198851+ 2120591119885

2) 119909 (119896 + 1)

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) 120582lowast

119868119909 (119896 + 1) + 119909119879

(119896)

times (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879

times 120582lowast

119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

times 120582lowast

119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]

119879

120582lowast

119868

times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

120582lowast

119868

times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

+ 120582lowast

119909119879

(119860 + 119862119870)119879

(119860 + 119862119870) 119909 (119896)

+ 120582lowast

119909119879

(119896 minus 120591 (119896)) 119861119879

119861119909 (119896 minus 120591 (119896))

+ 120582lowast

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

119863119879

119863

times (

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

+5120582lowast

1198622

1199032

infin

(31)

Then from (18) to (31) we have

119864Δ119881 (119896) le 119864120585119879

(119896) [Ξ1015840

+ (radic2119883 0 0 0 0 0 0 0)119879 1

120582lowast

times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)

+ 1205881199032

infin

(32)

where

Ξ1015840

119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582

lowast

1198632

120585119879

(119896) = [119909119879

(119896) 119909119879

(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast

) 119909 (119896 minus 119889lowast

)

120579119879

(119896) ℎ119879

(119909 (119896))

minus1

sum

minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119891119879

]

(33)

If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that

119864Δ119881 (119896) le minus120576119864119909 (119896)2

+ 1205881199032

infin (34)

It is easy to derive that

119864119881 (119896) le 1205831119864119909 (119896)

2

+ 1205832

119896minus1

sum

119894=119896minus1205912

119864119909 (119894)2

+ 1205833

119896minus1

sum

119894=119896minus1198892

119864119909 (119894)2

(35)

8 Abstract and Applied Analysis

with

1205831= 120582max (119875)

1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)

+ 2120582max (1198764) + 4120591lowast2

120582max (1198851)

+ 4(1205912minus 1205911)2

120582max (1198852)

1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)

times (1198892minus 1198891)]

times Γ2

120582max (119877)

(36)

For any 120579 gt 1 it follows from (34) and (35) that

119864 [120579119895+1

119881 (119895 + 1) minus 120579119895

119881 (119895)]

= 120579119895+1

119864Δ119881 (119895) + 120579119895

(120579 minus 1) 119864119881 (119895)

le 120579119895 [

[

(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ (120579 minus 1) 1205832

119895minus1

sum

119894=119895minus1205912

119864119909 (119894)2

+ 1205881205791199032

infin

+ (120579 minus 1) 1205833

119895minus1

sum

119894=119895minus1198892

119864119909 (119894)2]

]

(37)

Summing up both sides of (37) from 0 to 119896minus1 we can obtain

120579119896

119864119881 (119896) minus 119864119881 (0)

le (1205831(120579 minus 1) minus 120576120579)

119896minus1

sum

119895=0

120579119895

1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ 1205832(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

+ 1205833(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

(38)

Also it is easy to compute that

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1205912

119894+1205912

sum

119895=0

+

119896minus1minus1205912

sum

119894=0

119894+1205912

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1205912

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 12059121205791205912 sup119904isin[minus120591

20]

119864119909 (119904)2

+ 12059121205791205912

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1198892

119894+1198892

sum

119895=0

+

119896minus1minus1198892

sum

119894=0

119894+1198892

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1198892

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 11988921205791198892 sup119904isin[minus1198892 0]

119864119909 (119904)2

+ 11988921205791198892

times

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(39)Substituting (39) into (38) leads to

120579119896

119864119881 (119896) minus 119864119881 (0)

le 1205781(120579) sup119904isin[minus1205912 0]

119864119909 (119904)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

+ 1205782(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

+ 1205783(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(40)

where 1205781(120579) = 120583

2(120579 minus 1)120591

21205791205912 + 1205833(120579 minus 1)119889

21205791198892 1205782(120579) = 120583

2(120579 minus

1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578

3(120579) = 120583

3(120579minus1)119889

21205791198892 +1205831(120579minus1)minus120576120579

Since 1205782(1) lt 0 120578

3(1) lt 0 there must exist a positive

1205790gt 1 such that 120578

2(1205790) lt 0 120578

3(1205790) lt 0 Then we have

119864119881 (119896)

le 1205781(1205790) (

1

1205790

)

119896

sup119904isin[minus12059120]

119864119909 (119904)2

+ (1

1205790

)

119896

119864119881 (0) + 120588

119896minus1

sum

119895=0

1

120579119896minus119895minus1

0

1199032

infin

+ 1205782(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

+ 1205783(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+120588

1205790minus 1

1199032

infin

(41)On the other hand by (16) we can get

119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2

(42)Combining (41) with (42) we have

119864119909 (119896)2

le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892

120582min (119875)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+1

120582min (119875)

120588

1205790minus 1

1199032

infin

(43)

Abstract and Applied Analysis 9

Thus

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198722

119864119909 (119896)2

le 1198731+ 11987321199032

infin (44)

where 1198731= 119872

2

((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))

sup119904isin[minusmax120591

211988920]119864119909(119904)

2 1198732

= (1120582min(119875))(120588(1205790 minus

1))1198722 By Definition 5 the nonlinear discrete-time

stochastic control system (1) is BIBO stability in meansquare This completes the proof

If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained

Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889

2

under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876

119894 119894 = 1 2 5119885

11198852 and119883with appropriate

dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868

Ξ

=

(((((((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))))))

)

le 0

(45)

where

Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 120577119868 + 2120591

lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860

minus 2119860119879

119883 minus 2119883119879

119860

Ξ22= 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(46)

Proof The proof is straightforward and hence omitted

Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)

Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay

Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591

1= 1205912

Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves

Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval

4 An Example

In this section a numerical example will be presented to showthe validity of the main results derived in Section 3

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

From (20) (22) and (23) we have

119864Δ1198815(119896) = 119864 [119909

119879

(119896) 1198765119909 (119896) minus 120579

119879

(119896) 1198765120579 (119896)

+ 1205912

120578119879

(119896) 1198852120578 (119896) + (

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

119879

times (

minus211988521198852

1198852

lowast minus1198852

0

lowast lowast minus1198852

)(

119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)

120579 (119896)

)

+ 120573(119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

))

119879

(minus11988521198852

lowast minus1198852

)

times (119909 (119896 minus 120591 (119896))

119909 (119896 minus 120591lowast

)) + (1 minus 120573)

times(119909 (119896 minus 120591 (119896))

120579 (119896))

119879

(minus11988521198852

lowast minus1198852

)

times(119909 (119896 minus 120591 (119896))

120579 (119896))]

(24)

When 1198891le 119889(119896) le 119889

lowast by Lemma 6 it is easy to get

119864Δ1198816(119896) = 119864[

[

(

minus1

sum

119894=minus119889(119896+1)

119896

sum

119895=119896+119894+1

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894

)

times ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

(

119896

sum

119897=119896+119895+1

minus

119896minus1

sum

119897=119896+119895

)

times ℎ119879

(119909 (119897)) 119877ℎ (119909 (119897)) ]

]

le 119864[

[

minus1

sum

119894=minus119889lowast

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

minus

minus1

sum

119894=minus119889(119896)

119896minus1

sum

119895=119896+119894+1

ℎ119879

(119909 (119895)) 119877ℎ (119909 (119895))

+

minus1

sum

119894=minus119889lowast

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1

sum

119894=minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))

+

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus

minus1198891minus1

sum

119894=minus119889lowast

minus1

sum

119895=119894+1

ℎ119879

(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]

]

le 119864[(119889lowast

+ 1198891minus 1) (119889

lowast

minus 1198891) + 2119889

lowast

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119889lowast(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(25)

When 119889lowast lt 119889(119896) le 1198892 similarly we can have

119864Δ1198816(119896) le 119864[

(119889lowast

+ 1198892minus 1) (119889

1minus 119889lowast

) + 21198892

2

times ℎ119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

1198892

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(26)

From (25) and (26) we have

119864Δ1198816(119896) le 119864[119887ℎ

119879

(119909 (119896)) 119877ℎ (119909 (119896))

minus1

119888(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

times 119877(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))]

(27)

From (7) it follows that

(ℎ119894(119909 (119896)) minus 120574

+

119894119909119894(119896))

times (ℎ119894(119909 (119896)) minus 120574

minus

119894119909119894(119896)) le 0 119894 = 1 2 119899

(28)

which are equivalent to

(119909 (119896)

ℎ (119909 (119896)))

119879

(120574minus

119894120574+

119894119890119894119890119879

119894minus120574minus

119894+ 120574+

119894

2119890119894119890119879

119894

lowast 119890119894119890119879

119894

)

times(119909 (119896)

ℎ (119909 (119896))) le 0

(29)

where 119890119894denotes the unit column vector having one element

on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582

1 1205822

120582119899 gt 0 it follows that

(119909 (119896)

ℎ (119909 (119896)))

119879

(minusΓ1Λ Γ2Λ

lowast minusΛ)(

119909 (119896)

ℎ (119909 (119896))) ge 0 (30)

Abstract and Applied Analysis 7

Note that by Lemma 7 we get

119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896) + 120591lowast2

120578119879

(119896) 1198851120578 (119896)

+120591120578119879

(119896) 1198852120578 (119896)]

= 119864 [120578119879

(119896) (119875 + 120591lowast2

1198851+ 1205911198852) 120578 (119896) + 2120578

119879

(119896) 119875119909 (119896)]

= 119864 [119909119879

(119896 + 1) (119875 + 120591lowast2

1198851+ 1205911198852) 119909 (119896 + 1)

minus 2119909119879

(119896 + 1) (120591lowast2

1198851+ 1205911198852) 119909 (119896)

+ 119909119879

(119896) (120591lowast2

1198851+ 1205911198852minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) (119875 + 2120591lowast2

1198851+ 2120591119885

2) 119909 (119896 + 1)

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) 120582lowast

119868119909 (119896 + 1) + 119909119879

(119896)

times (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879

times 120582lowast

119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

times 120582lowast

119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]

119879

120582lowast

119868

times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

120582lowast

119868

times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

+ 120582lowast

119909119879

(119860 + 119862119870)119879

(119860 + 119862119870) 119909 (119896)

+ 120582lowast

119909119879

(119896 minus 120591 (119896)) 119861119879

119861119909 (119896 minus 120591 (119896))

+ 120582lowast

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

119863119879

119863

times (

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

+5120582lowast

1198622

1199032

infin

(31)

Then from (18) to (31) we have

119864Δ119881 (119896) le 119864120585119879

(119896) [Ξ1015840

+ (radic2119883 0 0 0 0 0 0 0)119879 1

120582lowast

times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)

+ 1205881199032

infin

(32)

where

Ξ1015840

119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582

lowast

1198632

120585119879

(119896) = [119909119879

(119896) 119909119879

(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast

) 119909 (119896 minus 119889lowast

)

120579119879

(119896) ℎ119879

(119909 (119896))

minus1

sum

minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119891119879

]

(33)

If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that

119864Δ119881 (119896) le minus120576119864119909 (119896)2

+ 1205881199032

infin (34)

It is easy to derive that

119864119881 (119896) le 1205831119864119909 (119896)

2

+ 1205832

119896minus1

sum

119894=119896minus1205912

119864119909 (119894)2

+ 1205833

119896minus1

sum

119894=119896minus1198892

119864119909 (119894)2

(35)

8 Abstract and Applied Analysis

with

1205831= 120582max (119875)

1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)

+ 2120582max (1198764) + 4120591lowast2

120582max (1198851)

+ 4(1205912minus 1205911)2

120582max (1198852)

1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)

times (1198892minus 1198891)]

times Γ2

120582max (119877)

(36)

For any 120579 gt 1 it follows from (34) and (35) that

119864 [120579119895+1

119881 (119895 + 1) minus 120579119895

119881 (119895)]

= 120579119895+1

119864Δ119881 (119895) + 120579119895

(120579 minus 1) 119864119881 (119895)

le 120579119895 [

[

(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ (120579 minus 1) 1205832

119895minus1

sum

119894=119895minus1205912

119864119909 (119894)2

+ 1205881205791199032

infin

+ (120579 minus 1) 1205833

119895minus1

sum

119894=119895minus1198892

119864119909 (119894)2]

]

(37)

Summing up both sides of (37) from 0 to 119896minus1 we can obtain

120579119896

119864119881 (119896) minus 119864119881 (0)

le (1205831(120579 minus 1) minus 120576120579)

119896minus1

sum

119895=0

120579119895

1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ 1205832(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

+ 1205833(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

(38)

Also it is easy to compute that

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1205912

119894+1205912

sum

119895=0

+

119896minus1minus1205912

sum

119894=0

119894+1205912

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1205912

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 12059121205791205912 sup119904isin[minus120591

20]

119864119909 (119904)2

+ 12059121205791205912

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1198892

119894+1198892

sum

119895=0

+

119896minus1minus1198892

sum

119894=0

119894+1198892

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1198892

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 11988921205791198892 sup119904isin[minus1198892 0]

119864119909 (119904)2

+ 11988921205791198892

times

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(39)Substituting (39) into (38) leads to

120579119896

119864119881 (119896) minus 119864119881 (0)

le 1205781(120579) sup119904isin[minus1205912 0]

119864119909 (119904)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

+ 1205782(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

+ 1205783(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(40)

where 1205781(120579) = 120583

2(120579 minus 1)120591

21205791205912 + 1205833(120579 minus 1)119889

21205791198892 1205782(120579) = 120583

2(120579 minus

1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578

3(120579) = 120583

3(120579minus1)119889

21205791198892 +1205831(120579minus1)minus120576120579

Since 1205782(1) lt 0 120578

3(1) lt 0 there must exist a positive

1205790gt 1 such that 120578

2(1205790) lt 0 120578

3(1205790) lt 0 Then we have

119864119881 (119896)

le 1205781(1205790) (

1

1205790

)

119896

sup119904isin[minus12059120]

119864119909 (119904)2

+ (1

1205790

)

119896

119864119881 (0) + 120588

119896minus1

sum

119895=0

1

120579119896minus119895minus1

0

1199032

infin

+ 1205782(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

+ 1205783(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+120588

1205790minus 1

1199032

infin

(41)On the other hand by (16) we can get

119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2

(42)Combining (41) with (42) we have

119864119909 (119896)2

le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892

120582min (119875)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+1

120582min (119875)

120588

1205790minus 1

1199032

infin

(43)

Abstract and Applied Analysis 9

Thus

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198722

119864119909 (119896)2

le 1198731+ 11987321199032

infin (44)

where 1198731= 119872

2

((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))

sup119904isin[minusmax120591

211988920]119864119909(119904)

2 1198732

= (1120582min(119875))(120588(1205790 minus

1))1198722 By Definition 5 the nonlinear discrete-time

stochastic control system (1) is BIBO stability in meansquare This completes the proof

If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained

Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889

2

under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876

119894 119894 = 1 2 5119885

11198852 and119883with appropriate

dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868

Ξ

=

(((((((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))))))

)

le 0

(45)

where

Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 120577119868 + 2120591

lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860

minus 2119860119879

119883 minus 2119883119879

119860

Ξ22= 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(46)

Proof The proof is straightforward and hence omitted

Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)

Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay

Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591

1= 1205912

Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves

Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval

4 An Example

In this section a numerical example will be presented to showthe validity of the main results derived in Section 3

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Note that by Lemma 7 we get

119864 [120578119879

(119896) 119875120578 (119896) + 2120578119879

(119896) 119875119909 (119896) + 120591lowast2

120578119879

(119896) 1198851120578 (119896)

+120591120578119879

(119896) 1198852120578 (119896)]

= 119864 [120578119879

(119896) (119875 + 120591lowast2

1198851+ 1205911198852) 120578 (119896) + 2120578

119879

(119896) 119875119909 (119896)]

= 119864 [119909119879

(119896 + 1) (119875 + 120591lowast2

1198851+ 1205911198852) 119909 (119896 + 1)

minus 2119909119879

(119896 + 1) (120591lowast2

1198851+ 1205911198852) 119909 (119896)

+ 119909119879

(119896) (120591lowast2

1198851+ 1205911198852minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) (119875 + 2120591lowast2

1198851+ 2120591119885

2) 119909 (119896 + 1)

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

le 119864 [119909119879

(119896 + 1) 120582lowast

119868119909 (119896 + 1) + 119909119879

(119896)

times (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)]

= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879

times 120582lowast

119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

times 120582lowast

119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+ 119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894))

+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]

119879

120582lowast

119868

times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))

+119863

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]

+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879

120582lowast

119868

times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]

+ 119909119879

(119896) (2120591lowast2

1198851+ 2120591119885

2minus 119875) 119909 (119896)

+ 120582lowast

119909119879

(119860 + 119862119870)119879

(119860 + 119862119870) 119909 (119896)

+ 120582lowast

119909119879

(119896 minus 120591 (119896)) 119861119879

119861119909 (119896 minus 120591 (119896))

+ 120582lowast

(

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

119879

119863119879

119863

times (

minus1

sum

119894=minus119889(119896)

ℎ (119909 (119896 + 119894)))

+5120582lowast

1198622

1199032

infin

(31)

Then from (18) to (31) we have

119864Δ119881 (119896) le 119864120585119879

(119896) [Ξ1015840

+ (radic2119883 0 0 0 0 0 0 0)119879 1

120582lowast

times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)

+ 1205881199032

infin

(32)

where

Ξ1015840

119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582

lowast

1198632

120585119879

(119896) = [119909119879

(119896) 119909119879

(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast

) 119909 (119896 minus 119889lowast

)

120579119879

(119896) ℎ119879

(119909 (119896))

minus1

sum

minus119889(119896)

ℎ119879

(119909 (119896 + 119894)) 119891119879

]

(33)

If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that

119864Δ119881 (119896) le minus120576119864119909 (119896)2

+ 1205881199032

infin (34)

It is easy to derive that

119864119881 (119896) le 1205831119864119909 (119896)

2

+ 1205832

119896minus1

sum

119894=119896minus1205912

119864119909 (119894)2

+ 1205833

119896minus1

sum

119894=119896minus1198892

119864119909 (119894)2

(35)

8 Abstract and Applied Analysis

with

1205831= 120582max (119875)

1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)

+ 2120582max (1198764) + 4120591lowast2

120582max (1198851)

+ 4(1205912minus 1205911)2

120582max (1198852)

1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)

times (1198892minus 1198891)]

times Γ2

120582max (119877)

(36)

For any 120579 gt 1 it follows from (34) and (35) that

119864 [120579119895+1

119881 (119895 + 1) minus 120579119895

119881 (119895)]

= 120579119895+1

119864Δ119881 (119895) + 120579119895

(120579 minus 1) 119864119881 (119895)

le 120579119895 [

[

(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ (120579 minus 1) 1205832

119895minus1

sum

119894=119895minus1205912

119864119909 (119894)2

+ 1205881205791199032

infin

+ (120579 minus 1) 1205833

119895minus1

sum

119894=119895minus1198892

119864119909 (119894)2]

]

(37)

Summing up both sides of (37) from 0 to 119896minus1 we can obtain

120579119896

119864119881 (119896) minus 119864119881 (0)

le (1205831(120579 minus 1) minus 120576120579)

119896minus1

sum

119895=0

120579119895

1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ 1205832(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

+ 1205833(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

(38)

Also it is easy to compute that

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1205912

119894+1205912

sum

119895=0

+

119896minus1minus1205912

sum

119894=0

119894+1205912

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1205912

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 12059121205791205912 sup119904isin[minus120591

20]

119864119909 (119904)2

+ 12059121205791205912

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1198892

119894+1198892

sum

119895=0

+

119896minus1minus1198892

sum

119894=0

119894+1198892

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1198892

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 11988921205791198892 sup119904isin[minus1198892 0]

119864119909 (119904)2

+ 11988921205791198892

times

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(39)Substituting (39) into (38) leads to

120579119896

119864119881 (119896) minus 119864119881 (0)

le 1205781(120579) sup119904isin[minus1205912 0]

119864119909 (119904)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

+ 1205782(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

+ 1205783(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(40)

where 1205781(120579) = 120583

2(120579 minus 1)120591

21205791205912 + 1205833(120579 minus 1)119889

21205791198892 1205782(120579) = 120583

2(120579 minus

1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578

3(120579) = 120583

3(120579minus1)119889

21205791198892 +1205831(120579minus1)minus120576120579

Since 1205782(1) lt 0 120578

3(1) lt 0 there must exist a positive

1205790gt 1 such that 120578

2(1205790) lt 0 120578

3(1205790) lt 0 Then we have

119864119881 (119896)

le 1205781(1205790) (

1

1205790

)

119896

sup119904isin[minus12059120]

119864119909 (119904)2

+ (1

1205790

)

119896

119864119881 (0) + 120588

119896minus1

sum

119895=0

1

120579119896minus119895minus1

0

1199032

infin

+ 1205782(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

+ 1205783(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+120588

1205790minus 1

1199032

infin

(41)On the other hand by (16) we can get

119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2

(42)Combining (41) with (42) we have

119864119909 (119896)2

le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892

120582min (119875)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+1

120582min (119875)

120588

1205790minus 1

1199032

infin

(43)

Abstract and Applied Analysis 9

Thus

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198722

119864119909 (119896)2

le 1198731+ 11987321199032

infin (44)

where 1198731= 119872

2

((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))

sup119904isin[minusmax120591

211988920]119864119909(119904)

2 1198732

= (1120582min(119875))(120588(1205790 minus

1))1198722 By Definition 5 the nonlinear discrete-time

stochastic control system (1) is BIBO stability in meansquare This completes the proof

If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained

Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889

2

under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876

119894 119894 = 1 2 5119885

11198852 and119883with appropriate

dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868

Ξ

=

(((((((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))))))

)

le 0

(45)

where

Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 120577119868 + 2120591

lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860

minus 2119860119879

119883 minus 2119883119879

119860

Ξ22= 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(46)

Proof The proof is straightforward and hence omitted

Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)

Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay

Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591

1= 1205912

Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves

Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval

4 An Example

In this section a numerical example will be presented to showthe validity of the main results derived in Section 3

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

with

1205831= 120582max (119875)

1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)

+ 2120582max (1198764) + 4120591lowast2

120582max (1198851)

+ 4(1205912minus 1205911)2

120582max (1198852)

1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)

times (1198892minus 1198891)]

times Γ2

120582max (119877)

(36)

For any 120579 gt 1 it follows from (34) and (35) that

119864 [120579119895+1

119881 (119895 + 1) minus 120579119895

119881 (119895)]

= 120579119895+1

119864Δ119881 (119895) + 120579119895

(120579 minus 1) 119864119881 (119895)

le 120579119895 [

[

(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ (120579 minus 1) 1205832

119895minus1

sum

119894=119895minus1205912

119864119909 (119894)2

+ 1205881205791199032

infin

+ (120579 minus 1) 1205833

119895minus1

sum

119894=119895minus1198892

119864119909 (119894)2]

]

(37)

Summing up both sides of (37) from 0 to 119896minus1 we can obtain

120579119896

119864119881 (119896) minus 119864119881 (0)

le (1205831(120579 minus 1) minus 120576120579)

119896minus1

sum

119895=0

120579119895

1198641003817100381710038171003817119909 (119895)

1003817100381710038171003817

2

+ 1205832(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

+ 1205833(120579 minus 1)

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

(38)

Also it is easy to compute that

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1205912

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1205912

119894+1205912

sum

119895=0

+

119896minus1minus1205912

sum

119894=0

119894+1205912

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1205912

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 12059121205791205912 sup119904isin[minus120591

20]

119864119909 (119904)2

+ 12059121205791205912

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

119896minus1

sum

119895=0

119895minus1

sum

119894=119895minus1198892

120579119895

119864119909 (119894)2

le (

minus1

sum

119894=minus1198892

119894+1198892

sum

119895=0

+

119896minus1minus1198892

sum

119894=0

119894+1198892

sum

119895=119894+1

+

119896minus1

sum

119894=119896minus1198892

119896minus1

sum

119895=119894+1

)

times 120579119895

119864119909 (119894)2

le 11988921205791198892 sup119904isin[minus1198892 0]

119864119909 (119904)2

+ 11988921205791198892

times

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(39)Substituting (39) into (38) leads to

120579119896

119864119881 (119896) minus 119864119881 (0)

le 1205781(120579) sup119904isin[minus1205912 0]

119864119909 (119904)2

+ 120588

119896minus1

sum

119895=0

120579119895+1

1199032

infin

+ 1205782(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

+ 1205783(120579)

119896minus1

sum

119894=0

120579119894

119864119909 (119894)2

(40)

where 1205781(120579) = 120583

2(120579 minus 1)120591

21205791205912 + 1205833(120579 minus 1)119889

21205791198892 1205782(120579) = 120583

2(120579 minus

1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578

3(120579) = 120583

3(120579minus1)119889

21205791198892 +1205831(120579minus1)minus120576120579

Since 1205782(1) lt 0 120578

3(1) lt 0 there must exist a positive

1205790gt 1 such that 120578

2(1205790) lt 0 120578

3(1205790) lt 0 Then we have

119864119881 (119896)

le 1205781(1205790) (

1

1205790

)

119896

sup119904isin[minus12059120]

119864119909 (119904)2

+ (1

1205790

)

119896

119864119881 (0) + 120588

119896minus1

sum

119895=0

1

120579119896minus119895minus1

0

1199032

infin

+ 1205782(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

+ 1205783(1205790)

119896minus1

sum

119894=0

1

120579119896minus119894

0

119864119909 (119894)2

le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+120588

1205790minus 1

1199032

infin

(41)On the other hand by (16) we can get

119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2

(42)Combining (41) with (42) we have

119864119909 (119896)2

le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892

120582min (119875)

times sup119904isin[minusmax120591211988920]

119864119909 (119904)2

+1

120582min (119875)

120588

1205790minus 1

1199032

infin

(43)

Abstract and Applied Analysis 9

Thus

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198722

119864119909 (119896)2

le 1198731+ 11987321199032

infin (44)

where 1198731= 119872

2

((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))

sup119904isin[minusmax120591

211988920]119864119909(119904)

2 1198732

= (1120582min(119875))(120588(1205790 minus

1))1198722 By Definition 5 the nonlinear discrete-time

stochastic control system (1) is BIBO stability in meansquare This completes the proof

If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained

Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889

2

under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876

119894 119894 = 1 2 5119885

11198852 and119883with appropriate

dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868

Ξ

=

(((((((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))))))

)

le 0

(45)

where

Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 120577119868 + 2120591

lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860

minus 2119860119879

119883 minus 2119883119879

119860

Ξ22= 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(46)

Proof The proof is straightforward and hence omitted

Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)

Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay

Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591

1= 1205912

Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves

Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval

4 An Example

In this section a numerical example will be presented to showthe validity of the main results derived in Section 3

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

Thus

E1003817100381710038171003817119910 (119896)

1003817100381710038171003817

2

le 1198722

119864119909 (119896)2

le 1198731+ 11987321199032

infin (44)

where 1198731= 119872

2

((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))

sup119904isin[minusmax120591

211988920]119864119909(119904)

2 1198732

= (1120582min(119875))(120588(1205790 minus

1))1198722 By Definition 5 the nonlinear discrete-time

stochastic control system (1) is BIBO stability in meansquare This completes the proof

If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained

Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889

2

under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876

119894 119894 = 1 2 5119885

11198852 and119883with appropriate

dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold

119875 + 2 (120591lowast2

1198851+ 1205912

1198852) le 120582lowast

119868

Ξ

=

(((((((((((

(

Ξ11Ξ12

1198851

0 0 Γ2Λ Ξ17

Ξ18

radic2119883

lowast Ξ22Ξ23

0 Ξ25

0 119861119879

119863 119861119879

0

lowast lowast Ξ33

0 0 0 0 0 0

lowast lowast lowast minus1198762

0 0 0 0 0

lowast lowast lowast lowast Ξ55

0 0 0 0

lowast lowast lowast lowast lowast Ξ66

0 0 0

lowast lowast lowast lowast lowast lowast Ξ77

120582lowast

119863119879

0

lowast lowast lowast lowast lowast lowast lowast minus120577119868 0

lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast

119868

)))))))))))

)

le 0

(45)

where

Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876

3+ 1198764) minus 1198851

+ 1198765minus Γ1Λ + 120577119868 + 2120591

lowast2

1198851+ 21205912

1198852

+ 2120582lowast

1205721119868 + 1205721120577119868 minus 119875 + 2120582

lowast

119860119879

119860

minus 2119860119879

119883 minus 2119883119879

119860

Ξ22= 2120582lowast

119861119879

119861 minus 1198764minus 21198852minus 1205731198852

+ 2120582lowast

1205722119868 + 1205722120577119868 minus (1 minus 120573)119885

2

Ξ12= 120582lowast

119860119879

119861 minus 119883119879

119861

Ξ17= 120582lowast

119860119879

119863 minus 119883119879

119863

Ξ18= 120582lowast

119860119879

minus 119883119879

Ξ25= 1198852+ (1 minus 120573)119885

2

Ξ23= 1198852+ 1205731198852

Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852

Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885

2

Ξ66= 119887119877 minus Λ

Ξ77= 2120582lowast

119863119879

119863 minus1

119888119877

119883 = minus120582lowast

119862119870

(46)

Proof The proof is straightforward and hence omitted

Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)

Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay

Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591

1= 1205912

Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves

Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval

4 An Example

In this section a numerical example will be presented to showthe validity of the main results derived in Section 3

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

Table 1 For 120591(119896) = 1 2 3 4 5 120573

120591(119896) 1 2 3 4 5

120573 0 1 12 12 0

Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by

119860 = (minus01 0

01 minus02) 119861 = (

minus01 01

minus01 01)

119862 = (01 01

05 03) 119863 = (

01 01

0 02)

(47)

119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879

ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ

2(119904) = tanh(minus04119904) 120591

1= 1

1205912= 5 119889

1= 2 119889

2= 7

It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and

Γ1= (

minus064 0

0 0) Γ

2= (

0 0

0 minus02)

120591lowast

=1205911+ 1205912

2minusmin (minus1)1205911+1205912 0

2

119889lowast

=1205911+ 1205912

2+min (minus1)1205911+1205912 0

2

119887 = 9 119889

1le 119889 (119896) le 119889

lowast

20 119889lowast

lt 119889 (119896) le 1198892

119888 = 4 119889

1le 119889 (119896) le 119889

lowast

7 119889lowast

lt 119889 (119896) le 1198892

(48)

Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1

By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows

When 120573 = 1 119887 = 9 119888 = 4

119875 = (1450684 minus32651

minus32651 1477799) 119885

1= (

00478 00298

00298 00230)

1198852= (

03443 02142

02142 01663) 119876

1= (

28401 17808

17808 13624)

1198762= (

37464 23462

23462 17985) 119876

3= (

02731 01713

01713 01309)

1198764= (

153027 minus68645

minus68645 155746) 119876

5= (

31500 19740

19740 15118)

119877 = (1282646 minus921100

minus921100 2390910) 119873 = (

04512 0

0 343341)

119883 = (133495 31121

31121 07692) 119870 = (

minus14212 minus03303

05268 01218)

(49)

and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square

behavior as shown in Figure 1

0 20 40 60 80 100 1200

20

40

60

k

0 20 40 60 80 100 120minus05

0

05

1

Varia

tion

of120591k

Figure 1

5 Conclusions

In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults

Acknowledgments

The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)

References

[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012

[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012

[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin

control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012

[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012

[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012

[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011

[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012

[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009

[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010

[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009

[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009

[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009

[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008

[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009

[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013

[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008

[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009

[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009

[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008

[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995

[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000

[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003

[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of

delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004

[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002

[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005

[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008

[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011

[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009

[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009

[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003

[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010

[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of