convergence analysis of stochastic hybrid bidirectional associative memory neural networks with...

Physics Letters A 370 (2007) 423–432

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Convergence analysis of stochastic hybrid bidirectional associative memoryneural networks with delays

Li Wan a,b,∗, Qinghua Zhou c

a Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, Chinab Department of Mathematics and Physics, Wuhan University of Science and Engineering, Wuhan 430073, China

c Department of Mathematics, Nanjing University, Nanjing 210008, China

Received 10 September 2006; received in revised form 17 May 2007; accepted 30 May 2007

Available online 2 June 2007

Communicated by A.R. Bishop

Abstract

The stability property of stochastic hybrid bidirectional associate memory (BAM) neural networks with discrete delays is considered. Withoutassuming the symmetry of synaptic connection weights and the monotonicity and differentiability of activation functions, the delay-independentsufficient conditions to guarantee the exponential stability of the equilibrium solution for such networks are given by using the nonnegativesemimartingale convergence theorem.© 2007 Elsevier B.V. All rights reserved.

Keywords: Bidirectional associate memory; Exponential stability; Delay

1. Introduction

Kosko in [11–14] has reported a series of bidirectional associate memory neural networks. These models generalized the single-layer autoassociative Hebbian correlator to a two-layer pattern-matched heteroassociative network. Hence, this class of networkshas good application perspective in the area of pattern recognition, signal and image process, etc. Gopalsamy and He in [8] proposedBAM models with axonal signal transmission delays. It has obtained significant advances in many fields such as pattern recognition,automatic control, etc. The stability of BAM neural networks with delays has attracted considerable interest, see, for example [7,8,17–19,23–25,27,29–33], and references therein.

In recent years, the nonlinear and delay-type feedback template has been explored in a hybrid neural network model known asthe cellular neural network (CNN), see [10]. The presence of the term involving nonlinear feedback assumes, in addition to thedelayed propagation of signals, a set of local interactions in the network whose propagation time is instantaneous. This model hasbeen applied in mimicking the activity of vertebrate retina. Harmori and Roska in [10] has been studied simple effects such as retinaeffects, length tuning and directional selectivity. By studying the effects of amacrine cells, they have established a simplified modelusing the nonlinear and delay-type feedback simultaneously. Moreover, they have also proposed a new modified retina model, theCNNRET-2, which is capable of performing simple illusions without the errors reported in the literature. CNN models of motionillusions including the phi phenomenon provide simple explanations for apparently complex effects. Clearly, both delay-type andsimple nonlinear templates play an important role in neural networks. However, most papers on BAM with delays were restrictedto the pure-delay models, for instance, see [7,8,17,18,23,24,33], and references therein. Few paper studied the hybrid BAM neuralnetworks in which instantaneous signaling as well as delayed signaling occur, for example, see [20] and references therein.

* Corresponding author at: Department of Mathematics and Physics, Wuhan University of Science and Engineering, Wuhan 430073, China.E-mail addresses: [email protected] (L. Wan), [email protected] (Q. Zhou).

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424 L. Wan, Q. Zhou / Physics Letters A 370 (2007) 423–432

Although discrete time delays in the delayed feedback neural networks having a small number of cells serve usually as goodapproximation of the prime models, a real system is usually affected by external perturbations which in many cases are of greatuncertainty and hence may be treated as random, as pointed out by [9] that in real nervous systems synaptic transmission is a noisyprocess brought on by random fluctuations from the release of neurotransmitters, and other probabilistic causes. Therefore, it issignificant and of prime importance to consider stochastic effects to the stability property of the hybrid BAM neural networks withdelays.

To the best of our knowledge, however, there are few results about stochastic effects to the stability property of neural networkswith discrete delays in the literature today. Chua et al. in [3,4] proposed a novel class of information-processing systems, calledcellular neural networks (CNN), with application to image processing and pattern recognition, and a CNN model with additivenoise current sources inserted to the cell state capacitor in [5], which has been found to be extremely useful in some biologicallearning processes [2]. Cun et al. in [6] described a network where noise is injected into the first hidden layer (only), and use theresult to obtain error derivatives, thereby avoiding back-propagation. To date, the approaches to the mathematical incorporation ofsuch stochastic effects to the stability property of the neural networks are to use probabilistic threshold models (e.g., [9]), to viewneural networks as nonlinear dynamical systems with intrinsic noise, i.e., to include a representation of the inherent stochasticityin the neurodynamics (e.g., [6]), etc. [15,16] supposed that there exists a stochastic perturbation to the neural network with discretedelays, and initiated the study of stability and instability of stochastic neural networks. And then, [1] continued their researchto discuss almost sure exponential stability for a class of stochastic neural network with discrete delays, by using the nonnegativesemimartingale convergence theorem. [26,28] discussed exponential stability in mean square for a class of stochastic neural networkwith discrete delays.

In this Letter, we intend to investigate the exponential stability for stochastic hybrid BAM neural networks with delays, wherethe synaptic connection weights are assumed asymmetric and the nonlinear activation functions are not necessarily differentiable,monotonic and nondecreasing. First we introduce in Section 2 the stochastic discrete delayed hybrid BAM neural networks. Theeasily verifiable delay-independent sufficient conditions to guarantee the most surely exponential stability and pth moment expo-nential stability of the equilibria are given in Section 3.

2. Model description and preliminaries

Consider the stochastically perturbed hybrid BAM

dui(t) ={

−aiui(t) +m∑

j=1

bjifj

(vj (t)

) +m∑

j=1

bj igj

(vj (t − σji)

) + Ii

}dt +

m∑j=1

Sji

(vj (t)

)dwj (t),

dvj (t) ={

−cj vj (t) +n∑

i=1

dijfi

(ui(t)

) +n∑

i=1

dij gi

(ui(t − τij )

) + Jj

}dt +

n∑i=1

Sij

(ui(t)

)dwi(t),

ui(s) = ξi(s), −τ � s � 0, τ = max1�j�m,1�i�n

{τij },(1)vj (s) = ηj (s), −σ � s � 0, σ = max

1�j�m,1�i�n{σji},

for 1 � i � n, 1 � j � m, m � n and t � 0. In the above model, ui(t) and vj (t) denote membrane potentials of the ith neuronsfrom the neuronal field FX and FY , respectively; fi(·), fj (·), gi(·) and gj (·) denote nonlinear activation functions; ξi(t) and ηj (t)

denote the initial boundary value; τij and σji denote time delays required for neural processing and axonal transmission of signals;Ii and Jj denote external inputs to the neurons introduced from outside the network; positive constants ai and cj denote the rateswith which the ith unit from the neuronal field FX and FY , respectively, will reset their potentials to the resting state in isolationwhen disconnected from the network and the external stochastic perturbation; bji, bj i , dij , dij denote synaptic connection weights.Moreover, w(t) = (w1(t), . . . ,wn(t))

T is n-dimensional Brownian motion defined on a complete probability space (Ω,F ,P) witha natural filtration {Ft }t�0 generated by {w(s): 0 � s � t}, where we associate Ω with the canonical space generated by all {wi(t)},and denote by F the associated σ -algebra generated by {w(t)} with the probability measure P.

Note that, ξ = {(ξ1(s), . . . , ξn(s))T : −τ � s � 0} is C([−τ,0];R

n)-valued function and F0-measurable Rn-valued random

variable, where, for example, F0 = Fs on [−τ,0], and C([−τ,0];Rn) is the space of all continuous R

n-valued functions definedon [−τ,0] with a norm ‖ξ‖2 = sup−τ�t�0{

∑ni=1 |ξi(t)|2}. η = {(η1(s), . . . , ηm(s))T : −σ � s � 0} is similar to ξ .

Throughout this Letter, we have the following assumptions:

(A1) fi, fj , gi, gj , Sji, Sij : R → R satisfy∣∣fi(u) − fi(v)∣∣ � Pi |u − v|, ∣∣fj (u) − fj (v)

∣∣ � Qj |u − v|,∣∣gi(u) − gi(v)∣∣ � Li |u − v|, ∣∣gj (u) − gj (v)

∣∣ � Mj |u − v|,

L. Wan, Q. Zhou / Physics Letters A 370 (2007) 423–432 425

∣∣Sij (u) − Sij (v)∣∣ � Nij |u − v|, ∣∣Sji(u) − Sji(v)

∣∣ � Oji |u − v|,∣∣fi(u)∣∣ � Ai < +∞,

∣∣fj (u)∣∣ � Bj < +∞,∣∣gi(u)

∣∣ � Ci < +∞,∣∣gj (u)

∣∣ � Dj < +∞,

for 1 � i � n, 1 � j � m, u,v ∈ R, where Pi,Qj ,Li,Mj ,Ai,Bj ,Ci,Dj ,Nij ,Oji are positive constants.

(A2)

ai >

m∑j=1

(|dij |Pi + |dij |Li

), 1 � i � n,

cj >

n∑i=1

(|bji |Qj + |bj i |Mj

), 1 � j � m.

It follows from [21] that under the condition (A1), Eq. (1) has the global solution on t � 0, which is denoted by (u(t; ξ), v(t;η))T ,where

u(t; ξ) = (u1(t; ξ1), . . . , un(t; ξn)

)T, v(t;η) = (

v1(t;η1), . . . , vm(t;ηm))T

,

or (u(t), v(t))T , if no confusion occurs. For the deterministic system

dui(t) ={

−aiui(t) +m∑

j=1

bjifj

(vj (t)

) +m∑

j=1

bj igj

(vj (t − σji)

) + Ii

}dt,

(2)dvj (t) ={

−cj vj (t) +n∑

i=1

dijfi

(ui(t)

) +n∑

i=1

dij gi

(ui(t − τij )

) + Jj

}dt,

from Theorem 1 of [20], we know under the assumptions (A1) and (A2), the deterministic system has a unique equilibrium(u∗, v∗)T , where u∗ = (u∗

1, . . . , u∗n)

T , v∗ = (v∗1 , . . . , v∗

m)T . We further assume

(A3) Sji

(v∗j

) = 0, Sij

(u∗

i

) = 0 for i = 1, . . . , n, j = 1, . . . ,m.

Then (u∗, v∗)T is a unique equilibrium of system (1) provided that system (1) satisfies (A1), (A2) and (A3). At this time, system (1)is equivalent to

d(ui(t) − u∗

i

) ={

−ai

[ui(t) − u∗

i

] +m∑

j=1

bji

[fj

(vj (t)

) − fj

(v∗j

)] +m∑

j=1

bj i

[gj

(vj

(t − σji

)) − gj

(v∗j

)]}dt

+m∑

j=1

Sji(vj (t))dwj (t),

(3)

d(vj (t) − v∗

j

) ={

−cj

[vj (t) − v∗

j

] +n∑

i=1

dij

[fi

(ui(t)

) − fi

(u∗

i

)] +n∑

i=1

dij

[gi

(ui(t − τij )

) − gi

(u∗

i

)]}dt

+n∑

i=1

Sij

(ui(t)

)dwi(t).

For completeness, we give the following definition, in which E denotes expectation with respect to P, ‖x‖2 = ∑ni=1 |xi |2, x ∈ R

n.

Definition 2.1. Eq. (3) is said to be almost surely exponentially stable if there exists a λ > 0 such that for any ξ and η

lim supt→∞

1

tlog

(∥∥u(t; ξ) − u∗∥∥2 + ∥∥v(t;η) − v∗∥∥2) � −λ, P-a.s.

Definition 2.2. Eq. (3) is said to be exponentially stable in mean square if there exists a pair of positive constants λ and C such thatfor any ξ and η

E∥∥u(t; ξ) − u∗∥∥2 + E

∥∥v(t;η) − v∗∥∥2 � C(E‖ξ − u∗‖2 + E‖η − v∗‖2)e−λt , t � 0.

In addition, we will need the following lemmas.


Lemma 2.3. (See [1].) Let A(t) and U(t) be two continuous adapted increasing processes on t � 0 with A(0) = U(0) = 0, a.s. LetM(t) be a real-valued continuous local martingale with M(0) = 0, a.s. Let ζ be a nonnegative F0-measurable random variablewith Eζ < ∞. Define X(t) = ζ + A(t) − U(t) + M(t) for t � 0. If X(t) is nonnegative, then

{lim

t→∞A(t) < ∞}

⊂{

limt→∞X(t) < ∞

}∩

{lim

t→∞U(t) < ∞}

a.s.,

where B ⊂ D a.s. denotes P(B ∩Dc) = 0. In particular, if limt→∞ A(t) < ∞ a.s., then for almost all ω ∈ Ω , limt→∞ X(t,ω) < ∞and limt→∞ U(t,ω) < ∞, i.e., both X(t) and U(t) converge to finite random variables.

Lemma 2.4. (See [22].) For a � 0, bk � 0, k = 1, . . . , l, the following inequality holds

a

l∏k=1

bqk

k � 1

r

l∑k=1

qkbrk + 1

rar ,

where qk > 0, k = 1, . . . , l, are some constants,∑l

k=1 qk = r − 1, r > 1.

3. Exponential stability of equilibrium solution

In this section, we will study almost surely exponential stability and exponential stability in mean square of equilibrium solutionfor system (1).

Theorem 3.1. If system (1) satisfies the assumptions (A1), (A2), (A3) and

(A4) There exist constants αk,j , βk,j , γk,i , δk,i ∈ R, qk > 0, k = 1,2, . . . , l, i = 1,2, . . . , n, j = 1,2, . . . ,m, such that

−2ai +m∑

j=1

[|bji |

l∑k=1

Q

2αk,jqk

j qk + |bj i |l∑

k=1

M

2βk,jqk

j qk + |dij |P 2γl+1,i

i + |dij |L2δl+1,i

i + N2ij

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[|dij |

l∑k=1

P

2γk,iqk

i qk + |dij |l∑

k=1

L

2δk,iqk

i qk + |bji |Q2αl+1,j

j + |bj i |M2βl+1,j

j + O2ji

]< 0, j = 1,2, . . . ,m,

where∑l+1

k=1 αk,j = ∑l+1k=1 βk,j = 1,

∑l+1k=1 γk,i = ∑l+1

k=1 δk,i = ∑lk=1 qk = 1.

Then Eq. (1) is almost surely exponentially stable and exponentially stable in mean square.

Proof. From (A4), there exists a positive constant λ such that

λ − 2ai +m∑

j=1

[|bji |

l∑k=1

Q

2αk,jqk

j qk + |bj i |l∑

k=1

M

2βk,jqk


i + |dij |L2δl+1,i

i + N2ij

]� 0, i = 1,2, . . . , n,

λ − 2cj +n∑

i=1

[|dij |

l∑k=1

P

2γk,iqk

i qk + |dij |l∑

k=1

L

2δk,iqk



j + O2ji

]� 0, j = 1,2, . . . ,m.

Let ui = ui − u∗i , vj = vj − v∗

j , i = 1,2, . . . , n, j = 1, . . . ,m. By Itô formula, we obtain

du2i (t) = 2

{−aiu

2i (t) +

m∑j=1

bji ui (t)[fj

(vj (t)

) − fj

(v∗j

)] +m∑

j=1

bj i ui (t)[gj

(vj (t − σji)

) − gj

(v∗j

)] + 1

2

m∑j=1

S2ji

(vj (t)

)}dt

+ 2ui (t)

m∑j=1

Sji

(vj (t)

)dwj (t),


and hence

(4)

eλt∣∣ui (t)

∣∣2 = ∣∣ui (0)∣∣2 +

t∫0

eλs2

{−aiu

2i (s) +

m∑j=1

bji ui (s)[fj

(vj (s)

) − fj

(v∗j

)]

+m∑

j=1

bj i ui (s)[gj

(vj (s − σji)

) − gj

(v∗j

)] + 1

2

m∑j=1

S2ji

(vj (s)

)}ds

+t∫

0

eλs2ui (s)

m∑j=1

Sji

(vj (s)

)dwj (s) + λ

t∫0

eλs∣∣ui (s)

∣∣2ds

�∣∣ui (0)

∣∣2 +t∫

0

eλs∣∣ui (s)

∣∣{λ∣∣ui (s)

∣∣ − 2ai

∣∣ui (s)∣∣ + 2

m∑j=1

|bji |Qj

∣∣vj (s)∣∣

+ 2m∑

j=1

|bj i |Mj

∣∣vj (s − σji)∣∣}ds +

t∫0

eλsm∑

j=1

O2jiv

2j (s) ds +

t∫0

eλs2ui (s)

m∑j=1

Sji

(vj (s)

)dwj (s).

From Lemma 2.4, we obtain

eλt∣∣ui (t)

∣∣2 �∣∣ui (0)

∣∣2 + (λ − 2ai)

t∫0

eλs∣∣ui (s)

∣∣2ds

+t∫

0

eλs

{m∑

j=1

|bji |[

l∑k=1

Q

2αk,jqk

j qk

∣∣ui (s)∣∣2 + Q

2αl+1,j

j

∣∣vj (s)∣∣2

]

+m∑

j=1

|bj i |[

l∑k=1

M

2βk,jqk

j qk

∣∣ui (s)∣∣2 + M

2βl+1,j

j

∣∣vj (s − σji)∣∣2

]}ds

+t∫

0

eλsm∑

j=1

O2ji

∣∣vj (s)∣∣2

ds +t∫

0

eλs2ui (s)

m∑j=1

Sji

(vj (s)

)dwj (s).

Note thatt∫

t−σji

eλs∣∣vj (s)

∣∣ds =t∫

−σji

eλs∣∣vj (s)

∣∣ds − e−λσji

t∫0

eλs∣∣vj (s − σji)

∣∣ds

�t∫

−σ

eλs∣∣vj (s)

∣∣ds − e−λσ

t∫0

eλs∣∣vj (s − σji)

∣∣ds.

Hence we have

eλt∣∣ui (t)

∣∣2 �∣∣ui (0)

∣∣2 + (λ − 2ai)

t∫0

eλs∣∣ui (s)

∣∣2ds

+t∫

0

eλs

{m∑

j=1

|bji |[

l∑k=1

Q

2αk,jqk

j qk

∣∣ui (s)∣∣2 + Q

2αl+1,j

j

∣∣vj (s)∣∣2

]

+m∑

j=1

|bj i |[

l∑k=1

M

2βk,jqk

j qk

∣∣ui (s)∣∣2 + M

2βl+1,j

j eλσ∣∣vj (s)

∣∣2

]}ds

+t∫eλs

m∑j=1

O2ji

∣∣vj (s)∣∣2

ds +t∫eλs2ui (s)

m∑j=1

Sji

(vj (s)

)dwj (s)

0 0


+0∫

−σ

eλsm∑

j=1

|bj i |M2βl+1,j

j eλσ∣∣vj (s)

∣∣2ds

= ∣∣ui (0)∣∣2 +

0∫−σ

eλs

m∑j=1

|bj i |M2βl+1,j

j eλσ∣∣vj (s)

∣∣2ds

+{

λ − 2ai +m∑

j=1

|bji |l∑

k=1

Q

2αk,jqk

j qk +m∑

j=1

|bj i |l∑

k=1

M

2βk,jqk

j qk

} t∫0

eλs∣∣ui (s)

∣∣2ds

+{

m∑j=1

|bji |Q2αl+1,j

j +m∑

j=1

|bj i |M2βl+1,j

j eλσ +m∑

j=1

O2ji

} t∫0

eλs∣∣vj (s)

∣∣2ds

+t∫

0

eλs2ui (s)

m∑j=1

Sji

(vj (s)

)dwj (s).

Similarly, we can obtain

eλt∣∣vj (t)

∣∣2 �∣∣vj (0)

∣∣2 +0∫

−τ

eλs

n∑i=1

|dij |L2δl+1,i

i eλτ∣∣ui (s)

∣∣2ds

+{

λ − 2cj +n∑

i=1

|dij |l∑

k=1

P

2γk,iqk

i qk +n∑

i=1

|dij |l∑

k=1

L

2δk,iqk

i qk

} t∫0

eλs∣∣vj (s)

∣∣2ds

+{

n∑i=1

|dij |P 2γl+1,i

i +n∑

i=1

|dij |L2δl+1,i

i eλτ +n∑

i=1

N2ij

} t∫0

eλs∣∣ui (s)

∣∣2ds

+t∫

0

eλs2vj (s)

n∑i=1

Sij

(ui(s)

)dwi(s).

Thus, we have

n∑i=1

eλt∣∣ui (t)

∣∣2 +m∑

j=1

eλt∣∣vj (t)

∣∣2

�n∑

i=1

∣∣ui (0)∣∣2 +

m∑j=1

∣∣vj (0)∣∣2 +

t∫0

eλsn∑

i=1

∣∣ui (s)∣∣2

×{

λ − 2ai +m∑

j=1

[|bji |

l∑k=1

Q

2αk,jqk

j qk + |bj i |l∑

k=1

M

2βk,jqk


i + |dij |L2δl+1,i

i eλτ + N2ij

]}ds

+t∫

0

eλs2n∑

i=1

m∑j=1

ui (s)Sji

(vj (s)

)dwj (s) +

t∫0

eλsm∑

j=1

∣∣vj (s)∣∣2

{λ − 2cj +

n∑i=1

[|dij |

l∑k=1

P

2γk,iqk

i qk + |dij |l∑

k=1

L

2δk,iqk

i qk

+ |bji |Q2αl+1,j


j eλσ + O2ji

]}ds +

t∫0

eλs2n∑

i=1

m∑j=1

vj (s)Sij

(ui(s)

)dwi(s)

(∗)+0∫

−σ

eλsn∑

i=1

m∑j=1

|bj i |M2βl+1,j

j eλσ∣∣vj (s)

∣∣2ds +

0∫−τ

eλsn∑

i=1

m∑j=1

|dij |L2δl+1,i

i eλτ∣∣ui (s)

∣∣2ds.


From Lemma 2.3, we obtain

eλt

[n∑

i=1

∣∣ui (t)∣∣2 +

m∑j=1

∣∣vj (t)∣∣2

]< ∞, P-a.s.,

which implies

lim supt→∞

1

tlog

(∥∥u(t; ξ) − u∗∥∥2 + ∥∥v(t;η) − v∗∥∥2) � −λ, P-a.s.

From (∗), we further obtain

E

[n∑

i=1

∣∣ui (t)∣∣2 +

m∑j=1

∣∣vj (t)∣∣2

]� e−λt

E

{n∑

i=1

∣∣ui (0)∣∣2 +

m∑j=1

∣∣vj (0)∣∣2 +

0∫−σ

eλsn∑

i=1

m∑j=1

|bj i |M2βl+1,j

j eλσ∣∣vj (s)

∣∣2ds

+0∫

−τ

eλs

n∑i=1

m∑j=1

|dij |L2δl+1,i

i eλτ∣∣ui (s)

∣∣2ds

}� Ce−λt

E

{n∑

i=1

∣∣ui (0)∣∣2 +

m∑j=1

∣∣vj (0)∣∣2

},

where C > 1 is some constant. The proof is complete. �Theorem 3.2. If system (1) satisfies the assumptions (A1), (A2), (A3) and

(A5) There exist constants αk,j , βk,j , γk,i , δk,i ∈ R, qk > 0, k = 1,2, . . . , l, i = 1,2, . . . , n, j = 1,2, . . . ,m, such that

−2ai +m∑

j=1

[Qj

l∑k=1

|bji |2αk,j

qk qk + Mj

l∑k=1

|bj i |2βk,j

qk qk + |dij |2γl+1,i Pi + |dij |2δl+1,i Li + N2ij

]< 0,

i = 1,2, . . . , n,

−2cj +n∑

i=1

[Pi

l∑k=1

|dij |2γk,iqk qk + Li

l∑k=1

|dij |2δk,iqk qk + |bji |2αl+1,j Qj + |bj i |2βl+1,j Mj + O2

ji

]< 0,

j = 1,2, . . . ,m,

where∑l+1

k=1 αk,j = ∑l+1k=1 βk,j = 1,

∑l+1k=1 γk,i = ∑l+1

k=1 δk,i = 1,∑l

k=1 qk = 1.


Proof. The proof is similar to that of Theorem 3.1. �Theorem 3.3. Suppose that system (1) satisfies the assumptions (A1), (A2), and (A3). Furthermore, if one of the following condi-tions holds

(H1)

−2ai +m∑

j=1

[|bji | + |bj i | + |dij |P 2i + |dij |L2

i + N2ij

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[|dij | + |dij | + |bji |Q2j + |bj i |M2

j + O2ji

]< 0, j = 1,2, . . . ,m.

(H2)

−2ai +m∑

j=1

[|bji |Q2j + |bj i |M2

j + |dij | + |dij | + N2ij

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[|dij |P 2i + |dij |L2

i + |bji | + |bj i | + O2ji

]< 0, j = 1,2, . . . ,m.


(H3)

−2ai +m∑

j=1

[Qj + Mj + |dij |2Pi + |dij |2Li + N2

ij

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[Pi + Li + |bji |2Qj + |bj i |2Mj + O2

ji

]< 0, j = 1,2, . . . ,m.

(H4)

−2ai +m∑

j=1

[Qj |bji |2 + Mj |bj i |2 + Pi + Li + N2

ij

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[Pi |dij |2 + Li |dij |2 + Qj + Mj + O2

ji

]< 0, j = 1,2, . . . ,m.


Proof. Let l = 1, αk,j = βk,j = γk,i = δk,i = 0, αl+1,j = βl+1,j = γl+1,i = δl+1,i = 1, k = 1,2, . . . , l, i = 1,2, . . . , n, j =1,2, . . . ,m, in (A4), then (A4) turns to (H1).

Let l = 1, αk,j = βk,j = γk,i = δk,i = 1, αl+1,j = βl+1,j = γl+1,i = δl+1,i = 0, k = 1,2, . . . , l, i = 1,2, . . . , n, j = 1,2, . . . ,m,in (A4), then (A4) turns to (H2).

Let l = 1, αk,j = βk,j = γk,i = δk,i = 0, αl+1,j = βl+1,j = γl+1,i = δl+1,i = 1, k = 1,2, . . . , l, i = 1,2, . . . , n, j = 1,2, . . . ,m,in (A5), then (A5) turns to (H3).

Let l = 1, αk,j = βk,j = γk,i = δk,i = 1, αl+1,j = βl+1,j = γl+1,i = δl+1,i = 0, k = 1,2, . . . , l, i = 1,2, . . . , n, j = 1,2, . . . ,m,in (A5), then (A5) turns to (H4). �Theorem 3.4. If system (1) satisfies the assumptions (A1), (A2), (A3) and

(A6)

−2ai +m∑

j=1

[|bji |Qj + |bj i |Mj + |dij |Pi + |dij |Li + N2ij

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[|dij |Pi + |dij |Li + |bji |Qj + |bj i |Mj + O2ji

]< 0, j = 1,2, . . . ,m.


Proof. From (4), we have

eλt∣∣ui (t)

∣∣2 �∣∣ui (0)

∣∣2 +t∫

0

eλs∣∣ui (s)

∣∣2(λ − 2ai) ds +

t∫0

eλs

{m∑

j=1

|bji |Qj

[∣∣ui (s)∣∣2 + ∣∣vj (s)

∣∣2]

+m∑

j=1

|bj i |Mj

[∣∣ui (s)∣∣2 + ∣∣vj (s − σji)

∣∣2]}ds +

m∑j=1

O2ji

t∫0

eλs∣∣vj (s)

∣∣2ds

+t∫

0

eλs2ui (s)

m∑j=1

Sji

(vj (s)

)dwj (s)

= ∣∣ui (0)∣∣2 +

t∫eλs

∣∣ui (s)∣∣2

{(λ − 2ai) +

m∑j=1

|bji |Qj +m∑

j=1

|bj i |Mj

}ds

0


+t∫

0

eλs

{m∑

j=1

[|bji |Qj +

m∑j=1

O2ji

]∣∣vj (s)∣∣2 +

m∑j=1

|bj i |Mj

∣∣vj (s − σji)∣∣2

}ds

+t∫

0

eλs2ui (s)

m∑j=1

Sji

(vj (s)

)dwj (s).

The rest is similar to the proof of Theorem 3.1. �As the corollary of the above results, we have

Corollary 3.5. Suppose that system (2) satisfies the assumptions (A1) and (A2). Then the equilibrium (u∗, v∗)T of system (2) isglobally exponentially stable provided that one of the following conditions holds:

(A4′) There exist constants αk,j , βk,j , γk,i , δk,i ∈ R, qk > 0, k = 1,2, . . . , l, i = 1,2, . . . , n, j = 1,2, . . . ,m, such that

−2ai +m∑

j=1

[|bji |

l∑k=1

Q

2αk,jqk

j qk + |bj i |l∑

k=1

M

2βk,jqk


i + |dij |L2δl+1,i

i

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[|dij |

l∑k=1

P

2γk,iqk

i qk + |dij |l∑

k=1

L

2δk,iqk



j

]< 0, j = 1,2, . . . ,m,

where∑l+1

k=1 αk,j = ∑l+1k=1 βk,j = 1,

∑l+1k=1 γk,i = ∑l+1

k=1 δk,i = ∑lk=1 qk = 1.

(A5′) There exist constants αk,j , βk,j , γk,i , δk,i ∈ R, qk > 0, k = 1,2, . . . , l, i = 1,2, . . . , n, j = 1,2, . . . ,m, such that

−2ai +m∑

j=1

[Qj

l∑k=1

|bji |2αk,j

qk qk + Mj

l∑k=1

|bj i |2βk,j

qk qk + |dij |2γl+1,i Pi + |dij |2δl+1,i Li

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[Pi

l∑k=1

|dij |2γk,iqk qk + Li

l∑k=1

|dij |2δk,iqk qk + |bji |2αl+1,j Qj + |bj i |2βl+1,j Mj

]< 0, j = 1,2, . . . ,m,

where∑l+1

k=1 αk,j = ∑l+1k=1 βk,j = 1,

∑l+1k=1 γk,i = ∑l+1

k=1 δk,i = 1,∑l

k=1 qk = 1.

(A6′)

−2ai +m∑

j=1

[|bji |Qj + |bj i |Mj + |dij |Pi + |dij |Li

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[|dij |Pi + |dij |Li + |bji |Qj + |bj i |Mj

]< 0, j = 1,2, . . . ,m.

(H1′)

−2ai +m∑

j=1

[|bji | + |bj i | + |dij |P 2i + |dij |L2

i

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[|dij | + |dij | + |bji |Q2j + |bj i |M2

j

]< 0, j = 1,2, . . . ,m.

(H2′)

−2ai +m∑

j=1

[|bji |Q2j + |bj i |M2

j + |dij | + |dij |]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[|dij |P 2i + |dij |L2

i + |bji | + |bj i |]< 0, j = 1,2, . . . ,m.


(H3′)

−2ai +m∑

j=1

[Qj + Mj + |dij |2Pi + |dij |2Li

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[Pi + Li + |bji |2Qj + |bj i |2Mj

]< 0, j = 1,2, . . . ,m.

(H4′)

−2ai +m∑

j=1

[Qj |bji |2 + Mj |bj i |2 + Pi + Li

]< 0, i = 1,2, . . . , n,

−2cj +n∑

i=1

[Pi |dij |2 + Li |dij |2 + Qj + Mj

]< 0, j = 1,2, . . . ,m.

Acknowledgements

This work was supported in part by the Natural Science Foundation of China (No. 10171059) and the Foundation for UniversityKey Teachers of the Ministry of Education of China.

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