research article approximate analyzing of labeled...

Research ArticleApproximate Analyzing of Labeled Transition Systems

Qiong Yu1 Shihan Yang2 and Jinzhao Wu3

1School of Science Guangxi University for Nationalities No 188 East Daxue Road Nanning Guangxi 530006 China2School of Information Science and Engineering Guangxi University for Nationalities No 188 East Daxue Road NanningGuangxi 530006 China3Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis No 188 East Daxue Road NanningGuangxi 530006 China

Correspondence should be addressed to Shihan Yang dryangshgmailcom

Received 11 June 2015 Accepted 6 July 2015

Academic Editor Xiaoyu Song

Copyright copy 2015 Qiong Yu et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As the most important formal semantic model labeled transition systems are widely used which can describe the generalconcurrent systems or control systems without disturbance However under normal circumstance transition systems are complexand difficult to use due to large amount of calculation and the state space explosion problems In order to overcome theseproblems approximate equivalent labeled transition systems are proposed by means of incomplete low-up matrix decompositionfactorizationThis technique can reduce the complexity of computation and calculate under the allowing errors As for continuous-time linear systems we develop a modeling method of approximated transition system based on the approximate solution ofmatrix which provides a facility for approximately formal semantic modeling for linear systems and to effectively analyze errorsAn example of application in the context of linear systems without disturbances is studied

1 Introduction

Transition system is an important formal model of concur-rent systems which is widely used in concurrency theorysuch as petri nets and process algebras Petri nets can beused as a graphical tool to simulate the dynamic behaviorand concurrent activities of the systems and also can beused as a mathematical tool to establish mathematical modeldescribing the behaviors of the systems It is a widelytransition system and gives a method to describe Petri netusing classical transition system [1] Apart from Petri nettheory there is a process algebra which is used to describeand study concurrent or distributed system behavior theoryby algebraic methods Thus many process algebraic systemsare proposed Among them the first is to propose thefollowing three process algebra systems CCS (Calculus ofCommunicating Systems) proposed by Milner [2 3] CSP(Communicating Sequential Processes) proposed by Hoare[4] and ACP (Algebra of Communicating Processes) pro-posed by Baeten andWeijland [5] In recent years some newprocess calculus systems have been proposed For example

probabilistic process algebra [6ndash12] and stochastic processalgebra [13] are proposed to describe possible or randominformation real-time process algebra [14ndash17] is proposed todescribe the action time In the field of process algebra asthe most important semantic model the transition system iswidely used especially labeled transition system [18] whichcan describe the general behavior of concurrent systems

Approximation of transition systems has recently beenintroduced as a powerful tool for the approximation of con-tinuous systemsThe notion of simulation is one such formalnotion of abstraction that has been used for reducing thecomplexity of finite state systems such as labeled transitionsystems Approximation of purely discrete systems has tra-ditionally been based on language inclusion and equivalencewith notions such as simulation or bisimulation relations [19]These concepts have been useful for simplifying problemssuch as safety verification or controller synthesis Howeverin the cases that the quality requirements of simulationor bisimulation are so exacting sometimes it is so difficultfor us to construct finite abstracts with the original systemsimulation or bisimulation Although finite abstract with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 963597 9 pageshttpdxdoiorg1011552015963597

2 Mathematical Problems in Engineering

the original system approximate simulation or approximatebisimulation is easy to construct these methods can onlyensure that the system approximately satisfies the given rangeand is not able to calculate the allowable error of transitionbetween states To solve this problem this paper puts forwarda design method for transition systemmdashapproximate simula-tion method based on ILU

As a particular class of preconditioners incomplete fac-torizations can be thought of as approximating the exact 119871119880factorization of a given matrix (eg computed via Gaussianelimination) by disallowing certain filling of this As opposedto other PDE-based preconditioners such as multigrid anddomain decomposition this class of preconditioners is pri-marily algebraic in nature and can be applied to any matricesin principle However one of the basic methods of precondi-tioned method is the use of incomplete 119871119880 decompositionThe most well known is the incomplete triangular decom-position ILU0 [20] that has nothing more than zero-elementfilling Since ILU0 unchanged the structure of sparse matrixthe application of ILU0 is very convenient However becauseof this it is so rough approximation of the coefficient matrixthat the effectiveness of ILU0 is limited To improve theapproximation filling with nonzero elements is allowed [2122] While the ILU preconditioner works quite well for manyproblems it does not perform well for some PDE problems[23] Then the generalization of this modified ILU (MILU)preconditioner is proposed by Gustafsson [24] MILU is toadd the fill-ins abandoned by ILU back to the diagonal typewhich makes up for inaccuracies caused by discarded Andits accuracy can be calculated However mathematically thetransition between states can be described by matrix andthen the allowable error of the approximate transition isneeded to be characterized Furthermore the allowable errorcan be defined by the MILU accuracy

In the rest of this paper Section 2 reviews the approximatesimulation relations of transition systems In Section 3 theapproximate and incomplete factorizations are recalled InSection 4 the approximate equivalent of labeled transitionsystems is proposed in detail Section 5 proposes the theoryof the approximate labeled transition systems In Section 6 acase is studied Then Section 7 draws the conclusion

2 Approximation of Transition Systems

21 Labeled Transition Systems We know that labeled tran-sition systems allow modeling in a unified frameworkdiscrete continuous and hybrid systems The main results[25] are reviewed here

Definition 1 A labeled transition system with observation isa tuple 119879 = (119876 Σ rarr 119876

0 Π(sdot)) that consists of

(i) a set 119876 of states(ii) a set Σ of labels(iii) a transition relation rarrsube 119876 times Σ times 119876(iv) a set 1198760

sube 119876 of initial states(v) a set Π of observations(vi) an observation map(sdot) 119876 rarr Π

A state trajectory of 119879 is a sequence of transitions

1199020 120590

0

997888rarr 1199021 120590

1

997888rarr 1199022 120590

2

997888rarr sdot sdot sdot

where 1199020 isin 1198760 119902119894

isin 119876 120590119894minus1

isin Σ 119894 = 1 2 (1)

For a given initial state and sequence of labels theremay exist several state trajectories of 119879 Thus the systemswe consider are possibly nondeterministic The associatedexternal trajectory

1205870 120590

0

997888rarr 1205871 120590

1

997888rarr 1205872 120590

2


where 120587119894 = (119902119894) 119894 = 0 1 2 (2)

describes the evolution of the observations under the dynam-ics of the labeled transition system

The set of external trajectories of the labeled system 119879 iscalled the language of 119879 and is denoted by 119871(119879) The subsetof Π reachable by the external trajectories of 119879 is denoted byReach(119879)

Reach (119879) = 120587 isinΠ | exist1205870 120590

0

997888rarr 1205871 120590

1

997888rarr 1205872 120590

2


isin 119871 (119879) exist119895 isinN 120587119895

=120587

(3)

where N is the set of positive integersAn important problem for transition systems is the safety

verification problem which consists in checking whether thereachable set Reach(119879) intersects a set of observations Π

119880

associated with unsafe states

22 Approximate Simulation Relations Exact simulationrelations between two labeled transition systems require thattheir observations are (and remain) identical Approximatesimulation relations are less rigid since they only require thatthe observations of both systems are (and remain) arbitrarilyclose

Let 1198791 = (1198761 Σ1 rarr 1 11987601 Π1(sdot)1) and 1198792 =

(1198762 Σ2 rarr 2 11987602 Π2(sdot)2) be two labeled transition systems

with the same set of labels (Σ1 = Σ2 = Σ) and the same setof observations (Π1 = Π2 = Π) (ie 1198791 and 1198792 are elementsof 119879(ΣΠ)) Let us assume that the set of observation Π is ametric space 119889

Πdenotes the metric of Π

Definition 2 A metric of a set 119860 is a positive function 119889

119860 times 119860 rarr R cup +infin whereR is the set of real numbers suchthat the three following properties hold for all 119909 119910 119911 isin 119860

(1) 119889(119909 119910) = 0 if and only if 119909 = 119910(2) 119889(119909 119910) = 119889(119910 119909)(3) 119889(119909 119911) le 119889(119909 119910) + 119889(119910 119911)

If the second property is replaced by 119909 = 119910 rArr 119889(119909 119910) =

0 then 119889 is called a pseudometric If the third property isdropped then 119889 is called a directed metric

Mathematical Problems in Engineering 3

Definition 3 A relation 119878120575sube 1198761 times 1198762 is a 120575-approximate

simulation relation of 1198791 by 1198792 if for all (1199021 1199022) isin 119878120575

(1) 119889Π((1199021)1(1199022)2) le 120575

(2) forall1199021120590

997888rarr11199021015840

1 exist1199022120590

997888rarr21199021015840

2 such that (11990210158401 1199021015840

2) isin 119878120575

Note that in there rarr 1 describes a transition relation of1198791 rarr 2 describes a transition relation of 1198792 119902

1015840

1 representsthe next state of 1199021 119902

1015840

2 represents the next state of 1199022 and120590 isin Σ And that for 120575 = 0 we have the usual notion of exactsimulation relation

Definition 4 1198792 approximately simulates 1198791 with the preci-sion 120575 (noted 1198791 ⪯120575 1198792) if there exists 119878

120575 a 120575-approximate

simulation relation of1198791 by1198792 such that for all 1199021 isin 11987601 there

exists 1199022 isin 11987602 such that (1199021 1199022) isin 119878120575

If1198792 approximately simulates1198791 with the precision 120575 thenthe language of 1198791 is approximated with precision 120575 by thelanguage of 1198792

Theorem 5 If 1198791 ⪯120575 1198792 then for all external trajectories of 1198791

120587011205900

997888rarr 120587111205901

997888rarr 120587211205902

997888rarr sdot sdot sdot (4)

there exists an external trajectory of 1198792 with the same sequenceof labels

120587021205900

997888rarr 120587121205901

997888rarr 120587221205902


such that for all 119894 isin N 119889Π(120587119894

1 120587119894

2) le 120575

In Theorem 5 if 1198792 approximately simulates 1198791 withthe precision 120575 there must exist a state of 1198792 approximatesimulation the state of 1198791 But it is very difficult for us tocalculate the degree of approximation

3 Approximate and Incomplete Factorizations

The general problem of finding a preconditioner for a largesparse linear system 119860119909 = 119887 is to find a matrix 119872 (thepreconditioner) However the matrix 119872 should be with thefollowing properties

(i) 119872 is a nonsingular matrix(ii) 119872 is a good approximation to 119860 in some sense(iii) the system 119872119909 = 119887 is much easier to solve than the

original system 119860119909 = 119887(iv) the construction of 119872 has some memory and CPU

needs protection(v) the condition number of the119872minus1119860 is much less than

that of 119860 and its minimum singular value becomeslarger

That is we are looking for a nearby problem whichis easier to solve than the given one The idea is tolook for a matrix 119872 such that the original linear system119860119909 = 119887 is transformed into an equivalent linear system

119872minus1119860119909 = 119872

minus1119887 Mathematically we will solve the precon-

ditioned system 119872minus1119860119909 = 119872

minus1119887 (or the symmetric version

(119872minus12

119860119872minus12

)(11987212119909) = 119872

minus12119887 when both 119860 and 119872 are

symmetric positive definite) by standard iterative methods(such as the conjugate gradient method) in which only theactions of 119860 and119872minus1 are needed

There are several preconditions matrix splitting pre-conditioner polynomial preconditioners incomplete factor-ization preconditioners approximate inverse precondition-ers and multilevel preconditioners Here we mainly focuson incomplete factorization preconditioners which can beapplied to a general purpose They can be thought of asmodifications of Gaussian Elimination in which some sparsemode is required as the decomposition of matrices such asmaking some of the special position of the elements be zeroand also abandoning some absolute value in the process ofdecomposition of minor elements in order to ensure thatsparse decomposition

31 Incomplete Low-Up Matrix Decomposition (ILU) Themost common type of incomplete factorization is based ontaking a set 119878 of matrix positions and keeping all positionsoutside this set equal to zero during the factorization Theresulting factorization is incomplete in the sense that fill issuppressed

Definition 6 A matrix 119860 if 119860 = 119871119880 where 119871 (119880) is thelower (upper) triangular sparse matrix is called a completedecomposition of the matrix 119860 that is 119871119880 [26]

Definition 7 A matrix 119860 if 119860 = 119871119880 + 119877 where 119871 (119880)is the lower (upper) triangular sparse matrix is called anincomplete decomposition of the matrix 119860 that is ILU Here119877 is called error matrix [27]

Let the allowable fill-in positions be given by the index set119878 that is

(1) 119897119894119895= 0 if 119895 gt 119894 or (119894 119895) isin 119878 119906

119894119895= 0 if 119894 gt 119895 or (119894 119895) isin 119878

A commonly used strategy is to define 119878 by

(2) 119878 = (119894 119895) | 119886119894119895

= 0

That is the only nonzeros allowed in the 119871119880 factors arethose for which the corresponding entries in 119860 are nonzeroLet the preconditioner 119872 be defined by the product of theresulting 119871119880 factors that is119872 = 119871119880 If119872 becomes a goodpreconditioner itmust be a good approximation to119860 in somemeasure 119860 typical strategy is to require the entries of 119872 tomatch those of 119860 on the set 119878

(3) 119898119894119895= 119886119894119895if (119894 119895) isin 119878

Even though conditions (1) and (3) together are sufficient(for certain classes of matrices) to determine the nonzeroentries of 119871 and 119880 directly it is more natural and simplerto compute these entries based on a simple modification ofthe Gaussian elimination algorithmMatrix119860 is decomposedinto 119871 and 119880 by the incomplete 119871119880 And the correspondingentries of 119871 and 119880 are the same as those obtained by settingthose entries of the full 119871119880 factors which belong to 119878


(1) for 119903 = 1 Step 119894 until 119899 minus 1 do(2) 119889 = 1119886

119903119903

(3) for 119894 = (119903 + 1) Step (1) until 119899 do(4) if (119894 119903) isin 119878 then(5) 119890 = 119889119886

119894119903 119886119894119903= 119890

(6) for 119895 = (119903 + 1) Step (1) until 119899 do(7) if (119894 119895) isin 119878 and (119903 119895) isin 119878 then(8) 119886

119894119895= 119886119894119895minus 119890119886119903119895

(9) end if(10) end for (119895 minus loop)(11) end if(12) end for (119894 minus loop)(13) end for (119903 minus loop)

Algorithm 1 Algorithm ILU

However the incomplete 119871119880 factors of 119860 respectively withthe lower and upper triangular part of the matrix 119860 have thesame nonzero structure

The constructing process of ILU precondition is in theprocess of Gaussian elimination by discarding partially filledelements to get sparse triangular matrixes 119871 and 119880 such that119872 = 119871119880 From the decomposition of ILU we can see that thepreconditioners 119872 and 119860 have the same nonzero structureTherefore the error matrix 119877 = 119860 minus 119871119880 = 119860 minus 119872 and themain difference from the usual Gaussian elimination lie inhow to discard fill-ins in the Gaussian elimination

32 Modified Incomplete Low-Up Matrix Decomposition(MILU) While the ILU preconditioner works quite well formany problems it does not perform well for some problemsWe will next describe the generalization of this modified ILU(MILU) preconditioner to a general matrix 119860 [28]

The basic idea is that in condition (3) for ILU thecondition119898

119894119894= 119886119894119894is removed and a new row sum condition

is added That is (3) is replaced by

(4) sum119899119895=1119898119894119895 = sum

119899

119895=1 119886119894119895 forall119894 and 119898119894119895 = 119886119894119895

if 119894 = 119895 and(119894 119895) isin 119878

Again for certain classes of matrices conditions (4) and(1) are sufficient to determine the119871119880 factors inMILUdirectlyHowever in practice it is easier to compute these 119871119880 factorsby a modification of the ILU algorithm instead of droppingthe disallowed fill-ins in the ILU algorithm these terms areadded to the diagonal of the same row Furthermore the ideaof MILU is to make up for inaccuracies caused by discardedby using fill-ins plus backup to cover the diagonal type

Incomplete factorization preconditioners can be por-trayed accuracy and stability Here accuracy refers to thedegree of preconditioner 119872 and matrix 119860 can be measuredby the size of 119872 minus 119860

119865 And stability refers to the degree of

preconditioner119872 and the unit matrix 119864 can be measured bythe size of 119864 minus 119872

minus1119860119865(for the left preconditioner) Below

we give algorithm ILU and algorithmMILU [29]From the analysis of Algorithms 1 and 2 the difference

between algorithm ILU andMILU is that certain fill-ins givenup in algorithm ILU are back to the main diagonal

(1) for 119903 = 1 Step (1) until 119899 minus 1 do(2) 119889 = 1119886

119903119903


119894119903 119886119894119903= 119890

(6) for 119895 = (119903 + 1) Step (1) until 119899 do(7) if (119903 119895) isin 119878 then(8) if (119894 119895) isin 119878 then(9) 119886

119894119895= 119886119894119895minus 119890119886119903119895

(10) else(11) 119886

119894119894= 119886119894119894minus 119890119886119903119895

(12) end if(13) end if(14) end for (119895 minus loop)(15) end if(16) end for (119894 minus loop)(17) end for (119903 minus loop)

Algorithm 2 Algorithm MILU

4 Equivalent Labeled Transition Systems

In labeled transition systems we can assume that Σ is a finiteset of labels a label120590 isin Σ describes an action rarrsube 119876timesΣtimes119876 isa transition relation where119876 is a set of states Each transitionrarr is a tuple ⟨119902 120590 1199021015840⟩ where 119902 and 1199021015840 represent the pre- andpoststates of the transition respectively A state trajectory oflabeled transition systems is a sequence of transitions

1199020 120590

0

997888rarr 1199021 120590

1

997888rarr 1199022 120590

2


where 1199020 isin 1198760 is the initial state and 1198760sube 119876 is the initial set

of statesFor a given initial state and sequence of labels there

may be several state trajectories of labeled transition systemsThus the systems we consider are possibly nondeterministic

Here we mainly consider the set Σ of labels Let label 120590 isinΣ has the following form

1 = 11988611 (119905) 1199091 + 11988612 (119905) 1199092 + sdot sdot sdot + 1198861119899 (119905) 119909119899

2 = 11988621 (119905) 1199091 + 11988622 (119905) 1199092 + sdot sdot sdot + 1198862119899 (119905) 119909119899

119899= 1198861198991 (119905) 1199091 + 1198861198992 (119905) 1199092 + sdot sdot sdot + 119886119899119899 (119905) 119909119899

(7)

It can be abbreviated as = 119860(119905)119909 where =

(1 2 sdot sdot sdot 119899)119879 119909 = (1199091 1199092 sdot sdot sdot 119909

119899)119879 and

119860 (119905) =(

11988611 (119905) 11988612 (t) sdot sdot sdot 1198861119899 (119905)

11988621 (119905) 11988622 (119905) sdot sdot sdot 1198862119899 (119905)

1198861198991 (119905) 119886

1198992 (119905) sdot sdot sdot 119886119899119899(119905)

) (8)


We can obtain the preconditioner119872(119905) of matrix 119860(119905) bymeans of ILU We assume the preconditioner119872(119905) is

119872(119905) =(

11988711 (119905) 11988712 (119905) sdot sdot sdot 1198871119899 (119905)

11988721 (119905) 11988722 (119905) sdot sdot sdot 1198872119899 (119905)

1198871198991 (119905) 119887

1198992 (119905) sdot sdot sdot 119887119899119899(119905)

) (9)

According to Section 3 we can know that the equivalentequation of = 119860(119905)119909 is 119872minus1(119905)119860(119905)119909 = 119872

minus1(119905) Let

1205901015840 be symbol of 119872minus1(119905)119860(119905)119909 = 119872

minus1(119905) That is 1205901015840

119872minus1(119905)119860(119905)119909 = 119872

minus1(119905) So label 120590 and label 1205901015840 are

equivalent

Definition 8 Label 1205901015840 119872minus1(119905)119860(119905)119909 = 119872

minus1(119905) is the

equivalent label of label 120590 = 119860(119905)119909 where 119909 and arethe pre- and poststate values of the transition if matrix119872(119905)

is the preconditioner matrix of matrix 119860(119905)The labeled transition system can only be used for

symbolic reasoning Two labels are equivalent if they havethe same form syntax and semantics Here Definition 8gives the equivalent of two labels It has the characteristic thatlabelsrsquo equations are equivalent instead of having the sameform syntax and semantics

For all labels 120590119894isin Σ 119894 isin N where N is the set of

positive integers 120590119894 = 119860

119894(119905)119909 its equivalent label is

1205901015840

119894 119872minus1119894(119905)119860119894(119905)119909 = 119872

minus1119894(119905) where matrix 119872

119894(119905) is the

preconditioner matrix of matrix 119860119894(119905) Let Σ1015840 = 120590

1015840

119894| 119894 isin N

Definition 9 Set Σ = 120590119894| 119894 isin N of labels and set Σ1015840 = 120590

1015840

119894|

119894 isin N are equivalent if forall120590119894isin Σ there exists 119895 isin Nmaking 120590

119894

and 120590119895equivalent

A labeled transition system with observation is a tuple119879 = (119876 Σrarr 1198760

Π(sdot)) Let1198791015840 = (119876 Σ1015840

rarr 1198760 Π(sdot))

Definition 10 Labeled transition system with observation119879 and labeled transition system with observation 119879

1015840 areequivalent if set Σ of labels and set Σ1015840 of labels are equivalentOne labeled transition system is called equivalent system ofanother labeled transition system

Here we only study labels of labeled transition systems

5 Approximate Labeled Transition System

In general we know that a labeled transition system withobservations is used to prescribe control systems In thissection our goal is to study = 119860119909 (the transitionrelationship between states)

The purpose of this paper is to characterize approximatesimilar transition systems that are generated by continuous-time linear systems which are conceptually similar to thetransition systems generated by discrete-time systems Webegin with studying continuous-time linear systems whosedynamics are closer to transition systems due to the existence

of an atomic time step [30] Consider a continuous-timelinear system without disturbances

119862 = 119860119909 (10)

where with time 119905 isin R+ state 119909(119905) isin R119899 and matrix 119860 of

approximate dimension given an initial condition 1199090

Definition 11 Consider continuous-time system 119862 given by = 119860119909 and observation set Π = 119861

119901 The transition system119879119903+

119862= (119876 Σ rarr Π(sdot)) generated by 119862 and Π consists of(i) state space 119876 = R119899(ii) label Σ = R

+

(iii) transition relation rarrsube 119876 times R+times 119876 defined as 119909 119905997888rarr

1199091015840

hArr exist119906[0119905] with 119909

1015840

= Φ119862(119905 119909 119906

[0119905])(iv) observation Π = 119861

119901(v) observation map(sdot) 119876 rarr ΠUsually the linear systems are presented by ordinary

differential equations partial differential equations and dif-ference equations

For 119905 isin R+ the transition relation is given by 119902 119905997888rarr 1199021015840 if

and only if there exists a function 119909(sdot) such that 119909(0) = 119902119909(119905) = 119902

1015840 and for almost all 119904 isin [0 119905] (119904) isin 119865(119909(119904)) where119865 is a valued map

51 Approximate Similar States Consider the followingcontinuous-time linear system of state 119902 translated to anotherstate 119902

1015840 without disturbances where the continuous-timelinear system is a nondeterministic continuous system Thatis 119902 120590997888rarr1119902

1015840

1 = 11988611 (119905) 1199091 + 11988612 (119905) 1199092 + sdot sdot sdot + 1198861119899 (119905) 119909119899

2 = 11988621 (119905) 1199091 + 11988622 (119905) 1199092 + sdot sdot sdot + 1198862119899 (119905) 119909119899

119899= 1198861198991 (119905) 1199091 + 1198861198992 (119905) 1199092 + sdot sdot sdot + 119886119899119899 (119905) 119909119899

(11)

and state value of the initial state 119902 is 119909(0) = (1198871 1198872 sdot sdot sdot 119887119899)

119905 isin [0 1199050]The state is observed through the variable 1205871(119905) = 1199091(119905)

and the transition relation is expressed as rarr 1We know that the equation is the linear differential

equations We introduce the following mark

119860 (119905) =(

11988611 (119905) 11988612 (119905) sdot sdot sdot 1198861119899 (119905)

11988621 (119905) 11988622 (119905) sdot sdot sdot 1198862119899 (119905)

1198861198991 (119905) 119886

1198992 (119905) sdot sdot sdot 119886119899119899(119905)

) (12)

where 119860(119905) is an 119899 by 119899 matrix The above equations can beexpressed as

= 119860 (119905) 119909 (13)

where 119909 = (1199091 1199092 sdot sdot sdot 119909119899)119879 = (1 2 sdot sdot sdot

119899)119879


However 119860(119905) is a large sparse matrix for most mathe-matical models of practical problems in the real world so it isvery difficult for us to obtain the solution of the equation =119860(119905)119909Thenwe use ILU to calculate the approximate solutionThe preconditioner119872 = 119871119880 where 119871 is a triangular matrixand 119880 is an upper triangular matrix Hence the originallinear system = 119860(119905)119909 is transformed into an equivalentlinear system 119872

minus1119860119909 = 119872


(119872minus12

119860119872minus12

)(11987212119909) = 119872


symmetric positive definite) Because system = 119860(119905)119909 is aportrait of the transition process from the state 119902 to anotherstate 1199021015840 the elements of 119860(119905) present the dynamic changesfrom state 119902 to another state 1199021015840 The preconditioner119872 is theapproximate dynamic changed from the state 119902 to anotherstate 1199021015840 Hence119860minus119872 an error matrix has the meaning thatstate 119902 can be translated into another state 1199021015840 in the error andits degree of approximation is 119872 minus 119860

119865 Then we can define

the approximate distance between the two states

Proposition 12 119889(119909 119910) = 119909 minus 119910119865is the distance between 119909

and 119910 where 119909 119910 isin 119883 and119883 is a linear space

Proof In fact for the norm axioms forall119909 119910 119911 isin 119883 119889(119909 119910) =119909 minus 119910

119865ge 0 and 119889(119909 119910) = 0 if and only if 119909 minus 119910

119865= 0 that

is 119909 = 119910What is more

119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119910 minus 119909

1003817100381710038171003817119865= 119889 (119910 119909)

119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119909 minus 119911 + 119911 minus119910

1003817100381710038171003817119865

le 119909minus 119911119865+1003817100381710038171003817119911 minus 119910

1003817100381710038171003817119865= 119889 (119909 119911) + 119889 (119911 119910)

(14)

Definition 13 forall119902 1199021015840 isin 119876 the approximate distance between119902 and 1199021015840 is

119889119876(119902 1199021015840

) = 119872minus119860119865 (15)

Let us assume that 1199021 1199022 1199023 are elements of 119876 and thestate trajectory is

11990211205901997888rarr 1199022

1205902997888rarr 1199023 (16)

The mathematical relationship of 11990211205901997888rarr 1199022 1199022

1205902997888rarr 1199023

is respective = 1198601(119905)119909 = 1198602(119905)119911 This system is acontinuous transition so a transition from state 1199021 to state1199023 can be expressed as 1199021

12059021205901997888997888997888rarr 1199023 This can be represented by

using Figure 1 A line with an arrow indicates state transitionwhere the end of a line without arrow points to the prestateand the end of a line with arrow points to the poststate a labelmarks on the line

Call 119889119876(119902 1199021015840

) = 119872minus119860119865as the degree of approximation

of state 119902 translated into state 1199021015840 Furthermore because 119909(0) =119902 119909(1199050) = 119902

1015840 the solution 119910 of the equivalent linear system119872minus1119860119909 = 119872

minus1 is the approximate solution of the original

q1

q1

q2

q2

q3

q3

1205901

1205901

1205902

1205902

12059021205901

Figure 1 Continuous transition of states

linear system = 119860(119905)119909 where 119905 isin [0 1199050] 119910(0) = 119902 119910(1199050) =1199011015840 The mathematical relationship of 119902 120590997888rarr 119901

1015840 is

119910 = 119860 (119905) 119910 (17)

where 119910 = (1199101 1199102 sdot sdot sdot 119910119899)119879 119910 = ( 1199101 1199102 sdot sdot sdot 119910

119899)119879 and state

value of initial state 119910(0) = (1198871 1198872 sdot sdot sdot 119887119899)

The state is observed through the variable 1205872(119905) = 1199101(119905)and the transition relation is expressed as rarr 2 Thereforethe accuracy between the exact solution 119909 and approximatesolution 119910 of the linear systems is 120575 = 119872 minus 119860

119865 That is

120575 = lim119905rarr+infin

119909 minus 119910119865= 119872 minus 119860

119865 And 119889

Π(1205871(119905) 1205872(119905)) =

1199091(119905) minus 1199101(119905)119865 le 119909 minus 119910119865= 120575

Theorem 14 Let 119910 be the approximate solution of = 119860(119905)119909then 119910 = 119860(119905)119910 simulates approximately = 119860(119905)119909 withprecision lim

119905rarr+infin119860(119905)

119865119872 minus 119860

119865

Proof

lim119905rarr+infin

1003817100381710038171003817 minus1199101003817100381710038171003817119865

= lim119905rarr+infin

1003817100381710038171003817119860 (119905) 119909 minus119860 (119905) 1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) (119909 minus 119910)1003817100381710038171003817119865

le lim119905rarr+infin

119860 (119905)119865

1003817100381710038171003817119909 minus1199101003817100381710038171003817119865


119860 (119905)119865

lim119905rarr+infin

1003817100381710038171003817119909 minus 1199101003817100381710038171003817119865


119860 (119905)119865119872minus119860

119865

(18)

So 119910 = 119860(119905)119910 simulates approximately = 119860(119905)119909 Andtherefore we can say the state 1199011015840 simulates approximately thestate 1199021015840

Here approximate simulation of states is controlled bylabels where labels are special Furthermore the degree ofapproximate similar states can be obtained by the approxi-mate solution of labelsrsquo mathematical models

52 Approximate Similar Linear Systems We have given anapproximate simulation method for a state Now we considerapproximate simulation of linear systems


Let the transition system generated by 119862 and Π beexpressed as 119879119903+

119862= (119876 Σ rarr Π(sdot)) Furthermore for any

state in119876we have given its approximate state that isforall119902119894isin 119876

119894 = 1 2 119902119894is approximately simulated by state 1199021015840

119894bymeans

of ILU and the equivalent label of label 120590119894 = 119860

119894(119905)119909

is 1205901015840119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the

preconditioner of matrix 119860119894(119905) Let 1198761015840 = 119902

1015840

119894| 119894 = 1 2

Σ1015840

= 1205901015840

119894| 119894 = 1 2 and then 1198791015840 = (119876

1015840

Σ1015840

rarr Π(sdot))

Definition 15 1198791015840 approximately simulates 119879 if there exists

1199021015840

119894and 120590

1015840

119894 119894 = 1 2 where state 1199021015840

119894is an approximate

simulation state of state 119902119894 label 1205901015840

119894is the equivalent label of

label 120590119894 One labeled transition system is called approximate

system of another labeled transition system

Theorem 16 The precision of the 1198791015840 approximate simulation119879 is

120575 = max119894

( lim119905rarr+infin

1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (19)

According to Theorem 14 we know that a state 1199021015840

can simulate approximately the state 119902 with the precisionlim119905rarr+infin

119860(119905)119865119872 minus 119860

119865 Thus for each 119902

119894isin 119876

exist1199021015840

119894isin 1198761015840 simulates approximately state 119902

119894with the precision

lim119905rarr+infin

119860119894(119905)119865119872119894minus 119860119894119865 This means that each element

of set 119876 has its approximate element with correspondingpercision But the set 1198761015840 generally approximates to the set 119876with some other precision Fortunately we know that in thissituation the precision between the set 119876 and the set 1198761015840 canbe defined by themaximumpercision of their elments So theprecision between set 1198761015840 of approximate states and set 119876 ofstates is

120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (20)

But because the precision of the 1198791015840 approximate simulation119879 is caused by set 1198761015840 of approximate state Theorem 16 isreasonable

6 A Case Study

Now let us show the approximate theorem by taking exampleof some of the computations by considering linear systemswithout disturbances Consider the linear systems = 119860119909where

(

1

2

3

) = (

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

) (21)

and the initial state is 119909(0) = (1 0 0)

The linear system is approximately simulated bymeans ofILU Firstly we let

119860 = (

1 2 11 minus1 12 0 1

) (22)

where 119860 is the matrix of the equations The 119871119880 factors of 119860are

1198711015840

= (

1 0 01 1 0

2 43

1)

1198801015840

= (

1 2 10 minus3 00 0 minus1

)

(23)

In terms of Algorithm 1 the ILU algorithm gives

119871 = (

1 0 01 1 02 0 1

)

119880 = (


)

(24)

which is different from setting the (3 2) element of the 119871119880factors to zero

Now we use the method of ILU to solve the problemThepreconditioner

119872 = 119871119880 = (

1 0 01 1 02 0 1

)(


)

= (

1 2 11 minus1 12 4 1

)

(25)

Then mathematically the equivalent system of the orig-inal linear system = 119860119909 is the preconditioned system119872minus1 = 119872

minus1119860119909 That is

(

1 2 11 minus1 12 4 1

)

minus1

(

1

2

3

)

= (

1 2 11 minus1 12 4 1

)

minus1

(

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

)

(26)


In the example we use ILU method to approximatesimulation states and the precise of approximate simulationstates can be calculated That is

119872minus119860 = (

1 2 11 minus1 12 4 1

)minus(

1 2 11 minus1 12 0 1

) = (

0 0 00 0 00 4 0

)

119872minus119860119865

= radic(02 + 02 + 02) + (02 + 02 + 02) + (02 + 42 + 02)

= 4

119860119865

= radic(12 + 22 + 12) + (12 + (minus1)2 + 12) + (22 + 02 + 12)

= radic14

(27)

So the precision of approximate simulation states is 120575 =

119860119865119872minus119860

119865= 4radic14 asymp 149666where the precision retains

four significant numbers after the decimal point If we wantto use MILU to solve the problem of approximate simulationstates we just deal with 119860 that is using Algorithm 2 todecomposition 119860

7 Conclusions

In this paper we extend the notion of approximate simu-lation relations The semantic analysis of equivalent labeledtransition systems is developed by equivalent labels And aneffective characterization of approximate simulation relationsof states based on ILU is also developed where error aboutthe state set approximate is controlled Furthermore thetechnique can be used to approximately simulate labeledtransition systems generated by continuous-time systemFinally a case study of application in the context of linearsystems without disturbances is shown

The approximate simulation relation is a key performanceconsideration of systems Our future work will focus onthese two items in two steps firstly developing methodsto approximate simulation linear systems with disturbancessecondly solving the nonlinear systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China under Grants nos 11371003 and 11461006 theNatural Science Foundation of Guangxi under Grants nos2014GXNSFAA118359 and 2012GXNSFGA060003 the Sci-ence andTechnology Foundation ofGuangxi underGrant no10169-1 and the Scientific Research Project no 201012MS274from the Guangxi Education Department

References

[1] J Kleijn M Koutny and M Pietkiewicz-Koutny ldquoRegionsof Petri nets with async connectionsrdquo Theoretical ComputerScience vol 454 pp 189ndash198 2012

[2] RMilnerACalculus of Communicating Systems Springer NewYork NY USA 1980

[3] RMilnerCommunication andConcurrency PrenticeHall NewYork NY USA 1989

[4] C A R Hoare Communicating Sequential Processes PrenticeHall 1985

[5] J C Baeten and W P Weijland Process Algebra CambridgeUniversity Press Cambridge UK 1990

[6] S Tini ldquoNon-expansive 120598-bisimulations for probabilistic pro-cessesrdquo Theoretical Computer Science vol 411 no 22-24 pp2202ndash2222 2010

[7] S Kramer C Palamidessi R Segala A Turrini and C BraunldquoA quantitative doxastic logic for probabilistic processes andapplications to information-hidingrdquo Journal of Applied Non-Classical Logics vol 19 no 4 pp 489ndash516 2009

[8] S Georgievska and S Andova ldquoTesting reactive probabilisticprocessesrdquo Electronic Proceedings in Theoretical Computer Sci-ence vol 28 pp 99ndash113 2010

[9] M Schroder and A Simpson ldquoRepresenting probability mea-sures using probabilistic processesrdquo Journal of Complexity vol22 no 6 pp 768ndash782 2006

[10] S Oh ldquoGeostatistical integration of seismic velocity andresistivity data for probabilistic evaluation of rock qualityrdquoEnvironmental Earth Sciences vol 69 no 3 pp 939ndash945 2013

[11] M Hennessy ldquoExploring probabilistic bisimulations part IrdquoFormalAspects of Computing vol 24 no 4ndash6 pp 749ndash768 2012

[12] S Andova S Georgievska and N Trcka ldquoBranching bisimu-lation congruence for probabilistic systemsrdquo Theoretical Com-puter Science vol 413 no 1 pp 58ndash72 2012

[13] WDu and S Tang ldquoOn the evolution rule for regional industrialstructure based on the stochastic process theoryrdquoManagementScience and Engineering vol 5 no 3 pp 55ndash60 2011

[14] A Hermann Reynisson M Sirjan L Aceto et al ldquoModellingand simulation of asynchronous real-time systems using timedRebecardquo Science of Computer Programming vol 89 pp 41ndash682014

[15] R Cardell-Oliver ldquoConformance tests for real-time systemswith timed automata specificationsrdquo Formal Aspects of Comput-ing vol 12 no 5 pp 350ndash366 2000

[16] M M Jaghoori F S de Boer T Chothia and M SirjanildquoSchedulability of asynchronous real-time concurrent objectsrdquoJournal of Logic and Algebraic Programming vol 78 no 5 pp402ndash416 2009

[17] K Lampka S Perathoner and L Thiele ldquoAnalytic real-timeanalysis and timed automata a hybrid methodology for theperformance analysis of embedded real-time systemsrdquo DesignAutomation for Embedded Systems vol 14 no 3 pp 193ndash2272010

[18] A Girard and G J Pappas ldquoApproximation metrics for discreteand continuous systemsrdquo Tech Rep Department of Computeramp Information Science 2005

[19] E M Clarke O Grumberg and D A Peled Model CheckingMIT Press 2000

[20] R Kechroud A Soulaimani Y Saad and S Gowda ldquoPrecondi-tioning techniques for the solution of the Helmholtz equationby the finite element methodrdquo Mathematics and Computers inSimulation vol 65 no 4-5 pp 303ndash321 2004


[21] E F DrsquoAzevedo P A Forsyth andW-P Tang ldquoTowards a cost-effective ILUpreconditioner with high level fillrdquoBIT vol 32 no3 pp 442ndash463 1992

[22] D P Young R G Melvin F T Johnson J E Bussoletti L BWigton and S S Samant ldquoApplication of sparse matrix solversas effective preconditionersrdquo SIAM Journal on Scientific andStatistical Computing vol 10 no 6 pp 1186ndash1199 1989

[23] T Dupont R P Kendall and J Rachford ldquoAn approximate fac-torization procedure for solving self-adjoint elliptic differenceequationsrdquo SIAM Journal onNumerical Analysis vol 5 pp 559ndash573 1968

[24] T Gustafsson ldquoA class of first-order factorizationmethodsrdquoBITNumerical Mathematics vol 18 pp 142ndash156 1978

[25] A Girard and G J Pappas ldquoApproximation metrics fordiscrete and continuous systemsrdquo Tech Rep MS-CIS-05-10Department of Computer Information Systems University ofPennsylvania 2005

[26] M A F Araghi and A Fallahzadeh ldquoInherited LU factorizationfor solving fuzzy system of linear equationsrdquo Soft Computingvol 17 no 1 pp 159ndash163 2013

[27] A El Kacimi andO Laghrouche ldquoWavelet based ILU precondi-tioners for the numerical solution by PUFEMof high frequencyelastic wave scatteringrdquo Journal of Computational Physics vol230 no 8 pp 3119ndash3134 2011

[28] G Ewald Combinatorial Convexity and Algebraic GeometrySpringer 1996

[29] O Axelsson Iterative Solution Methods Cambridge UniversityPress 1994

[30] G J Pappas ldquoBisimilar linear systemsrdquo Automatica vol 39 no12 pp 2035ndash2047 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the original system approximate simulation or approximatebisimulation is easy to construct these methods can onlyensure that the system approximately satisfies the given rangeand is not able to calculate the allowable error of transitionbetween states To solve this problem this paper puts forwarda design method for transition systemmdashapproximate simula-tion method based on ILU

As a particular class of preconditioners incomplete fac-torizations can be thought of as approximating the exact 119871119880factorization of a given matrix (eg computed via Gaussianelimination) by disallowing certain filling of this As opposedto other PDE-based preconditioners such as multigrid anddomain decomposition this class of preconditioners is pri-marily algebraic in nature and can be applied to any matricesin principle However one of the basic methods of precondi-tioned method is the use of incomplete 119871119880 decompositionThe most well known is the incomplete triangular decom-position ILU0 [20] that has nothing more than zero-elementfilling Since ILU0 unchanged the structure of sparse matrixthe application of ILU0 is very convenient However becauseof this it is so rough approximation of the coefficient matrixthat the effectiveness of ILU0 is limited To improve theapproximation filling with nonzero elements is allowed [2122] While the ILU preconditioner works quite well for manyproblems it does not perform well for some PDE problems[23] Then the generalization of this modified ILU (MILU)preconditioner is proposed by Gustafsson [24] MILU is toadd the fill-ins abandoned by ILU back to the diagonal typewhich makes up for inaccuracies caused by discarded Andits accuracy can be calculated However mathematically thetransition between states can be described by matrix andthen the allowable error of the approximate transition isneeded to be characterized Furthermore the allowable errorcan be defined by the MILU accuracy

In the rest of this paper Section 2 reviews the approximatesimulation relations of transition systems In Section 3 theapproximate and incomplete factorizations are recalled InSection 4 the approximate equivalent of labeled transitionsystems is proposed in detail Section 5 proposes the theoryof the approximate labeled transition systems In Section 6 acase is studied Then Section 7 draws the conclusion

2 Approximation of Transition Systems

21 Labeled Transition Systems We know that labeled tran-sition systems allow modeling in a unified frameworkdiscrete continuous and hybrid systems The main results[25] are reviewed here

Definition 1 A labeled transition system with observation isa tuple 119879 = (119876 Σ rarr 119876

0 Π(sdot)) that consists of

(i) a set 119876 of states(ii) a set Σ of labels(iii) a transition relation rarrsube 119876 times Σ times 119876(iv) a set 1198760

sube 119876 of initial states(v) a set Π of observations(vi) an observation map(sdot) 119876 rarr Π

A state trajectory of 119879 is a sequence of transitions

1199020 120590

0

997888rarr 1199021 120590

1

997888rarr 1199022 120590

2


where 1199020 isin 1198760 119902119894

isin 119876 120590119894minus1

isin Σ 119894 = 1 2 (1)

For a given initial state and sequence of labels theremay exist several state trajectories of 119879 Thus the systemswe consider are possibly nondeterministic The associatedexternal trajectory

1205870 120590

0

997888rarr 1205871 120590

1

997888rarr 1205872 120590

2


where 120587119894 = (119902119894) 119894 = 0 1 2 (2)

describes the evolution of the observations under the dynam-ics of the labeled transition system

The set of external trajectories of the labeled system 119879 iscalled the language of 119879 and is denoted by 119871(119879) The subsetof Π reachable by the external trajectories of 119879 is denoted byReach(119879)

Reach (119879) = 120587 isinΠ | exist1205870 120590

0

997888rarr 1205871 120590

1

997888rarr 1205872 120590

2


isin 119871 (119879) exist119895 isinN 120587119895

=120587

(3)

where N is the set of positive integersAn important problem for transition systems is the safety

verification problem which consists in checking whether thereachable set Reach(119879) intersects a set of observations Π

119880

associated with unsafe states

22 Approximate Simulation Relations Exact simulationrelations between two labeled transition systems require thattheir observations are (and remain) identical Approximatesimulation relations are less rigid since they only require thatthe observations of both systems are (and remain) arbitrarilyclose

Let 1198791 = (1198761 Σ1 rarr 1 11987601 Π1(sdot)1) and 1198792 =

(1198762 Σ2 rarr 2 11987602 Π2(sdot)2) be two labeled transition systems

with the same set of labels (Σ1 = Σ2 = Σ) and the same setof observations (Π1 = Π2 = Π) (ie 1198791 and 1198792 are elementsof 119879(ΣΠ)) Let us assume that the set of observation Π is ametric space 119889

Πdenotes the metric of Π

Definition 2 A metric of a set 119860 is a positive function 119889

119860 times 119860 rarr R cup +infin whereR is the set of real numbers suchthat the three following properties hold for all 119909 119910 119911 isin 119860

(1) 119889(119909 119910) = 0 if and only if 119909 = 119910(2) 119889(119909 119910) = 119889(119910 119909)(3) 119889(119909 119911) le 119889(119909 119910) + 119889(119910 119911)

If the second property is replaced by 119909 = 119910 rArr 119889(119909 119910) =

0 then 119889 is called a pseudometric If the third property isdropped then 119889 is called a directed metric




(1) 119889Π((1199021)1(1199022)2) le 120575

(2) forall1199021120590

997888rarr11199021015840

1 exist1199022120590

997888rarr21199021015840

2 such that (11990210158401 1199021015840

2) isin 119878120575


1015840


1015840








120587011205900

997888rarr 120587111205901

997888rarr 120587211205902



120587021205900

997888rarr 120587121205901

997888rarr 120587221205902



1 120587119894

2) le 120575









119872minus1119860119909 = 119872




(119872minus12

119860119872minus12

)(11987212119909) = 119872








(1) 119897119894119895= 0 if 119895 gt 119894 or (119894 119895) isin 119878 119906

119894119895= 0 if 119894 gt 119895 or (119894 119895) isin 119878


(2) 119878 = (119894 119895) | 119886119894119895

= 0


(3) 119898119894119895= 119886119894119895if (119894 119895) isin 119878




119903119903


119894119903 119886119894119903= 119890


119894119895= 119886119894119895minus 119890119886119903119895









(4) sum119899119895=1119898119894119895 = sum

119899

119895=1 119886119894119895 forall119894 and 119898119894119895 = 119886119894119895

if 119894 = 119895 and(119894 119895) isin 119878









119903119903


119894119903 119886119894119903= 119890


119894119895= 119886119894119895minus 119890119886119903119895

(10) else(11) 119886

119894119894= 119886119894119894minus 119890119886119903119895





1199020 120590

0

997888rarr 1199021 120590

1

997888rarr 1199022 120590

2






1 = 11988611 (119905) 1199091 + 11988612 (119905) 1199092 + sdot sdot sdot + 1198861119899 (119905) 119909119899

2 = 11988621 (119905) 1199091 + 11988622 (119905) 1199092 + sdot sdot sdot + 1198862119899 (119905) 119909119899

119899= 1198861198991 (119905) 1199091 + 1198861198992 (119905) 1199092 + sdot sdot sdot + 119886119899119899 (119905) 119909119899

(7)



119899)119879 and

119860 (119905) =(

11988611 (119905) 11988612 (t) sdot sdot sdot 1198861119899 (119905)

11988621 (119905) 11988622 (119905) sdot sdot sdot 1198862119899 (119905)

1198861198991 (119905) 119886

1198992 (119905) sdot sdot sdot 119886119899119899(119905)

) (8)



119872(119905) =(

11988711 (119905) 11988712 (119905) sdot sdot sdot 1198871119899 (119905)

11988721 (119905) 11988722 (119905) sdot sdot sdot 1198872119899 (119905)

1198871198991 (119905) 119887

1198992 (119905) sdot sdot sdot 119887119899119899(119905)

) (9)


minus1(119905) Let


minus1(119905) That is 1205901015840

119872minus1(119905)119860(119905)119909 = 119872


equivalent









1205901015840

119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 isin N


1015840

119894|


119894



Π(sdot)) Let1198791015840 = (119876 Σ1015840









119862 = 119860119909 (10)






+


1199091015840

hArr exist119906[0119905] with 119909

1015840

= Φ119862(119905 119909 119906

[0119905])(iv) observation Π = 119861








1015840

1 = 11988611 (119905) 1199091 + 11988612 (119905) 1199092 + sdot sdot sdot + 1198861119899 (119905) 119909119899

2 = 11988621 (119905) 1199091 + 11988622 (119905) 1199092 + sdot sdot sdot + 1198862119899 (119905) 119909119899

119899= 1198861198991 (119905) 1199091 + 1198861198992 (119905) 1199092 + sdot sdot sdot + 119886119899119899 (119905) 119909119899

(11)





119860 (119905) =(

11988611 (119905) 11988612 (119905) sdot sdot sdot 1198861119899 (119905)

11988621 (119905) 11988622 (119905) sdot sdot sdot 1198862119899 (119905)

1198861198991 (119905) 119886

1198992 (119905) sdot sdot sdot 119886119899119899(119905)

) (12)


= 119860 (119905) 119909 (13)


119899)119879



minus1119860119909 = 119872


(119872minus12

119860119872minus12

)(11987212119909) = 119872









119865= 0 that


119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119910 minus 119909

1003817100381710038171003817119865= 119889 (119910 119909)

119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119909 minus 119911 + 119911 minus119910

1003817100381710038171003817119865

le 119909minus 119911119865+1003817100381710038171003817119911 minus 119910

1003817100381710038171003817119865= 119889 (119909 119911) + 119889 (119911 119910)

(14)


119889119876(119902 1199021015840

) = 119872minus119860119865 (15)


11990211205901997888rarr 1199022

1205902997888rarr 1199023 (16)


1205902997888rarr 1199023




Call 119889119876(119902 1199021015840





q1

q1

q2

q2

q3

q3

1205901

1205901

1205902

1205902

12059021205901



1015840 is

119910 = 119860 (119905) 119910 (17)


119899)119879 and state



119865 That is


119909 minus 119910119865= 119872 minus 119860

119865 And 119889

Π(1205871(119905) 1205872(119905)) =

1199091(119905) minus 1199101(119905)119865 le 119909 minus 119910119865= 120575


119905rarr+infin119860(119905)

119865119872 minus 119860

119865

Proof

lim119905rarr+infin

1003817100381710038171003817 minus1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) 119909 minus119860 (119905) 1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) (119909 minus 119910)1003817100381710038171003817119865


119860 (119905)119865

1003817100381710038171003817119909 minus1199101003817100381710038171003817119865


119860 (119905)119865

lim119905rarr+infin

1003817100381710038171003817119909 minus 1199101003817100381710038171003817119865


119860 (119905)119865119872minus119860

119865

(18)









119894bymeans


119894(119905)119909

is 1205901015840119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 = 1 2

Σ1015840

= 1205901015840

119894| 119894 = 1 2 and then 1198791015840 = (119876

1015840

Σ1015840

rarr Π(sdot))


1199021015840

119894and 120590

1015840

119894 119894 = 1 2 where state 1199021015840







120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (19)



119860(119905)119865119872 minus 119860


119894isin 119876

exist1199021015840



lim119905rarr+infin



120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (20)


6 A Case Study


(

1

2

3

) = (

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

) (21)



119860 = (

1 2 11 minus1 12 0 1

) (22)


1198711015840

= (

1 0 01 1 0

2 43

1)

1198801015840

= (


)

(23)


119871 = (

1 0 01 1 02 0 1

)

119880 = (


)

(24)



119872 = 119871119880 = (

1 0 01 1 02 0 1

)(


)

= (

1 2 11 minus1 12 4 1

)

(25)



(

1 2 11 minus1 12 4 1

)

minus1

(

1

2

3

)

= (

1 2 11 minus1 12 4 1

)

minus1

(

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

)

(26)



119872minus119860 = (

1 2 11 minus1 12 4 1

)minus(

1 2 11 minus1 12 0 1

) = (

0 0 00 0 00 4 0

)

119872minus119860119865

= radic(02 + 02 + 02) + (02 + 02 + 02) + (02 + 42 + 02)

= 4

119860119865

= radic(12 + 22 + 12) + (12 + (minus1)2 + 12) + (22 + 02 + 12)

= radic14

(27)


119860119865119872minus119860



7 Conclusions





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(1) 119889Π((1199021)1(1199022)2) le 120575

(2) forall1199021120590

997888rarr11199021015840

1 exist1199022120590

997888rarr21199021015840

2 such that (11990210158401 1199021015840

2) isin 119878120575


1015840


1015840








120587011205900

997888rarr 120587111205901

997888rarr 120587211205902



120587021205900

997888rarr 120587121205901

997888rarr 120587221205902



1 120587119894

2) le 120575









119872minus1119860119909 = 119872




(119872minus12

119860119872minus12

)(11987212119909) = 119872








(1) 119897119894119895= 0 if 119895 gt 119894 or (119894 119895) isin 119878 119906

119894119895= 0 if 119894 gt 119895 or (119894 119895) isin 119878


(2) 119878 = (119894 119895) | 119886119894119895

= 0


(3) 119898119894119895= 119886119894119895if (119894 119895) isin 119878




119903119903


119894119903 119886119894119903= 119890


119894119895= 119886119894119895minus 119890119886119903119895









(4) sum119899119895=1119898119894119895 = sum

119899

119895=1 119886119894119895 forall119894 and 119898119894119895 = 119886119894119895

if 119894 = 119895 and(119894 119895) isin 119878









119903119903


119894119903 119886119894119903= 119890


119894119895= 119886119894119895minus 119890119886119903119895

(10) else(11) 119886

119894119894= 119886119894119894minus 119890119886119903119895





1199020 120590

0

997888rarr 1199021 120590

1

997888rarr 1199022 120590

2






1 = 11988611 (119905) 1199091 + 11988612 (119905) 1199092 + sdot sdot sdot + 1198861119899 (119905) 119909119899

2 = 11988621 (119905) 1199091 + 11988622 (119905) 1199092 + sdot sdot sdot + 1198862119899 (119905) 119909119899

119899= 1198861198991 (119905) 1199091 + 1198861198992 (119905) 1199092 + sdot sdot sdot + 119886119899119899 (119905) 119909119899

(7)



119899)119879 and

119860 (119905) =(

11988611 (119905) 11988612 (t) sdot sdot sdot 1198861119899 (119905)

11988621 (119905) 11988622 (119905) sdot sdot sdot 1198862119899 (119905)

1198861198991 (119905) 119886

1198992 (119905) sdot sdot sdot 119886119899119899(119905)

) (8)



119872(119905) =(

11988711 (119905) 11988712 (119905) sdot sdot sdot 1198871119899 (119905)

11988721 (119905) 11988722 (119905) sdot sdot sdot 1198872119899 (119905)

1198871198991 (119905) 119887

1198992 (119905) sdot sdot sdot 119887119899119899(119905)

) (9)


minus1(119905) Let


minus1(119905) That is 1205901015840

119872minus1(119905)119860(119905)119909 = 119872


equivalent









1205901015840

119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 isin N


1015840

119894|


119894



Π(sdot)) Let1198791015840 = (119876 Σ1015840









119862 = 119860119909 (10)






+


1199091015840

hArr exist119906[0119905] with 119909

1015840

= Φ119862(119905 119909 119906

[0119905])(iv) observation Π = 119861








1015840

1 = 11988611 (119905) 1199091 + 11988612 (119905) 1199092 + sdot sdot sdot + 1198861119899 (119905) 119909119899

2 = 11988621 (119905) 1199091 + 11988622 (119905) 1199092 + sdot sdot sdot + 1198862119899 (119905) 119909119899

119899= 1198861198991 (119905) 1199091 + 1198861198992 (119905) 1199092 + sdot sdot sdot + 119886119899119899 (119905) 119909119899

(11)





119860 (119905) =(

11988611 (119905) 11988612 (119905) sdot sdot sdot 1198861119899 (119905)

11988621 (119905) 11988622 (119905) sdot sdot sdot 1198862119899 (119905)

1198861198991 (119905) 119886

1198992 (119905) sdot sdot sdot 119886119899119899(119905)

) (12)


= 119860 (119905) 119909 (13)


119899)119879



minus1119860119909 = 119872


(119872minus12

119860119872minus12

)(11987212119909) = 119872









119865= 0 that


119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119910 minus 119909

1003817100381710038171003817119865= 119889 (119910 119909)

119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119909 minus 119911 + 119911 minus119910

1003817100381710038171003817119865

le 119909minus 119911119865+1003817100381710038171003817119911 minus 119910

1003817100381710038171003817119865= 119889 (119909 119911) + 119889 (119911 119910)

(14)


119889119876(119902 1199021015840

) = 119872minus119860119865 (15)


11990211205901997888rarr 1199022

1205902997888rarr 1199023 (16)


1205902997888rarr 1199023




Call 119889119876(119902 1199021015840





q1

q1

q2

q2

q3

q3

1205901

1205901

1205902

1205902

12059021205901



1015840 is

119910 = 119860 (119905) 119910 (17)


119899)119879 and state



119865 That is


119909 minus 119910119865= 119872 minus 119860

119865 And 119889

Π(1205871(119905) 1205872(119905)) =

1199091(119905) minus 1199101(119905)119865 le 119909 minus 119910119865= 120575


119905rarr+infin119860(119905)

119865119872 minus 119860

119865

Proof

lim119905rarr+infin

1003817100381710038171003817 minus1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) 119909 minus119860 (119905) 1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) (119909 minus 119910)1003817100381710038171003817119865


119860 (119905)119865

1003817100381710038171003817119909 minus1199101003817100381710038171003817119865


119860 (119905)119865

lim119905rarr+infin

1003817100381710038171003817119909 minus 1199101003817100381710038171003817119865


119860 (119905)119865119872minus119860

119865

(18)









119894bymeans


119894(119905)119909

is 1205901015840119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 = 1 2

Σ1015840

= 1205901015840

119894| 119894 = 1 2 and then 1198791015840 = (119876

1015840

Σ1015840

rarr Π(sdot))


1199021015840

119894and 120590

1015840

119894 119894 = 1 2 where state 1199021015840







120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (19)



119860(119905)119865119872 minus 119860


119894isin 119876

exist1199021015840



lim119905rarr+infin



120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (20)


6 A Case Study


(

1

2

3

) = (

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

) (21)



119860 = (

1 2 11 minus1 12 0 1

) (22)


1198711015840

= (

1 0 01 1 0

2 43

1)

1198801015840

= (


)

(23)


119871 = (

1 0 01 1 02 0 1

)

119880 = (


)

(24)



119872 = 119871119880 = (

1 0 01 1 02 0 1

)(


)

= (

1 2 11 minus1 12 4 1

)

(25)



(

1 2 11 minus1 12 4 1

)

minus1

(

1

2

3

)

= (

1 2 11 minus1 12 4 1

)

minus1

(

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

)

(26)



119872minus119860 = (

1 2 11 minus1 12 4 1

)minus(

1 2 11 minus1 12 0 1

) = (

0 0 00 0 00 4 0

)

119872minus119860119865

= radic(02 + 02 + 02) + (02 + 02 + 02) + (02 + 42 + 02)

= 4

119860119865

= radic(12 + 22 + 12) + (12 + (minus1)2 + 12) + (22 + 02 + 12)

= radic14

(27)


119860119865119872minus119860



7 Conclusions





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903119903


119894119903 119886119894119903= 119890


119894119895= 119886119894119895minus 119890119886119903119895









(4) sum119899119895=1119898119894119895 = sum

119899

119895=1 119886119894119895 forall119894 and 119898119894119895 = 119886119894119895

if 119894 = 119895 and(119894 119895) isin 119878









119903119903


119894119903 119886119894119903= 119890


119894119895= 119886119894119895minus 119890119886119903119895

(10) else(11) 119886

119894119894= 119886119894119894minus 119890119886119903119895





1199020 120590

0

997888rarr 1199021 120590

1

997888rarr 1199022 120590

2






1 = 11988611 (119905) 1199091 + 11988612 (119905) 1199092 + sdot sdot sdot + 1198861119899 (119905) 119909119899

2 = 11988621 (119905) 1199091 + 11988622 (119905) 1199092 + sdot sdot sdot + 1198862119899 (119905) 119909119899

119899= 1198861198991 (119905) 1199091 + 1198861198992 (119905) 1199092 + sdot sdot sdot + 119886119899119899 (119905) 119909119899

(7)



119899)119879 and

119860 (119905) =(

11988611 (119905) 11988612 (t) sdot sdot sdot 1198861119899 (119905)

11988621 (119905) 11988622 (119905) sdot sdot sdot 1198862119899 (119905)

1198861198991 (119905) 119886

1198992 (119905) sdot sdot sdot 119886119899119899(119905)

) (8)



119872(119905) =(

11988711 (119905) 11988712 (119905) sdot sdot sdot 1198871119899 (119905)

11988721 (119905) 11988722 (119905) sdot sdot sdot 1198872119899 (119905)

1198871198991 (119905) 119887

1198992 (119905) sdot sdot sdot 119887119899119899(119905)

) (9)


minus1(119905) Let


minus1(119905) That is 1205901015840

119872minus1(119905)119860(119905)119909 = 119872


equivalent









1205901015840

119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 isin N


1015840

119894|


119894



Π(sdot)) Let1198791015840 = (119876 Σ1015840









119862 = 119860119909 (10)






+


1199091015840

hArr exist119906[0119905] with 119909

1015840

= Φ119862(119905 119909 119906

[0119905])(iv) observation Π = 119861








1015840

1 = 11988611 (119905) 1199091 + 11988612 (119905) 1199092 + sdot sdot sdot + 1198861119899 (119905) 119909119899

2 = 11988621 (119905) 1199091 + 11988622 (119905) 1199092 + sdot sdot sdot + 1198862119899 (119905) 119909119899

119899= 1198861198991 (119905) 1199091 + 1198861198992 (119905) 1199092 + sdot sdot sdot + 119886119899119899 (119905) 119909119899

(11)





119860 (119905) =(

11988611 (119905) 11988612 (119905) sdot sdot sdot 1198861119899 (119905)

11988621 (119905) 11988622 (119905) sdot sdot sdot 1198862119899 (119905)

1198861198991 (119905) 119886

1198992 (119905) sdot sdot sdot 119886119899119899(119905)

) (12)


= 119860 (119905) 119909 (13)


119899)119879



minus1119860119909 = 119872


(119872minus12

119860119872minus12

)(11987212119909) = 119872









119865= 0 that


119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119910 minus 119909

1003817100381710038171003817119865= 119889 (119910 119909)

119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119909 minus 119911 + 119911 minus119910

1003817100381710038171003817119865

le 119909minus 119911119865+1003817100381710038171003817119911 minus 119910

1003817100381710038171003817119865= 119889 (119909 119911) + 119889 (119911 119910)

(14)


119889119876(119902 1199021015840

) = 119872minus119860119865 (15)


11990211205901997888rarr 1199022

1205902997888rarr 1199023 (16)


1205902997888rarr 1199023




Call 119889119876(119902 1199021015840





q1

q1

q2

q2

q3

q3

1205901

1205901

1205902

1205902

12059021205901



1015840 is

119910 = 119860 (119905) 119910 (17)


119899)119879 and state



119865 That is


119909 minus 119910119865= 119872 minus 119860

119865 And 119889

Π(1205871(119905) 1205872(119905)) =

1199091(119905) minus 1199101(119905)119865 le 119909 minus 119910119865= 120575


119905rarr+infin119860(119905)

119865119872 minus 119860

119865

Proof

lim119905rarr+infin

1003817100381710038171003817 minus1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) 119909 minus119860 (119905) 1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) (119909 minus 119910)1003817100381710038171003817119865


119860 (119905)119865

1003817100381710038171003817119909 minus1199101003817100381710038171003817119865


119860 (119905)119865

lim119905rarr+infin

1003817100381710038171003817119909 minus 1199101003817100381710038171003817119865


119860 (119905)119865119872minus119860

119865

(18)









119894bymeans


119894(119905)119909

is 1205901015840119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 = 1 2

Σ1015840

= 1205901015840

119894| 119894 = 1 2 and then 1198791015840 = (119876

1015840

Σ1015840

rarr Π(sdot))


1199021015840

119894and 120590

1015840

119894 119894 = 1 2 where state 1199021015840







120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (19)



119860(119905)119865119872 minus 119860


119894isin 119876

exist1199021015840



lim119905rarr+infin



120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (20)


6 A Case Study


(

1

2

3

) = (

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

) (21)



119860 = (

1 2 11 minus1 12 0 1

) (22)


1198711015840

= (

1 0 01 1 0

2 43

1)

1198801015840

= (


)

(23)


119871 = (

1 0 01 1 02 0 1

)

119880 = (


)

(24)



119872 = 119871119880 = (

1 0 01 1 02 0 1

)(


)

= (

1 2 11 minus1 12 4 1

)

(25)



(

1 2 11 minus1 12 4 1

)

minus1

(

1

2

3

)

= (

1 2 11 minus1 12 4 1

)

minus1

(

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

)

(26)



119872minus119860 = (

1 2 11 minus1 12 4 1

)minus(

1 2 11 minus1 12 0 1

) = (

0 0 00 0 00 4 0

)

119872minus119860119865

= radic(02 + 02 + 02) + (02 + 02 + 02) + (02 + 42 + 02)

= 4

119860119865

= radic(12 + 22 + 12) + (12 + (minus1)2 + 12) + (22 + 02 + 12)

= radic14

(27)


119860119865119872minus119860



7 Conclusions





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119872(119905) =(

11988711 (119905) 11988712 (119905) sdot sdot sdot 1198871119899 (119905)

11988721 (119905) 11988722 (119905) sdot sdot sdot 1198872119899 (119905)

1198871198991 (119905) 119887

1198992 (119905) sdot sdot sdot 119887119899119899(119905)

) (9)


minus1(119905) Let


minus1(119905) That is 1205901015840

119872minus1(119905)119860(119905)119909 = 119872


equivalent









1205901015840

119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 isin N


1015840

119894|


119894



Π(sdot)) Let1198791015840 = (119876 Σ1015840









119862 = 119860119909 (10)






+


1199091015840

hArr exist119906[0119905] with 119909

1015840

= Φ119862(119905 119909 119906

[0119905])(iv) observation Π = 119861








1015840

1 = 11988611 (119905) 1199091 + 11988612 (119905) 1199092 + sdot sdot sdot + 1198861119899 (119905) 119909119899

2 = 11988621 (119905) 1199091 + 11988622 (119905) 1199092 + sdot sdot sdot + 1198862119899 (119905) 119909119899

119899= 1198861198991 (119905) 1199091 + 1198861198992 (119905) 1199092 + sdot sdot sdot + 119886119899119899 (119905) 119909119899

(11)





119860 (119905) =(

11988611 (119905) 11988612 (119905) sdot sdot sdot 1198861119899 (119905)

11988621 (119905) 11988622 (119905) sdot sdot sdot 1198862119899 (119905)

1198861198991 (119905) 119886

1198992 (119905) sdot sdot sdot 119886119899119899(119905)

) (12)


= 119860 (119905) 119909 (13)


119899)119879



minus1119860119909 = 119872


(119872minus12

119860119872minus12

)(11987212119909) = 119872









119865= 0 that


119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119910 minus 119909

1003817100381710038171003817119865= 119889 (119910 119909)

119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119909 minus 119911 + 119911 minus119910

1003817100381710038171003817119865

le 119909minus 119911119865+1003817100381710038171003817119911 minus 119910

1003817100381710038171003817119865= 119889 (119909 119911) + 119889 (119911 119910)

(14)


119889119876(119902 1199021015840

) = 119872minus119860119865 (15)


11990211205901997888rarr 1199022

1205902997888rarr 1199023 (16)


1205902997888rarr 1199023




Call 119889119876(119902 1199021015840





q1

q1

q2

q2

q3

q3

1205901

1205901

1205902

1205902

12059021205901



1015840 is

119910 = 119860 (119905) 119910 (17)


119899)119879 and state



119865 That is


119909 minus 119910119865= 119872 minus 119860

119865 And 119889

Π(1205871(119905) 1205872(119905)) =

1199091(119905) minus 1199101(119905)119865 le 119909 minus 119910119865= 120575


119905rarr+infin119860(119905)

119865119872 minus 119860

119865

Proof

lim119905rarr+infin

1003817100381710038171003817 minus1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) 119909 minus119860 (119905) 1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) (119909 minus 119910)1003817100381710038171003817119865


119860 (119905)119865

1003817100381710038171003817119909 minus1199101003817100381710038171003817119865


119860 (119905)119865

lim119905rarr+infin

1003817100381710038171003817119909 minus 1199101003817100381710038171003817119865


119860 (119905)119865119872minus119860

119865

(18)









119894bymeans


119894(119905)119909

is 1205901015840119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 = 1 2

Σ1015840

= 1205901015840

119894| 119894 = 1 2 and then 1198791015840 = (119876

1015840

Σ1015840

rarr Π(sdot))


1199021015840

119894and 120590

1015840

119894 119894 = 1 2 where state 1199021015840







120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (19)



119860(119905)119865119872 minus 119860


119894isin 119876

exist1199021015840



lim119905rarr+infin



120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (20)


6 A Case Study


(

1

2

3

) = (

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

) (21)



119860 = (

1 2 11 minus1 12 0 1

) (22)


1198711015840

= (

1 0 01 1 0

2 43

1)

1198801015840

= (


)

(23)


119871 = (

1 0 01 1 02 0 1

)

119880 = (


)

(24)



119872 = 119871119880 = (

1 0 01 1 02 0 1

)(


)

= (

1 2 11 minus1 12 4 1

)

(25)



(

1 2 11 minus1 12 4 1

)

minus1

(

1

2

3

)

= (

1 2 11 minus1 12 4 1

)

minus1

(

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

)

(26)



119872minus119860 = (

1 2 11 minus1 12 4 1

)minus(

1 2 11 minus1 12 0 1

) = (

0 0 00 0 00 4 0

)

119872minus119860119865

= radic(02 + 02 + 02) + (02 + 02 + 02) + (02 + 42 + 02)

= 4

119860119865

= radic(12 + 22 + 12) + (12 + (minus1)2 + 12) + (22 + 02 + 12)

= radic14

(27)


119860119865119872minus119860



7 Conclusions





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus1119860119909 = 119872


(119872minus12

119860119872minus12

)(11987212119909) = 119872









119865= 0 that


119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119910 minus 119909

1003817100381710038171003817119865= 119889 (119910 119909)

119889 (119909 119910) =1003817100381710038171003817119909 minus119910

1003817100381710038171003817119865=1003817100381710038171003817119909 minus 119911 + 119911 minus119910

1003817100381710038171003817119865

le 119909minus 119911119865+1003817100381710038171003817119911 minus 119910

1003817100381710038171003817119865= 119889 (119909 119911) + 119889 (119911 119910)

(14)


119889119876(119902 1199021015840

) = 119872minus119860119865 (15)


11990211205901997888rarr 1199022

1205902997888rarr 1199023 (16)


1205902997888rarr 1199023




Call 119889119876(119902 1199021015840





q1

q1

q2

q2

q3

q3

1205901

1205901

1205902

1205902

12059021205901



1015840 is

119910 = 119860 (119905) 119910 (17)


119899)119879 and state



119865 That is


119909 minus 119910119865= 119872 minus 119860

119865 And 119889

Π(1205871(119905) 1205872(119905)) =

1199091(119905) minus 1199101(119905)119865 le 119909 minus 119910119865= 120575


119905rarr+infin119860(119905)

119865119872 minus 119860

119865

Proof

lim119905rarr+infin

1003817100381710038171003817 minus1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) 119909 minus119860 (119905) 1199101003817100381710038171003817119865


1003817100381710038171003817119860 (119905) (119909 minus 119910)1003817100381710038171003817119865


119860 (119905)119865

1003817100381710038171003817119909 minus1199101003817100381710038171003817119865


119860 (119905)119865

lim119905rarr+infin

1003817100381710038171003817119909 minus 1199101003817100381710038171003817119865


119860 (119905)119865119872minus119860

119865

(18)









119894bymeans


119894(119905)119909

is 1205901015840119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 = 1 2

Σ1015840

= 1205901015840

119894| 119894 = 1 2 and then 1198791015840 = (119876

1015840

Σ1015840

rarr Π(sdot))


1199021015840

119894and 120590

1015840

119894 119894 = 1 2 where state 1199021015840







120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (19)



119860(119905)119865119872 minus 119860


119894isin 119876

exist1199021015840



lim119905rarr+infin



120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (20)


6 A Case Study


(

1

2

3

) = (

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

) (21)



119860 = (

1 2 11 minus1 12 0 1

) (22)


1198711015840

= (

1 0 01 1 0

2 43

1)

1198801015840

= (


)

(23)


119871 = (

1 0 01 1 02 0 1

)

119880 = (


)

(24)



119872 = 119871119880 = (

1 0 01 1 02 0 1

)(


)

= (

1 2 11 minus1 12 4 1

)

(25)



(

1 2 11 minus1 12 4 1

)

minus1

(

1

2

3

)

= (

1 2 11 minus1 12 4 1

)

minus1

(

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

)

(26)



119872minus119860 = (

1 2 11 minus1 12 4 1

)minus(

1 2 11 minus1 12 0 1

) = (

0 0 00 0 00 4 0

)

119872minus119860119865

= radic(02 + 02 + 02) + (02 + 02 + 02) + (02 + 42 + 02)

= 4

119860119865

= radic(12 + 22 + 12) + (12 + (minus1)2 + 12) + (22 + 02 + 12)

= radic14

(27)


119860119865119872minus119860



7 Conclusions





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119894bymeans


119894(119905)119909

is 1205901015840119894 119872minus1119894(119905)119860119894(119905)119909 = 119872


119894(119905) is the


1015840

119894| 119894 = 1 2

Σ1015840

= 1205901015840

119894| 119894 = 1 2 and then 1198791015840 = (119876

1015840

Σ1015840

rarr Π(sdot))


1199021015840

119894and 120590

1015840

119894 119894 = 1 2 where state 1199021015840







120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (19)



119860(119905)119865119872 minus 119860


119894isin 119876

exist1199021015840



lim119905rarr+infin



120575 = max119894


1003817100381710038171003817119860 119894 (119905)1003817100381710038171003817119865

1003817100381710038171003817119872119894 minus119860 1198941003817100381710038171003817119865)

119894 = 1 2 (20)


6 A Case Study


(

1

2

3

) = (

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

) (21)



119860 = (

1 2 11 minus1 12 0 1

) (22)


1198711015840

= (

1 0 01 1 0

2 43

1)

1198801015840

= (


)

(23)


119871 = (

1 0 01 1 02 0 1

)

119880 = (


)

(24)



119872 = 119871119880 = (

1 0 01 1 02 0 1

)(


)

= (

1 2 11 minus1 12 4 1

)

(25)



(

1 2 11 minus1 12 4 1

)

minus1

(

1

2

3

)

= (

1 2 11 minus1 12 4 1

)

minus1

(

1 2 11 minus1 12 0 1

)(

1199091

1199092

1199093

)

(26)



119872minus119860 = (

1 2 11 minus1 12 4 1

)minus(

1 2 11 minus1 12 0 1

) = (

0 0 00 0 00 4 0

)

119872minus119860119865

= radic(02 + 02 + 02) + (02 + 02 + 02) + (02 + 42 + 02)

= 4

119860119865

= radic(12 + 22 + 12) + (12 + (minus1)2 + 12) + (22 + 02 + 12)

= radic14

(27)


119860119865119872minus119860



7 Conclusions





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119872minus119860 = (

1 2 11 minus1 12 4 1

)minus(

1 2 11 minus1 12 0 1

) = (

0 0 00 0 00 4 0

)

119872minus119860119865

= radic(02 + 02 + 02) + (02 + 02 + 02) + (02 + 42 + 02)

= 4

119860119865

= radic(12 + 22 + 12) + (12 + (minus1)2 + 12) + (22 + 02 + 12)

= radic14

(27)


119860119865119872minus119860



7 Conclusions





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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