Research Article Stability and Linear Quadratic Differential Games of Discrete-Time Markovian Jump Linear Systems with State-Dependent Noise Huiying Sun, Meng Li, Shenglin Ji, and Long Yan College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China Correspondence should be addressed to Huiying Sun; [email protected] Received 7 July 2014; Accepted 6 September 2014; Published 23 November 2014 Academic Editor: Ramachandran Raja Copyright © 2014 Huiying Sun et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We mainly consider the stability of discrete-time Markovian jump linear systems with state-dependent noise as well as its linear quadratic (LQ) differential games. A necessary and sufficient condition involved with the connection between stochastic - stability of Markovian jump linear systems with state-dependent noise and Lyapunov equation is proposed. And using the theory of stochastic -stability, we give the optimal strategies and the optimal cost values for infinite horizon LQ stochastic differential games. It is demonstrated that the solutions of infinite horizon LQ stochastic differential games are concerned with four coupled generalized algebraic Riccati equations (GAREs). Finally, an iterative algorithm is presented to solve the four coupled GAREs and a simulation example is given to illustrate the effectiveness of it. 1. Introduction In this paper we discuss linear systems with Markovian jump and state-dependent noise. Here the discrete-time stochastic linear systems subject to abrupt parameter changes can be modeled by a discrete-time finite-state Markov chain. ey are a special sort of hybrid systems with both modes and state variables. Since the class of systems was firstly introduced in early 1960s, the hybrid systems driven by continuous-time Markov chains have been broadly employed to model many practical systems which may experience abrupt changes in system structure and parameters such as solar-powered systems, power systems, battle management in command, and control and communication systems [1–4]. In the past several decades, considerable attention has been focused on the analysis and synthesis of linear systems with Markovian jump, including stability analysis, state feedback, and output feedback controller design, filter design, and so forth [5–11]. e stability theory of linear systems with Markovian jump and state-dependent noise, here we also say Marko- vian jump stochastic linear systems (MJSLS for short), is rather complex in that there exist some stability concepts. Particularly, the study of stability about these systems has attracted the attention of many researchers [12–17]. e very important stability notions are mean-square stability, moment stability, and almost sure stability. Mean-square stability deals with the asymptotic convergence to zero of the second moment of the state norm. ere are some necessary and sufficient conditions for mean-square stability involving either the solution of the coupled Lyapunov equations or the location in the complex plane of the eigenvalues of suitable augmented matrices [12, 13]. Moment stability, -moment stability, requires the convergence to zero of -moment of the state norm (mean-square stability is just a particular case for =2). Although there exist some practical sufficient conditions, a simple necessary and sufficient condition test- ing -moment stability is not available (except for =2). Almost sure stability holds if the sample path of the state converges to zero with probability one. e checking about almost sure stability involves the determination of the sign of the top Lyapunov exponent, which is usually a rather difficult task [14, 15]. Contrary to deterministic systems, for which all Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 265621, 11 pages http://dx.doi.org/10.1155/2014/265621
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Research ArticleStability and Linear Quadratic Differential Gamesof Discrete-Time Markovian Jump Linear Systems withState-Dependent Noise
Huiying Sun Meng Li Shenglin Ji and Long Yan
College of Electrical Engineering and Automation Shandong University of Science and Technology Qingdao 266590 China
Correspondence should be addressed to Huiying Sun sunhyinggmailcom
Received 7 July 2014 Accepted 6 September 2014 Published 23 November 2014
Academic Editor Ramachandran Raja
Copyright copy 2014 Huiying Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We mainly consider the stability of discrete-time Markovian jump linear systems with state-dependent noise as well as its linearquadratic (LQ) differential games A necessary and sufficient condition involved with the connection between stochastic 119879
119899-
stability of Markovian jump linear systems with state-dependent noise and Lyapunov equation is proposed And using the theoryof stochastic 119879
119899-stability we give the optimal strategies and the optimal cost values for infinite horizon LQ stochastic differential
games It is demonstrated that the solutions of infinite horizon LQ stochastic differential games are concerned with four coupledgeneralized algebraic Riccati equations (GAREs) Finally an iterative algorithm is presented to solve the four coupled GAREs anda simulation example is given to illustrate the effectiveness of it
1 Introduction
In this paper we discuss linear systems with Markovian jumpand state-dependent noise Here the discrete-time stochasticlinear systems subject to abrupt parameter changes can bemodeled by a discrete-time finite-state Markov chain Theyare a special sort of hybrid systems with bothmodes and statevariables Since the class of systems was firstly introduced inearly 1960s the hybrid systems driven by continuous-timeMarkov chains have been broadly employed to model manypractical systems which may experience abrupt changesin system structure and parameters such as solar-poweredsystems power systems battle management in commandand control and communication systems [1ndash4] In the pastseveral decades considerable attention has been focused onthe analysis and synthesis of linear systems with Markovianjump including stability analysis state feedback and outputfeedback controller design filter design and so forth [5ndash11]
The stability theory of linear systems with Markovianjump and state-dependent noise here we also say Marko-vian jump stochastic linear systems (MJSLS for short) is
rather complex in that there exist some stability conceptsParticularly the study of stability about these systems hasattracted the attention of many researchers [12ndash17] Thevery important stability notions are mean-square stabilitymoment stability and almost sure stability Mean-squarestability deals with the asymptotic convergence to zero of thesecond moment of the state norm There are some necessaryand sufficient conditions for mean-square stability involvingeither the solution of the coupled Lyapunov equations or thelocation in the complex plane of the eigenvalues of suitableaugmented matrices [12 13] Moment stability 120575-momentstability requires the convergence to zero of 120575-moment ofthe state norm (mean-square stability is just a particular casefor 120575 = 2) Although there exist some practical sufficientconditions a simple necessary and sufficient condition test-ing 120575-moment stability is not available (except for 120575 = 2)Almost sure stability holds if the sample path of the stateconverges to zero with probability one The checking aboutalmost sure stability involves the determination of the sign ofthe top Lyapunov exponent which is usually a rather difficulttask [14 15] Contrary to deterministic systems for which all
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 265621 11 pageshttpdxdoiorg1011552014265621
2 Mathematical Problems in Engineering
moments are stable whenever the sample path is stable themoment stability for stochastic systems implies almost surestability but not vice versa as pointed out by [16]
On the other hand the LQ differential games have manyapplications in the economy military and intelligent robotsSince the book [18] entitled ldquoDifferential Gamesrdquo written byDr Isaacs came out theory and application of differentialgames have been developed greatly A differential game isa mathematical model that represents a conflict betweendifferent agentswhich control a dynamical system and each ofthem is trying to minimize his individual objective functionby giving a control to the system In fact many situationsin industry economies management and elsewhere arecharacterized by multiple decision makers and enduringconsequences of decisions which can be treated as dynamicgames Particularly applications of differential games arewidely researched in LQ control problem [19ndash23] By solvingthe LQ control problems players can avoid most of theadditional cost incurred by this perturbation The authorsconsider the zero-sum infinite-horizon and LQ differentialgames in [19] A sufficient condition for the LQ differentialgames is applied to the119867
infinoptimization problem in [20]The
authors in [21] study the problemof designing suboptimal LQdifferential games with multiple players In [22] the authorsstudy the LQ nonzero sum stochastic differential gamesproblem with random jump A leader-follower stochasticdifferential game is considered with the state equation beinga linear Ito-type stochastic differential equation and the costfunctionals being quadratic [23]
As far as we know there are few researchers paying atten-tion to the discrete-time stochastic LQ differential gamesespecially MJSLS Thus it is significant to consider thesesystems In [24] we have considered stochastic differentialgames in infinite-time horizon By introducing stochasticexact observability and stochastic exact detectability the opti-mal strategies (Nash equilibrium strategies) and the optimalcost values have been given In [25] we have consideredLQ differential games in finite horizon for discrete-timestochastic systems with Markovian jump parameters andmultiplicative noise Furthermore a suboptimal solution ofthe LQ differential games is proposed based on a convexoptimization approach On the basis of [26] in the paperwe further investigate the LQ differential games for discrete-time MJSLS with a finite number of jump times It gives theoptimal strategies and the optimal cost values for infinitehorizon LQ stochastic differential gameswhich are associatedwith the four coupled GAREs Generally it is difficult to solvethe four coupled GAREs analytically Here we will solve theLQ differential games by means of stochastic 119879
119899-stability and
we will employ a recursive procedure to solve the coupledGAREs
The paper is organized as follows In Section 2 some basicdefinitions are recalled A necessary and sufficient conditionin relation to the connection between the 119879
119899-stability for
MJSLS and Lyapunov equation is presented In Section 3 weformulate LQ differential games with quadratic cost functionfor the MJSLS and propose the stochastic 119879
119899-stability of
discrete-time MJSLS with a finite number of jump timeswhich is essential to obtain the main results An iterative
algorithm for solving the four coupled GAREs is put forwardand an illustrative example is also displayed in Section 4Section 5 ends this paper with some concluding remarks
For convenience the paper adopts the following basicnotations It uses R119899 to denote the linear space of all 119899-dimensional real vectors R
119898times119899denotes the linear space of
all 119898 times 119899 real matrices N = 0 1 2 1198801015840 indicates the
transpose of matrix 119880 and 119880 ge 0 (119880 gt 0) represents anonnegative definite matrix (positive definite) The standardvector norm inR119899 is indicated by sdot and the correspondinginduced norm of matrix 119880 by 119880 119871
2
(infinR119896) represents thespace of R119896-valued square integrable random vectors and120575sdot
is the Dirac measure Finally we write 119864119896[sdot] instead of
119864[sdot | 119909119896 120579119896] and we define the following operator 120576
119894(119880) =
sum119895 =119894
119901119894119895119880119895for 119880 = 119880
1015840
ge 0
2 Stochastic 119879119899-Stability for Discrete-Time
MJSLS with a Finite Number of Jump Times
Let (ΩF 119875 F119896) be a given fundamental probability space
where there exist a Markov chain 120579119896and a sequence of real
random variables 119908(119896) F119896denotes the 120590-algebra generated
by 120579119896and 119908(119896) that isF
119896= 120590120579
119904 119908(119904) | 119904 = 0 1 2 119896 sub
F Consider the following MJSLS defined on the space(ΩF 119875 F
119896)
119909 (119896 + 1) = 119860120579119896
119909 (119896) + 119860120579119896
119909 (119896) 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
(1)
where 119909(119896) 120579119896 119896 isin N are the states of process with values
inR119899 timesX 120579119896 119896 isin N is a time homogeneous Markov chain
taking values in a finite set X = 1 2 119873 with initialdistribution 120583 and transition probability matrix P = [119901
119894119895]
where
119901119894119895
= 119875 (120579119896+1
= 119895 | 120579119896
= 119894) forall119894 119895 isin X 119896 isin N (2)
The set X comprises the various operation modes of thesystem (1) For each 120579
119896= 119894 isin X the matrices 119860
120579119896isin R119899times119899
and 119860120579119896
isin R119899times119899
(associated with ldquo119894thrdquo mode) will be assignedas 119860120579119896
= 119860119894and 119860
120579119896= 119860119894in someplace of the paper
119908(119896) 119896 isin N is a sequence of real random variables whichis also a wide sense stationary second-order process with119864[119908(119896)] = 0 and 119864[119908(119894)119908(119895)] = 120575
119894119895 where 120575
119894119895refers to a
Kronecker function that is 120575119894119895
= 1 if 119894 = 119895 and 120575119894119895
= 0 if 119894 = 119895TheMJSLS as defined is trivially a strongMarkov processWeassume that 120579
119896is independent of119908(119896) Tomake the operation
more convenient let 119909(119896) = 119909119896 When system (1) is stable we
also say (119860119894 119860119894) stable for short
Although several concepts of stochastic stability can befound in the literature in this paper the stochastic stabilityconcept associated with the stopping times for MJSLS isresearched The stopping times in relation to jump times aredefined as follows
1198790
= 0
119879119899
= min 119896 gt 119879119899minus1
120579119896
= 120579119879119899minus1
119899 = 1 2 119873
(3)
Mathematical Problems in Engineering 3
The stopping time may represent interesting situations fromthe point of view of applications For instance it can be theaccumulated nth failure and repair of the system In anothersituation the stopping time can represent the occurrence of aldquocrucial failurerdquo (which may happen after a random numberof failures)
This class of stochastic systems is associated with sys-tems subject to failures in their components or connectionsaccording to a Markov chain The situation that we areinterested in arises when one wishes to study the stabilityof such a system until the occurrence of a fixed number 119873
of failures and repairs The paper recognizes the sequenceof the stopping times containing the successive times of theoccurrence of such failures and then it studies the stability ofsystem (1) according to these stopping times
Definition 1 (see [27]) Consider a stopping time 119879119899 The
MJSLS (1) is
(i) stochastically 119879119899-stable if for each initial condition 119909
0
and initial distribution 120583
119864 [
infin
sum
119896=0
1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] lt infin (4)
(ii) mean-square 119879119899-stable if for each initial condition 119909
0
and initial distribution 120583
lim119896rarrinfin
119864 [1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] = 0 (5)
Lemma 2 (see [27]) For all 119898 ge 1 and 119894 119895 isin X
119875 (1198791
= 119898 120579119898
= 119895 | 1205790
= 119894)
= 119901119894119895120575119898=1
119908ℎ119890119899 119901119894119894
= 0
119901119898minus1
119894119894119901119894119895120575119898gt1
119908ℎ119890119899 0 lt 119901119894119894
lt 1
(6)
Remark 3 119875(1198791
= 1 | 1205790
= 119894) = 1 and 119875(1198791
= +infin | 1205790
= 119894) =
1 whenever 119901119894119894
= 0 and 119901119894119894
= 1 respectively That is in anycase system will jump to another state
Next we will give an important theorem that will be usedlater
Theorem 4 The MJSLS (1) is 119879119899-stable if and only if for any
given set ofmatrices119882119894gt 0 there exists a unique set ofmatrices
119871119894gt 0 satisfying the Lyapunov equations
119901119894119894
(1198601015840
119894119871119894119860119894+ 1198601015840
119894119871119894119860119894) minus 119871119894+ 119882119894= 0 119894 isin X (7)
Proof Sufficiency In the proof we employ an inductionargument on the stopping times 119879
119899 First define the function
119881119896
(119909 119894) = 1199091015840
119896(1198751198941205751198791gt119896
+ 1198661198941205751198791=119896
) 119909119896 (8)
where 119866119894gt 0 and 119875
119894gt 0 is the solution of
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 119882119894+ 1198601015840
119894120576119894(119866) 119860
119894
+ 1198601015840
119894120576119894(119866) 119860
119894= 0 119894 isin X
(9)
The existence of such119875119894gt 0 relies on (7) Hence to functional
119881119896(119909 119894) in the following operation we can derive
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
= 119864119896
[1199091015840
119896+1(119875120579119896+11205751198791gt119896+1
+ 119866120579119896+11205751198791=119896+1
) 119909119896+1
minus 1199091015840
119896(1198751205791198961205751198791ge119896+1
+ 1198661205791198961205751198791=119896
) 119909119896]
= 119864119896
[(1199091015840
119896+1119875120579119896+1
119909119896+1
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791gt119896+1
]
+ 119864119896
[(1199091015840
119896+1119866120579119896+1
119909119896+1
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791=119896+1
]
minus 1199091015840
119896119866120579119896
1199091198961205751198791=119896
= 119864119896
(1199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896
+ 1199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896119908119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896119908119896
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791gt119896+1
+ 119864119896
(1199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896
+ 1199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896119908119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896119908119896
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791=119896+1
minus 1199091015840
119896119866120579119896
1199091198961205751198791=119896
(10)
We know that 119864[119908(119896)] = 0 and 119864[119908(119894)119908(119895)] = 120575119894119895 where
120575119894119895
= 1 if 119894 = 119895 and 120575119894119895
= 0 if 119894 = 119895 Calculating the expectedvalues above we can obtain that
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
= 1199091015840
119896[119901120579119896120579119896
(1198601015840
120579119896
119875120579119896
119860120579119896
+ 1198601015840
120579119896
119875120579119896
119860120579119896
) + 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
+ 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
minus 119875120579119896
] 1199091198961205751198791gt119896
minus 1199091015840
119896119866120579119896
1199091198961205751198791=119896
(11)
due to119875(1198791
= 119896+1 | 120579119896) = 1minus119901
120579119896120579119896 119875(1198791
gt 119896+1 | 120579119896) = 119901120579119896120579119896
and 120576120579119896
(119866) = Σ120579119896+1 =120579119896
119901120579119896120579119896+1
119866120579119896+1
The above relation can be rewritten as
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
= minus1199091015840
119896(1198821205791198961205751198791gt119896
+ 1198661205791198961205751198791=119896
) 119909119896
(12)
where
minus119882120579119896
= 119901120579119896120579119896
(1198601015840
120579119896
119875120579119896
119860120579119896
+ 1198601015840
120579119896
119875120579119896
119860120579119896
)
+ 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
+ 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
minus 119875120579119896
(13)
4 Mathematical Problems in Engineering
Now let us observe that119873
sum
119896=0
1198640
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
=
119873
sum
119896=0
1198640
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
(14)
By applying (12) and considering that 119882119894gt 0 119866
119894gt 0 for each
initial condition 1199090and initial distribution 120583 then we have
1198640
[119881119896+1
(119909119896+1
120579119896+1
)] minus 1198810
(1199090 1205790)
= minus
119873
sum
119896=0
1198640
[1199091015840
119896(1198821205791198961205751198791gt119896
+ 1198661205791198961205751198791=119896
) 1199091198961205751198791ge119896
]
le minus
119873
sum
119896=0
1205741198640
[1003817100381710038171003817119909119896
1003817100381710038171003817
2
1205751198791ge119896
]
(15)
for some 120574 gt 0 Because 1198640[119881119896(119909119896 120579119896)] ge 0 forall119896 ge 0 then
lim119896rarrinfin
1198640[119881119896+1
(119909119896+1
120579119896+1
)] = 0 by (5) Finally it is easy toverify that for any 119879
1
lim sup119873rarrinfin
119873
sum
119896=0
1198640
[1003817100381710038171003817119909119896
1003817100381710038171003817
2
1205751198791ge119896
] le1
1205741198810
(1199090 1205790) lt infin (16)
holds from (15) Therefore for any 1198791 MJSLS (1) is stable
according to (i) of Definition 1Now using an induction argument we assume that for
some 119899 the inequality
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899ge119896
] lt 1199091015840
01198751205790
1199090
(17)
holds and thus by setting 119876 equiv 119868 119864[119909119879119899
2
120575119879119899ge119896
] lt infinHowever
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
]
= lim sup119873rarrinfin
119864 [
[
119873
sum
119896=0
1199091015840
119896119876120579119896
119909119896120575119879119899gt119896
+
119873
sum
119896=119879119899
1199091015840
119896119876120579119896
119909119896120575119879119899le119896le119879119899+1
]
]
(18)
Notice that using the strong Markov property and thehomogeneity property the second term conditioned to theknowledge of (119909
119879119899 120579119879119899) can be written as
lim sup119873rarrinfin
119864 [
[
119873
sum
119896=119879119899
1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
| 119909119879119899
120579119879119899
]
]
= lim sup119873rarrinfin
119864 [
119873minus119879119899
sum
119896=0
1199091015840
119896119876120579119896
119909119896120575119896le1198791
| 1199090
= 119909119879119899
1205790
= 120579119879119899
]
lt 1199091015840
119879119899
119875120579119879119899
119909119879119899
(19)
So one can conclude from (18) and (19) that
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
]
lt lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899gt119896
+ 1199091015840
119879119899
119875120579119879119899
119909119879119899
]
= lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899ge119896
+ 1199091015840
119879119899
(119875120579119879119899
minus 119876120579119879119899
) 119909119879119899
]
lt 21199091015840
01198751205790
1199090
(20)
Therefore for any 119879119899
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] lt infin (21)
indicate that the MJSLS (1) is 119879119899-stable
Necessity As in the previous part define the functional
1199091015840
01198751205790
1199090
= 1198640
[
infin
sum
119896=0
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
] (22)
for all (1199090 1205790) isin R119899 times X Therefore
1199091015840
11198751205791
1199091
= 1198641198791
[
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
] 1205751198791gt1
(23)
The right-hand side of (22) can be expressed as
1198640
1199091015840
01198821205790
1199090
+ 1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791ge1
]
(24)
In addition
1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791ge1
]
= 1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791gt1
+ 1199091015840
1198791
1198661205791198791
11990911987911205751198791=1
]
(25)
Thus based on the strong Markov property applying homo-geneity in (25) and introducing it in (24) we arrive at
1199091015840
01198751205790
1199090
= 1199091015840
01198821205790
1199090
+ 1198640
[1199091015840
11198751205791
11990911205751198791gt1
+ 1199091015840
1198791
1198661205791198791
11990911987911205751198791=1
]
(26)
Mathematical Problems in Engineering 5
Since 1199090is arbitrary and calculating the expected values
above (26) implies that
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 1198601015840
119894120576119894(119866) 119860
119894+ 1198601015840
119894120576119894(119866) 119860
119894
= minus119882119894
(27)
using the fact that 119875(1198791
= 119896 + 1 | 120579119896
= 119894) = 1 minus 119901119894119894and 119875(119879
1gt
119896+1 | 120579119896
= 119894) = 119901119894119894Thus from the Lyapunov stability theory
the existence of the set 119871119894
gt 0 satisfying (7) is guaranteedcompleting the proof for 119899 = 1
Now for the general case from the stochastically 119879119899-
stable of the system we can obtain that
119864 [
infin
sum
119896=0
1199091015840
119896119882120579119896
119909119896120575119879119899gt119896
+ 1199091015840
119879119899
119866120579119879119899
119909119879119899
] lt infin (28)
And from the strong Markov property we can deduce that
119864119879119899
[
[
infin
sum
119896=119879119899
1199091015840
119896119882120579119896
119909119896120575119879119899+1gt119896
+ 1199091015840
119879119899+1
119866120579119879119899+1
119909119879119899+1
]
]
lt infin (29)
for 119899 = 0 1 119873minus1 By the homogeneity property it followsthat (29) is equivalent to (22) with 119909
0= 119909119879119899
and 1205790
= 120579119879119899
and the existence of a set of matrices 119871119894
gt 0 satisfying (7) isassured Then the proof of Theorem 4 is completed
3 LQ Differential Games for MJSLS witha Finite Number of Jump Times
31 Problem Formulation Now we study the LQ differentialgames for discrete-time MJSLS Comparing with system (1)consider the following discrete-time MJSLS with a finitenumber of jump times
119909 (119896 + 1) = 119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)
+ [119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
119910120591
(119896) = 119876120591
120579119896
119909 (119896) 120591 = 1 2
(30)
119910120591
(119896) isin R119898 are the measurement outputs for each playerHere (119906(119896) V(119896)) isin R119903 times R119903 represent the system controlinputs The matrices (119861
120579119896 119861120579119896
119862120579119896
119862120579119896
119876120591
120579119896
) isin R119899times119903
times R119899times119903
times
R119899times119903
times R119899times119903
times R119898times119899
(associated with ldquo119894thrdquo mode) will beassigned as (119861
119894 119861119894 119862119894 119862119894 119876120591
119894) for each 120579
119896= 119894 isin X
Throughout this paper we choose the infinite horizonquadratic cost functions associated with each player
119869120591
(119906 V) =
infin
sum
119896=0
119864 [119909 (119896)1015840
(119876120591
120579119896
)1015840
119876120591
120579119896
119909 (119896)
+ 119906 (119896)1015840
119877120591
120579119896
119906 (119896) + V (119896)1015840
119878120591
120579119896
V (119896) ]
120591 = 1 2
(31)
The weighting matrices 119876120591
120579119896
= 119876120591
119894ge 0 119877
120591
120579119896
= 119877120591
119894gt 0 isin R
119903times119903
and 119878120591
120579119896
= 119878120591
119894gt 0 isin R
119903times119903
So we are looking for actions that satisfy simultaneously
1198691
(119906lowast
Vlowast) le 1198691
(119906lowast
V) 1198692
(119906lowast
Vlowast) le 1198692
(119906 Vlowast) (32)
where (119906lowast
(119896) Vlowast(119896)) isin 1198712
(infinR119903119906) times 1198712
(infinR119903V)To ensure the finiteness of the infinite-time cost function
we restrain the admissible control set to the constant linearfeedback strategies that is 119906(119896) = 119870
1
120579119896
119909(119896) V(119896) = 1198702
120579119896
119909(119896)where 119870
1
120579119896
and 1198702
120579119896
are constant matrices of appropriatedimensions and (119870
1
120579119896
1198702
120579119896
) belong to
K = 119870 = (1198701
120579119896
1198702
120579119896
) | system (30) can be stabilized
with 119906 (119896) = 1198701
120579119896
119909 (119896)
V (119905) = 1198702
120579119896
119909 (119896)
(33)
We say that the optimization problem is well posedand the 119906(119896) and V(119896) have the following two additionalproperties
119864 [|119906 (119896)|2
] lt infin 119864 [|V (119896)|2
] lt infin 119896 isin N (34)
The optimal strategies 119906lowast and Vlowast determined by (32) are
also called the Nash equilibrium strategies (119906lowast
Vlowast) In orderto guarantee the unique global Nash game solutions both theplayers are only allowed to take constant feedback controlsNext we focus on finding the optimal strategies
32Main Results First we give an important lemma that willbe used later If the system (1) is 119879
119899-stable we can obtain the
following result for the discrete-time MJSLS (30)
Lemma 5 If [119860120579119896
119860120579119896
] is 119879119899-stable then so is [119860
120579119896+ 119861120579119896
1198701
120579119896
+
119862120579119896
1198702
120579119896
119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
] where (1198701
120579119896
1198702
120579119896
) isin K
Proof SufficiencyThe proof employs an induction argumenton the stopping times 119879
119899 First define the function
119881119896
(119909 119894) = 1199091015840
119896(1198751198941205751198791gt119896
+ 1198661198941205751198791=119896
) 119909119896 (35)
where 119866119894gt 0 and 119875
119894gt 0 is the solution of
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 119882119894+ 1198601015840
119894120576119894(119866) 119860
119894
+ 1198601015840
119894120576119894(119866) 119860
119894= 0 119894 isin X
(36)
6 Mathematical Problems in Engineering
The existence of such 119875119894
gt 0 relies on (7) Hence tothe function 119881
119896(119909 119894) and the system (30) in the following
operation we acquire that
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896) minus 119881119896
(119909 (119896) 120579119896)]
= 119909 (119896)1015840
119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
minus119875120579119896
119909 (119896) 1205751198791gt119896
minus 119909 (119896)1015840
119866120579119896
119909 (119896) 1205751198791=119896
(37)
Compared with (12) we know
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896+1
) minus 119881119896
(119909 (119896) 120579119896)]
= minus119909 (119896)1015840
(119882120579119896120575119879119899gt119896
+ 119866120579119896120575119879119899=119896
) 119909 (119896)
(38)
where
minus119882120579119896
= minus119875120579119896
+ 119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
(39)
Considering that 119882120579119896
gt 0 119866120579119896
gt 0 we can obtain (19)And because 119864[119881
119896(119909(119896) 120579
119896)] ge 0 forall119896 ge 0 for each initial
condition 1199090 from (40) it is easy to verify (21) Therefore
the MJSLS (30) is 119879119899-stable
Theorem 6 For system (30) suppose the following coupledequations admit the solutions (119871
1
119894 1198712
119894 1198701
119894 1198702
119894) with 119871
1
119894gt 0
1198712
119894gt 0
minus 1198711
119894+ 119901119894119894
[(119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)
+ (119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)] + 119876
11015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894minus 11987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894= 0
1198671
119894(1198711
119894) gt 0
(40)
1198701
119894= minus119867
2
119894(1198712
119894)minus1
1198704
119894 (41)
minus 1198712
119894+ 119901119894119894
[(119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)
+ (119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)] + 119876
21015840
1198941198762
119894
+ 11987021015840
1198941198782
1198941198702
119894minus 11987041015840
1198941198672
119894(1198712
119894)minus1
1198704
119894= 0
1198672
119894(1198712
119894) gt 0
(42)
1198702
119894= minus119867
1
119894(1198711
119894)minus1
1198703
119894 (43)
where
1198671
119894(1198711
119894) = 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894)
1198703
119894= 119901119894119894
[1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894) + 1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894)]
1198672
119894(1198712
119894) = 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894)
1198704
119894= 119901119894119894
[1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894) + 1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894)]
(44)
If (119860119894 119860119894) is 119879119899-stable then
(i) (1198701
119894 1198702
119894) isin K
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Proof In the deduction of Lemma 5 we can prove that (i) iscorrect Next what we have to do is to prove (ii) and (iii)In the light of the Lyapunov equation (7) and any given set
Mathematical Problems in Engineering 7
of matrices 119882119894in Theorem 4 it is easy to get the following
equations for system (30)
119901119894119894
[(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987611015840
1198941198761
119894+ 11987011015840
1198941198771
1198941198701
119894+ 11987021015840
1198941198781
1198941198702
119894= 1198711
119894
1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894) gt 0
(45)
119901119894119894
[ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987621015840
1198941198762
119894+ 11987011015840
1198941198772
1198941198701
119894+ 11987021015840
1198941198782
1198941198702
119894= 1198712
119894
1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894) gt 0
(46)
By rearranging (45) and (46) (40) and (42) can be obtainedrespectively
Noting 119906lowast
(119896) = 1198701
119894119909(119896) and by substituting 119906
lowast
(119896) into(30) it is easy to get the following system
119909 (119896 + 1) = (119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)
+ [(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
(47)
Then considering the scalar function 119885(119909119896) = 119909
1015840
1198961198711
120579119896
119909119896 we
have
119864119896
[Δ119885 (119909119896)]
= 119864119896
[119885 (119909119896+1
) minus 119885 (119909119896)]
= 119864119896
[1199091015840
119896+11198711
120579119896+1
119909119896+1
minus 1199091015840
1198961198711
120579119896
119909119896]
= 119864119896
minus1199091015840
1198961198711
120579119896
119909119896
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
(48)
Due toinfin
sum
119896=0
119864119896
[Δ119885 (119909119896)]
= 119864119896
[
infin
sum
119896=0
Δ119885 (119909119896)] = 119864
119896[119885 (119909infin
) minus 119885 (1199090)] = minus119909
1015840
01198711
1198941199090
(49)
by (40) and a completing squares technique (31) can bederived that
1198691
(119906lowast
V)
=
infin
sum
119896=0
119864119896
[1199091015840
119896(11987611015840
120579119896
1198761
120579119896
+ 11987011015840
120579119896
1198771
120579119896
1198701
120579119896
) 119909119896
+ V10158401198961198781
120579119896
V119896]
+
infin
sum
119896=0
119864119896
[Δ119885 (119909119896)] + 119909
1015840
01198711
1198941199090
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
1199091015840
119896[minus1198711
119894+ 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 11987611015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894] 119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V1015840119896
(1198781
119894+ 1199011198941198941198621015840
1198941198711
119894119862119894+ 1199011198941198941198621015840
1198941198711
119894119862119894) V119896
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[1199091015840
11989611987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V10158401198961198671
119894(1198711
119894) V119896]
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[V (119896) minus 1198702
119894119909 (119896)]
1015840
1198671
119894(1198711
119894) [V (119896) minus 119870
2
119894119909 (119896)]
ge 1199091015840
01198711
1198941199090 120591 = 1
(50)
Then by (32) it follows that Vlowast(119896) = 1198702
119894119909(119896) and 119869
1
(119906lowast
Vlowast) =
1199091015840
01198711
1198941199090 Finally by substituting Vlowast(119896) into (30) in the same
way as before we have 119906lowast
(119896) = 1198701
119894119909(119896) and 119869
2
(119906lowast
Vlowast) =
1199091015840
01198712
1198941199090
Theorem 7 If (119860119894 119860119894) is 119879
119899-stable and for system (30)
assume that (40)ndash(43) admit the solution (1198711
119894 1198712
119894 1198701
119894 1198702
119894) with
(1198701
119894 1198702
119894) isin K then
(i) 1198711
119894gt 0 1198712
119894gt 0
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Remark 8 When 119908(119896) equiv 0 these results still hold inthe paper Only for the reason of simplicity in (1) and(30) we assume the state 119909(119905) and control inputs (119906(119905) V(119905))depend on the same noise 119908(119896) If they rely on the different
8 Mathematical Problems in Engineering
noises (1199081(119896) 119908
2(119896)) then new results will be yielded The
discussion is omitted
4 Iterative Algorithm and Simulation
41 An Iterative Algorithm In this section an iterative algo-rithm is proposed to solve the four coupled GAREs (40)ndash(43) Infinite horizon Riccati equations are hard to be solvedhence the particular problems can be solved via finite horizonequations 119873 represents the finite number of iterations in thefollowing equations
1198711
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
1198711
119894
119873
(119896 + 1)
times (119860119894+ 1198611198941198701
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
times 1198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896))
+ 11987611015840
1198941198761
119894+ 1198701
119894
119873
(119896)1015840
1198771
1198941198701
119894
119873
(119896)
minus 1198703
119894
119873
(119896)1015840
1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896)
1198711
119894
119873
(119896 + 1) = 0
1198671
119894(1198711
119894
119873
(119896 + 1)) gt 0
(51)
1198701
119894
119873
(119896) = minus1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896) (52)
1198712
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
1198712
119894
119873
(119896 + 1)
times (119860119894+ 1198621198941198702
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
times 1198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896))
+ 11987621015840
1198941198762
119894+ 1198702
119894
119873
(119896)1015840
1198782
1198941198702
119894
119873
(119896)
minus 1198704
119894
119873
(119896)1015840
1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896)
1198712
119894
119873
(119896 + 1) = 0
1198672
119894(1198712
119894
119873
(119896 + 1)) gt 0
(53)
1198702
119894
119873
(119896) = minus1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896) (54)
where
1198671
119894(1198711
119894
119873
(119896 + 1))
= 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894)
1198672
119894(1198712
119894
119873
(119896 + 1))
= 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894
+ 1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894)
1198703
119894
119873
(119896) = 119901119894119894
[1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))]
1198704
119894
119873
(119896) = 119901119894119894
[1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))
+1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))]
(55)
An iterative process for solving (40)ndash(43) based on the aboverecursions is presented as follows
(a) Given appropriate natural number 119873 and the initialconditions 119871
1
119894
119873
(119873+1) = 0 1198712119894
119873
(119873+1) = 0 1198701119894
119873
(119873+
1) = 0 and 1198702
119894
119873
(119873 + 1) = 0
(b) Through the numerical values of 1198711
119894
119873
(119873+1) 1198712119894
119873
(119873+
1)1198701119894
119873
(119873+1) and1198702
119894
119873
(119873+1) we have1198671
119894(1198711
119894
119873
(119873+
1))1198672119894(1198712
119894
119873
(119873+1))1198703119894
119873
(119873) and1198704
119894
119873
(119873) accordingto (55)
(c) 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873) can be respectively computedby (52) and (54) Then 119871
1
119894
119873
(119873) and 1198712
119894
119873
(119873) can alsobe respectively obtained by (51) and (53)
(d) Let 1198711
119894
119873
(119873 + 1) = 1198711
119894
119873
(119873) 1198712
119894
119873
(119873 + 1) = 1198712
119894
119873
(119873)1198701
119894
119873
(119873 + 1) = 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873 + 1) = 1198702
119894
119873
(119873)
(e) Then 119873 = 119873 minus 1 Repeat steps (b)ndash(d) until thenumber of iterations is 119873 + 1 We can finally obtainthe numerical values of 119871
1
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) and1198702
119894
119873
(0)
As in [28] under the assumptions of stabilizability for any1199090
isin R119899
lim119873rarrinfin
1199091015840
01198711
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198691119873
(119906lowast
119873 V) = min 119869
1infin
(119906lowast
V) = 1199091015840
01198711
1198941199090
lim119873rarrinfin
1199091015840
01198712
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198692119873
(119906 Vlowast119873
) = min 1198692infin
(119906 Vlowast) = 1199091015840
01198712
1198941199090
lim119873rarrinfin
1198701
119894
119873
(0) = 1198701
119894 lim
119873rarrinfin
1198702
119894
119873
(0) = 1198702
119894
(56)
Mathematical Problems in Engineering 9
Therefore
lim119873rarrinfin
(1198711
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) 1198702
119894
119873
(0))
= (1198711
119894 1198712
119894 1198701
119894 1198702
119894)
(57)
where (1198711
119894 1198712
119894 1198701
119894 1198702
119894) are the solutions of (40)ndash(43)
42 A Simulation Example To verify the efficiency of theabove iterative algorithm we consider the following 2-Dexample In the system (30) we set 120579
119896= 119894 isin X = 1 2
119877120591
119894= 119878120591
119894= 1 (120591 = 1 2)
1198601
= [065 0
0 09] 119860
1= [
045 0
0 055]
1198611
= [06
055] 119861
1= [
045
085]
1198621
= [075
055] 119862
1= [
05
085]
1198761
1= [
055 0
0 065] 119876
2
1= [
075 0
0 025]
1198602
= [075 0
0 07] 119860
2= [
035 0
0 045]
1198612
= [05
045] 119861
2= [
055
085]
1198622
= [065
055] 119862
2= [
04
085]
1198761
2= [
035 0
0 045] 119876
2
2= [
055 0
0 035]
(58)
For convenience let 11990111
= 04 11990122
= 05 and 119873 = 50When 120579
119896= 1 by applying the above iterative algorithm we
obtain the solutions of the four coupled equations (51)ndash(54)as follows
1198711
1
119873
(0) = [1198711
1(1 1) 119871
1
1(1 2)
1198711
1(2 1) 119871
1
1(2 2)
] = [04023 minus00588
minus00588 06820]
1198712
1
119873
(0) = [1198712
1(1 1) 119871
2
1(1 2)
1198712
1(2 1) 119871
2
1(2 2)
] = [07111 minus00331
minus00331 01487]
1198701
1
119873
(0) = [1198701
1(1 1) 119870
1
1(1 2)] = [minus01245 minus00053]
1198702
1
119873
(0) = [1198702
1(1 1) 119870
2
1(1 2)] = [minus00390 minus01739]
(59)
(1198711
1
119873
(0) 1198712
1
119873
(0) 1198701
1
119873
(0) 1198702
1
119873
(0)) are also the solutionsof (40)ndash(43) according to (57) By the solutions itshows that 119871
1
1gt 0 and 119871
2
1gt 0 The evolution of
(1198711
1
119873
(119896) 1198712
1
119873
(119896) 1198701
1
119873
(119896) 1198702
1
119873
(119896)) is exhibited in Figures 1and 2 And the figures clearly illustrate the convergence andspeediness of the backward iterations When 120579
119896= 2 it is easy
0 10 20 30 40 50minus01
0
01
02
03
04
05
06
07
08
N
L11(1 1)
L11(2 1)
L11(2 2)
L21(1 1)
L21(2 1)
L21(2 2)
Figure 1 Evolution of 1198711
1
119873
(119896) and 1198712
1
119873
(119896)
0 10 20 30 40 50minus018
minus016
minus014
minus012
minus01
minus008
minus006
minus004
minus002
0
N
K11(1 1)
K11(1 2)
K21(1 1)
K21(1 2)
Figure 2 Evolution of 1198701
1
119873
(119896) and 1198702
1
119873
(119896)
to get (1198711
2
119873
(0) 1198712
2
119873
(0) 1198701
2
119873
(0) 1198702
2
119873
(0)) that are also thesolutions of (40)ndash(43) And 119871
1
2gt 0 and 119871
2
2gt 0 Because it is
the same as the above process (120579119896
= 1) we do not introduceit again due to space limitations
5 Conclusions
In this paper we have discussed the 119879119899-stability for the
discrete-time MJSLS with a finite number of jump timesand its infinite horizon LQ differential games Based on therelations between the Lyapunov equation and the stabil-ity of discrete-time MJSLS we have obtained some useful
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
moments are stable whenever the sample path is stable themoment stability for stochastic systems implies almost surestability but not vice versa as pointed out by [16]
On the other hand the LQ differential games have manyapplications in the economy military and intelligent robotsSince the book [18] entitled ldquoDifferential Gamesrdquo written byDr Isaacs came out theory and application of differentialgames have been developed greatly A differential game isa mathematical model that represents a conflict betweendifferent agentswhich control a dynamical system and each ofthem is trying to minimize his individual objective functionby giving a control to the system In fact many situationsin industry economies management and elsewhere arecharacterized by multiple decision makers and enduringconsequences of decisions which can be treated as dynamicgames Particularly applications of differential games arewidely researched in LQ control problem [19ndash23] By solvingthe LQ control problems players can avoid most of theadditional cost incurred by this perturbation The authorsconsider the zero-sum infinite-horizon and LQ differentialgames in [19] A sufficient condition for the LQ differentialgames is applied to the119867
infinoptimization problem in [20]The
authors in [21] study the problemof designing suboptimal LQdifferential games with multiple players In [22] the authorsstudy the LQ nonzero sum stochastic differential gamesproblem with random jump A leader-follower stochasticdifferential game is considered with the state equation beinga linear Ito-type stochastic differential equation and the costfunctionals being quadratic [23]
As far as we know there are few researchers paying atten-tion to the discrete-time stochastic LQ differential gamesespecially MJSLS Thus it is significant to consider thesesystems In [24] we have considered stochastic differentialgames in infinite-time horizon By introducing stochasticexact observability and stochastic exact detectability the opti-mal strategies (Nash equilibrium strategies) and the optimalcost values have been given In [25] we have consideredLQ differential games in finite horizon for discrete-timestochastic systems with Markovian jump parameters andmultiplicative noise Furthermore a suboptimal solution ofthe LQ differential games is proposed based on a convexoptimization approach On the basis of [26] in the paperwe further investigate the LQ differential games for discrete-time MJSLS with a finite number of jump times It gives theoptimal strategies and the optimal cost values for infinitehorizon LQ stochastic differential gameswhich are associatedwith the four coupled GAREs Generally it is difficult to solvethe four coupled GAREs analytically Here we will solve theLQ differential games by means of stochastic 119879
119899-stability and
we will employ a recursive procedure to solve the coupledGAREs
The paper is organized as follows In Section 2 some basicdefinitions are recalled A necessary and sufficient conditionin relation to the connection between the 119879
119899-stability for
MJSLS and Lyapunov equation is presented In Section 3 weformulate LQ differential games with quadratic cost functionfor the MJSLS and propose the stochastic 119879
119899-stability of
discrete-time MJSLS with a finite number of jump timeswhich is essential to obtain the main results An iterative
algorithm for solving the four coupled GAREs is put forwardand an illustrative example is also displayed in Section 4Section 5 ends this paper with some concluding remarks
For convenience the paper adopts the following basicnotations It uses R119899 to denote the linear space of all 119899-dimensional real vectors R
119898times119899denotes the linear space of
all 119898 times 119899 real matrices N = 0 1 2 1198801015840 indicates the
transpose of matrix 119880 and 119880 ge 0 (119880 gt 0) represents anonnegative definite matrix (positive definite) The standardvector norm inR119899 is indicated by sdot and the correspondinginduced norm of matrix 119880 by 119880 119871
2
(infinR119896) represents thespace of R119896-valued square integrable random vectors and120575sdot
is the Dirac measure Finally we write 119864119896[sdot] instead of
119864[sdot | 119909119896 120579119896] and we define the following operator 120576
119894(119880) =
sum119895 =119894
119901119894119895119880119895for 119880 = 119880
1015840
ge 0
2 Stochastic 119879119899-Stability for Discrete-Time
MJSLS with a Finite Number of Jump Times
Let (ΩF 119875 F119896) be a given fundamental probability space
where there exist a Markov chain 120579119896and a sequence of real
random variables 119908(119896) F119896denotes the 120590-algebra generated
by 120579119896and 119908(119896) that isF
119896= 120590120579
119904 119908(119904) | 119904 = 0 1 2 119896 sub
F Consider the following MJSLS defined on the space(ΩF 119875 F
119896)
119909 (119896 + 1) = 119860120579119896
119909 (119896) + 119860120579119896
119909 (119896) 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
(1)
where 119909(119896) 120579119896 119896 isin N are the states of process with values
inR119899 timesX 120579119896 119896 isin N is a time homogeneous Markov chain
taking values in a finite set X = 1 2 119873 with initialdistribution 120583 and transition probability matrix P = [119901
119894119895]
where
119901119894119895
= 119875 (120579119896+1
= 119895 | 120579119896
= 119894) forall119894 119895 isin X 119896 isin N (2)
The set X comprises the various operation modes of thesystem (1) For each 120579
119896= 119894 isin X the matrices 119860
120579119896isin R119899times119899
and 119860120579119896
isin R119899times119899
(associated with ldquo119894thrdquo mode) will be assignedas 119860120579119896
= 119860119894and 119860
120579119896= 119860119894in someplace of the paper
119908(119896) 119896 isin N is a sequence of real random variables whichis also a wide sense stationary second-order process with119864[119908(119896)] = 0 and 119864[119908(119894)119908(119895)] = 120575
119894119895 where 120575
119894119895refers to a
Kronecker function that is 120575119894119895
= 1 if 119894 = 119895 and 120575119894119895
= 0 if 119894 = 119895TheMJSLS as defined is trivially a strongMarkov processWeassume that 120579
119896is independent of119908(119896) Tomake the operation
more convenient let 119909(119896) = 119909119896 When system (1) is stable we
also say (119860119894 119860119894) stable for short
Although several concepts of stochastic stability can befound in the literature in this paper the stochastic stabilityconcept associated with the stopping times for MJSLS isresearched The stopping times in relation to jump times aredefined as follows
1198790
= 0
119879119899
= min 119896 gt 119879119899minus1
120579119896
= 120579119879119899minus1
119899 = 1 2 119873
(3)
Mathematical Problems in Engineering 3
The stopping time may represent interesting situations fromthe point of view of applications For instance it can be theaccumulated nth failure and repair of the system In anothersituation the stopping time can represent the occurrence of aldquocrucial failurerdquo (which may happen after a random numberof failures)
This class of stochastic systems is associated with sys-tems subject to failures in their components or connectionsaccording to a Markov chain The situation that we areinterested in arises when one wishes to study the stabilityof such a system until the occurrence of a fixed number 119873
of failures and repairs The paper recognizes the sequenceof the stopping times containing the successive times of theoccurrence of such failures and then it studies the stability ofsystem (1) according to these stopping times
Definition 1 (see [27]) Consider a stopping time 119879119899 The
MJSLS (1) is
(i) stochastically 119879119899-stable if for each initial condition 119909
0
and initial distribution 120583
119864 [
infin
sum
119896=0
1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] lt infin (4)
(ii) mean-square 119879119899-stable if for each initial condition 119909
0
and initial distribution 120583
lim119896rarrinfin
119864 [1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] = 0 (5)
Lemma 2 (see [27]) For all 119898 ge 1 and 119894 119895 isin X
119875 (1198791
= 119898 120579119898
= 119895 | 1205790
= 119894)
= 119901119894119895120575119898=1
119908ℎ119890119899 119901119894119894
= 0
119901119898minus1
119894119894119901119894119895120575119898gt1
119908ℎ119890119899 0 lt 119901119894119894
lt 1
(6)
Remark 3 119875(1198791
= 1 | 1205790
= 119894) = 1 and 119875(1198791
= +infin | 1205790
= 119894) =
1 whenever 119901119894119894
= 0 and 119901119894119894
= 1 respectively That is in anycase system will jump to another state
Next we will give an important theorem that will be usedlater
Theorem 4 The MJSLS (1) is 119879119899-stable if and only if for any
given set ofmatrices119882119894gt 0 there exists a unique set ofmatrices
119871119894gt 0 satisfying the Lyapunov equations
119901119894119894
(1198601015840
119894119871119894119860119894+ 1198601015840
119894119871119894119860119894) minus 119871119894+ 119882119894= 0 119894 isin X (7)
Proof Sufficiency In the proof we employ an inductionargument on the stopping times 119879
119899 First define the function
119881119896
(119909 119894) = 1199091015840
119896(1198751198941205751198791gt119896
+ 1198661198941205751198791=119896
) 119909119896 (8)
where 119866119894gt 0 and 119875
119894gt 0 is the solution of
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 119882119894+ 1198601015840
119894120576119894(119866) 119860
119894
+ 1198601015840
119894120576119894(119866) 119860
119894= 0 119894 isin X
(9)
The existence of such119875119894gt 0 relies on (7) Hence to functional
119881119896(119909 119894) in the following operation we can derive
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
= 119864119896
[1199091015840
119896+1(119875120579119896+11205751198791gt119896+1
+ 119866120579119896+11205751198791=119896+1
) 119909119896+1
minus 1199091015840
119896(1198751205791198961205751198791ge119896+1
+ 1198661205791198961205751198791=119896
) 119909119896]
= 119864119896
[(1199091015840
119896+1119875120579119896+1
119909119896+1
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791gt119896+1
]
+ 119864119896
[(1199091015840
119896+1119866120579119896+1
119909119896+1
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791=119896+1
]
minus 1199091015840
119896119866120579119896
1199091198961205751198791=119896
= 119864119896
(1199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896
+ 1199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896119908119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896119908119896
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791gt119896+1
+ 119864119896
(1199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896
+ 1199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896119908119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896119908119896
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791=119896+1
minus 1199091015840
119896119866120579119896
1199091198961205751198791=119896
(10)
We know that 119864[119908(119896)] = 0 and 119864[119908(119894)119908(119895)] = 120575119894119895 where
120575119894119895
= 1 if 119894 = 119895 and 120575119894119895
= 0 if 119894 = 119895 Calculating the expectedvalues above we can obtain that
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
= 1199091015840
119896[119901120579119896120579119896
(1198601015840
120579119896
119875120579119896
119860120579119896
+ 1198601015840
120579119896
119875120579119896
119860120579119896
) + 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
+ 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
minus 119875120579119896
] 1199091198961205751198791gt119896
minus 1199091015840
119896119866120579119896
1199091198961205751198791=119896
(11)
due to119875(1198791
= 119896+1 | 120579119896) = 1minus119901
120579119896120579119896 119875(1198791
gt 119896+1 | 120579119896) = 119901120579119896120579119896
and 120576120579119896
(119866) = Σ120579119896+1 =120579119896
119901120579119896120579119896+1
119866120579119896+1
The above relation can be rewritten as
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
= minus1199091015840
119896(1198821205791198961205751198791gt119896
+ 1198661205791198961205751198791=119896
) 119909119896
(12)
where
minus119882120579119896
= 119901120579119896120579119896
(1198601015840
120579119896
119875120579119896
119860120579119896
+ 1198601015840
120579119896
119875120579119896
119860120579119896
)
+ 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
+ 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
minus 119875120579119896
(13)
4 Mathematical Problems in Engineering
Now let us observe that119873
sum
119896=0
1198640
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
=
119873
sum
119896=0
1198640
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
(14)
By applying (12) and considering that 119882119894gt 0 119866
119894gt 0 for each
initial condition 1199090and initial distribution 120583 then we have
1198640
[119881119896+1
(119909119896+1
120579119896+1
)] minus 1198810
(1199090 1205790)
= minus
119873
sum
119896=0
1198640
[1199091015840
119896(1198821205791198961205751198791gt119896
+ 1198661205791198961205751198791=119896
) 1199091198961205751198791ge119896
]
le minus
119873
sum
119896=0
1205741198640
[1003817100381710038171003817119909119896
1003817100381710038171003817
2
1205751198791ge119896
]
(15)
for some 120574 gt 0 Because 1198640[119881119896(119909119896 120579119896)] ge 0 forall119896 ge 0 then
lim119896rarrinfin
1198640[119881119896+1
(119909119896+1
120579119896+1
)] = 0 by (5) Finally it is easy toverify that for any 119879
1
lim sup119873rarrinfin
119873
sum
119896=0
1198640
[1003817100381710038171003817119909119896
1003817100381710038171003817
2
1205751198791ge119896
] le1
1205741198810
(1199090 1205790) lt infin (16)
holds from (15) Therefore for any 1198791 MJSLS (1) is stable
according to (i) of Definition 1Now using an induction argument we assume that for
some 119899 the inequality
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899ge119896
] lt 1199091015840
01198751205790
1199090
(17)
holds and thus by setting 119876 equiv 119868 119864[119909119879119899
2
120575119879119899ge119896
] lt infinHowever
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
]
= lim sup119873rarrinfin
119864 [
[
119873
sum
119896=0
1199091015840
119896119876120579119896
119909119896120575119879119899gt119896
+
119873
sum
119896=119879119899
1199091015840
119896119876120579119896
119909119896120575119879119899le119896le119879119899+1
]
]
(18)
Notice that using the strong Markov property and thehomogeneity property the second term conditioned to theknowledge of (119909
119879119899 120579119879119899) can be written as
lim sup119873rarrinfin
119864 [
[
119873
sum
119896=119879119899
1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
| 119909119879119899
120579119879119899
]
]
= lim sup119873rarrinfin
119864 [
119873minus119879119899
sum
119896=0
1199091015840
119896119876120579119896
119909119896120575119896le1198791
| 1199090
= 119909119879119899
1205790
= 120579119879119899
]
lt 1199091015840
119879119899
119875120579119879119899
119909119879119899
(19)
So one can conclude from (18) and (19) that
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
]
lt lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899gt119896
+ 1199091015840
119879119899
119875120579119879119899
119909119879119899
]
= lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899ge119896
+ 1199091015840
119879119899
(119875120579119879119899
minus 119876120579119879119899
) 119909119879119899
]
lt 21199091015840
01198751205790
1199090
(20)
Therefore for any 119879119899
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] lt infin (21)
indicate that the MJSLS (1) is 119879119899-stable
Necessity As in the previous part define the functional
1199091015840
01198751205790
1199090
= 1198640
[
infin
sum
119896=0
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
] (22)
for all (1199090 1205790) isin R119899 times X Therefore
1199091015840
11198751205791
1199091
= 1198641198791
[
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
] 1205751198791gt1
(23)
The right-hand side of (22) can be expressed as
1198640
1199091015840
01198821205790
1199090
+ 1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791ge1
]
(24)
In addition
1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791ge1
]
= 1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791gt1
+ 1199091015840
1198791
1198661205791198791
11990911987911205751198791=1
]
(25)
Thus based on the strong Markov property applying homo-geneity in (25) and introducing it in (24) we arrive at
1199091015840
01198751205790
1199090
= 1199091015840
01198821205790
1199090
+ 1198640
[1199091015840
11198751205791
11990911205751198791gt1
+ 1199091015840
1198791
1198661205791198791
11990911987911205751198791=1
]
(26)
Mathematical Problems in Engineering 5
Since 1199090is arbitrary and calculating the expected values
above (26) implies that
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 1198601015840
119894120576119894(119866) 119860
119894+ 1198601015840
119894120576119894(119866) 119860
119894
= minus119882119894
(27)
using the fact that 119875(1198791
= 119896 + 1 | 120579119896
= 119894) = 1 minus 119901119894119894and 119875(119879
1gt
119896+1 | 120579119896
= 119894) = 119901119894119894Thus from the Lyapunov stability theory
the existence of the set 119871119894
gt 0 satisfying (7) is guaranteedcompleting the proof for 119899 = 1
Now for the general case from the stochastically 119879119899-
stable of the system we can obtain that
119864 [
infin
sum
119896=0
1199091015840
119896119882120579119896
119909119896120575119879119899gt119896
+ 1199091015840
119879119899
119866120579119879119899
119909119879119899
] lt infin (28)
And from the strong Markov property we can deduce that
119864119879119899
[
[
infin
sum
119896=119879119899
1199091015840
119896119882120579119896
119909119896120575119879119899+1gt119896
+ 1199091015840
119879119899+1
119866120579119879119899+1
119909119879119899+1
]
]
lt infin (29)
for 119899 = 0 1 119873minus1 By the homogeneity property it followsthat (29) is equivalent to (22) with 119909
0= 119909119879119899
and 1205790
= 120579119879119899
and the existence of a set of matrices 119871119894
gt 0 satisfying (7) isassured Then the proof of Theorem 4 is completed
3 LQ Differential Games for MJSLS witha Finite Number of Jump Times
31 Problem Formulation Now we study the LQ differentialgames for discrete-time MJSLS Comparing with system (1)consider the following discrete-time MJSLS with a finitenumber of jump times
119909 (119896 + 1) = 119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)
+ [119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
119910120591
(119896) = 119876120591
120579119896
119909 (119896) 120591 = 1 2
(30)
119910120591
(119896) isin R119898 are the measurement outputs for each playerHere (119906(119896) V(119896)) isin R119903 times R119903 represent the system controlinputs The matrices (119861
120579119896 119861120579119896
119862120579119896
119862120579119896
119876120591
120579119896
) isin R119899times119903
times R119899times119903
times
R119899times119903
times R119899times119903
times R119898times119899
(associated with ldquo119894thrdquo mode) will beassigned as (119861
119894 119861119894 119862119894 119862119894 119876120591
119894) for each 120579
119896= 119894 isin X
Throughout this paper we choose the infinite horizonquadratic cost functions associated with each player
119869120591
(119906 V) =
infin
sum
119896=0
119864 [119909 (119896)1015840
(119876120591
120579119896
)1015840
119876120591
120579119896
119909 (119896)
+ 119906 (119896)1015840
119877120591
120579119896
119906 (119896) + V (119896)1015840
119878120591
120579119896
V (119896) ]
120591 = 1 2
(31)
The weighting matrices 119876120591
120579119896
= 119876120591
119894ge 0 119877
120591
120579119896
= 119877120591
119894gt 0 isin R
119903times119903
and 119878120591
120579119896
= 119878120591
119894gt 0 isin R
119903times119903
So we are looking for actions that satisfy simultaneously
1198691
(119906lowast
Vlowast) le 1198691
(119906lowast
V) 1198692
(119906lowast
Vlowast) le 1198692
(119906 Vlowast) (32)
where (119906lowast
(119896) Vlowast(119896)) isin 1198712
(infinR119903119906) times 1198712
(infinR119903V)To ensure the finiteness of the infinite-time cost function
we restrain the admissible control set to the constant linearfeedback strategies that is 119906(119896) = 119870
1
120579119896
119909(119896) V(119896) = 1198702
120579119896
119909(119896)where 119870
1
120579119896
and 1198702
120579119896
are constant matrices of appropriatedimensions and (119870
1
120579119896
1198702
120579119896
) belong to
K = 119870 = (1198701
120579119896
1198702
120579119896
) | system (30) can be stabilized
with 119906 (119896) = 1198701
120579119896
119909 (119896)
V (119905) = 1198702
120579119896
119909 (119896)
(33)
We say that the optimization problem is well posedand the 119906(119896) and V(119896) have the following two additionalproperties
119864 [|119906 (119896)|2
] lt infin 119864 [|V (119896)|2
] lt infin 119896 isin N (34)
The optimal strategies 119906lowast and Vlowast determined by (32) are
also called the Nash equilibrium strategies (119906lowast
Vlowast) In orderto guarantee the unique global Nash game solutions both theplayers are only allowed to take constant feedback controlsNext we focus on finding the optimal strategies
32Main Results First we give an important lemma that willbe used later If the system (1) is 119879
119899-stable we can obtain the
following result for the discrete-time MJSLS (30)
Lemma 5 If [119860120579119896
119860120579119896
] is 119879119899-stable then so is [119860
120579119896+ 119861120579119896
1198701
120579119896
+
119862120579119896
1198702
120579119896
119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
] where (1198701
120579119896
1198702
120579119896
) isin K
Proof SufficiencyThe proof employs an induction argumenton the stopping times 119879
119899 First define the function
119881119896
(119909 119894) = 1199091015840
119896(1198751198941205751198791gt119896
+ 1198661198941205751198791=119896
) 119909119896 (35)
where 119866119894gt 0 and 119875
119894gt 0 is the solution of
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 119882119894+ 1198601015840
119894120576119894(119866) 119860
119894
+ 1198601015840
119894120576119894(119866) 119860
119894= 0 119894 isin X
(36)
6 Mathematical Problems in Engineering
The existence of such 119875119894
gt 0 relies on (7) Hence tothe function 119881
119896(119909 119894) and the system (30) in the following
operation we acquire that
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896) minus 119881119896
(119909 (119896) 120579119896)]
= 119909 (119896)1015840
119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
minus119875120579119896
119909 (119896) 1205751198791gt119896
minus 119909 (119896)1015840
119866120579119896
119909 (119896) 1205751198791=119896
(37)
Compared with (12) we know
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896+1
) minus 119881119896
(119909 (119896) 120579119896)]
= minus119909 (119896)1015840
(119882120579119896120575119879119899gt119896
+ 119866120579119896120575119879119899=119896
) 119909 (119896)
(38)
where
minus119882120579119896
= minus119875120579119896
+ 119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
(39)
Considering that 119882120579119896
gt 0 119866120579119896
gt 0 we can obtain (19)And because 119864[119881
119896(119909(119896) 120579
119896)] ge 0 forall119896 ge 0 for each initial
condition 1199090 from (40) it is easy to verify (21) Therefore
the MJSLS (30) is 119879119899-stable
Theorem 6 For system (30) suppose the following coupledequations admit the solutions (119871
1
119894 1198712
119894 1198701
119894 1198702
119894) with 119871
1
119894gt 0
1198712
119894gt 0
minus 1198711
119894+ 119901119894119894
[(119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)
+ (119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)] + 119876
11015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894minus 11987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894= 0
1198671
119894(1198711
119894) gt 0
(40)
1198701
119894= minus119867
2
119894(1198712
119894)minus1
1198704
119894 (41)
minus 1198712
119894+ 119901119894119894
[(119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)
+ (119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)] + 119876
21015840
1198941198762
119894
+ 11987021015840
1198941198782
1198941198702
119894minus 11987041015840
1198941198672
119894(1198712
119894)minus1
1198704
119894= 0
1198672
119894(1198712
119894) gt 0
(42)
1198702
119894= minus119867
1
119894(1198711
119894)minus1
1198703
119894 (43)
where
1198671
119894(1198711
119894) = 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894)
1198703
119894= 119901119894119894
[1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894) + 1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894)]
1198672
119894(1198712
119894) = 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894)
1198704
119894= 119901119894119894
[1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894) + 1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894)]
(44)
If (119860119894 119860119894) is 119879119899-stable then
(i) (1198701
119894 1198702
119894) isin K
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Proof In the deduction of Lemma 5 we can prove that (i) iscorrect Next what we have to do is to prove (ii) and (iii)In the light of the Lyapunov equation (7) and any given set
Mathematical Problems in Engineering 7
of matrices 119882119894in Theorem 4 it is easy to get the following
equations for system (30)
119901119894119894
[(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987611015840
1198941198761
119894+ 11987011015840
1198941198771
1198941198701
119894+ 11987021015840
1198941198781
1198941198702
119894= 1198711
119894
1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894) gt 0
(45)
119901119894119894
[ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987621015840
1198941198762
119894+ 11987011015840
1198941198772
1198941198701
119894+ 11987021015840
1198941198782
1198941198702
119894= 1198712
119894
1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894) gt 0
(46)
By rearranging (45) and (46) (40) and (42) can be obtainedrespectively
Noting 119906lowast
(119896) = 1198701
119894119909(119896) and by substituting 119906
lowast
(119896) into(30) it is easy to get the following system
119909 (119896 + 1) = (119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)
+ [(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
(47)
Then considering the scalar function 119885(119909119896) = 119909
1015840
1198961198711
120579119896
119909119896 we
have
119864119896
[Δ119885 (119909119896)]
= 119864119896
[119885 (119909119896+1
) minus 119885 (119909119896)]
= 119864119896
[1199091015840
119896+11198711
120579119896+1
119909119896+1
minus 1199091015840
1198961198711
120579119896
119909119896]
= 119864119896
minus1199091015840
1198961198711
120579119896
119909119896
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
(48)
Due toinfin
sum
119896=0
119864119896
[Δ119885 (119909119896)]
= 119864119896
[
infin
sum
119896=0
Δ119885 (119909119896)] = 119864
119896[119885 (119909infin
) minus 119885 (1199090)] = minus119909
1015840
01198711
1198941199090
(49)
by (40) and a completing squares technique (31) can bederived that
1198691
(119906lowast
V)
=
infin
sum
119896=0
119864119896
[1199091015840
119896(11987611015840
120579119896
1198761
120579119896
+ 11987011015840
120579119896
1198771
120579119896
1198701
120579119896
) 119909119896
+ V10158401198961198781
120579119896
V119896]
+
infin
sum
119896=0
119864119896
[Δ119885 (119909119896)] + 119909
1015840
01198711
1198941199090
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
1199091015840
119896[minus1198711
119894+ 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 11987611015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894] 119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V1015840119896
(1198781
119894+ 1199011198941198941198621015840
1198941198711
119894119862119894+ 1199011198941198941198621015840
1198941198711
119894119862119894) V119896
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[1199091015840
11989611987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V10158401198961198671
119894(1198711
119894) V119896]
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[V (119896) minus 1198702
119894119909 (119896)]
1015840
1198671
119894(1198711
119894) [V (119896) minus 119870
2
119894119909 (119896)]
ge 1199091015840
01198711
1198941199090 120591 = 1
(50)
Then by (32) it follows that Vlowast(119896) = 1198702
119894119909(119896) and 119869
1
(119906lowast
Vlowast) =
1199091015840
01198711
1198941199090 Finally by substituting Vlowast(119896) into (30) in the same
way as before we have 119906lowast
(119896) = 1198701
119894119909(119896) and 119869
2
(119906lowast
Vlowast) =
1199091015840
01198712
1198941199090
Theorem 7 If (119860119894 119860119894) is 119879
119899-stable and for system (30)
assume that (40)ndash(43) admit the solution (1198711
119894 1198712
119894 1198701
119894 1198702
119894) with
(1198701
119894 1198702
119894) isin K then
(i) 1198711
119894gt 0 1198712
119894gt 0
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Remark 8 When 119908(119896) equiv 0 these results still hold inthe paper Only for the reason of simplicity in (1) and(30) we assume the state 119909(119905) and control inputs (119906(119905) V(119905))depend on the same noise 119908(119896) If they rely on the different
8 Mathematical Problems in Engineering
noises (1199081(119896) 119908
2(119896)) then new results will be yielded The
discussion is omitted
4 Iterative Algorithm and Simulation
41 An Iterative Algorithm In this section an iterative algo-rithm is proposed to solve the four coupled GAREs (40)ndash(43) Infinite horizon Riccati equations are hard to be solvedhence the particular problems can be solved via finite horizonequations 119873 represents the finite number of iterations in thefollowing equations
1198711
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
1198711
119894
119873
(119896 + 1)
times (119860119894+ 1198611198941198701
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
times 1198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896))
+ 11987611015840
1198941198761
119894+ 1198701
119894
119873
(119896)1015840
1198771
1198941198701
119894
119873
(119896)
minus 1198703
119894
119873
(119896)1015840
1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896)
1198711
119894
119873
(119896 + 1) = 0
1198671
119894(1198711
119894
119873
(119896 + 1)) gt 0
(51)
1198701
119894
119873
(119896) = minus1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896) (52)
1198712
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
1198712
119894
119873
(119896 + 1)
times (119860119894+ 1198621198941198702
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
times 1198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896))
+ 11987621015840
1198941198762
119894+ 1198702
119894
119873
(119896)1015840
1198782
1198941198702
119894
119873
(119896)
minus 1198704
119894
119873
(119896)1015840
1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896)
1198712
119894
119873
(119896 + 1) = 0
1198672
119894(1198712
119894
119873
(119896 + 1)) gt 0
(53)
1198702
119894
119873
(119896) = minus1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896) (54)
where
1198671
119894(1198711
119894
119873
(119896 + 1))
= 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894)
1198672
119894(1198712
119894
119873
(119896 + 1))
= 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894
+ 1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894)
1198703
119894
119873
(119896) = 119901119894119894
[1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))]
1198704
119894
119873
(119896) = 119901119894119894
[1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))
+1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))]
(55)
An iterative process for solving (40)ndash(43) based on the aboverecursions is presented as follows
(a) Given appropriate natural number 119873 and the initialconditions 119871
1
119894
119873
(119873+1) = 0 1198712119894
119873
(119873+1) = 0 1198701119894
119873
(119873+
1) = 0 and 1198702
119894
119873
(119873 + 1) = 0
(b) Through the numerical values of 1198711
119894
119873
(119873+1) 1198712119894
119873
(119873+
1)1198701119894
119873
(119873+1) and1198702
119894
119873
(119873+1) we have1198671
119894(1198711
119894
119873
(119873+
1))1198672119894(1198712
119894
119873
(119873+1))1198703119894
119873
(119873) and1198704
119894
119873
(119873) accordingto (55)
(c) 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873) can be respectively computedby (52) and (54) Then 119871
1
119894
119873
(119873) and 1198712
119894
119873
(119873) can alsobe respectively obtained by (51) and (53)
(d) Let 1198711
119894
119873
(119873 + 1) = 1198711
119894
119873
(119873) 1198712
119894
119873
(119873 + 1) = 1198712
119894
119873
(119873)1198701
119894
119873
(119873 + 1) = 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873 + 1) = 1198702
119894
119873
(119873)
(e) Then 119873 = 119873 minus 1 Repeat steps (b)ndash(d) until thenumber of iterations is 119873 + 1 We can finally obtainthe numerical values of 119871
1
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) and1198702
119894
119873
(0)
As in [28] under the assumptions of stabilizability for any1199090
isin R119899
lim119873rarrinfin
1199091015840
01198711
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198691119873
(119906lowast
119873 V) = min 119869
1infin
(119906lowast
V) = 1199091015840
01198711
1198941199090
lim119873rarrinfin
1199091015840
01198712
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198692119873
(119906 Vlowast119873
) = min 1198692infin
(119906 Vlowast) = 1199091015840
01198712
1198941199090
lim119873rarrinfin
1198701
119894
119873
(0) = 1198701
119894 lim
119873rarrinfin
1198702
119894
119873
(0) = 1198702
119894
(56)
Mathematical Problems in Engineering 9
Therefore
lim119873rarrinfin
(1198711
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) 1198702
119894
119873
(0))
= (1198711
119894 1198712
119894 1198701
119894 1198702
119894)
(57)
where (1198711
119894 1198712
119894 1198701
119894 1198702
119894) are the solutions of (40)ndash(43)
42 A Simulation Example To verify the efficiency of theabove iterative algorithm we consider the following 2-Dexample In the system (30) we set 120579
119896= 119894 isin X = 1 2
119877120591
119894= 119878120591
119894= 1 (120591 = 1 2)
1198601
= [065 0
0 09] 119860
1= [
045 0
0 055]
1198611
= [06
055] 119861
1= [
045
085]
1198621
= [075
055] 119862
1= [
05
085]
1198761
1= [
055 0
0 065] 119876
2
1= [
075 0
0 025]
1198602
= [075 0
0 07] 119860
2= [
035 0
0 045]
1198612
= [05
045] 119861
2= [
055
085]
1198622
= [065
055] 119862
2= [
04
085]
1198761
2= [
035 0
0 045] 119876
2
2= [
055 0
0 035]
(58)
For convenience let 11990111
= 04 11990122
= 05 and 119873 = 50When 120579
119896= 1 by applying the above iterative algorithm we
obtain the solutions of the four coupled equations (51)ndash(54)as follows
1198711
1
119873
(0) = [1198711
1(1 1) 119871
1
1(1 2)
1198711
1(2 1) 119871
1
1(2 2)
] = [04023 minus00588
minus00588 06820]
1198712
1
119873
(0) = [1198712
1(1 1) 119871
2
1(1 2)
1198712
1(2 1) 119871
2
1(2 2)
] = [07111 minus00331
minus00331 01487]
1198701
1
119873
(0) = [1198701
1(1 1) 119870
1
1(1 2)] = [minus01245 minus00053]
1198702
1
119873
(0) = [1198702
1(1 1) 119870
2
1(1 2)] = [minus00390 minus01739]
(59)
(1198711
1
119873
(0) 1198712
1
119873
(0) 1198701
1
119873
(0) 1198702
1
119873
(0)) are also the solutionsof (40)ndash(43) according to (57) By the solutions itshows that 119871
1
1gt 0 and 119871
2
1gt 0 The evolution of
(1198711
1
119873
(119896) 1198712
1
119873
(119896) 1198701
1
119873
(119896) 1198702
1
119873
(119896)) is exhibited in Figures 1and 2 And the figures clearly illustrate the convergence andspeediness of the backward iterations When 120579
119896= 2 it is easy
0 10 20 30 40 50minus01
0
01
02
03
04
05
06
07
08
N
L11(1 1)
L11(2 1)
L11(2 2)
L21(1 1)
L21(2 1)
L21(2 2)
Figure 1 Evolution of 1198711
1
119873
(119896) and 1198712
1
119873
(119896)
0 10 20 30 40 50minus018
minus016
minus014
minus012
minus01
minus008
minus006
minus004
minus002
0
N
K11(1 1)
K11(1 2)
K21(1 1)
K21(1 2)
Figure 2 Evolution of 1198701
1
119873
(119896) and 1198702
1
119873
(119896)
to get (1198711
2
119873
(0) 1198712
2
119873
(0) 1198701
2
119873
(0) 1198702
2
119873
(0)) that are also thesolutions of (40)ndash(43) And 119871
1
2gt 0 and 119871
2
2gt 0 Because it is
the same as the above process (120579119896
= 1) we do not introduceit again due to space limitations
5 Conclusions
In this paper we have discussed the 119879119899-stability for the
discrete-time MJSLS with a finite number of jump timesand its infinite horizon LQ differential games Based on therelations between the Lyapunov equation and the stabil-ity of discrete-time MJSLS we have obtained some useful
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
The stopping time may represent interesting situations fromthe point of view of applications For instance it can be theaccumulated nth failure and repair of the system In anothersituation the stopping time can represent the occurrence of aldquocrucial failurerdquo (which may happen after a random numberof failures)
This class of stochastic systems is associated with sys-tems subject to failures in their components or connectionsaccording to a Markov chain The situation that we areinterested in arises when one wishes to study the stabilityof such a system until the occurrence of a fixed number 119873
of failures and repairs The paper recognizes the sequenceof the stopping times containing the successive times of theoccurrence of such failures and then it studies the stability ofsystem (1) according to these stopping times
Definition 1 (see [27]) Consider a stopping time 119879119899 The
MJSLS (1) is
(i) stochastically 119879119899-stable if for each initial condition 119909
0
and initial distribution 120583
119864 [
infin
sum
119896=0
1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] lt infin (4)
(ii) mean-square 119879119899-stable if for each initial condition 119909
0
and initial distribution 120583
lim119896rarrinfin
119864 [1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] = 0 (5)
Lemma 2 (see [27]) For all 119898 ge 1 and 119894 119895 isin X
119875 (1198791
= 119898 120579119898
= 119895 | 1205790
= 119894)
= 119901119894119895120575119898=1
119908ℎ119890119899 119901119894119894
= 0
119901119898minus1
119894119894119901119894119895120575119898gt1
119908ℎ119890119899 0 lt 119901119894119894
lt 1
(6)
Remark 3 119875(1198791
= 1 | 1205790
= 119894) = 1 and 119875(1198791
= +infin | 1205790
= 119894) =
1 whenever 119901119894119894
= 0 and 119901119894119894
= 1 respectively That is in anycase system will jump to another state
Next we will give an important theorem that will be usedlater
Theorem 4 The MJSLS (1) is 119879119899-stable if and only if for any
given set ofmatrices119882119894gt 0 there exists a unique set ofmatrices
119871119894gt 0 satisfying the Lyapunov equations
119901119894119894
(1198601015840
119894119871119894119860119894+ 1198601015840
119894119871119894119860119894) minus 119871119894+ 119882119894= 0 119894 isin X (7)
Proof Sufficiency In the proof we employ an inductionargument on the stopping times 119879
119899 First define the function
119881119896
(119909 119894) = 1199091015840
119896(1198751198941205751198791gt119896
+ 1198661198941205751198791=119896
) 119909119896 (8)
where 119866119894gt 0 and 119875
119894gt 0 is the solution of
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 119882119894+ 1198601015840
119894120576119894(119866) 119860
119894
+ 1198601015840
119894120576119894(119866) 119860
119894= 0 119894 isin X
(9)
The existence of such119875119894gt 0 relies on (7) Hence to functional
119881119896(119909 119894) in the following operation we can derive
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
= 119864119896
[1199091015840
119896+1(119875120579119896+11205751198791gt119896+1
+ 119866120579119896+11205751198791=119896+1
) 119909119896+1
minus 1199091015840
119896(1198751205791198961205751198791ge119896+1
+ 1198661205791198961205751198791=119896
) 119909119896]
= 119864119896
[(1199091015840
119896+1119875120579119896+1
119909119896+1
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791gt119896+1
]
+ 119864119896
[(1199091015840
119896+1119866120579119896+1
119909119896+1
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791=119896+1
]
minus 1199091015840
119896119866120579119896
1199091198961205751198791=119896
= 119864119896
(1199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896
+ 1199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896119908119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119875120579119896+1
119860120579119896
119909119896119908119896
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791gt119896+1
+ 119864119896
(1199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896
+ 1199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896119908119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896
+ 1199081015840
1198961199091015840
1198961198601015840
120579119896
119866120579119896+1
119860120579119896
119909119896119908119896
minus 1199091015840
119896119875120579119896
119909119896) 1205751198791=119896+1
minus 1199091015840
119896119866120579119896
1199091198961205751198791=119896
(10)
We know that 119864[119908(119896)] = 0 and 119864[119908(119894)119908(119895)] = 120575119894119895 where
120575119894119895
= 1 if 119894 = 119895 and 120575119894119895
= 0 if 119894 = 119895 Calculating the expectedvalues above we can obtain that
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
= 1199091015840
119896[119901120579119896120579119896
(1198601015840
120579119896
119875120579119896
119860120579119896
+ 1198601015840
120579119896
119875120579119896
119860120579119896
) + 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
+ 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
minus 119875120579119896
] 1199091198961205751198791gt119896
minus 1199091015840
119896119866120579119896
1199091198961205751198791=119896
(11)
due to119875(1198791
= 119896+1 | 120579119896) = 1minus119901
120579119896120579119896 119875(1198791
gt 119896+1 | 120579119896) = 119901120579119896120579119896
and 120576120579119896
(119866) = Σ120579119896+1 =120579119896
119901120579119896120579119896+1
119866120579119896+1
The above relation can be rewritten as
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
= minus1199091015840
119896(1198821205791198961205751198791gt119896
+ 1198661205791198961205751198791=119896
) 119909119896
(12)
where
minus119882120579119896
= 119901120579119896120579119896
(1198601015840
120579119896
119875120579119896
119860120579119896
+ 1198601015840
120579119896
119875120579119896
119860120579119896
)
+ 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
+ 1198601015840
120579119896
120576120579119896
(119866) 119860120579119896
minus 119875120579119896
(13)
4 Mathematical Problems in Engineering
Now let us observe that119873
sum
119896=0
1198640
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
=
119873
sum
119896=0
1198640
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
(14)
By applying (12) and considering that 119882119894gt 0 119866
119894gt 0 for each
initial condition 1199090and initial distribution 120583 then we have
1198640
[119881119896+1
(119909119896+1
120579119896+1
)] minus 1198810
(1199090 1205790)
= minus
119873
sum
119896=0
1198640
[1199091015840
119896(1198821205791198961205751198791gt119896
+ 1198661205791198961205751198791=119896
) 1199091198961205751198791ge119896
]
le minus
119873
sum
119896=0
1205741198640
[1003817100381710038171003817119909119896
1003817100381710038171003817
2
1205751198791ge119896
]
(15)
for some 120574 gt 0 Because 1198640[119881119896(119909119896 120579119896)] ge 0 forall119896 ge 0 then
lim119896rarrinfin
1198640[119881119896+1
(119909119896+1
120579119896+1
)] = 0 by (5) Finally it is easy toverify that for any 119879
1
lim sup119873rarrinfin
119873
sum
119896=0
1198640
[1003817100381710038171003817119909119896
1003817100381710038171003817
2
1205751198791ge119896
] le1
1205741198810
(1199090 1205790) lt infin (16)
holds from (15) Therefore for any 1198791 MJSLS (1) is stable
according to (i) of Definition 1Now using an induction argument we assume that for
some 119899 the inequality
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899ge119896
] lt 1199091015840
01198751205790
1199090
(17)
holds and thus by setting 119876 equiv 119868 119864[119909119879119899
2
120575119879119899ge119896
] lt infinHowever
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
]
= lim sup119873rarrinfin
119864 [
[
119873
sum
119896=0
1199091015840
119896119876120579119896
119909119896120575119879119899gt119896
+
119873
sum
119896=119879119899
1199091015840
119896119876120579119896
119909119896120575119879119899le119896le119879119899+1
]
]
(18)
Notice that using the strong Markov property and thehomogeneity property the second term conditioned to theknowledge of (119909
119879119899 120579119879119899) can be written as
lim sup119873rarrinfin
119864 [
[
119873
sum
119896=119879119899
1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
| 119909119879119899
120579119879119899
]
]
= lim sup119873rarrinfin
119864 [
119873minus119879119899
sum
119896=0
1199091015840
119896119876120579119896
119909119896120575119896le1198791
| 1199090
= 119909119879119899
1205790
= 120579119879119899
]
lt 1199091015840
119879119899
119875120579119879119899
119909119879119899
(19)
So one can conclude from (18) and (19) that
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
]
lt lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899gt119896
+ 1199091015840
119879119899
119875120579119879119899
119909119879119899
]
= lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899ge119896
+ 1199091015840
119879119899
(119875120579119879119899
minus 119876120579119879119899
) 119909119879119899
]
lt 21199091015840
01198751205790
1199090
(20)
Therefore for any 119879119899
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] lt infin (21)
indicate that the MJSLS (1) is 119879119899-stable
Necessity As in the previous part define the functional
1199091015840
01198751205790
1199090
= 1198640
[
infin
sum
119896=0
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
] (22)
for all (1199090 1205790) isin R119899 times X Therefore
1199091015840
11198751205791
1199091
= 1198641198791
[
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
] 1205751198791gt1
(23)
The right-hand side of (22) can be expressed as
1198640
1199091015840
01198821205790
1199090
+ 1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791ge1
]
(24)
In addition
1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791ge1
]
= 1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791gt1
+ 1199091015840
1198791
1198661205791198791
11990911987911205751198791=1
]
(25)
Thus based on the strong Markov property applying homo-geneity in (25) and introducing it in (24) we arrive at
1199091015840
01198751205790
1199090
= 1199091015840
01198821205790
1199090
+ 1198640
[1199091015840
11198751205791
11990911205751198791gt1
+ 1199091015840
1198791
1198661205791198791
11990911987911205751198791=1
]
(26)
Mathematical Problems in Engineering 5
Since 1199090is arbitrary and calculating the expected values
above (26) implies that
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 1198601015840
119894120576119894(119866) 119860
119894+ 1198601015840
119894120576119894(119866) 119860
119894
= minus119882119894
(27)
using the fact that 119875(1198791
= 119896 + 1 | 120579119896
= 119894) = 1 minus 119901119894119894and 119875(119879
1gt
119896+1 | 120579119896
= 119894) = 119901119894119894Thus from the Lyapunov stability theory
the existence of the set 119871119894
gt 0 satisfying (7) is guaranteedcompleting the proof for 119899 = 1
Now for the general case from the stochastically 119879119899-
stable of the system we can obtain that
119864 [
infin
sum
119896=0
1199091015840
119896119882120579119896
119909119896120575119879119899gt119896
+ 1199091015840
119879119899
119866120579119879119899
119909119879119899
] lt infin (28)
And from the strong Markov property we can deduce that
119864119879119899
[
[
infin
sum
119896=119879119899
1199091015840
119896119882120579119896
119909119896120575119879119899+1gt119896
+ 1199091015840
119879119899+1
119866120579119879119899+1
119909119879119899+1
]
]
lt infin (29)
for 119899 = 0 1 119873minus1 By the homogeneity property it followsthat (29) is equivalent to (22) with 119909
0= 119909119879119899
and 1205790
= 120579119879119899
and the existence of a set of matrices 119871119894
gt 0 satisfying (7) isassured Then the proof of Theorem 4 is completed
3 LQ Differential Games for MJSLS witha Finite Number of Jump Times
31 Problem Formulation Now we study the LQ differentialgames for discrete-time MJSLS Comparing with system (1)consider the following discrete-time MJSLS with a finitenumber of jump times
119909 (119896 + 1) = 119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)
+ [119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
119910120591
(119896) = 119876120591
120579119896
119909 (119896) 120591 = 1 2
(30)
119910120591
(119896) isin R119898 are the measurement outputs for each playerHere (119906(119896) V(119896)) isin R119903 times R119903 represent the system controlinputs The matrices (119861
120579119896 119861120579119896
119862120579119896
119862120579119896
119876120591
120579119896
) isin R119899times119903
times R119899times119903
times
R119899times119903
times R119899times119903
times R119898times119899
(associated with ldquo119894thrdquo mode) will beassigned as (119861
119894 119861119894 119862119894 119862119894 119876120591
119894) for each 120579
119896= 119894 isin X
Throughout this paper we choose the infinite horizonquadratic cost functions associated with each player
119869120591
(119906 V) =
infin
sum
119896=0
119864 [119909 (119896)1015840
(119876120591
120579119896
)1015840
119876120591
120579119896
119909 (119896)
+ 119906 (119896)1015840
119877120591
120579119896
119906 (119896) + V (119896)1015840
119878120591
120579119896
V (119896) ]
120591 = 1 2
(31)
The weighting matrices 119876120591
120579119896
= 119876120591
119894ge 0 119877
120591
120579119896
= 119877120591
119894gt 0 isin R
119903times119903
and 119878120591
120579119896
= 119878120591
119894gt 0 isin R
119903times119903
So we are looking for actions that satisfy simultaneously
1198691
(119906lowast
Vlowast) le 1198691
(119906lowast
V) 1198692
(119906lowast
Vlowast) le 1198692
(119906 Vlowast) (32)
where (119906lowast
(119896) Vlowast(119896)) isin 1198712
(infinR119903119906) times 1198712
(infinR119903V)To ensure the finiteness of the infinite-time cost function
we restrain the admissible control set to the constant linearfeedback strategies that is 119906(119896) = 119870
1
120579119896
119909(119896) V(119896) = 1198702
120579119896
119909(119896)where 119870
1
120579119896
and 1198702
120579119896
are constant matrices of appropriatedimensions and (119870
1
120579119896
1198702
120579119896
) belong to
K = 119870 = (1198701
120579119896
1198702
120579119896
) | system (30) can be stabilized
with 119906 (119896) = 1198701
120579119896
119909 (119896)
V (119905) = 1198702
120579119896
119909 (119896)
(33)
We say that the optimization problem is well posedand the 119906(119896) and V(119896) have the following two additionalproperties
119864 [|119906 (119896)|2
] lt infin 119864 [|V (119896)|2
] lt infin 119896 isin N (34)
The optimal strategies 119906lowast and Vlowast determined by (32) are
also called the Nash equilibrium strategies (119906lowast
Vlowast) In orderto guarantee the unique global Nash game solutions both theplayers are only allowed to take constant feedback controlsNext we focus on finding the optimal strategies
32Main Results First we give an important lemma that willbe used later If the system (1) is 119879
119899-stable we can obtain the
following result for the discrete-time MJSLS (30)
Lemma 5 If [119860120579119896
119860120579119896
] is 119879119899-stable then so is [119860
120579119896+ 119861120579119896
1198701
120579119896
+
119862120579119896
1198702
120579119896
119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
] where (1198701
120579119896
1198702
120579119896
) isin K
Proof SufficiencyThe proof employs an induction argumenton the stopping times 119879
119899 First define the function
119881119896
(119909 119894) = 1199091015840
119896(1198751198941205751198791gt119896
+ 1198661198941205751198791=119896
) 119909119896 (35)
where 119866119894gt 0 and 119875
119894gt 0 is the solution of
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 119882119894+ 1198601015840
119894120576119894(119866) 119860
119894
+ 1198601015840
119894120576119894(119866) 119860
119894= 0 119894 isin X
(36)
6 Mathematical Problems in Engineering
The existence of such 119875119894
gt 0 relies on (7) Hence tothe function 119881
119896(119909 119894) and the system (30) in the following
operation we acquire that
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896) minus 119881119896
(119909 (119896) 120579119896)]
= 119909 (119896)1015840
119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
minus119875120579119896
119909 (119896) 1205751198791gt119896
minus 119909 (119896)1015840
119866120579119896
119909 (119896) 1205751198791=119896
(37)
Compared with (12) we know
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896+1
) minus 119881119896
(119909 (119896) 120579119896)]
= minus119909 (119896)1015840
(119882120579119896120575119879119899gt119896
+ 119866120579119896120575119879119899=119896
) 119909 (119896)
(38)
where
minus119882120579119896
= minus119875120579119896
+ 119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
(39)
Considering that 119882120579119896
gt 0 119866120579119896
gt 0 we can obtain (19)And because 119864[119881
119896(119909(119896) 120579
119896)] ge 0 forall119896 ge 0 for each initial
condition 1199090 from (40) it is easy to verify (21) Therefore
the MJSLS (30) is 119879119899-stable
Theorem 6 For system (30) suppose the following coupledequations admit the solutions (119871
1
119894 1198712
119894 1198701
119894 1198702
119894) with 119871
1
119894gt 0
1198712
119894gt 0
minus 1198711
119894+ 119901119894119894
[(119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)
+ (119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)] + 119876
11015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894minus 11987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894= 0
1198671
119894(1198711
119894) gt 0
(40)
1198701
119894= minus119867
2
119894(1198712
119894)minus1
1198704
119894 (41)
minus 1198712
119894+ 119901119894119894
[(119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)
+ (119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)] + 119876
21015840
1198941198762
119894
+ 11987021015840
1198941198782
1198941198702
119894minus 11987041015840
1198941198672
119894(1198712
119894)minus1
1198704
119894= 0
1198672
119894(1198712
119894) gt 0
(42)
1198702
119894= minus119867
1
119894(1198711
119894)minus1
1198703
119894 (43)
where
1198671
119894(1198711
119894) = 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894)
1198703
119894= 119901119894119894
[1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894) + 1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894)]
1198672
119894(1198712
119894) = 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894)
1198704
119894= 119901119894119894
[1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894) + 1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894)]
(44)
If (119860119894 119860119894) is 119879119899-stable then
(i) (1198701
119894 1198702
119894) isin K
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Proof In the deduction of Lemma 5 we can prove that (i) iscorrect Next what we have to do is to prove (ii) and (iii)In the light of the Lyapunov equation (7) and any given set
Mathematical Problems in Engineering 7
of matrices 119882119894in Theorem 4 it is easy to get the following
equations for system (30)
119901119894119894
[(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987611015840
1198941198761
119894+ 11987011015840
1198941198771
1198941198701
119894+ 11987021015840
1198941198781
1198941198702
119894= 1198711
119894
1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894) gt 0
(45)
119901119894119894
[ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987621015840
1198941198762
119894+ 11987011015840
1198941198772
1198941198701
119894+ 11987021015840
1198941198782
1198941198702
119894= 1198712
119894
1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894) gt 0
(46)
By rearranging (45) and (46) (40) and (42) can be obtainedrespectively
Noting 119906lowast
(119896) = 1198701
119894119909(119896) and by substituting 119906
lowast
(119896) into(30) it is easy to get the following system
119909 (119896 + 1) = (119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)
+ [(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
(47)
Then considering the scalar function 119885(119909119896) = 119909
1015840
1198961198711
120579119896
119909119896 we
have
119864119896
[Δ119885 (119909119896)]
= 119864119896
[119885 (119909119896+1
) minus 119885 (119909119896)]
= 119864119896
[1199091015840
119896+11198711
120579119896+1
119909119896+1
minus 1199091015840
1198961198711
120579119896
119909119896]
= 119864119896
minus1199091015840
1198961198711
120579119896
119909119896
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
(48)
Due toinfin
sum
119896=0
119864119896
[Δ119885 (119909119896)]
= 119864119896
[
infin
sum
119896=0
Δ119885 (119909119896)] = 119864
119896[119885 (119909infin
) minus 119885 (1199090)] = minus119909
1015840
01198711
1198941199090
(49)
by (40) and a completing squares technique (31) can bederived that
1198691
(119906lowast
V)
=
infin
sum
119896=0
119864119896
[1199091015840
119896(11987611015840
120579119896
1198761
120579119896
+ 11987011015840
120579119896
1198771
120579119896
1198701
120579119896
) 119909119896
+ V10158401198961198781
120579119896
V119896]
+
infin
sum
119896=0
119864119896
[Δ119885 (119909119896)] + 119909
1015840
01198711
1198941199090
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
1199091015840
119896[minus1198711
119894+ 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 11987611015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894] 119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V1015840119896
(1198781
119894+ 1199011198941198941198621015840
1198941198711
119894119862119894+ 1199011198941198941198621015840
1198941198711
119894119862119894) V119896
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[1199091015840
11989611987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V10158401198961198671
119894(1198711
119894) V119896]
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[V (119896) minus 1198702
119894119909 (119896)]
1015840
1198671
119894(1198711
119894) [V (119896) minus 119870
2
119894119909 (119896)]
ge 1199091015840
01198711
1198941199090 120591 = 1
(50)
Then by (32) it follows that Vlowast(119896) = 1198702
119894119909(119896) and 119869
1
(119906lowast
Vlowast) =
1199091015840
01198711
1198941199090 Finally by substituting Vlowast(119896) into (30) in the same
way as before we have 119906lowast
(119896) = 1198701
119894119909(119896) and 119869
2
(119906lowast
Vlowast) =
1199091015840
01198712
1198941199090
Theorem 7 If (119860119894 119860119894) is 119879
119899-stable and for system (30)
assume that (40)ndash(43) admit the solution (1198711
119894 1198712
119894 1198701
119894 1198702
119894) with
(1198701
119894 1198702
119894) isin K then
(i) 1198711
119894gt 0 1198712
119894gt 0
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Remark 8 When 119908(119896) equiv 0 these results still hold inthe paper Only for the reason of simplicity in (1) and(30) we assume the state 119909(119905) and control inputs (119906(119905) V(119905))depend on the same noise 119908(119896) If they rely on the different
8 Mathematical Problems in Engineering
noises (1199081(119896) 119908
2(119896)) then new results will be yielded The
discussion is omitted
4 Iterative Algorithm and Simulation
41 An Iterative Algorithm In this section an iterative algo-rithm is proposed to solve the four coupled GAREs (40)ndash(43) Infinite horizon Riccati equations are hard to be solvedhence the particular problems can be solved via finite horizonequations 119873 represents the finite number of iterations in thefollowing equations
1198711
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
1198711
119894
119873
(119896 + 1)
times (119860119894+ 1198611198941198701
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
times 1198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896))
+ 11987611015840
1198941198761
119894+ 1198701
119894
119873
(119896)1015840
1198771
1198941198701
119894
119873
(119896)
minus 1198703
119894
119873
(119896)1015840
1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896)
1198711
119894
119873
(119896 + 1) = 0
1198671
119894(1198711
119894
119873
(119896 + 1)) gt 0
(51)
1198701
119894
119873
(119896) = minus1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896) (52)
1198712
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
1198712
119894
119873
(119896 + 1)
times (119860119894+ 1198621198941198702
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
times 1198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896))
+ 11987621015840
1198941198762
119894+ 1198702
119894
119873
(119896)1015840
1198782
1198941198702
119894
119873
(119896)
minus 1198704
119894
119873
(119896)1015840
1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896)
1198712
119894
119873
(119896 + 1) = 0
1198672
119894(1198712
119894
119873
(119896 + 1)) gt 0
(53)
1198702
119894
119873
(119896) = minus1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896) (54)
where
1198671
119894(1198711
119894
119873
(119896 + 1))
= 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894)
1198672
119894(1198712
119894
119873
(119896 + 1))
= 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894
+ 1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894)
1198703
119894
119873
(119896) = 119901119894119894
[1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))]
1198704
119894
119873
(119896) = 119901119894119894
[1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))
+1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))]
(55)
An iterative process for solving (40)ndash(43) based on the aboverecursions is presented as follows
(a) Given appropriate natural number 119873 and the initialconditions 119871
1
119894
119873
(119873+1) = 0 1198712119894
119873
(119873+1) = 0 1198701119894
119873
(119873+
1) = 0 and 1198702
119894
119873
(119873 + 1) = 0
(b) Through the numerical values of 1198711
119894
119873
(119873+1) 1198712119894
119873
(119873+
1)1198701119894
119873
(119873+1) and1198702
119894
119873
(119873+1) we have1198671
119894(1198711
119894
119873
(119873+
1))1198672119894(1198712
119894
119873
(119873+1))1198703119894
119873
(119873) and1198704
119894
119873
(119873) accordingto (55)
(c) 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873) can be respectively computedby (52) and (54) Then 119871
1
119894
119873
(119873) and 1198712
119894
119873
(119873) can alsobe respectively obtained by (51) and (53)
(d) Let 1198711
119894
119873
(119873 + 1) = 1198711
119894
119873
(119873) 1198712
119894
119873
(119873 + 1) = 1198712
119894
119873
(119873)1198701
119894
119873
(119873 + 1) = 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873 + 1) = 1198702
119894
119873
(119873)
(e) Then 119873 = 119873 minus 1 Repeat steps (b)ndash(d) until thenumber of iterations is 119873 + 1 We can finally obtainthe numerical values of 119871
1
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) and1198702
119894
119873
(0)
As in [28] under the assumptions of stabilizability for any1199090
isin R119899
lim119873rarrinfin
1199091015840
01198711
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198691119873
(119906lowast
119873 V) = min 119869
1infin
(119906lowast
V) = 1199091015840
01198711
1198941199090
lim119873rarrinfin
1199091015840
01198712
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198692119873
(119906 Vlowast119873
) = min 1198692infin
(119906 Vlowast) = 1199091015840
01198712
1198941199090
lim119873rarrinfin
1198701
119894
119873
(0) = 1198701
119894 lim
119873rarrinfin
1198702
119894
119873
(0) = 1198702
119894
(56)
Mathematical Problems in Engineering 9
Therefore
lim119873rarrinfin
(1198711
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) 1198702
119894
119873
(0))
= (1198711
119894 1198712
119894 1198701
119894 1198702
119894)
(57)
where (1198711
119894 1198712
119894 1198701
119894 1198702
119894) are the solutions of (40)ndash(43)
42 A Simulation Example To verify the efficiency of theabove iterative algorithm we consider the following 2-Dexample In the system (30) we set 120579
119896= 119894 isin X = 1 2
119877120591
119894= 119878120591
119894= 1 (120591 = 1 2)
1198601
= [065 0
0 09] 119860
1= [
045 0
0 055]
1198611
= [06
055] 119861
1= [
045
085]
1198621
= [075
055] 119862
1= [
05
085]
1198761
1= [
055 0
0 065] 119876
2
1= [
075 0
0 025]
1198602
= [075 0
0 07] 119860
2= [
035 0
0 045]
1198612
= [05
045] 119861
2= [
055
085]
1198622
= [065
055] 119862
2= [
04
085]
1198761
2= [
035 0
0 045] 119876
2
2= [
055 0
0 035]
(58)
For convenience let 11990111
= 04 11990122
= 05 and 119873 = 50When 120579
119896= 1 by applying the above iterative algorithm we
obtain the solutions of the four coupled equations (51)ndash(54)as follows
1198711
1
119873
(0) = [1198711
1(1 1) 119871
1
1(1 2)
1198711
1(2 1) 119871
1
1(2 2)
] = [04023 minus00588
minus00588 06820]
1198712
1
119873
(0) = [1198712
1(1 1) 119871
2
1(1 2)
1198712
1(2 1) 119871
2
1(2 2)
] = [07111 minus00331
minus00331 01487]
1198701
1
119873
(0) = [1198701
1(1 1) 119870
1
1(1 2)] = [minus01245 minus00053]
1198702
1
119873
(0) = [1198702
1(1 1) 119870
2
1(1 2)] = [minus00390 minus01739]
(59)
(1198711
1
119873
(0) 1198712
1
119873
(0) 1198701
1
119873
(0) 1198702
1
119873
(0)) are also the solutionsof (40)ndash(43) according to (57) By the solutions itshows that 119871
1
1gt 0 and 119871
2
1gt 0 The evolution of
(1198711
1
119873
(119896) 1198712
1
119873
(119896) 1198701
1
119873
(119896) 1198702
1
119873
(119896)) is exhibited in Figures 1and 2 And the figures clearly illustrate the convergence andspeediness of the backward iterations When 120579
119896= 2 it is easy
0 10 20 30 40 50minus01
0
01
02
03
04
05
06
07
08
N
L11(1 1)
L11(2 1)
L11(2 2)
L21(1 1)
L21(2 1)
L21(2 2)
Figure 1 Evolution of 1198711
1
119873
(119896) and 1198712
1
119873
(119896)
0 10 20 30 40 50minus018
minus016
minus014
minus012
minus01
minus008
minus006
minus004
minus002
0
N
K11(1 1)
K11(1 2)
K21(1 1)
K21(1 2)
Figure 2 Evolution of 1198701
1
119873
(119896) and 1198702
1
119873
(119896)
to get (1198711
2
119873
(0) 1198712
2
119873
(0) 1198701
2
119873
(0) 1198702
2
119873
(0)) that are also thesolutions of (40)ndash(43) And 119871
1
2gt 0 and 119871
2
2gt 0 Because it is
the same as the above process (120579119896
= 1) we do not introduceit again due to space limitations
5 Conclusions
In this paper we have discussed the 119879119899-stability for the
discrete-time MJSLS with a finite number of jump timesand its infinite horizon LQ differential games Based on therelations between the Lyapunov equation and the stabil-ity of discrete-time MJSLS we have obtained some useful
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Now let us observe that119873
sum
119896=0
1198640
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
=
119873
sum
119896=0
1198640
119864119896
[119881119896+1
(119909119896+1
120579119896+1
) minus 119881119896
(119909119896 120579119896)]
(14)
By applying (12) and considering that 119882119894gt 0 119866
119894gt 0 for each
initial condition 1199090and initial distribution 120583 then we have
1198640
[119881119896+1
(119909119896+1
120579119896+1
)] minus 1198810
(1199090 1205790)
= minus
119873
sum
119896=0
1198640
[1199091015840
119896(1198821205791198961205751198791gt119896
+ 1198661205791198961205751198791=119896
) 1199091198961205751198791ge119896
]
le minus
119873
sum
119896=0
1205741198640
[1003817100381710038171003817119909119896
1003817100381710038171003817
2
1205751198791ge119896
]
(15)
for some 120574 gt 0 Because 1198640[119881119896(119909119896 120579119896)] ge 0 forall119896 ge 0 then
lim119896rarrinfin
1198640[119881119896+1
(119909119896+1
120579119896+1
)] = 0 by (5) Finally it is easy toverify that for any 119879
1
lim sup119873rarrinfin
119873
sum
119896=0
1198640
[1003817100381710038171003817119909119896
1003817100381710038171003817
2
1205751198791ge119896
] le1
1205741198810
(1199090 1205790) lt infin (16)
holds from (15) Therefore for any 1198791 MJSLS (1) is stable
according to (i) of Definition 1Now using an induction argument we assume that for
some 119899 the inequality
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899ge119896
] lt 1199091015840
01198751205790
1199090
(17)
holds and thus by setting 119876 equiv 119868 119864[119909119879119899
2
120575119879119899ge119896
] lt infinHowever
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
]
= lim sup119873rarrinfin
119864 [
[
119873
sum
119896=0
1199091015840
119896119876120579119896
119909119896120575119879119899gt119896
+
119873
sum
119896=119879119899
1199091015840
119896119876120579119896
119909119896120575119879119899le119896le119879119899+1
]
]
(18)
Notice that using the strong Markov property and thehomogeneity property the second term conditioned to theknowledge of (119909
119879119899 120579119879119899) can be written as
lim sup119873rarrinfin
119864 [
[
119873
sum
119896=119879119899
1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
| 119909119879119899
120579119879119899
]
]
= lim sup119873rarrinfin
119864 [
119873minus119879119899
sum
119896=0
1199091015840
119896119876120579119896
119909119896120575119896le1198791
| 1199090
= 119909119879119899
1205790
= 120579119879119899
]
lt 1199091015840
119879119899
119875120579119879119899
119909119879119899
(19)
So one can conclude from (18) and (19) that
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899+1ge119896
]
lt lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899gt119896
+ 1199091015840
119879119899
119875120579119879119899
119909119879119899
]
= lim sup119873rarrinfin
119873
sum
119896=0
119864 [1199091015840
119896119876120579119896
119909119896120575119879119899ge119896
+ 1199091015840
119879119899
(119875120579119879119899
minus 119876120579119879119899
) 119909119879119899
]
lt 21199091015840
01198751205790
1199090
(20)
Therefore for any 119879119899
lim sup119873rarrinfin
119873
sum
119896=0
119864 [1003817100381710038171003817119909119896
1003817100381710038171003817
2
120575119879119899ge119896
] lt infin (21)
indicate that the MJSLS (1) is 119879119899-stable
Necessity As in the previous part define the functional
1199091015840
01198751205790
1199090
= 1198640
[
infin
sum
119896=0
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
] (22)
for all (1199090 1205790) isin R119899 times X Therefore
1199091015840
11198751205791
1199091
= 1198641198791
[
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
] 1205751198791gt1
(23)
The right-hand side of (22) can be expressed as
1198640
1199091015840
01198821205790
1199090
+ 1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791ge1
]
(24)
In addition
1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791ge1
]
= 1198641198791
[(
infin
sum
119896=1
1199091015840
119896119882120579119896
1199091198961205751198791gt119896
+ 1199091015840
1198791
1198661205791198791
1199091198791
) 1205751198791gt1
+ 1199091015840
1198791
1198661205791198791
11990911987911205751198791=1
]
(25)
Thus based on the strong Markov property applying homo-geneity in (25) and introducing it in (24) we arrive at
1199091015840
01198751205790
1199090
= 1199091015840
01198821205790
1199090
+ 1198640
[1199091015840
11198751205791
11990911205751198791gt1
+ 1199091015840
1198791
1198661205791198791
11990911987911205751198791=1
]
(26)
Mathematical Problems in Engineering 5
Since 1199090is arbitrary and calculating the expected values
above (26) implies that
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 1198601015840
119894120576119894(119866) 119860
119894+ 1198601015840
119894120576119894(119866) 119860
119894
= minus119882119894
(27)
using the fact that 119875(1198791
= 119896 + 1 | 120579119896
= 119894) = 1 minus 119901119894119894and 119875(119879
1gt
119896+1 | 120579119896
= 119894) = 119901119894119894Thus from the Lyapunov stability theory
the existence of the set 119871119894
gt 0 satisfying (7) is guaranteedcompleting the proof for 119899 = 1
Now for the general case from the stochastically 119879119899-
stable of the system we can obtain that
119864 [
infin
sum
119896=0
1199091015840
119896119882120579119896
119909119896120575119879119899gt119896
+ 1199091015840
119879119899
119866120579119879119899
119909119879119899
] lt infin (28)
And from the strong Markov property we can deduce that
119864119879119899
[
[
infin
sum
119896=119879119899
1199091015840
119896119882120579119896
119909119896120575119879119899+1gt119896
+ 1199091015840
119879119899+1
119866120579119879119899+1
119909119879119899+1
]
]
lt infin (29)
for 119899 = 0 1 119873minus1 By the homogeneity property it followsthat (29) is equivalent to (22) with 119909
0= 119909119879119899
and 1205790
= 120579119879119899
and the existence of a set of matrices 119871119894
gt 0 satisfying (7) isassured Then the proof of Theorem 4 is completed
3 LQ Differential Games for MJSLS witha Finite Number of Jump Times
31 Problem Formulation Now we study the LQ differentialgames for discrete-time MJSLS Comparing with system (1)consider the following discrete-time MJSLS with a finitenumber of jump times
119909 (119896 + 1) = 119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)
+ [119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
119910120591
(119896) = 119876120591
120579119896
119909 (119896) 120591 = 1 2
(30)
119910120591
(119896) isin R119898 are the measurement outputs for each playerHere (119906(119896) V(119896)) isin R119903 times R119903 represent the system controlinputs The matrices (119861
120579119896 119861120579119896
119862120579119896
119862120579119896
119876120591
120579119896
) isin R119899times119903
times R119899times119903
times
R119899times119903
times R119899times119903
times R119898times119899
(associated with ldquo119894thrdquo mode) will beassigned as (119861
119894 119861119894 119862119894 119862119894 119876120591
119894) for each 120579
119896= 119894 isin X
Throughout this paper we choose the infinite horizonquadratic cost functions associated with each player
119869120591
(119906 V) =
infin
sum
119896=0
119864 [119909 (119896)1015840
(119876120591
120579119896
)1015840
119876120591
120579119896
119909 (119896)
+ 119906 (119896)1015840
119877120591
120579119896
119906 (119896) + V (119896)1015840
119878120591
120579119896
V (119896) ]
120591 = 1 2
(31)
The weighting matrices 119876120591
120579119896
= 119876120591
119894ge 0 119877
120591
120579119896
= 119877120591
119894gt 0 isin R
119903times119903
and 119878120591
120579119896
= 119878120591
119894gt 0 isin R
119903times119903
So we are looking for actions that satisfy simultaneously
1198691
(119906lowast
Vlowast) le 1198691
(119906lowast
V) 1198692
(119906lowast
Vlowast) le 1198692
(119906 Vlowast) (32)
where (119906lowast
(119896) Vlowast(119896)) isin 1198712
(infinR119903119906) times 1198712
(infinR119903V)To ensure the finiteness of the infinite-time cost function
we restrain the admissible control set to the constant linearfeedback strategies that is 119906(119896) = 119870
1
120579119896
119909(119896) V(119896) = 1198702
120579119896
119909(119896)where 119870
1
120579119896
and 1198702
120579119896
are constant matrices of appropriatedimensions and (119870
1
120579119896
1198702
120579119896
) belong to
K = 119870 = (1198701
120579119896
1198702
120579119896
) | system (30) can be stabilized
with 119906 (119896) = 1198701
120579119896
119909 (119896)
V (119905) = 1198702
120579119896
119909 (119896)
(33)
We say that the optimization problem is well posedand the 119906(119896) and V(119896) have the following two additionalproperties
119864 [|119906 (119896)|2
] lt infin 119864 [|V (119896)|2
] lt infin 119896 isin N (34)
The optimal strategies 119906lowast and Vlowast determined by (32) are
also called the Nash equilibrium strategies (119906lowast
Vlowast) In orderto guarantee the unique global Nash game solutions both theplayers are only allowed to take constant feedback controlsNext we focus on finding the optimal strategies
32Main Results First we give an important lemma that willbe used later If the system (1) is 119879
119899-stable we can obtain the
following result for the discrete-time MJSLS (30)
Lemma 5 If [119860120579119896
119860120579119896
] is 119879119899-stable then so is [119860
120579119896+ 119861120579119896
1198701
120579119896
+
119862120579119896
1198702
120579119896
119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
] where (1198701
120579119896
1198702
120579119896
) isin K
Proof SufficiencyThe proof employs an induction argumenton the stopping times 119879
119899 First define the function
119881119896
(119909 119894) = 1199091015840
119896(1198751198941205751198791gt119896
+ 1198661198941205751198791=119896
) 119909119896 (35)
where 119866119894gt 0 and 119875
119894gt 0 is the solution of
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 119882119894+ 1198601015840
119894120576119894(119866) 119860
119894
+ 1198601015840
119894120576119894(119866) 119860
119894= 0 119894 isin X
(36)
6 Mathematical Problems in Engineering
The existence of such 119875119894
gt 0 relies on (7) Hence tothe function 119881
119896(119909 119894) and the system (30) in the following
operation we acquire that
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896) minus 119881119896
(119909 (119896) 120579119896)]
= 119909 (119896)1015840
119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
minus119875120579119896
119909 (119896) 1205751198791gt119896
minus 119909 (119896)1015840
119866120579119896
119909 (119896) 1205751198791=119896
(37)
Compared with (12) we know
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896+1
) minus 119881119896
(119909 (119896) 120579119896)]
= minus119909 (119896)1015840
(119882120579119896120575119879119899gt119896
+ 119866120579119896120575119879119899=119896
) 119909 (119896)
(38)
where
minus119882120579119896
= minus119875120579119896
+ 119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
(39)
Considering that 119882120579119896
gt 0 119866120579119896
gt 0 we can obtain (19)And because 119864[119881
119896(119909(119896) 120579
119896)] ge 0 forall119896 ge 0 for each initial
condition 1199090 from (40) it is easy to verify (21) Therefore
the MJSLS (30) is 119879119899-stable
Theorem 6 For system (30) suppose the following coupledequations admit the solutions (119871
1
119894 1198712
119894 1198701
119894 1198702
119894) with 119871
1
119894gt 0
1198712
119894gt 0
minus 1198711
119894+ 119901119894119894
[(119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)
+ (119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)] + 119876
11015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894minus 11987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894= 0
1198671
119894(1198711
119894) gt 0
(40)
1198701
119894= minus119867
2
119894(1198712
119894)minus1
1198704
119894 (41)
minus 1198712
119894+ 119901119894119894
[(119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)
+ (119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)] + 119876
21015840
1198941198762
119894
+ 11987021015840
1198941198782
1198941198702
119894minus 11987041015840
1198941198672
119894(1198712
119894)minus1
1198704
119894= 0
1198672
119894(1198712
119894) gt 0
(42)
1198702
119894= minus119867
1
119894(1198711
119894)minus1
1198703
119894 (43)
where
1198671
119894(1198711
119894) = 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894)
1198703
119894= 119901119894119894
[1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894) + 1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894)]
1198672
119894(1198712
119894) = 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894)
1198704
119894= 119901119894119894
[1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894) + 1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894)]
(44)
If (119860119894 119860119894) is 119879119899-stable then
(i) (1198701
119894 1198702
119894) isin K
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Proof In the deduction of Lemma 5 we can prove that (i) iscorrect Next what we have to do is to prove (ii) and (iii)In the light of the Lyapunov equation (7) and any given set
Mathematical Problems in Engineering 7
of matrices 119882119894in Theorem 4 it is easy to get the following
equations for system (30)
119901119894119894
[(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987611015840
1198941198761
119894+ 11987011015840
1198941198771
1198941198701
119894+ 11987021015840
1198941198781
1198941198702
119894= 1198711
119894
1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894) gt 0
(45)
119901119894119894
[ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987621015840
1198941198762
119894+ 11987011015840
1198941198772
1198941198701
119894+ 11987021015840
1198941198782
1198941198702
119894= 1198712
119894
1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894) gt 0
(46)
By rearranging (45) and (46) (40) and (42) can be obtainedrespectively
Noting 119906lowast
(119896) = 1198701
119894119909(119896) and by substituting 119906
lowast
(119896) into(30) it is easy to get the following system
119909 (119896 + 1) = (119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)
+ [(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
(47)
Then considering the scalar function 119885(119909119896) = 119909
1015840
1198961198711
120579119896
119909119896 we
have
119864119896
[Δ119885 (119909119896)]
= 119864119896
[119885 (119909119896+1
) minus 119885 (119909119896)]
= 119864119896
[1199091015840
119896+11198711
120579119896+1
119909119896+1
minus 1199091015840
1198961198711
120579119896
119909119896]
= 119864119896
minus1199091015840
1198961198711
120579119896
119909119896
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
(48)
Due toinfin
sum
119896=0
119864119896
[Δ119885 (119909119896)]
= 119864119896
[
infin
sum
119896=0
Δ119885 (119909119896)] = 119864
119896[119885 (119909infin
) minus 119885 (1199090)] = minus119909
1015840
01198711
1198941199090
(49)
by (40) and a completing squares technique (31) can bederived that
1198691
(119906lowast
V)
=
infin
sum
119896=0
119864119896
[1199091015840
119896(11987611015840
120579119896
1198761
120579119896
+ 11987011015840
120579119896
1198771
120579119896
1198701
120579119896
) 119909119896
+ V10158401198961198781
120579119896
V119896]
+
infin
sum
119896=0
119864119896
[Δ119885 (119909119896)] + 119909
1015840
01198711
1198941199090
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
1199091015840
119896[minus1198711
119894+ 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 11987611015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894] 119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V1015840119896
(1198781
119894+ 1199011198941198941198621015840
1198941198711
119894119862119894+ 1199011198941198941198621015840
1198941198711
119894119862119894) V119896
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[1199091015840
11989611987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V10158401198961198671
119894(1198711
119894) V119896]
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[V (119896) minus 1198702
119894119909 (119896)]
1015840
1198671
119894(1198711
119894) [V (119896) minus 119870
2
119894119909 (119896)]
ge 1199091015840
01198711
1198941199090 120591 = 1
(50)
Then by (32) it follows that Vlowast(119896) = 1198702
119894119909(119896) and 119869
1
(119906lowast
Vlowast) =
1199091015840
01198711
1198941199090 Finally by substituting Vlowast(119896) into (30) in the same
way as before we have 119906lowast
(119896) = 1198701
119894119909(119896) and 119869
2
(119906lowast
Vlowast) =
1199091015840
01198712
1198941199090
Theorem 7 If (119860119894 119860119894) is 119879
119899-stable and for system (30)
assume that (40)ndash(43) admit the solution (1198711
119894 1198712
119894 1198701
119894 1198702
119894) with
(1198701
119894 1198702
119894) isin K then
(i) 1198711
119894gt 0 1198712
119894gt 0
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Remark 8 When 119908(119896) equiv 0 these results still hold inthe paper Only for the reason of simplicity in (1) and(30) we assume the state 119909(119905) and control inputs (119906(119905) V(119905))depend on the same noise 119908(119896) If they rely on the different
8 Mathematical Problems in Engineering
noises (1199081(119896) 119908
2(119896)) then new results will be yielded The
discussion is omitted
4 Iterative Algorithm and Simulation
41 An Iterative Algorithm In this section an iterative algo-rithm is proposed to solve the four coupled GAREs (40)ndash(43) Infinite horizon Riccati equations are hard to be solvedhence the particular problems can be solved via finite horizonequations 119873 represents the finite number of iterations in thefollowing equations
1198711
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
1198711
119894
119873
(119896 + 1)
times (119860119894+ 1198611198941198701
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
times 1198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896))
+ 11987611015840
1198941198761
119894+ 1198701
119894
119873
(119896)1015840
1198771
1198941198701
119894
119873
(119896)
minus 1198703
119894
119873
(119896)1015840
1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896)
1198711
119894
119873
(119896 + 1) = 0
1198671
119894(1198711
119894
119873
(119896 + 1)) gt 0
(51)
1198701
119894
119873
(119896) = minus1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896) (52)
1198712
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
1198712
119894
119873
(119896 + 1)
times (119860119894+ 1198621198941198702
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
times 1198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896))
+ 11987621015840
1198941198762
119894+ 1198702
119894
119873
(119896)1015840
1198782
1198941198702
119894
119873
(119896)
minus 1198704
119894
119873
(119896)1015840
1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896)
1198712
119894
119873
(119896 + 1) = 0
1198672
119894(1198712
119894
119873
(119896 + 1)) gt 0
(53)
1198702
119894
119873
(119896) = minus1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896) (54)
where
1198671
119894(1198711
119894
119873
(119896 + 1))
= 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894)
1198672
119894(1198712
119894
119873
(119896 + 1))
= 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894
+ 1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894)
1198703
119894
119873
(119896) = 119901119894119894
[1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))]
1198704
119894
119873
(119896) = 119901119894119894
[1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))
+1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))]
(55)
An iterative process for solving (40)ndash(43) based on the aboverecursions is presented as follows
(a) Given appropriate natural number 119873 and the initialconditions 119871
1
119894
119873
(119873+1) = 0 1198712119894
119873
(119873+1) = 0 1198701119894
119873
(119873+
1) = 0 and 1198702
119894
119873
(119873 + 1) = 0
(b) Through the numerical values of 1198711
119894
119873
(119873+1) 1198712119894
119873
(119873+
1)1198701119894
119873
(119873+1) and1198702
119894
119873
(119873+1) we have1198671
119894(1198711
119894
119873
(119873+
1))1198672119894(1198712
119894
119873
(119873+1))1198703119894
119873
(119873) and1198704
119894
119873
(119873) accordingto (55)
(c) 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873) can be respectively computedby (52) and (54) Then 119871
1
119894
119873
(119873) and 1198712
119894
119873
(119873) can alsobe respectively obtained by (51) and (53)
(d) Let 1198711
119894
119873
(119873 + 1) = 1198711
119894
119873
(119873) 1198712
119894
119873
(119873 + 1) = 1198712
119894
119873
(119873)1198701
119894
119873
(119873 + 1) = 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873 + 1) = 1198702
119894
119873
(119873)
(e) Then 119873 = 119873 minus 1 Repeat steps (b)ndash(d) until thenumber of iterations is 119873 + 1 We can finally obtainthe numerical values of 119871
1
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) and1198702
119894
119873
(0)
As in [28] under the assumptions of stabilizability for any1199090
isin R119899
lim119873rarrinfin
1199091015840
01198711
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198691119873
(119906lowast
119873 V) = min 119869
1infin
(119906lowast
V) = 1199091015840
01198711
1198941199090
lim119873rarrinfin
1199091015840
01198712
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198692119873
(119906 Vlowast119873
) = min 1198692infin
(119906 Vlowast) = 1199091015840
01198712
1198941199090
lim119873rarrinfin
1198701
119894
119873
(0) = 1198701
119894 lim
119873rarrinfin
1198702
119894
119873
(0) = 1198702
119894
(56)
Mathematical Problems in Engineering 9
Therefore
lim119873rarrinfin
(1198711
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) 1198702
119894
119873
(0))
= (1198711
119894 1198712
119894 1198701
119894 1198702
119894)
(57)
where (1198711
119894 1198712
119894 1198701
119894 1198702
119894) are the solutions of (40)ndash(43)
42 A Simulation Example To verify the efficiency of theabove iterative algorithm we consider the following 2-Dexample In the system (30) we set 120579
119896= 119894 isin X = 1 2
119877120591
119894= 119878120591
119894= 1 (120591 = 1 2)
1198601
= [065 0
0 09] 119860
1= [
045 0
0 055]
1198611
= [06
055] 119861
1= [
045
085]
1198621
= [075
055] 119862
1= [
05
085]
1198761
1= [
055 0
0 065] 119876
2
1= [
075 0
0 025]
1198602
= [075 0
0 07] 119860
2= [
035 0
0 045]
1198612
= [05
045] 119861
2= [
055
085]
1198622
= [065
055] 119862
2= [
04
085]
1198761
2= [
035 0
0 045] 119876
2
2= [
055 0
0 035]
(58)
For convenience let 11990111
= 04 11990122
= 05 and 119873 = 50When 120579
119896= 1 by applying the above iterative algorithm we
obtain the solutions of the four coupled equations (51)ndash(54)as follows
1198711
1
119873
(0) = [1198711
1(1 1) 119871
1
1(1 2)
1198711
1(2 1) 119871
1
1(2 2)
] = [04023 minus00588
minus00588 06820]
1198712
1
119873
(0) = [1198712
1(1 1) 119871
2
1(1 2)
1198712
1(2 1) 119871
2
1(2 2)
] = [07111 minus00331
minus00331 01487]
1198701
1
119873
(0) = [1198701
1(1 1) 119870
1
1(1 2)] = [minus01245 minus00053]
1198702
1
119873
(0) = [1198702
1(1 1) 119870
2
1(1 2)] = [minus00390 minus01739]
(59)
(1198711
1
119873
(0) 1198712
1
119873
(0) 1198701
1
119873
(0) 1198702
1
119873
(0)) are also the solutionsof (40)ndash(43) according to (57) By the solutions itshows that 119871
1
1gt 0 and 119871
2
1gt 0 The evolution of
(1198711
1
119873
(119896) 1198712
1
119873
(119896) 1198701
1
119873
(119896) 1198702
1
119873
(119896)) is exhibited in Figures 1and 2 And the figures clearly illustrate the convergence andspeediness of the backward iterations When 120579
119896= 2 it is easy
0 10 20 30 40 50minus01
0
01
02
03
04
05
06
07
08
N
L11(1 1)
L11(2 1)
L11(2 2)
L21(1 1)
L21(2 1)
L21(2 2)
Figure 1 Evolution of 1198711
1
119873
(119896) and 1198712
1
119873
(119896)
0 10 20 30 40 50minus018
minus016
minus014
minus012
minus01
minus008
minus006
minus004
minus002
0
N
K11(1 1)
K11(1 2)
K21(1 1)
K21(1 2)
Figure 2 Evolution of 1198701
1
119873
(119896) and 1198702
1
119873
(119896)
to get (1198711
2
119873
(0) 1198712
2
119873
(0) 1198701
2
119873
(0) 1198702
2
119873
(0)) that are also thesolutions of (40)ndash(43) And 119871
1
2gt 0 and 119871
2
2gt 0 Because it is
the same as the above process (120579119896
= 1) we do not introduceit again due to space limitations
5 Conclusions
In this paper we have discussed the 119879119899-stability for the
discrete-time MJSLS with a finite number of jump timesand its infinite horizon LQ differential games Based on therelations between the Lyapunov equation and the stabil-ity of discrete-time MJSLS we have obtained some useful
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Since 1199090is arbitrary and calculating the expected values
above (26) implies that
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 1198601015840
119894120576119894(119866) 119860
119894+ 1198601015840
119894120576119894(119866) 119860
119894
= minus119882119894
(27)
using the fact that 119875(1198791
= 119896 + 1 | 120579119896
= 119894) = 1 minus 119901119894119894and 119875(119879
1gt
119896+1 | 120579119896
= 119894) = 119901119894119894Thus from the Lyapunov stability theory
the existence of the set 119871119894
gt 0 satisfying (7) is guaranteedcompleting the proof for 119899 = 1
Now for the general case from the stochastically 119879119899-
stable of the system we can obtain that
119864 [
infin
sum
119896=0
1199091015840
119896119882120579119896
119909119896120575119879119899gt119896
+ 1199091015840
119879119899
119866120579119879119899
119909119879119899
] lt infin (28)
And from the strong Markov property we can deduce that
119864119879119899
[
[
infin
sum
119896=119879119899
1199091015840
119896119882120579119896
119909119896120575119879119899+1gt119896
+ 1199091015840
119879119899+1
119866120579119879119899+1
119909119879119899+1
]
]
lt infin (29)
for 119899 = 0 1 119873minus1 By the homogeneity property it followsthat (29) is equivalent to (22) with 119909
0= 119909119879119899
and 1205790
= 120579119879119899
and the existence of a set of matrices 119871119894
gt 0 satisfying (7) isassured Then the proof of Theorem 4 is completed
3 LQ Differential Games for MJSLS witha Finite Number of Jump Times
31 Problem Formulation Now we study the LQ differentialgames for discrete-time MJSLS Comparing with system (1)consider the following discrete-time MJSLS with a finitenumber of jump times
119909 (119896 + 1) = 119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)
+ [119860120579119896
119909 (119896) + 119861120579119896
119906 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
119910120591
(119896) = 119876120591
120579119896
119909 (119896) 120591 = 1 2
(30)
119910120591
(119896) isin R119898 are the measurement outputs for each playerHere (119906(119896) V(119896)) isin R119903 times R119903 represent the system controlinputs The matrices (119861
120579119896 119861120579119896
119862120579119896
119862120579119896
119876120591
120579119896
) isin R119899times119903
times R119899times119903
times
R119899times119903
times R119899times119903
times R119898times119899
(associated with ldquo119894thrdquo mode) will beassigned as (119861
119894 119861119894 119862119894 119862119894 119876120591
119894) for each 120579
119896= 119894 isin X
Throughout this paper we choose the infinite horizonquadratic cost functions associated with each player
119869120591
(119906 V) =
infin
sum
119896=0
119864 [119909 (119896)1015840
(119876120591
120579119896
)1015840
119876120591
120579119896
119909 (119896)
+ 119906 (119896)1015840
119877120591
120579119896
119906 (119896) + V (119896)1015840
119878120591
120579119896
V (119896) ]
120591 = 1 2
(31)
The weighting matrices 119876120591
120579119896
= 119876120591
119894ge 0 119877
120591
120579119896
= 119877120591
119894gt 0 isin R
119903times119903
and 119878120591
120579119896
= 119878120591
119894gt 0 isin R
119903times119903
So we are looking for actions that satisfy simultaneously
1198691
(119906lowast
Vlowast) le 1198691
(119906lowast
V) 1198692
(119906lowast
Vlowast) le 1198692
(119906 Vlowast) (32)
where (119906lowast
(119896) Vlowast(119896)) isin 1198712
(infinR119903119906) times 1198712
(infinR119903V)To ensure the finiteness of the infinite-time cost function
we restrain the admissible control set to the constant linearfeedback strategies that is 119906(119896) = 119870
1
120579119896
119909(119896) V(119896) = 1198702
120579119896
119909(119896)where 119870
1
120579119896
and 1198702
120579119896
are constant matrices of appropriatedimensions and (119870
1
120579119896
1198702
120579119896
) belong to
K = 119870 = (1198701
120579119896
1198702
120579119896
) | system (30) can be stabilized
with 119906 (119896) = 1198701
120579119896
119909 (119896)
V (119905) = 1198702
120579119896
119909 (119896)
(33)
We say that the optimization problem is well posedand the 119906(119896) and V(119896) have the following two additionalproperties
119864 [|119906 (119896)|2
] lt infin 119864 [|V (119896)|2
] lt infin 119896 isin N (34)
The optimal strategies 119906lowast and Vlowast determined by (32) are
also called the Nash equilibrium strategies (119906lowast
Vlowast) In orderto guarantee the unique global Nash game solutions both theplayers are only allowed to take constant feedback controlsNext we focus on finding the optimal strategies
32Main Results First we give an important lemma that willbe used later If the system (1) is 119879
119899-stable we can obtain the
following result for the discrete-time MJSLS (30)
Lemma 5 If [119860120579119896
119860120579119896
] is 119879119899-stable then so is [119860
120579119896+ 119861120579119896
1198701
120579119896
+
119862120579119896
1198702
120579119896
119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
] where (1198701
120579119896
1198702
120579119896
) isin K
Proof SufficiencyThe proof employs an induction argumenton the stopping times 119879
119899 First define the function
119881119896
(119909 119894) = 1199091015840
119896(1198751198941205751198791gt119896
+ 1198661198941205751198791=119896
) 119909119896 (35)
where 119866119894gt 0 and 119875
119894gt 0 is the solution of
119901119894119894
(1198601015840
119894119875119894119860119894+ 1198601015840
119894119875119894119860119894) minus 119875119894+ 119882119894+ 1198601015840
119894120576119894(119866) 119860
119894
+ 1198601015840
119894120576119894(119866) 119860
119894= 0 119894 isin X
(36)
6 Mathematical Problems in Engineering
The existence of such 119875119894
gt 0 relies on (7) Hence tothe function 119881
119896(119909 119894) and the system (30) in the following
operation we acquire that
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896) minus 119881119896
(119909 (119896) 120579119896)]
= 119909 (119896)1015840
119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
minus119875120579119896
119909 (119896) 1205751198791gt119896
minus 119909 (119896)1015840
119866120579119896
119909 (119896) 1205751198791=119896
(37)
Compared with (12) we know
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896+1
) minus 119881119896
(119909 (119896) 120579119896)]
= minus119909 (119896)1015840
(119882120579119896120575119879119899gt119896
+ 119866120579119896120575119879119899=119896
) 119909 (119896)
(38)
where
minus119882120579119896
= minus119875120579119896
+ 119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
(39)
Considering that 119882120579119896
gt 0 119866120579119896
gt 0 we can obtain (19)And because 119864[119881
119896(119909(119896) 120579
119896)] ge 0 forall119896 ge 0 for each initial
condition 1199090 from (40) it is easy to verify (21) Therefore
the MJSLS (30) is 119879119899-stable
Theorem 6 For system (30) suppose the following coupledequations admit the solutions (119871
1
119894 1198712
119894 1198701
119894 1198702
119894) with 119871
1
119894gt 0
1198712
119894gt 0
minus 1198711
119894+ 119901119894119894
[(119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)
+ (119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)] + 119876
11015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894minus 11987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894= 0
1198671
119894(1198711
119894) gt 0
(40)
1198701
119894= minus119867
2
119894(1198712
119894)minus1
1198704
119894 (41)
minus 1198712
119894+ 119901119894119894
[(119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)
+ (119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)] + 119876
21015840
1198941198762
119894
+ 11987021015840
1198941198782
1198941198702
119894minus 11987041015840
1198941198672
119894(1198712
119894)minus1
1198704
119894= 0
1198672
119894(1198712
119894) gt 0
(42)
1198702
119894= minus119867
1
119894(1198711
119894)minus1
1198703
119894 (43)
where
1198671
119894(1198711
119894) = 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894)
1198703
119894= 119901119894119894
[1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894) + 1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894)]
1198672
119894(1198712
119894) = 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894)
1198704
119894= 119901119894119894
[1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894) + 1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894)]
(44)
If (119860119894 119860119894) is 119879119899-stable then
(i) (1198701
119894 1198702
119894) isin K
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Proof In the deduction of Lemma 5 we can prove that (i) iscorrect Next what we have to do is to prove (ii) and (iii)In the light of the Lyapunov equation (7) and any given set
Mathematical Problems in Engineering 7
of matrices 119882119894in Theorem 4 it is easy to get the following
equations for system (30)
119901119894119894
[(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987611015840
1198941198761
119894+ 11987011015840
1198941198771
1198941198701
119894+ 11987021015840
1198941198781
1198941198702
119894= 1198711
119894
1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894) gt 0
(45)
119901119894119894
[ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987621015840
1198941198762
119894+ 11987011015840
1198941198772
1198941198701
119894+ 11987021015840
1198941198782
1198941198702
119894= 1198712
119894
1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894) gt 0
(46)
By rearranging (45) and (46) (40) and (42) can be obtainedrespectively
Noting 119906lowast
(119896) = 1198701
119894119909(119896) and by substituting 119906
lowast
(119896) into(30) it is easy to get the following system
119909 (119896 + 1) = (119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)
+ [(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
(47)
Then considering the scalar function 119885(119909119896) = 119909
1015840
1198961198711
120579119896
119909119896 we
have
119864119896
[Δ119885 (119909119896)]
= 119864119896
[119885 (119909119896+1
) minus 119885 (119909119896)]
= 119864119896
[1199091015840
119896+11198711
120579119896+1
119909119896+1
minus 1199091015840
1198961198711
120579119896
119909119896]
= 119864119896
minus1199091015840
1198961198711
120579119896
119909119896
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
(48)
Due toinfin
sum
119896=0
119864119896
[Δ119885 (119909119896)]
= 119864119896
[
infin
sum
119896=0
Δ119885 (119909119896)] = 119864
119896[119885 (119909infin
) minus 119885 (1199090)] = minus119909
1015840
01198711
1198941199090
(49)
by (40) and a completing squares technique (31) can bederived that
1198691
(119906lowast
V)
=
infin
sum
119896=0
119864119896
[1199091015840
119896(11987611015840
120579119896
1198761
120579119896
+ 11987011015840
120579119896
1198771
120579119896
1198701
120579119896
) 119909119896
+ V10158401198961198781
120579119896
V119896]
+
infin
sum
119896=0
119864119896
[Δ119885 (119909119896)] + 119909
1015840
01198711
1198941199090
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
1199091015840
119896[minus1198711
119894+ 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 11987611015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894] 119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V1015840119896
(1198781
119894+ 1199011198941198941198621015840
1198941198711
119894119862119894+ 1199011198941198941198621015840
1198941198711
119894119862119894) V119896
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[1199091015840
11989611987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V10158401198961198671
119894(1198711
119894) V119896]
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[V (119896) minus 1198702
119894119909 (119896)]
1015840
1198671
119894(1198711
119894) [V (119896) minus 119870
2
119894119909 (119896)]
ge 1199091015840
01198711
1198941199090 120591 = 1
(50)
Then by (32) it follows that Vlowast(119896) = 1198702
119894119909(119896) and 119869
1
(119906lowast
Vlowast) =
1199091015840
01198711
1198941199090 Finally by substituting Vlowast(119896) into (30) in the same
way as before we have 119906lowast
(119896) = 1198701
119894119909(119896) and 119869
2
(119906lowast
Vlowast) =
1199091015840
01198712
1198941199090
Theorem 7 If (119860119894 119860119894) is 119879
119899-stable and for system (30)
assume that (40)ndash(43) admit the solution (1198711
119894 1198712
119894 1198701
119894 1198702
119894) with
(1198701
119894 1198702
119894) isin K then
(i) 1198711
119894gt 0 1198712
119894gt 0
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Remark 8 When 119908(119896) equiv 0 these results still hold inthe paper Only for the reason of simplicity in (1) and(30) we assume the state 119909(119905) and control inputs (119906(119905) V(119905))depend on the same noise 119908(119896) If they rely on the different
8 Mathematical Problems in Engineering
noises (1199081(119896) 119908
2(119896)) then new results will be yielded The
discussion is omitted
4 Iterative Algorithm and Simulation
41 An Iterative Algorithm In this section an iterative algo-rithm is proposed to solve the four coupled GAREs (40)ndash(43) Infinite horizon Riccati equations are hard to be solvedhence the particular problems can be solved via finite horizonequations 119873 represents the finite number of iterations in thefollowing equations
1198711
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
1198711
119894
119873
(119896 + 1)
times (119860119894+ 1198611198941198701
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
times 1198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896))
+ 11987611015840
1198941198761
119894+ 1198701
119894
119873
(119896)1015840
1198771
1198941198701
119894
119873
(119896)
minus 1198703
119894
119873
(119896)1015840
1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896)
1198711
119894
119873
(119896 + 1) = 0
1198671
119894(1198711
119894
119873
(119896 + 1)) gt 0
(51)
1198701
119894
119873
(119896) = minus1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896) (52)
1198712
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
1198712
119894
119873
(119896 + 1)
times (119860119894+ 1198621198941198702
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
times 1198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896))
+ 11987621015840
1198941198762
119894+ 1198702
119894
119873
(119896)1015840
1198782
1198941198702
119894
119873
(119896)
minus 1198704
119894
119873
(119896)1015840
1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896)
1198712
119894
119873
(119896 + 1) = 0
1198672
119894(1198712
119894
119873
(119896 + 1)) gt 0
(53)
1198702
119894
119873
(119896) = minus1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896) (54)
where
1198671
119894(1198711
119894
119873
(119896 + 1))
= 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894)
1198672
119894(1198712
119894
119873
(119896 + 1))
= 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894
+ 1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894)
1198703
119894
119873
(119896) = 119901119894119894
[1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))]
1198704
119894
119873
(119896) = 119901119894119894
[1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))
+1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))]
(55)
An iterative process for solving (40)ndash(43) based on the aboverecursions is presented as follows
(a) Given appropriate natural number 119873 and the initialconditions 119871
1
119894
119873
(119873+1) = 0 1198712119894
119873
(119873+1) = 0 1198701119894
119873
(119873+
1) = 0 and 1198702
119894
119873
(119873 + 1) = 0
(b) Through the numerical values of 1198711
119894
119873
(119873+1) 1198712119894
119873
(119873+
1)1198701119894
119873
(119873+1) and1198702
119894
119873
(119873+1) we have1198671
119894(1198711
119894
119873
(119873+
1))1198672119894(1198712
119894
119873
(119873+1))1198703119894
119873
(119873) and1198704
119894
119873
(119873) accordingto (55)
(c) 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873) can be respectively computedby (52) and (54) Then 119871
1
119894
119873
(119873) and 1198712
119894
119873
(119873) can alsobe respectively obtained by (51) and (53)
(d) Let 1198711
119894
119873
(119873 + 1) = 1198711
119894
119873
(119873) 1198712
119894
119873
(119873 + 1) = 1198712
119894
119873
(119873)1198701
119894
119873
(119873 + 1) = 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873 + 1) = 1198702
119894
119873
(119873)
(e) Then 119873 = 119873 minus 1 Repeat steps (b)ndash(d) until thenumber of iterations is 119873 + 1 We can finally obtainthe numerical values of 119871
1
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) and1198702
119894
119873
(0)
As in [28] under the assumptions of stabilizability for any1199090
isin R119899
lim119873rarrinfin
1199091015840
01198711
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198691119873
(119906lowast
119873 V) = min 119869
1infin
(119906lowast
V) = 1199091015840
01198711
1198941199090
lim119873rarrinfin
1199091015840
01198712
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198692119873
(119906 Vlowast119873
) = min 1198692infin
(119906 Vlowast) = 1199091015840
01198712
1198941199090
lim119873rarrinfin
1198701
119894
119873
(0) = 1198701
119894 lim
119873rarrinfin
1198702
119894
119873
(0) = 1198702
119894
(56)
Mathematical Problems in Engineering 9
Therefore
lim119873rarrinfin
(1198711
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) 1198702
119894
119873
(0))
= (1198711
119894 1198712
119894 1198701
119894 1198702
119894)
(57)
where (1198711
119894 1198712
119894 1198701
119894 1198702
119894) are the solutions of (40)ndash(43)
42 A Simulation Example To verify the efficiency of theabove iterative algorithm we consider the following 2-Dexample In the system (30) we set 120579
119896= 119894 isin X = 1 2
119877120591
119894= 119878120591
119894= 1 (120591 = 1 2)
1198601
= [065 0
0 09] 119860
1= [
045 0
0 055]
1198611
= [06
055] 119861
1= [
045
085]
1198621
= [075
055] 119862
1= [
05
085]
1198761
1= [
055 0
0 065] 119876
2
1= [
075 0
0 025]
1198602
= [075 0
0 07] 119860
2= [
035 0
0 045]
1198612
= [05
045] 119861
2= [
055
085]
1198622
= [065
055] 119862
2= [
04
085]
1198761
2= [
035 0
0 045] 119876
2
2= [
055 0
0 035]
(58)
For convenience let 11990111
= 04 11990122
= 05 and 119873 = 50When 120579
119896= 1 by applying the above iterative algorithm we
obtain the solutions of the four coupled equations (51)ndash(54)as follows
1198711
1
119873
(0) = [1198711
1(1 1) 119871
1
1(1 2)
1198711
1(2 1) 119871
1
1(2 2)
] = [04023 minus00588
minus00588 06820]
1198712
1
119873
(0) = [1198712
1(1 1) 119871
2
1(1 2)
1198712
1(2 1) 119871
2
1(2 2)
] = [07111 minus00331
minus00331 01487]
1198701
1
119873
(0) = [1198701
1(1 1) 119870
1
1(1 2)] = [minus01245 minus00053]
1198702
1
119873
(0) = [1198702
1(1 1) 119870
2
1(1 2)] = [minus00390 minus01739]
(59)
(1198711
1
119873
(0) 1198712
1
119873
(0) 1198701
1
119873
(0) 1198702
1
119873
(0)) are also the solutionsof (40)ndash(43) according to (57) By the solutions itshows that 119871
1
1gt 0 and 119871
2
1gt 0 The evolution of
(1198711
1
119873
(119896) 1198712
1
119873
(119896) 1198701
1
119873
(119896) 1198702
1
119873
(119896)) is exhibited in Figures 1and 2 And the figures clearly illustrate the convergence andspeediness of the backward iterations When 120579
119896= 2 it is easy
0 10 20 30 40 50minus01
0
01
02
03
04
05
06
07
08
N
L11(1 1)
L11(2 1)
L11(2 2)
L21(1 1)
L21(2 1)
L21(2 2)
Figure 1 Evolution of 1198711
1
119873
(119896) and 1198712
1
119873
(119896)
0 10 20 30 40 50minus018
minus016
minus014
minus012
minus01
minus008
minus006
minus004
minus002
0
N
K11(1 1)
K11(1 2)
K21(1 1)
K21(1 2)
Figure 2 Evolution of 1198701
1
119873
(119896) and 1198702
1
119873
(119896)
to get (1198711
2
119873
(0) 1198712
2
119873
(0) 1198701
2
119873
(0) 1198702
2
119873
(0)) that are also thesolutions of (40)ndash(43) And 119871
1
2gt 0 and 119871
2
2gt 0 Because it is
the same as the above process (120579119896
= 1) we do not introduceit again due to space limitations
5 Conclusions
In this paper we have discussed the 119879119899-stability for the
discrete-time MJSLS with a finite number of jump timesand its infinite horizon LQ differential games Based on therelations between the Lyapunov equation and the stabil-ity of discrete-time MJSLS we have obtained some useful
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
The existence of such 119875119894
gt 0 relies on (7) Hence tothe function 119881
119896(119909 119894) and the system (30) in the following
operation we acquire that
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896) minus 119881119896
(119909 (119896) 120579119896)]
= 119909 (119896)1015840
119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
minus119875120579119896
119909 (119896) 1205751198791gt119896
minus 119909 (119896)1015840
119866120579119896
119909 (119896) 1205751198791=119896
(37)
Compared with (12) we know
119864119896
[119881119896+1
(119909 (119896 + 1) 120579119896+1
) minus 119881119896
(119909 (119896) 120579119896)]
= minus119909 (119896)1015840
(119882120579119896120575119879119899gt119896
+ 119866120579119896120575119879119899=119896
) 119909 (119896)
(38)
where
minus119882120579119896
= minus119875120579119896
+ 119901120579119896120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
times 119875120579119896
(119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
) ]
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
+ (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)1015840
120576120579119896
(119866)
times (119860120579119896
+ 119861120579119896
1198701
120579119896
+ 119862120579119896
1198702
120579119896
)
(39)
Considering that 119882120579119896
gt 0 119866120579119896
gt 0 we can obtain (19)And because 119864[119881
119896(119909(119896) 120579
119896)] ge 0 forall119896 ge 0 for each initial
condition 1199090 from (40) it is easy to verify (21) Therefore
the MJSLS (30) is 119879119899-stable
Theorem 6 For system (30) suppose the following coupledequations admit the solutions (119871
1
119894 1198712
119894 1198701
119894 1198702
119894) with 119871
1
119894gt 0
1198712
119894gt 0
minus 1198711
119894+ 119901119894119894
[(119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)
+ (119860119894+ 1198611198941198701
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894)] + 119876
11015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894minus 11987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894= 0
1198671
119894(1198711
119894) gt 0
(40)
1198701
119894= minus119867
2
119894(1198712
119894)minus1
1198704
119894 (41)
minus 1198712
119894+ 119901119894119894
[(119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)
+ (119860119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198621198941198702
119894)] + 119876
21015840
1198941198762
119894
+ 11987021015840
1198941198782
1198941198702
119894minus 11987041015840
1198941198672
119894(1198712
119894)minus1
1198704
119894= 0
1198672
119894(1198712
119894) gt 0
(42)
1198702
119894= minus119867
1
119894(1198711
119894)minus1
1198703
119894 (43)
where
1198671
119894(1198711
119894) = 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894)
1198703
119894= 119901119894119894
[1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894) + 1198621015840
1198941198711
119894(119860119894+ 1198611198941198701
119894)]
1198672
119894(1198712
119894) = 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894)
1198704
119894= 119901119894119894
[1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894) + 1198611015840
1198941198712
119894(119860119894+ 1198621198941198702
119894)]
(44)
If (119860119894 119860119894) is 119879119899-stable then
(i) (1198701
119894 1198702
119894) isin K
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Proof In the deduction of Lemma 5 we can prove that (i) iscorrect Next what we have to do is to prove (ii) and (iii)In the light of the Lyapunov equation (7) and any given set
Mathematical Problems in Engineering 7
of matrices 119882119894in Theorem 4 it is easy to get the following
equations for system (30)
119901119894119894
[(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987611015840
1198941198761
119894+ 11987011015840
1198941198771
1198941198701
119894+ 11987021015840
1198941198781
1198941198702
119894= 1198711
119894
1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894) gt 0
(45)
119901119894119894
[ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987621015840
1198941198762
119894+ 11987011015840
1198941198772
1198941198701
119894+ 11987021015840
1198941198782
1198941198702
119894= 1198712
119894
1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894) gt 0
(46)
By rearranging (45) and (46) (40) and (42) can be obtainedrespectively
Noting 119906lowast
(119896) = 1198701
119894119909(119896) and by substituting 119906
lowast
(119896) into(30) it is easy to get the following system
119909 (119896 + 1) = (119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)
+ [(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
(47)
Then considering the scalar function 119885(119909119896) = 119909
1015840
1198961198711
120579119896
119909119896 we
have
119864119896
[Δ119885 (119909119896)]
= 119864119896
[119885 (119909119896+1
) minus 119885 (119909119896)]
= 119864119896
[1199091015840
119896+11198711
120579119896+1
119909119896+1
minus 1199091015840
1198961198711
120579119896
119909119896]
= 119864119896
minus1199091015840
1198961198711
120579119896
119909119896
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
(48)
Due toinfin
sum
119896=0
119864119896
[Δ119885 (119909119896)]
= 119864119896
[
infin
sum
119896=0
Δ119885 (119909119896)] = 119864
119896[119885 (119909infin
) minus 119885 (1199090)] = minus119909
1015840
01198711
1198941199090
(49)
by (40) and a completing squares technique (31) can bederived that
1198691
(119906lowast
V)
=
infin
sum
119896=0
119864119896
[1199091015840
119896(11987611015840
120579119896
1198761
120579119896
+ 11987011015840
120579119896
1198771
120579119896
1198701
120579119896
) 119909119896
+ V10158401198961198781
120579119896
V119896]
+
infin
sum
119896=0
119864119896
[Δ119885 (119909119896)] + 119909
1015840
01198711
1198941199090
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
1199091015840
119896[minus1198711
119894+ 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 11987611015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894] 119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V1015840119896
(1198781
119894+ 1199011198941198941198621015840
1198941198711
119894119862119894+ 1199011198941198941198621015840
1198941198711
119894119862119894) V119896
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[1199091015840
11989611987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V10158401198961198671
119894(1198711
119894) V119896]
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[V (119896) minus 1198702
119894119909 (119896)]
1015840
1198671
119894(1198711
119894) [V (119896) minus 119870
2
119894119909 (119896)]
ge 1199091015840
01198711
1198941199090 120591 = 1
(50)
Then by (32) it follows that Vlowast(119896) = 1198702
119894119909(119896) and 119869
1
(119906lowast
Vlowast) =
1199091015840
01198711
1198941199090 Finally by substituting Vlowast(119896) into (30) in the same
way as before we have 119906lowast
(119896) = 1198701
119894119909(119896) and 119869
2
(119906lowast
Vlowast) =
1199091015840
01198712
1198941199090
Theorem 7 If (119860119894 119860119894) is 119879
119899-stable and for system (30)
assume that (40)ndash(43) admit the solution (1198711
119894 1198712
119894 1198701
119894 1198702
119894) with
(1198701
119894 1198702
119894) isin K then
(i) 1198711
119894gt 0 1198712
119894gt 0
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Remark 8 When 119908(119896) equiv 0 these results still hold inthe paper Only for the reason of simplicity in (1) and(30) we assume the state 119909(119905) and control inputs (119906(119905) V(119905))depend on the same noise 119908(119896) If they rely on the different
8 Mathematical Problems in Engineering
noises (1199081(119896) 119908
2(119896)) then new results will be yielded The
discussion is omitted
4 Iterative Algorithm and Simulation
41 An Iterative Algorithm In this section an iterative algo-rithm is proposed to solve the four coupled GAREs (40)ndash(43) Infinite horizon Riccati equations are hard to be solvedhence the particular problems can be solved via finite horizonequations 119873 represents the finite number of iterations in thefollowing equations
1198711
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
1198711
119894
119873
(119896 + 1)
times (119860119894+ 1198611198941198701
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
times 1198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896))
+ 11987611015840
1198941198761
119894+ 1198701
119894
119873
(119896)1015840
1198771
1198941198701
119894
119873
(119896)
minus 1198703
119894
119873
(119896)1015840
1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896)
1198711
119894
119873
(119896 + 1) = 0
1198671
119894(1198711
119894
119873
(119896 + 1)) gt 0
(51)
1198701
119894
119873
(119896) = minus1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896) (52)
1198712
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
1198712
119894
119873
(119896 + 1)
times (119860119894+ 1198621198941198702
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
times 1198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896))
+ 11987621015840
1198941198762
119894+ 1198702
119894
119873
(119896)1015840
1198782
1198941198702
119894
119873
(119896)
minus 1198704
119894
119873
(119896)1015840
1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896)
1198712
119894
119873
(119896 + 1) = 0
1198672
119894(1198712
119894
119873
(119896 + 1)) gt 0
(53)
1198702
119894
119873
(119896) = minus1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896) (54)
where
1198671
119894(1198711
119894
119873
(119896 + 1))
= 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894)
1198672
119894(1198712
119894
119873
(119896 + 1))
= 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894
+ 1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894)
1198703
119894
119873
(119896) = 119901119894119894
[1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))]
1198704
119894
119873
(119896) = 119901119894119894
[1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))
+1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))]
(55)
An iterative process for solving (40)ndash(43) based on the aboverecursions is presented as follows
(a) Given appropriate natural number 119873 and the initialconditions 119871
1
119894
119873
(119873+1) = 0 1198712119894
119873
(119873+1) = 0 1198701119894
119873
(119873+
1) = 0 and 1198702
119894
119873
(119873 + 1) = 0
(b) Through the numerical values of 1198711
119894
119873
(119873+1) 1198712119894
119873
(119873+
1)1198701119894
119873
(119873+1) and1198702
119894
119873
(119873+1) we have1198671
119894(1198711
119894
119873
(119873+
1))1198672119894(1198712
119894
119873
(119873+1))1198703119894
119873
(119873) and1198704
119894
119873
(119873) accordingto (55)
(c) 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873) can be respectively computedby (52) and (54) Then 119871
1
119894
119873
(119873) and 1198712
119894
119873
(119873) can alsobe respectively obtained by (51) and (53)
(d) Let 1198711
119894
119873
(119873 + 1) = 1198711
119894
119873
(119873) 1198712
119894
119873
(119873 + 1) = 1198712
119894
119873
(119873)1198701
119894
119873
(119873 + 1) = 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873 + 1) = 1198702
119894
119873
(119873)
(e) Then 119873 = 119873 minus 1 Repeat steps (b)ndash(d) until thenumber of iterations is 119873 + 1 We can finally obtainthe numerical values of 119871
1
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) and1198702
119894
119873
(0)
As in [28] under the assumptions of stabilizability for any1199090
isin R119899
lim119873rarrinfin
1199091015840
01198711
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198691119873
(119906lowast
119873 V) = min 119869
1infin
(119906lowast
V) = 1199091015840
01198711
1198941199090
lim119873rarrinfin
1199091015840
01198712
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198692119873
(119906 Vlowast119873
) = min 1198692infin
(119906 Vlowast) = 1199091015840
01198712
1198941199090
lim119873rarrinfin
1198701
119894
119873
(0) = 1198701
119894 lim
119873rarrinfin
1198702
119894
119873
(0) = 1198702
119894
(56)
Mathematical Problems in Engineering 9
Therefore
lim119873rarrinfin
(1198711
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) 1198702
119894
119873
(0))
= (1198711
119894 1198712
119894 1198701
119894 1198702
119894)
(57)
where (1198711
119894 1198712
119894 1198701
119894 1198702
119894) are the solutions of (40)ndash(43)
42 A Simulation Example To verify the efficiency of theabove iterative algorithm we consider the following 2-Dexample In the system (30) we set 120579
119896= 119894 isin X = 1 2
119877120591
119894= 119878120591
119894= 1 (120591 = 1 2)
1198601
= [065 0
0 09] 119860
1= [
045 0
0 055]
1198611
= [06
055] 119861
1= [
045
085]
1198621
= [075
055] 119862
1= [
05
085]
1198761
1= [
055 0
0 065] 119876
2
1= [
075 0
0 025]
1198602
= [075 0
0 07] 119860
2= [
035 0
0 045]
1198612
= [05
045] 119861
2= [
055
085]
1198622
= [065
055] 119862
2= [
04
085]
1198761
2= [
035 0
0 045] 119876
2
2= [
055 0
0 035]
(58)
For convenience let 11990111
= 04 11990122
= 05 and 119873 = 50When 120579
119896= 1 by applying the above iterative algorithm we
obtain the solutions of the four coupled equations (51)ndash(54)as follows
1198711
1
119873
(0) = [1198711
1(1 1) 119871
1
1(1 2)
1198711
1(2 1) 119871
1
1(2 2)
] = [04023 minus00588
minus00588 06820]
1198712
1
119873
(0) = [1198712
1(1 1) 119871
2
1(1 2)
1198712
1(2 1) 119871
2
1(2 2)
] = [07111 minus00331
minus00331 01487]
1198701
1
119873
(0) = [1198701
1(1 1) 119870
1
1(1 2)] = [minus01245 minus00053]
1198702
1
119873
(0) = [1198702
1(1 1) 119870
2
1(1 2)] = [minus00390 minus01739]
(59)
(1198711
1
119873
(0) 1198712
1
119873
(0) 1198701
1
119873
(0) 1198702
1
119873
(0)) are also the solutionsof (40)ndash(43) according to (57) By the solutions itshows that 119871
1
1gt 0 and 119871
2
1gt 0 The evolution of
(1198711
1
119873
(119896) 1198712
1
119873
(119896) 1198701
1
119873
(119896) 1198702
1
119873
(119896)) is exhibited in Figures 1and 2 And the figures clearly illustrate the convergence andspeediness of the backward iterations When 120579
119896= 2 it is easy
0 10 20 30 40 50minus01
0
01
02
03
04
05
06
07
08
N
L11(1 1)
L11(2 1)
L11(2 2)
L21(1 1)
L21(2 1)
L21(2 2)
Figure 1 Evolution of 1198711
1
119873
(119896) and 1198712
1
119873
(119896)
0 10 20 30 40 50minus018
minus016
minus014
minus012
minus01
minus008
minus006
minus004
minus002
0
N
K11(1 1)
K11(1 2)
K21(1 1)
K21(1 2)
Figure 2 Evolution of 1198701
1
119873
(119896) and 1198702
1
119873
(119896)
to get (1198711
2
119873
(0) 1198712
2
119873
(0) 1198701
2
119873
(0) 1198702
2
119873
(0)) that are also thesolutions of (40)ndash(43) And 119871
1
2gt 0 and 119871
2
2gt 0 Because it is
the same as the above process (120579119896
= 1) we do not introduceit again due to space limitations
5 Conclusions
In this paper we have discussed the 119879119899-stability for the
discrete-time MJSLS with a finite number of jump timesand its infinite horizon LQ differential games Based on therelations between the Lyapunov equation and the stabil-ity of discrete-time MJSLS we have obtained some useful
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
of matrices 119882119894in Theorem 4 it is easy to get the following
equations for system (30)
119901119894119894
[(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198711
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987611015840
1198941198761
119894+ 11987011015840
1198941198771
1198941198701
119894+ 11987021015840
1198941198781
1198941198702
119894= 1198711
119894
1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894119862119894+ 1198621015840
1198941198711
119894119862119894) gt 0
(45)
119901119894119894
[ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)
+ (119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)1015840
1198712
119894(119860119894+ 1198611198941198701
119894+ 1198621198941198702
119894)]
+ 11987621015840
1198941198762
119894+ 11987011015840
1198941198772
1198941198701
119894+ 11987021015840
1198941198782
1198941198702
119894= 1198712
119894
1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894119861119894+ 1198611015840
1198941198712
119894119861119894) gt 0
(46)
By rearranging (45) and (46) (40) and (42) can be obtainedrespectively
Noting 119906lowast
(119896) = 1198701
119894119909(119896) and by substituting 119906
lowast
(119896) into(30) it is easy to get the following system
119909 (119896 + 1) = (119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)
+ [(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909 (119896) + 119862120579119896V (119896)] 119908 (119896)
119909 (0) = 1199090
isin R119899
119896 isin N
(47)
Then considering the scalar function 119885(119909119896) = 119909
1015840
1198961198711
120579119896
119909119896 we
have
119864119896
[Δ119885 (119909119896)]
= 119864119896
[119885 (119909119896+1
) minus 119885 (119909119896)]
= 119864119896
[1199091015840
119896+11198711
120579119896+1
119909119896+1
minus 1199091015840
1198961198711
120579119896
119909119896]
= 119864119896
minus1199091015840
1198961198711
120579119896
119909119896
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
+ 119901119894119894
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]1015840
times 1198711
120579119896
[(119860120579119896
+ 119861120579119896
1198701
120579119896
) 119909119896
+ 119862120579119896V119896]
(48)
Due toinfin
sum
119896=0
119864119896
[Δ119885 (119909119896)]
= 119864119896
[
infin
sum
119896=0
Δ119885 (119909119896)] = 119864
119896[119885 (119909infin
) minus 119885 (1199090)] = minus119909
1015840
01198711
1198941199090
(49)
by (40) and a completing squares technique (31) can bederived that
1198691
(119906lowast
V)
=
infin
sum
119896=0
119864119896
[1199091015840
119896(11987611015840
120579119896
1198761
120579119896
+ 11987011015840
120579119896
1198771
120579119896
1198701
120579119896
) 119909119896
+ V10158401198961198781
120579119896
V119896]
+
infin
sum
119896=0
119864119896
[Δ119885 (119909119896)] + 119909
1015840
01198711
1198941199090
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
1199091015840
119896[minus1198711
119894+ 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 119901119894119894
(119860119894+ 1198611198941198701
119894)1015840
times 1198711
119894(119860119894+ 1198611198941198701
119894) + 11987611015840
1198941198761
119894
+ 11987011015840
1198941198771
1198941198701
119894] 119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V1015840119896
(1198781
119894+ 1199011198941198941198621015840
1198941198711
119894119862119894+ 1199011198941198941198621015840
1198941198711
119894119862119894) V119896
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[1199091015840
11989611987031015840
1198941198671
119894(1198711
119894)minus1
1198703
119894119909119896
+ 1199091015840
11989611987031015840
119894V119896
+ V10158401198961198703
119894119909119896
+ V10158401198961198671
119894(1198711
119894) V119896]
= 1199091015840
01198711
1198941199090
+
infin
sum
119896=0
119864119896
[V (119896) minus 1198702
119894119909 (119896)]
1015840
1198671
119894(1198711
119894) [V (119896) minus 119870
2
119894119909 (119896)]
ge 1199091015840
01198711
1198941199090 120591 = 1
(50)
Then by (32) it follows that Vlowast(119896) = 1198702
119894119909(119896) and 119869
1
(119906lowast
Vlowast) =
1199091015840
01198711
1198941199090 Finally by substituting Vlowast(119896) into (30) in the same
way as before we have 119906lowast
(119896) = 1198701
119894119909(119896) and 119869
2
(119906lowast
Vlowast) =
1199091015840
01198712
1198941199090
Theorem 7 If (119860119894 119860119894) is 119879
119899-stable and for system (30)
assume that (40)ndash(43) admit the solution (1198711
119894 1198712
119894 1198701
119894 1198702
119894) with
(1198701
119894 1198702
119894) isin K then
(i) 1198711
119894gt 0 1198712
119894gt 0
(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (119906
lowast
(119896) Vlowast(119896)) with119906lowast
(119896) = 1198701
119894119909(119896) Vlowast(119896) = 119870
2
119894119909(119896)
(iii) the optimal cost functions incurred by playing strategies(119906lowast
(119896) Vlowast(119896)) are 119869120591
= 1199091015840
0119871120591
1198941199090
(120591 = 1 2)
Remark 8 When 119908(119896) equiv 0 these results still hold inthe paper Only for the reason of simplicity in (1) and(30) we assume the state 119909(119905) and control inputs (119906(119905) V(119905))depend on the same noise 119908(119896) If they rely on the different
8 Mathematical Problems in Engineering
noises (1199081(119896) 119908
2(119896)) then new results will be yielded The
discussion is omitted
4 Iterative Algorithm and Simulation
41 An Iterative Algorithm In this section an iterative algo-rithm is proposed to solve the four coupled GAREs (40)ndash(43) Infinite horizon Riccati equations are hard to be solvedhence the particular problems can be solved via finite horizonequations 119873 represents the finite number of iterations in thefollowing equations
1198711
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
1198711
119894
119873
(119896 + 1)
times (119860119894+ 1198611198941198701
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
times 1198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896))
+ 11987611015840
1198941198761
119894+ 1198701
119894
119873
(119896)1015840
1198771
1198941198701
119894
119873
(119896)
minus 1198703
119894
119873
(119896)1015840
1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896)
1198711
119894
119873
(119896 + 1) = 0
1198671
119894(1198711
119894
119873
(119896 + 1)) gt 0
(51)
1198701
119894
119873
(119896) = minus1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896) (52)
1198712
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
1198712
119894
119873
(119896 + 1)
times (119860119894+ 1198621198941198702
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
times 1198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896))
+ 11987621015840
1198941198762
119894+ 1198702
119894
119873
(119896)1015840
1198782
1198941198702
119894
119873
(119896)
minus 1198704
119894
119873
(119896)1015840
1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896)
1198712
119894
119873
(119896 + 1) = 0
1198672
119894(1198712
119894
119873
(119896 + 1)) gt 0
(53)
1198702
119894
119873
(119896) = minus1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896) (54)
where
1198671
119894(1198711
119894
119873
(119896 + 1))
= 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894)
1198672
119894(1198712
119894
119873
(119896 + 1))
= 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894
+ 1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894)
1198703
119894
119873
(119896) = 119901119894119894
[1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))]
1198704
119894
119873
(119896) = 119901119894119894
[1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))
+1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))]
(55)
An iterative process for solving (40)ndash(43) based on the aboverecursions is presented as follows
(a) Given appropriate natural number 119873 and the initialconditions 119871
1
119894
119873
(119873+1) = 0 1198712119894
119873
(119873+1) = 0 1198701119894
119873
(119873+
1) = 0 and 1198702
119894
119873
(119873 + 1) = 0
(b) Through the numerical values of 1198711
119894
119873
(119873+1) 1198712119894
119873
(119873+
1)1198701119894
119873
(119873+1) and1198702
119894
119873
(119873+1) we have1198671
119894(1198711
119894
119873
(119873+
1))1198672119894(1198712
119894
119873
(119873+1))1198703119894
119873
(119873) and1198704
119894
119873
(119873) accordingto (55)
(c) 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873) can be respectively computedby (52) and (54) Then 119871
1
119894
119873
(119873) and 1198712
119894
119873
(119873) can alsobe respectively obtained by (51) and (53)
(d) Let 1198711
119894
119873
(119873 + 1) = 1198711
119894
119873
(119873) 1198712
119894
119873
(119873 + 1) = 1198712
119894
119873
(119873)1198701
119894
119873
(119873 + 1) = 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873 + 1) = 1198702
119894
119873
(119873)
(e) Then 119873 = 119873 minus 1 Repeat steps (b)ndash(d) until thenumber of iterations is 119873 + 1 We can finally obtainthe numerical values of 119871
1
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) and1198702
119894
119873
(0)
As in [28] under the assumptions of stabilizability for any1199090
isin R119899
lim119873rarrinfin
1199091015840
01198711
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198691119873
(119906lowast
119873 V) = min 119869
1infin
(119906lowast
V) = 1199091015840
01198711
1198941199090
lim119873rarrinfin
1199091015840
01198712
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198692119873
(119906 Vlowast119873
) = min 1198692infin
(119906 Vlowast) = 1199091015840
01198712
1198941199090
lim119873rarrinfin
1198701
119894
119873
(0) = 1198701
119894 lim
119873rarrinfin
1198702
119894
119873
(0) = 1198702
119894
(56)
Mathematical Problems in Engineering 9
Therefore
lim119873rarrinfin
(1198711
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) 1198702
119894
119873
(0))
= (1198711
119894 1198712
119894 1198701
119894 1198702
119894)
(57)
where (1198711
119894 1198712
119894 1198701
119894 1198702
119894) are the solutions of (40)ndash(43)
42 A Simulation Example To verify the efficiency of theabove iterative algorithm we consider the following 2-Dexample In the system (30) we set 120579
119896= 119894 isin X = 1 2
119877120591
119894= 119878120591
119894= 1 (120591 = 1 2)
1198601
= [065 0
0 09] 119860
1= [
045 0
0 055]
1198611
= [06
055] 119861
1= [
045
085]
1198621
= [075
055] 119862
1= [
05
085]
1198761
1= [
055 0
0 065] 119876
2
1= [
075 0
0 025]
1198602
= [075 0
0 07] 119860
2= [
035 0
0 045]
1198612
= [05
045] 119861
2= [
055
085]
1198622
= [065
055] 119862
2= [
04
085]
1198761
2= [
035 0
0 045] 119876
2
2= [
055 0
0 035]
(58)
For convenience let 11990111
= 04 11990122
= 05 and 119873 = 50When 120579
119896= 1 by applying the above iterative algorithm we
obtain the solutions of the four coupled equations (51)ndash(54)as follows
1198711
1
119873
(0) = [1198711
1(1 1) 119871
1
1(1 2)
1198711
1(2 1) 119871
1
1(2 2)
] = [04023 minus00588
minus00588 06820]
1198712
1
119873
(0) = [1198712
1(1 1) 119871
2
1(1 2)
1198712
1(2 1) 119871
2
1(2 2)
] = [07111 minus00331
minus00331 01487]
1198701
1
119873
(0) = [1198701
1(1 1) 119870
1
1(1 2)] = [minus01245 minus00053]
1198702
1
119873
(0) = [1198702
1(1 1) 119870
2
1(1 2)] = [minus00390 minus01739]
(59)
(1198711
1
119873
(0) 1198712
1
119873
(0) 1198701
1
119873
(0) 1198702
1
119873
(0)) are also the solutionsof (40)ndash(43) according to (57) By the solutions itshows that 119871
1
1gt 0 and 119871
2
1gt 0 The evolution of
(1198711
1
119873
(119896) 1198712
1
119873
(119896) 1198701
1
119873
(119896) 1198702
1
119873
(119896)) is exhibited in Figures 1and 2 And the figures clearly illustrate the convergence andspeediness of the backward iterations When 120579
119896= 2 it is easy
0 10 20 30 40 50minus01
0
01
02
03
04
05
06
07
08
N
L11(1 1)
L11(2 1)
L11(2 2)
L21(1 1)
L21(2 1)
L21(2 2)
Figure 1 Evolution of 1198711
1
119873
(119896) and 1198712
1
119873
(119896)
0 10 20 30 40 50minus018
minus016
minus014
minus012
minus01
minus008
minus006
minus004
minus002
0
N
K11(1 1)
K11(1 2)
K21(1 1)
K21(1 2)
Figure 2 Evolution of 1198701
1
119873
(119896) and 1198702
1
119873
(119896)
to get (1198711
2
119873
(0) 1198712
2
119873
(0) 1198701
2
119873
(0) 1198702
2
119873
(0)) that are also thesolutions of (40)ndash(43) And 119871
1
2gt 0 and 119871
2
2gt 0 Because it is
the same as the above process (120579119896
= 1) we do not introduceit again due to space limitations
5 Conclusions
In this paper we have discussed the 119879119899-stability for the
discrete-time MJSLS with a finite number of jump timesand its infinite horizon LQ differential games Based on therelations between the Lyapunov equation and the stabil-ity of discrete-time MJSLS we have obtained some useful
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
noises (1199081(119896) 119908
2(119896)) then new results will be yielded The
discussion is omitted
4 Iterative Algorithm and Simulation
41 An Iterative Algorithm In this section an iterative algo-rithm is proposed to solve the four coupled GAREs (40)ndash(43) Infinite horizon Riccati equations are hard to be solvedhence the particular problems can be solved via finite horizonequations 119873 represents the finite number of iterations in thefollowing equations
1198711
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
1198711
119894
119873
(119896 + 1)
times (119860119894+ 1198611198941198701
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198611198941198701
119894
119873
(119896))
1015840
times 1198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896))
+ 11987611015840
1198941198761
119894+ 1198701
119894
119873
(119896)1015840
1198771
1198941198701
119894
119873
(119896)
minus 1198703
119894
119873
(119896)1015840
1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896)
1198711
119894
119873
(119896 + 1) = 0
1198671
119894(1198711
119894
119873
(119896 + 1)) gt 0
(51)
1198701
119894
119873
(119896) = minus1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896) (52)
1198712
119894
119873
(119896) = 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
1198712
119894
119873
(119896 + 1)
times (119860119894+ 1198621198941198702
119894
119873
(119896)) + 119901119894119894
(119860119894+ 1198621198941198702
119894
119873
(119896))
1015840
times 1198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896))
+ 11987621015840
1198941198762
119894+ 1198702
119894
119873
(119896)1015840
1198782
1198941198702
119894
119873
(119896)
minus 1198704
119894
119873
(119896)1015840
1198672
119894(1198712
119894
119873
(119896 + 1))
minus1
1198704
119894
119873
(119896)
1198712
119894
119873
(119896 + 1) = 0
1198672
119894(1198712
119894
119873
(119896 + 1)) gt 0
(53)
1198702
119894
119873
(119896) = minus1198671
119894(1198711
119894
119873
(119896 + 1))
minus1
1198703
119894
119873
(119896) (54)
where
1198671
119894(1198711
119894
119873
(119896 + 1))
= 1198781
119894+ 119901119894119894
(1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) 119862119894)
1198672
119894(1198712
119894
119873
(119896 + 1))
= 1198772
119894+ 119901119894119894
(1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894
+ 1198611015840
1198941198712
119894
119873
(119896 + 1) 119861119894)
1198703
119894
119873
(119896) = 119901119894119894
[1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))
+ 1198621015840
1198941198711
119894
119873
(119896 + 1) (119860119894+ 1198611198941198701
119894
119873
(119896 + 1))]
1198704
119894
119873
(119896) = 119901119894119894
[1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))
+1198611015840
1198941198712
119894
119873
(119896 + 1) (119860119894+ 1198621198941198702
119894
119873
(119896 + 1))]
(55)
An iterative process for solving (40)ndash(43) based on the aboverecursions is presented as follows
(a) Given appropriate natural number 119873 and the initialconditions 119871
1
119894
119873
(119873+1) = 0 1198712119894
119873
(119873+1) = 0 1198701119894
119873
(119873+
1) = 0 and 1198702
119894
119873
(119873 + 1) = 0
(b) Through the numerical values of 1198711
119894
119873
(119873+1) 1198712119894
119873
(119873+
1)1198701119894
119873
(119873+1) and1198702
119894
119873
(119873+1) we have1198671
119894(1198711
119894
119873
(119873+
1))1198672119894(1198712
119894
119873
(119873+1))1198703119894
119873
(119873) and1198704
119894
119873
(119873) accordingto (55)
(c) 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873) can be respectively computedby (52) and (54) Then 119871
1
119894
119873
(119873) and 1198712
119894
119873
(119873) can alsobe respectively obtained by (51) and (53)
(d) Let 1198711
119894
119873
(119873 + 1) = 1198711
119894
119873
(119873) 1198712
119894
119873
(119873 + 1) = 1198712
119894
119873
(119873)1198701
119894
119873
(119873 + 1) = 1198701
119894
119873
(119873) and 1198702
119894
119873
(119873 + 1) = 1198702
119894
119873
(119873)
(e) Then 119873 = 119873 minus 1 Repeat steps (b)ndash(d) until thenumber of iterations is 119873 + 1 We can finally obtainthe numerical values of 119871
1
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) and1198702
119894
119873
(0)
As in [28] under the assumptions of stabilizability for any1199090
isin R119899
lim119873rarrinfin
1199091015840
01198711
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198691119873
(119906lowast
119873 V) = min 119869
1infin
(119906lowast
V) = 1199091015840
01198711
1198941199090
lim119873rarrinfin
1199091015840
01198712
119894
119873
(0) 1199090
= lim119873rarrinfin
min 1198692119873
(119906 Vlowast119873
) = min 1198692infin
(119906 Vlowast) = 1199091015840
01198712
1198941199090
lim119873rarrinfin
1198701
119894
119873
(0) = 1198701
119894 lim
119873rarrinfin
1198702
119894
119873
(0) = 1198702
119894
(56)
Mathematical Problems in Engineering 9
Therefore
lim119873rarrinfin
(1198711
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) 1198702
119894
119873
(0))
= (1198711
119894 1198712
119894 1198701
119894 1198702
119894)
(57)
where (1198711
119894 1198712
119894 1198701
119894 1198702
119894) are the solutions of (40)ndash(43)
42 A Simulation Example To verify the efficiency of theabove iterative algorithm we consider the following 2-Dexample In the system (30) we set 120579
119896= 119894 isin X = 1 2
119877120591
119894= 119878120591
119894= 1 (120591 = 1 2)
1198601
= [065 0
0 09] 119860
1= [
045 0
0 055]
1198611
= [06
055] 119861
1= [
045
085]
1198621
= [075
055] 119862
1= [
05
085]
1198761
1= [
055 0
0 065] 119876
2
1= [
075 0
0 025]
1198602
= [075 0
0 07] 119860
2= [
035 0
0 045]
1198612
= [05
045] 119861
2= [
055
085]
1198622
= [065
055] 119862
2= [
04
085]
1198761
2= [
035 0
0 045] 119876
2
2= [
055 0
0 035]
(58)
For convenience let 11990111
= 04 11990122
= 05 and 119873 = 50When 120579
119896= 1 by applying the above iterative algorithm we
obtain the solutions of the four coupled equations (51)ndash(54)as follows
1198711
1
119873
(0) = [1198711
1(1 1) 119871
1
1(1 2)
1198711
1(2 1) 119871
1
1(2 2)
] = [04023 minus00588
minus00588 06820]
1198712
1
119873
(0) = [1198712
1(1 1) 119871
2
1(1 2)
1198712
1(2 1) 119871
2
1(2 2)
] = [07111 minus00331
minus00331 01487]
1198701
1
119873
(0) = [1198701
1(1 1) 119870
1
1(1 2)] = [minus01245 minus00053]
1198702
1
119873
(0) = [1198702
1(1 1) 119870
2
1(1 2)] = [minus00390 minus01739]
(59)
(1198711
1
119873
(0) 1198712
1
119873
(0) 1198701
1
119873
(0) 1198702
1
119873
(0)) are also the solutionsof (40)ndash(43) according to (57) By the solutions itshows that 119871
1
1gt 0 and 119871
2
1gt 0 The evolution of
(1198711
1
119873
(119896) 1198712
1
119873
(119896) 1198701
1
119873
(119896) 1198702
1
119873
(119896)) is exhibited in Figures 1and 2 And the figures clearly illustrate the convergence andspeediness of the backward iterations When 120579
119896= 2 it is easy
0 10 20 30 40 50minus01
0
01
02
03
04
05
06
07
08
N
L11(1 1)
L11(2 1)
L11(2 2)
L21(1 1)
L21(2 1)
L21(2 2)
Figure 1 Evolution of 1198711
1
119873
(119896) and 1198712
1
119873
(119896)
0 10 20 30 40 50minus018
minus016
minus014
minus012
minus01
minus008
minus006
minus004
minus002
0
N
K11(1 1)
K11(1 2)
K21(1 1)
K21(1 2)
Figure 2 Evolution of 1198701
1
119873
(119896) and 1198702
1
119873
(119896)
to get (1198711
2
119873
(0) 1198712
2
119873
(0) 1198701
2
119873
(0) 1198702
2
119873
(0)) that are also thesolutions of (40)ndash(43) And 119871
1
2gt 0 and 119871
2
2gt 0 Because it is
the same as the above process (120579119896
= 1) we do not introduceit again due to space limitations
5 Conclusions
In this paper we have discussed the 119879119899-stability for the
discrete-time MJSLS with a finite number of jump timesand its infinite horizon LQ differential games Based on therelations between the Lyapunov equation and the stabil-ity of discrete-time MJSLS we have obtained some useful
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Therefore
lim119873rarrinfin
(1198711
119894
119873
(0) 1198712
119894
119873
(0) 1198701
119894
119873
(0) 1198702
119894
119873
(0))
= (1198711
119894 1198712
119894 1198701
119894 1198702
119894)
(57)
where (1198711
119894 1198712
119894 1198701
119894 1198702
119894) are the solutions of (40)ndash(43)
42 A Simulation Example To verify the efficiency of theabove iterative algorithm we consider the following 2-Dexample In the system (30) we set 120579
119896= 119894 isin X = 1 2
119877120591
119894= 119878120591
119894= 1 (120591 = 1 2)
1198601
= [065 0
0 09] 119860
1= [
045 0
0 055]
1198611
= [06
055] 119861
1= [
045
085]
1198621
= [075
055] 119862
1= [
05
085]
1198761
1= [
055 0
0 065] 119876
2
1= [
075 0
0 025]
1198602
= [075 0
0 07] 119860
2= [
035 0
0 045]
1198612
= [05
045] 119861
2= [
055
085]
1198622
= [065
055] 119862
2= [
04
085]
1198761
2= [
035 0
0 045] 119876
2
2= [
055 0
0 035]
(58)
For convenience let 11990111
= 04 11990122
= 05 and 119873 = 50When 120579
119896= 1 by applying the above iterative algorithm we
obtain the solutions of the four coupled equations (51)ndash(54)as follows
1198711
1
119873
(0) = [1198711
1(1 1) 119871
1
1(1 2)
1198711
1(2 1) 119871
1
1(2 2)
] = [04023 minus00588
minus00588 06820]
1198712
1
119873
(0) = [1198712
1(1 1) 119871
2
1(1 2)
1198712
1(2 1) 119871
2
1(2 2)
] = [07111 minus00331
minus00331 01487]
1198701
1
119873
(0) = [1198701
1(1 1) 119870
1
1(1 2)] = [minus01245 minus00053]
1198702
1
119873
(0) = [1198702
1(1 1) 119870
2
1(1 2)] = [minus00390 minus01739]
(59)
(1198711
1
119873
(0) 1198712
1
119873
(0) 1198701
1
119873
(0) 1198702
1
119873
(0)) are also the solutionsof (40)ndash(43) according to (57) By the solutions itshows that 119871
1
1gt 0 and 119871
2
1gt 0 The evolution of
(1198711
1
119873
(119896) 1198712
1
119873
(119896) 1198701
1
119873
(119896) 1198702
1
119873
(119896)) is exhibited in Figures 1and 2 And the figures clearly illustrate the convergence andspeediness of the backward iterations When 120579
119896= 2 it is easy
0 10 20 30 40 50minus01
0
01
02
03
04
05
06
07
08
N
L11(1 1)
L11(2 1)
L11(2 2)
L21(1 1)
L21(2 1)
L21(2 2)
Figure 1 Evolution of 1198711
1
119873
(119896) and 1198712
1
119873
(119896)
0 10 20 30 40 50minus018
minus016
minus014
minus012
minus01
minus008
minus006
minus004
minus002
0
N
K11(1 1)
K11(1 2)
K21(1 1)
K21(1 2)
Figure 2 Evolution of 1198701
1
119873
(119896) and 1198702
1
119873
(119896)
to get (1198711
2
119873
(0) 1198712
2
119873
(0) 1198701
2
119873
(0) 1198702
2
119873
(0)) that are also thesolutions of (40)ndash(43) And 119871
1
2gt 0 and 119871
2
2gt 0 Because it is
the same as the above process (120579119896
= 1) we do not introduceit again due to space limitations
5 Conclusions
In this paper we have discussed the 119879119899-stability for the
discrete-time MJSLS with a finite number of jump timesand its infinite horizon LQ differential games Based on therelations between the Lyapunov equation and the stabil-ity of discrete-time MJSLS we have obtained some useful
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
theorems on finding the solutions of the LQ differentialgames Moreover an iterative algorithm has been presentedfor the solvability of the four coupled equations Finally anumerical example is offered to demonstrate the efficiencyof the algorithm Exact observability and119882-observability fordiscrete-timeMJSLS are investigated by [29 30] On the basisof exact observability and 119882-observability infinite horizonstochastic differential games should be discussed and we willdo further research in the future
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61304080 and 61174078) a Projectof Shandong Province Higher Educational Science and Tech-nology Program (no J12LN14) the Research Fund for theTaishan Scholar Project of Shandong Province of China andthe State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources (no LAPS13018)
References
[1] M Mariton Jump Linear Systems in Automatic Control CRCPress 1990
[2] M K Ghosh A Arapostathis and S I Marcus ldquoOptimalcontrol of switching diffusions with application to flexible man-ufacturing systemsrdquo SIAM Journal onControl andOptimizationvol 31 no 5 pp 1183ndash1204 1993
[3] E K Boukas Z K Liu and G X Liu ldquoDelay-dependent robuststability and 119867
infincontrol of jump linear systems with time-
delayrdquo International Journal of Control vol 74 no 4 pp 329ndash340 2001
[4] X R Mao ldquoExponential stability of stochastic delay intervalsystems with Markovian switchingrdquo IEEE Transactions onAutomatic Control vol 47 no 10 pp 1604ndash1612 2002
[5] T Morozan ldquoStability and control for linear systems with jumpMarkov perturbationsrdquo Stochastic Analysis and Applicationsvol 13 no 1 pp 91ndash110 1995
[6] O L Costa and M D Fragoso ldquoDiscrete-time LQ-optimalcontrol problems for infinite Markov jump parameter systemsrdquoIEEE Transactions on Automatic Control vol 40 no 12 pp2076ndash2088 1995
[7] R Rakkiyappan Q Zhu and A Chandrasekar ldquoStability ofstochastic neural networks of neutral type with Markovianjumping parameters a delay-fractioning approachrdquo Journal ofthe Franklin Institute vol 351 no 3 pp 1553ndash1570 2014
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] Y Zhang P Shi S KiongNguang andH R Karimi ldquoObserver-based finite-time fuzzy 119867
infincontrol for discrete-time systems
with stochastic jumps and time-delaysrdquo Signal Processing vol97 pp 252ndash261 2014
[10] Y Wei J Qiu H R Karimi and M Wang ldquoFiltering designfor two-dimensionalMarkovian jump systems with state-delaysand deficient mode informationrdquo Information Sciences vol 269pp 316ndash331 2014
[11] H Dong Z Wang D W Ho and H Gao ldquoRobust 119867infin
filtering for Markovian jump systems with randomly occurringnonlinearities and sensor saturation the finite-horizon caserdquoIEEE Transactions on Signal Processing vol 59 no 7 pp 3048ndash3057 2011
[12] Y Ji H J Chizeck X Feng and K A Loparo ldquoStability andcontrol of discrete-time jump linear systemsrdquo Control Theoryand Advanced Technology vol 7 no 2 pp 247ndash270 1991
[13] X Feng K A Loparo Y Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992
[14] Z G Li Y C Soh and C Y Wen ldquoSufficient conditions foralmost sure stability of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 45 no 7 pp 1325ndash1329 2000
[15] Y Fang and K A Loparo ldquoOn the relationship between thesample path and moment Lyapunov exponents for jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 9pp 1556ndash1560 2002
[16] F Kozin ldquoA survey of stability of stochastic systemsrdquo Automat-ica vol 5 pp 95ndash112 1969
[17] Q X Zhu and J Cao ldquoStability analysis of markovian jumpstochastic BAM neural networks with impulse control andmixed time delaysrdquo IEEE Transactions on Neural Networks andLearning Systems vol 23 no 3 pp 467ndash479 2012
[18] R IsaacsDifferential Games JohnWiley amp Sons NewYork NYUSA 1965
[19] A A Stoorvogel ldquoThe singular zero-sum differential gamewith stability using119867
infincontrol theoryrdquoMathematics of Control
Signals and Systems vol 4 no 2 pp 121ndash138 1991[20] V Turetsky ldquoDifferential game solubility condition in 119867
infinopti-
mizationrdquo Nonsmooth and Discondinuous Problems of Controland Optimization pp 209ndash214 1998
[21] Z Wu and Z Y Yu ldquoLinear quadratic nonzero-sum differentialgames with random jumpsrdquo Applied Mathematics and Mechan-ics vol 26 no 8 pp 1034ndash1039 2005
[22] X-H Nian ldquoSuboptimal strategies of linear quadratic closed-loop differential games an BMI approachrdquo Acta AutomaticaSinica vol 31 no 2 pp 216ndash222 2005
[23] J Yong ldquoA leader-follower stochastic linear quadratic differen-tial gamerdquo SIAM Journal on Control and Optimization vol 41no 4 pp 1015ndash1041 2002
[24] H Y Sun M Li andW H Zhang ldquoLinear-quadratic stochasticdifferential game infinite-time caserdquo ICIC Express Letters vol5 no 4 pp 1449ndash1454 2011
[25] H Sun L Jiang andW Zhang ldquoFeedback control on nash equi-librium for discrete-time stochastic systems with markovianjumps finite-horizon caserdquo International Journal of ControlAutomation and Systems vol 10 no 5 pp 940ndash946 2012
[26] H Y Sun C Y Feng and L Y Jiang ldquoLinear quadraticdifferential games for discrete-timesMarkovian jump stochasticlinear systems infinite-horizon caserdquo in Proceedings of the 30thChinese Control Conference (CCC 11) pp 1983ndash1986 YantaiChina July 2011
[27] J B do Val C Nespoli and Y R Caceres ldquoStochastic stabilityfor Markovian jump linear systems associated with a finitenumber of jump timesrdquo Journal of Mathematical Analysis andApplications vol 285 no 2 pp 551ndash563 2003
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
[28] W H Zhang Y L Huang and H S Zhang ldquoStochastic 1198672119867infin
control for discrete-time systems with state and disturbancedependent noiserdquo Automatica vol 43 no 3 pp 513ndash521 2007
[29] T Hou Stability and robust H2Hinfin
control for discrete-timeMarkov jump systems [PhD dissertation] Shandong Universityof Science and Technology Qingdao China 2010
[30] W H Zhang and C Tan ldquoOn detectability and observabilityof discrete-time stochastic Markov jump systems with state-dependent noiserdquo Asian Journal of Control vol 15 no 5 pp1366ndash1375 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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