Research Article Stability and Linear Quadratic Differential Games … · 2019. 7. 31. · Research Article Stability and Linear Quadratic Differential Games of Discrete-Time Markovian

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; [email protected]

Received 7 July 2014; Accepted 6 September 2014; Published 23 November 2014

Academic Editor: Ramachandran Raja

Copyright © 2014 Huiying Sun et al. This 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 linearquadratic (LQ) differential games. A necessary and sufficient condition involved with the connection between stochastic 𝑇

𝑛-

stability of Markovian jump linear systems with state-dependent noise and Lyapunov equation is proposed. And using the theoryof stochastic 𝑇

𝑛-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 command,and control and communication systems [1–4]. 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 [5–11].

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 concepts.Particularly, the study of stability about these systems hasattracted the attention of many researchers [12–17]. Thevery important stability notions are mean-square stability,moment 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, 𝛿-momentstability, requires the convergence to zero of 𝛿-moment ofthe state norm (mean-square stability is just a particular casefor 𝛿 = 2). Although there exist some practical sufficientconditions, a simple necessary and sufficient condition test-ing 𝛿-moment stability is not available (except for 𝛿 = 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 pageshttp://dx.doi.org/10.1155/2014/265621

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 robots.Since the book [18], entitled “Differential Games,” 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 [19–23]. 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 the𝐻

∞optimization 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 Itô-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 games,especially 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 𝑇

𝑛-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 𝑇

𝑛-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 𝑇

𝑛-stability of

discrete-time MJSLS with a finite number of jump times,which is essential to obtain the main results. An iterative

algorithm for solving the four coupled GAREs is put forward,and an illustrative example is also displayed in Section 4.Section 5 ends this paper with some concluding remarks.

For convenience, the paper adopts the following basicnotations. It uses R𝑛 to denote the linear space of all 𝑛-dimensional real vectors. R

𝑚×𝑛denotes the linear space of

all 𝑚 × 𝑛 real matrices. N = {0, 1, 2, . . .}. 𝑈 indicates thetranspose of matrix 𝑈 and 𝑈 ≥ 0 (𝑈 > 0) represents anonnegative definite matrix (positive definite). The standardvector norm inR𝑛 is indicated by ‖ ⋅ ‖ and the correspondinginduced norm of matrix 𝑈 by ‖𝑈‖. 𝐿2(∞,R𝑘) represents thespace of R𝑘-valued, square integrable random vectors and𝛿{⋅}

is the Dirac measure. Finally, we write 𝐸𝑘[⋅] instead of

𝐸[⋅ | 𝑥𝑘, 𝜃𝑘], and we define the following operator: 𝜀

𝑖(𝑈) =

∑𝑗 ̸=𝑖

𝑝𝑖𝑗𝑈𝑗for 𝑈 = 𝑈 ≥ 0.

2. Stochastic 𝑇𝑛-Stability for Discrete-Time

MJSLS with a Finite Number of Jump Times

Let (Ω,F, 𝑃; {F𝑘}) be a given fundamental probability space

where there exist a Markov chain 𝜃𝑘and a sequence of real

random variables 𝑤(𝑘). F𝑘denotes the 𝜎-algebra generated

by 𝜃𝑘and 𝑤(𝑘); that is,F

𝑘= 𝜎{𝜃

𝑠, 𝑤(𝑠) | 𝑠 = 0, 1, 2, . . . , 𝑘} ⊂

F. Consider the following MJSLS defined on the space(Ω,F, 𝑃; {F

𝑘}):

𝑥 (𝑘 + 1) = 𝐴𝜃𝑘

𝑥 (𝑘) + 𝐴𝜃𝑘

𝑥 (𝑘) 𝑤 (𝑘) ,

𝑥 (0) = 𝑥0

∈ R𝑛

, 𝑘 ∈ N,(1)

where {𝑥(𝑘), 𝜃𝑘; 𝑘 ∈ N} are the states of process with values

inR𝑛 ×X; {𝜃𝑘; 𝑘 ∈ N} is a time homogeneous Markov chain

taking values in a finite set X = {1, 2, . . . , 𝑁}, with initialdistribution 𝜇 and transition probability matrix P = [𝑝

𝑖𝑗],

where

𝑝𝑖𝑗

:= 𝑃 (𝜃𝑘+1

= 𝑗 | 𝜃𝑘

= 𝑖) , ∀𝑖, 𝑗 ∈ X, 𝑘 ∈ N. (2)

The set X comprises the various operation modes of thesystem (1). For each 𝜃

𝑘= 𝑖 ∈ X, the matrices 𝐴

𝜃𝑘∈ R𝑛×𝑛

and 𝐴𝜃𝑘

∈ R𝑛×𝑛

(associated with “𝑖th” mode) will be assignedas 𝐴𝜃𝑘

:= 𝐴𝑖and 𝐴

𝜃𝑘:= 𝐴𝑖in someplace of the paper.

{𝑤(𝑘), 𝑘 ∈ N} is a sequence of real random variables, whichis also a wide sense stationary, second-order process with𝐸[𝑤(𝑘)] = 0 and 𝐸[𝑤(𝑖)𝑤(𝑗)] = 𝛿

𝑖𝑗, where 𝛿

𝑖𝑗refers to a

Kronecker function; that is, 𝛿𝑖𝑗

= 1 if 𝑖 = 𝑗 and 𝛿𝑖𝑗

= 0 if 𝑖 ̸= 𝑗.TheMJSLS as defined is trivially a strongMarkov process.Weassume that 𝜃

𝑘is independent of𝑤(𝑘). Tomake the operation

more convenient, let 𝑥(𝑘) = 𝑥𝑘. When system (1) is stable, we

also say (𝐴𝑖, 𝐴𝑖) 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:

𝑇0

= 0,

𝑇𝑛

= min {𝑘 > 𝑇𝑛−1

: 𝜃𝑘

̸= 𝜃𝑇𝑛−1

} , 𝑛 = 1, 2, . . . , 𝑁.

(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 a“crucial failure” (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 𝑁of 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 𝑇𝑛. The

MJSLS (1) is

(i) stochastically 𝑇𝑛-stable if for each initial condition 𝑥

0

and initial distribution 𝜇

𝐸 [

∞

∑

𝑘=0

𝑥𝑘

2

𝛿{𝑇𝑛≥𝑘}

] < ∞, (4)

(ii) mean-square 𝑇𝑛-stable if for each initial condition 𝑥

0

and initial distribution 𝜇

lim𝑘→∞

𝐸 [𝑥𝑘

2


] = 0. (5)

Lemma 2 (see [27]). For all 𝑚 ≥ 1 and 𝑖, 𝑗 ∈ X

𝑃 (𝑇1

= 𝑚, 𝜃𝑚

= 𝑗 | 𝜃0

= 𝑖)

= {𝑝𝑖𝑗𝛿{𝑚=1}

, 𝑤ℎ𝑒𝑛 𝑝𝑖𝑖

= 0,

𝑝𝑚−1

𝑖𝑖𝑝𝑖𝑗𝛿{𝑚>1}

, 𝑤ℎ𝑒𝑛 0 < 𝑝𝑖𝑖

< 1.

(6)

Remark 3. 𝑃(𝑇1

= 1 | 𝜃0

= 𝑖) = 1 and 𝑃(𝑇1

= +∞ | 𝜃0

= 𝑖) =

1, whenever 𝑝𝑖𝑖

= 0 and 𝑝𝑖𝑖

= 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 𝑇𝑛-stable if and only if, for any

given set ofmatrices𝑊𝑖> 0, there exists a unique set ofmatrices

𝐿𝑖> 0, satisfying the Lyapunov equations

𝑝𝑖𝑖

(𝐴

𝑖𝐿𝑖𝐴𝑖+ 𝐴

𝑖𝐿𝑖𝐴𝑖) − 𝐿𝑖+ 𝑊𝑖= 0, 𝑖 ∈ X. (7)

Proof. Sufficiency. In the proof we employ an inductionargument on the stopping times 𝑇

𝑛. First, define the function

𝑉𝑘

(𝑥, 𝑖) := 𝑥

𝑘(𝑃𝑖𝛿{𝑇1>𝑘}

+ 𝐺𝑖𝛿{𝑇1=𝑘}

) 𝑥𝑘, (8)

where 𝐺𝑖> 0, and 𝑃

𝑖> 0 is the solution of

𝑝𝑖𝑖

(𝐴

𝑖𝑃𝑖𝐴𝑖+ 𝐴

𝑖𝑃𝑖𝐴𝑖) − 𝑃𝑖+ 𝑊𝑖+ 𝐴

𝑖𝜀𝑖(𝐺) 𝐴

𝑖

+ 𝐴


𝑖= 0, 𝑖 ∈ X.

(9)

The existence of such𝑃𝑖> 0 relies on (7). Hence, to functional

𝑉𝑘(𝑥, 𝑖) in the following operation, we can derive

𝐸𝑘

[𝑉𝑘+1

(𝑥𝑘+1

, 𝜃𝑘+1

) − 𝑉𝑘

(𝑥𝑘, 𝜃𝑘)]

= 𝐸𝑘

[𝑥

𝑘+1(𝑃𝜃𝑘+1𝛿{𝑇1>𝑘+1}

+ 𝐺𝜃𝑘+1𝛿{𝑇1=𝑘+1}

) 𝑥𝑘+1

− 𝑥

𝑘(𝑃𝜃𝑘𝛿{𝑇1≥𝑘+1}

+ 𝐺𝜃𝑘𝛿{𝑇1=𝑘}

) 𝑥𝑘]

= 𝐸𝑘

[(𝑥

𝑘+1𝑃𝜃𝑘+1

𝑥𝑘+1

− 𝑥

𝑘𝑃𝜃𝑘

𝑥𝑘) 𝛿{𝑇1>𝑘+1}

]

+ 𝐸𝑘

[(𝑥

𝑘+1𝐺𝜃𝑘+1

𝑥𝑘+1

− 𝑥

𝑘𝑃𝜃𝑘

𝑥𝑘) 𝛿{𝑇1=𝑘+1}

]

− 𝑥

𝑘𝐺𝜃𝑘

𝑥𝑘𝛿{𝑇1=𝑘}

= 𝐸𝑘

(𝑥

𝑘𝐴

𝜃𝑘

𝑃𝜃𝑘+1

𝐴𝜃𝑘

𝑥𝑘

+ 𝑥

𝑘𝐴

𝜃𝑘

𝑃𝜃𝑘+1

𝐴𝜃𝑘

𝑥𝑘𝑤𝑘

+ 𝑤

𝑘𝑥

𝑘𝐴

𝜃𝑘

𝑃𝜃𝑘+1

𝐴𝜃𝑘

𝑥𝑘

+ 𝑤

𝑘𝑥

𝑘𝐴

𝜃𝑘

𝑃𝜃𝑘+1

𝐴𝜃𝑘

𝑥𝑘𝑤𝑘

− 𝑥

𝑘𝑃𝜃𝑘

𝑥𝑘) 𝛿{𝑇1>𝑘+1}

+ 𝐸𝑘

(𝑥

𝑘𝐴

𝜃𝑘

𝐺𝜃𝑘+1

𝐴𝜃𝑘

𝑥𝑘

+ 𝑥

𝑘𝐴

𝜃𝑘

𝐺𝜃𝑘+1

𝐴𝜃𝑘

𝑥𝑘𝑤𝑘

+ 𝑤

𝑘𝑥

𝑘𝐴

𝜃𝑘

𝐺𝜃𝑘+1

𝐴𝜃𝑘

𝑥𝑘

+ 𝑤

𝑘𝑥

𝑘𝐴

𝜃𝑘

𝐺𝜃𝑘+1

𝐴𝜃𝑘

𝑥𝑘𝑤𝑘

− 𝑥

𝑘𝑃𝜃𝑘

𝑥𝑘) 𝛿{𝑇1=𝑘+1}

− 𝑥

𝑘𝐺𝜃𝑘


.

(10)

We know that 𝐸[𝑤(𝑘)] = 0 and 𝐸[𝑤(𝑖)𝑤(𝑗)] = 𝛿𝑖𝑗, where

𝛿𝑖𝑗

= 1 if 𝑖 = 𝑗 and 𝛿𝑖𝑗

= 0 if 𝑖 ̸= 𝑗. Calculating the expectedvalues above, we can obtain that

𝐸𝑘

[𝑉𝑘+1

(𝑥𝑘+1

, 𝜃𝑘+1

) − 𝑉𝑘


= 𝑥

𝑘[𝑝𝜃𝑘𝜃𝑘

(𝐴

𝜃𝑘

𝑃𝜃𝑘

𝐴𝜃𝑘

+ 𝐴

𝜃𝑘

𝑃𝜃𝑘

𝐴𝜃𝑘

) + 𝐴

𝜃𝑘

𝜀𝜃𝑘

(𝐺) 𝐴𝜃𝑘

+ 𝐴

𝜃𝑘

𝜀𝜃𝑘

(𝐺) 𝐴𝜃𝑘

− 𝑃𝜃𝑘

] 𝑥𝑘𝛿{𝑇1>𝑘}

− 𝑥

𝑘𝐺𝜃𝑘


,

(11)

due to𝑃(𝑇1

= 𝑘+1 | 𝜃𝑘) = 1−𝑝

𝜃𝑘𝜃𝑘, 𝑃(𝑇1

> 𝑘+1 | 𝜃𝑘) = 𝑝𝜃𝑘𝜃𝑘

,and 𝜀𝜃𝑘

(𝐺) = Σ𝜃𝑘+1 ̸=𝜃𝑘

𝑝𝜃𝑘𝜃𝑘+1

𝐺𝜃𝑘+1

.The above relation can be rewritten as

𝐸𝑘

[𝑉𝑘+1

(𝑥𝑘+1

, 𝜃𝑘+1

) − 𝑉𝑘


= −𝑥

𝑘(𝑊𝜃𝑘𝛿{𝑇1>𝑘}


) 𝑥𝑘,

(12)

where

−𝑊𝜃𝑘

= 𝑝𝜃𝑘𝜃𝑘

(𝐴

𝜃𝑘

𝑃𝜃𝑘

𝐴𝜃𝑘

+ 𝐴

𝜃𝑘

𝑃𝜃𝑘

𝐴𝜃𝑘

)

+ 𝐴

𝜃𝑘

𝜀𝜃𝑘

(𝐺) 𝐴𝜃𝑘

+ 𝐴

𝜃𝑘

𝜀𝜃𝑘

(𝐺) 𝐴𝜃𝑘

− 𝑃𝜃𝑘

.

(13)


Now, let us observe that𝑁

∑

𝑘=0

𝐸0

[𝑉𝑘+1

(𝑥𝑘+1

, 𝜃𝑘+1

) − 𝑉𝑘


=

𝑁

∑

𝑘=0

𝐸0

{𝐸𝑘

[𝑉𝑘+1

(𝑥𝑘+1

, 𝜃𝑘+1

) − 𝑉𝑘

(𝑥𝑘, 𝜃𝑘)]} .

(14)

By applying (12) and considering that 𝑊𝑖> 0, 𝐺

𝑖> 0 for each

initial condition 𝑥0and initial distribution 𝜇, then we have

𝐸0

[𝑉𝑘+1

(𝑥𝑘+1

, 𝜃𝑘+1

)] − 𝑉0

(𝑥0, 𝜃0)

= −

𝑁

∑

𝑘=0

𝐸0

[𝑥

𝑘(𝑊𝜃𝑘𝛿{𝑇1>𝑘}


) 𝑥𝑘𝛿{𝑇1≥𝑘}

]

≤ −

𝑁

∑

𝑘=0

𝛾𝐸0

[𝑥𝑘

2

𝛿{𝑇1≥𝑘}

] ,

(15)

for some 𝛾 > 0. Because 𝐸0[𝑉𝑘(𝑥𝑘, 𝜃𝑘)] ≥ 0, ∀𝑘 ≥ 0, then

lim𝑘→∞

𝐸0[𝑉𝑘+1

(𝑥𝑘+1

, 𝜃𝑘+1

)] = 0 by (5). Finally, it is easy toverify that, for any 𝑇

1,

lim sup𝑁→∞

𝑁

∑

𝑘=0

𝐸0

[𝑥𝑘

2

𝛿{𝑇1≥𝑘}

] ≤1

𝛾𝑉0

(𝑥0, 𝜃0) < ∞ (16)

holds from (15). Therefore, for any 𝑇1, MJSLS (1) is stable

according to (i) of Definition 1.Now, using an induction argument, we assume that for

some 𝑛 the inequality

lim sup𝑁→∞

𝑁

∑

𝑘=0

𝐸 [𝑥

𝑘𝑄𝜃𝑘

𝑥𝑘𝛿{𝑇𝑛≥𝑘}

] < 𝑥

0𝑃𝜃0

𝑥0

(17)

holds and thus, by setting 𝑄 ≡ 𝐼, 𝐸[‖𝑥𝑇𝑛

‖2


] < ∞.However,

lim sup𝑁→∞

𝑁

∑

𝑘=0

𝐸 [𝑥

𝑘𝑄𝜃𝑘

𝑥𝑘𝛿{𝑇𝑛+1≥𝑘}

]

= lim sup𝑁→∞

𝐸 [

[

𝑁

∑

𝑘=0

𝑥

𝑘𝑄𝜃𝑘

𝑥𝑘𝛿{𝑇𝑛>𝑘}

+

𝑁

∑

𝑘=𝑇𝑛

𝑥

𝑘𝑄𝜃𝑘

𝑥𝑘𝛿{𝑇𝑛≤𝑘≤𝑇𝑛+1}

]

]

.

(18)

Notice that using the strong Markov property and thehomogeneity property, the second term conditioned to theknowledge of (𝑥

𝑇𝑛, 𝜃𝑇𝑛) can be written as

lim sup𝑁→∞

𝐸 [

[

𝑁

∑

𝑘=𝑇𝑛

𝑥

𝑘𝑄𝜃𝑘


| 𝑥𝑇𝑛

, 𝜃𝑇𝑛

]

]

= lim sup𝑁→∞

𝐸 [

𝑁−𝑇𝑛

∑

𝑘=0

𝑥

𝑘𝑄𝜃𝑘

𝑥𝑘𝛿{𝑘≤𝑇1}

| 𝑥0

= 𝑥𝑇𝑛

, 𝜃0

= 𝜃𝑇𝑛

]

< 𝑥

𝑇𝑛

𝑃𝜃𝑇𝑛

𝑥𝑇𝑛

.

(19)

So one can conclude from (18) and (19) that

lim sup𝑁→∞

𝑁

∑

𝑘=0

𝐸 [𝑥

𝑘𝑄𝜃𝑘


]

< lim sup𝑁→∞

𝑁

∑

𝑘=0

𝐸 [𝑥

𝑘𝑄𝜃𝑘


+ 𝑥

𝑇𝑛

𝑃𝜃𝑇𝑛

𝑥𝑇𝑛

]

= lim sup𝑁→∞

𝑁

∑

𝑘=0

𝐸 [𝑥

𝑘𝑄𝜃𝑘

𝑥𝑘𝛿{𝑇𝑛≥𝑘}

+ 𝑥

𝑇𝑛

(𝑃𝜃𝑇𝑛

− 𝑄𝜃𝑇𝑛

) 𝑥𝑇𝑛

]

< 2𝑥

0𝑃𝜃0

𝑥0.

(20)

Therefore, for any 𝑇𝑛,

lim sup𝑁→∞

𝑁

∑

𝑘=0

𝐸 [𝑥𝑘

2


] < ∞ (21)

indicate that the MJSLS (1) is 𝑇𝑛-stable.

Necessity. As in the previous part define the functional

𝑥

0𝑃𝜃0

𝑥0

:= 𝐸0

[

∞

∑

𝑘=0

𝑥

𝑘𝑊𝜃𝑘

𝑥𝑘𝛿{𝑇1>𝑘}

+ 𝑥

𝑇1

𝐺𝜃𝑇1

𝑥𝑇1

] , (22)

for all (𝑥0, 𝜃0) ∈ R𝑛 × X. Therefore,

𝑥

1𝑃𝜃1

𝑥1

= 𝐸𝑇1

[

∞

∑

𝑘=1

𝑥

𝑘𝑊𝜃𝑘


+ 𝑥

𝑇1

𝐺𝜃𝑇1

𝑥𝑇1

] 𝛿{𝑇1>1}

.

(23)

The right-hand side of (22) can be expressed as

𝐸0

{𝑥

0𝑊𝜃0

𝑥0

+ 𝐸𝑇1

[(

∞

∑

𝑘=1

𝑥

𝑘𝑊𝜃𝑘


+ 𝑥

𝑇1

𝐺𝜃𝑇1

𝑥𝑇1

) 𝛿{𝑇1≥1}

]} .

(24)

In addition,

𝐸𝑇1

[(

∞

∑

𝑘=1

𝑥

𝑘𝑊𝜃𝑘


+ 𝑥

𝑇1

𝐺𝜃𝑇1

𝑥𝑇1

) 𝛿{𝑇1≥1}

]

= 𝐸𝑇1

[(

∞

∑

𝑘=1

𝑥

𝑘𝑊𝜃𝑘


+ 𝑥

𝑇1

𝐺𝜃𝑇1

𝑥𝑇1

) 𝛿{𝑇1>1}

+ 𝑥

𝑇1

𝐺𝜃𝑇1

𝑥𝑇1𝛿{𝑇1=1}

] .

(25)

Thus, based on the strong Markov property, applying homo-geneity in (25) and introducing it in (24), we arrive at

𝑥

0𝑃𝜃0

𝑥0

= 𝑥

0𝑊𝜃0

𝑥0

+ 𝐸0

[𝑥

1𝑃𝜃1

𝑥1𝛿{𝑇1>1}

+ 𝑥

𝑇1

𝐺𝜃𝑇1

𝑥𝑇1𝛿{𝑇1=1}

] .

(26)


Since 𝑥0is arbitrary, and calculating the expected values

above, (26) implies that

𝑝𝑖𝑖

(𝐴


𝑖𝑃𝑖𝐴𝑖) − 𝑃𝑖+ 𝐴


𝑖+ 𝐴


𝑖

= −𝑊𝑖,

(27)

using the fact that 𝑃(𝑇1

= 𝑘 + 1 | 𝜃𝑘

= 𝑖) = 1 − 𝑝𝑖𝑖and 𝑃(𝑇

1>

𝑘+1 | 𝜃𝑘

= 𝑖) = 𝑝𝑖𝑖.Thus, from the Lyapunov stability theory,

the existence of the set 𝐿𝑖

> 0 satisfying (7) is guaranteed,completing the proof for 𝑛 = 1.

Now, for the general case, from the stochastically 𝑇𝑛-

stable of the system we can obtain that

𝐸 [

∞

∑

𝑘=0

𝑥

𝑘𝑊𝜃𝑘


+ 𝑥

𝑇𝑛

𝐺𝜃𝑇𝑛

𝑥𝑇𝑛

] < ∞. (28)

And from the strong Markov property, we can deduce that

𝐸𝑇𝑛

[

[

∞

∑

𝑘=𝑇𝑛

𝑥

𝑘𝑊𝜃𝑘

𝑥𝑘𝛿{𝑇𝑛+1>𝑘}

+ 𝑥

𝑇𝑛+1

𝐺𝜃𝑇𝑛+1

𝑥𝑇𝑛+1

]

]

< ∞, (29)

for 𝑛 = 0, 1, . . . , 𝑁−1. By the homogeneity property, it followsthat (29) is equivalent to (22) with 𝑥

0= 𝑥𝑇𝑛

and 𝜃0

= 𝜃𝑇𝑛,

and the existence of a set of matrices 𝐿𝑖

> 0 satisfying (7) isassured. Then, the proof of Theorem 4 is completed.

3. LQ Differential Games for MJSLS witha Finite Number of Jump Times

3.1. 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:

𝑥 (𝑘 + 1) = 𝐴𝜃𝑘

𝑥 (𝑘) + 𝐵𝜃𝑘

𝑢 (𝑘) + 𝐶𝜃𝑘V (𝑘)

+ [𝐴𝜃𝑘

𝑥 (𝑘) + 𝐵𝜃𝑘

𝑢 (𝑘) + 𝐶𝜃𝑘V (𝑘)] 𝑤 (𝑘) ,

𝑥 (0) = 𝑥0

∈ R𝑛

, 𝑘 ∈ N,

𝑦𝜏

(𝑘) = 𝑄𝜏

𝜃𝑘

𝑥 (𝑘) , 𝜏 = 1, 2.

(30)

𝑦𝜏

(𝑘) ∈ R𝑚 are the measurement outputs for each player.Here (𝑢(𝑘), V(𝑘)) ∈ R𝑟 × R𝑟 represent the system controlinputs. The matrices (𝐵

𝜃𝑘, 𝐵𝜃𝑘

, 𝐶𝜃𝑘

, 𝐶𝜃𝑘

, 𝑄𝜏

𝜃𝑘

) ∈ R𝑛×𝑟

× R𝑛×𝑟

×

R𝑛×𝑟

× R𝑛×𝑟

× R𝑚×𝑛

(associated with “𝑖th” mode) will beassigned as (𝐵

𝑖, 𝐵𝑖, 𝐶𝑖, 𝐶𝑖, 𝑄𝜏

𝑖) for each 𝜃

𝑘= 𝑖 ∈ X.

Throughout this paper, we choose the infinite horizonquadratic cost functions associated with each player:

𝐽𝜏

(𝑢, V) =∞

∑

𝑘=0

𝐸 [𝑥 (𝑘)

(𝑄𝜏

𝜃𝑘

)

𝑄𝜏

𝜃𝑘

𝑥 (𝑘)

+ 𝑢 (𝑘)

𝑅𝜏

𝜃𝑘

𝑢 (𝑘) + V (𝑘) 𝑆𝜏𝜃𝑘

V (𝑘) ] ,

𝜏 = 1, 2.

(31)

The weighting matrices 𝑄𝜏𝜃𝑘

= 𝑄𝜏

𝑖≥ 0, 𝑅

𝜏

𝜃𝑘

= 𝑅𝜏

𝑖> 0 ∈ R

𝑟×𝑟,

and 𝑆𝜏𝜃𝑘

= 𝑆𝜏

𝑖> 0 ∈ R

𝑟×𝑟.

So we are looking for actions that satisfy simultaneously

𝐽1

(𝑢∗

, V∗) ≤ 𝐽1 (𝑢∗, V) , 𝐽2 (𝑢∗, V∗) ≤ 𝐽2 (𝑢, V∗) , (32)

where (𝑢∗(𝑘), V∗(𝑘)) ∈ 𝐿2(∞,R𝑟𝑢) × 𝐿2(∞,R𝑟V).To ensure the finiteness of the infinite-time cost function,

we restrain the admissible control set to the constant linearfeedback strategies; that is, 𝑢(𝑘) = 𝐾1

𝜃𝑘

𝑥(𝑘), V(𝑘) = 𝐾2𝜃𝑘

𝑥(𝑘),where 𝐾1

𝜃𝑘

and 𝐾2𝜃𝑘

are constant matrices of appropriatedimensions, and (𝐾1

𝜃𝑘

, 𝐾2

𝜃𝑘

) belong to

K := {𝐾 = (𝐾1

𝜃𝑘

, 𝐾2

𝜃𝑘

) | system (30) can be stabilized

with 𝑢 (𝑘) = 𝐾1𝜃𝑘

𝑥 (𝑘) ,

V (𝑡) = 𝐾2𝜃𝑘

𝑥 (𝑘)} .

(33)

We say that the optimization problem is well posedand the 𝑢(𝑘) and V(𝑘) have the following two additionalproperties:

𝐸 [|𝑢 (𝑘)|2

] < ∞, 𝐸 [|V (𝑘)|2] < ∞, 𝑘 ∈ N. (34)

The optimal strategies 𝑢∗ and V∗ determined by (32) arealso called the Nash equilibrium strategies (𝑢∗, V∗). In orderto guarantee the unique global Nash game solutions, both theplayers are only allowed to take constant feedback controls.Next we focus on finding the optimal strategies.

3.2.Main Results. First, we give an important lemma that willbe used later. If the system (1) is 𝑇

𝑛-stable, we can obtain the

following result for the discrete-time MJSLS (30).

Lemma 5. If [𝐴𝜃𝑘

, 𝐴𝜃𝑘

] is 𝑇𝑛-stable, then so is [𝐴

𝜃𝑘+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+

𝐶𝜃𝑘

𝐾2

𝜃𝑘

, 𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

], where (𝐾1𝜃𝑘

, 𝐾2

𝜃𝑘

) ∈ K.

Proof. Sufficiency.The proof employs an induction argumenton the stopping times 𝑇

𝑛. First, define the function

𝑉𝑘

(𝑥, 𝑖) := 𝑥

𝑘(𝑃𝑖𝛿{𝑇1>𝑘}

+ 𝐺𝑖𝛿{𝑇1=𝑘}

) 𝑥𝑘, (35)

where 𝐺𝑖> 0, and 𝑃

𝑖> 0 is the solution of

𝑝𝑖𝑖

(𝐴


𝑖𝑃𝑖𝐴𝑖) − 𝑃𝑖+ 𝑊𝑖+ 𝐴


𝑖

+ 𝐴


𝑖= 0, 𝑖 ∈ X.

(36)


The existence of such 𝑃𝑖

> 0 relies on (7). Hence, tothe function 𝑉

𝑘(𝑥, 𝑖) and the system (30) in the following

operation, we acquire that

𝐸𝑘

[𝑉𝑘+1

(𝑥 (𝑘 + 1) , 𝜃𝑘) − 𝑉𝑘

(𝑥 (𝑘) , 𝜃𝑘)]

= 𝑥 (𝑘)

{𝑝𝜃𝑘𝜃𝑘

[(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

× 𝑃𝜃𝑘

(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

+ (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

× 𝑃𝜃𝑘

(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

) ]

+ (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

𝜀𝜃𝑘

(𝐺)

× (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

+ (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

𝜀𝜃𝑘

(𝐺)

× (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

−𝑃𝜃𝑘

} 𝑥 (𝑘) 𝛿{𝑇1>𝑘}

− 𝑥 (𝑘)

𝐺𝜃𝑘

𝑥 (𝑘) 𝛿{𝑇1=𝑘}

.

(37)

Compared with (12), we know

𝐸𝑘

[𝑉𝑘+1

(𝑥 (𝑘 + 1) , 𝜃𝑘+1

) − 𝑉𝑘

(𝑥 (𝑘) , 𝜃𝑘)]

= −𝑥 (𝑘)

(𝑊𝜃𝑘𝛿{𝑇𝑛>𝑘}

+ 𝐺𝜃𝑘𝛿{𝑇𝑛=𝑘}

) 𝑥 (𝑘) ,

(38)

where

−𝑊𝜃𝑘

= −𝑃𝜃𝑘

+ 𝑝𝜃𝑘𝜃𝑘

[(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

× 𝑃𝜃𝑘

(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

+ (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

× 𝑃𝜃𝑘

(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

) ]

+ (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

𝜀𝜃𝑘

(𝐺)

× (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

+ (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

)

𝜀𝜃𝑘

(𝐺)

× (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

+ 𝐶𝜃𝑘

𝐾2

𝜃𝑘

) .

(39)

Considering that 𝑊𝜃𝑘

> 0, 𝐺𝜃𝑘

> 0, we can obtain (19).And because 𝐸[𝑉

𝑘(𝑥(𝑘), 𝜃

𝑘)] ≥ 0, ∀𝑘 ≥ 0 for each initial

condition 𝑥0, from (40), it is easy to verify (21). Therefore,

the MJSLS (30) is 𝑇𝑛-stable.

Theorem 6. For system (30), suppose the following coupledequations admit the solutions (𝐿1

𝑖, 𝐿2

𝑖; 𝐾1

𝑖, 𝐾2

𝑖) with 𝐿1

𝑖> 0,

𝐿2

𝑖> 0:

− 𝐿1

𝑖+ 𝑝𝑖𝑖

[(𝐴𝑖+ 𝐵𝑖𝐾1

𝑖)

𝐿1

𝑖(𝐴𝑖+ 𝐵𝑖𝐾1

𝑖)

+ (𝐴𝑖+ 𝐵𝑖𝐾1

𝑖)

𝐿1


𝑖)] + 𝑄

1

𝑖𝑄1

𝑖,

+ 𝐾1

𝑖𝑅1

𝑖𝐾1

𝑖− 𝐾3

𝑖𝐻1

𝑖(𝐿1

𝑖)−1

𝐾3

𝑖= 0,

𝐻1

𝑖(𝐿1

𝑖) > 0,

(40)

𝐾1

𝑖= −𝐻

2

𝑖(𝐿2

𝑖)−1

𝐾4

𝑖, (41)

− 𝐿2

𝑖+ 𝑝𝑖𝑖

[(𝐴𝑖+ 𝐶𝑖𝐾2

𝑖)

𝐿2

𝑖(𝐴𝑖+ 𝐶𝑖𝐾2

𝑖)

+ (𝐴𝑖+ 𝐶𝑖𝐾2

𝑖)

𝐿2


𝑖)] + 𝑄

2

𝑖𝑄2

𝑖

+ 𝐾2

𝑖𝑆2

𝑖𝐾2

𝑖− 𝐾4

𝑖𝐻2

𝑖(𝐿2

𝑖)−1

𝐾4

𝑖= 0,

𝐻2

𝑖(𝐿2

𝑖) > 0,

(42)

𝐾2

𝑖= −𝐻

1

𝑖(𝐿1

𝑖)−1

𝐾3

𝑖, (43)

where

𝐻1

𝑖(𝐿1

𝑖) = 𝑆1

𝑖+ 𝑝𝑖𝑖

(𝐶

𝑖𝐿1

𝑖𝐶𝑖+ 𝐶

𝑖𝐿1

𝑖𝐶𝑖) ,

𝐾3

𝑖= 𝑝𝑖𝑖

[𝐶

𝑖𝐿1


𝑖) + 𝐶

𝑖𝐿1


𝑖)] ,

𝐻2

𝑖(𝐿2

𝑖) = 𝑅2

𝑖+ 𝑝𝑖𝑖

(𝐵

𝑖𝐿2

𝑖𝐵𝑖+ 𝐵

𝑖𝐿2

𝑖𝐵𝑖) ,

𝐾4

𝑖= 𝑝𝑖𝑖

[𝐵

𝑖𝐿2


𝑖) + 𝐵

𝑖𝐿2


𝑖)] .

(44)

If (𝐴𝑖, 𝐴𝑖) is 𝑇𝑛-stable, then

(i) (𝐾1𝑖, 𝐾2

𝑖) ∈ K;

(ii) the problem of infinite horizon stochastic differentialgames admits a pair of solutions (𝑢∗(𝑘), V∗(𝑘)) with𝑢∗

(𝑘) = 𝐾1

𝑖𝑥(𝑘), V∗(𝑘) = 𝐾2

𝑖𝑥(𝑘);

(iii) the optimal cost functions incurred by playing strategies(𝑢∗

(𝑘), V∗(𝑘)) are 𝐽𝜏 = 𝑥0𝐿𝜏

𝑖𝑥0

(𝜏 = 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


of matrices 𝑊𝑖in Theorem 4, it is easy to get the following

equations for system (30):

𝑝𝑖𝑖

[(𝐴𝑖+ 𝐵𝑖𝐾1

𝑖+ 𝐶𝑖𝐾2

𝑖)

𝐿1


𝑖+ 𝐶𝑖𝐾2

𝑖)


𝑖+ 𝐶𝑖𝐾2

𝑖)

𝐿1


𝑖+ 𝐶𝑖𝐾2

𝑖)]

+ 𝑄1

𝑖𝑄1

𝑖+ 𝐾1

𝑖𝑅1

𝑖𝐾1

𝑖+ 𝐾2

𝑖𝑆1

𝑖𝐾2

𝑖= 𝐿1

𝑖,

𝑆1

𝑖+ 𝑝𝑖𝑖

(𝐶

𝑖𝐿1

𝑖𝐶𝑖+ 𝐶

𝑖𝐿1

𝑖𝐶𝑖) > 0,

(45)

𝑝𝑖𝑖

[ (𝐴𝑖+ 𝐵𝑖𝐾1

𝑖+ 𝐶𝑖𝐾2

𝑖)

𝐿2


𝑖+ 𝐶𝑖𝐾2

𝑖)


𝑖+ 𝐶𝑖𝐾2

𝑖)

𝐿2


𝑖+ 𝐶𝑖𝐾2

𝑖)]

+ 𝑄2

𝑖𝑄2

𝑖+ 𝐾1

𝑖𝑅2

𝑖𝐾1

𝑖+ 𝐾2

𝑖𝑆2

𝑖𝐾2

𝑖= 𝐿2

𝑖,

𝑅2

𝑖+ 𝑝𝑖𝑖

(𝐵

𝑖𝐿2

𝑖𝐵𝑖+ 𝐵

𝑖𝐿2

𝑖𝐵𝑖) > 0.

(46)

By rearranging (45) and (46), (40) and (42) can be obtained,respectively.

Noting 𝑢∗(𝑘) = 𝐾1𝑖𝑥(𝑘), and by substituting 𝑢∗(𝑘) into

(30), it is easy to get the following system:

𝑥 (𝑘 + 1) = (𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

) 𝑥 (𝑘) + 𝐶𝜃𝑘V (𝑘)

+ [(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

) 𝑥 (𝑘) + 𝐶𝜃𝑘V (𝑘)] 𝑤 (𝑘) ,

𝑥 (0) = 𝑥0

∈ R𝑛

, 𝑘 ∈ N.

(47)

Then, considering the scalar function 𝑍(𝑥𝑘) := 𝑥

𝑘𝐿1

𝜃𝑘

𝑥𝑘, we

have

𝐸𝑘

[Δ𝑍 (𝑥𝑘)]

= 𝐸𝑘

[𝑍 (𝑥𝑘+1

) − 𝑍 (𝑥𝑘)]

= 𝐸𝑘

[𝑥

𝑘+1𝐿1

𝜃𝑘+1

𝑥𝑘+1

− 𝑥

𝑘𝐿1

𝜃𝑘

𝑥𝑘]

= 𝐸𝑘

{−𝑥

𝑘𝐿1

𝜃𝑘

𝑥𝑘

+ 𝑝𝑖𝑖

[(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

) 𝑥𝑘

+ 𝐶𝜃𝑘V𝑘]

× 𝐿1

𝜃𝑘

[(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

) 𝑥𝑘


+ 𝑝𝑖𝑖

[(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

) 𝑥𝑘


× 𝐿1

𝜃𝑘

[(𝐴𝜃𝑘

+ 𝐵𝜃𝑘

𝐾1

𝜃𝑘

) 𝑥𝑘

+ 𝐶𝜃𝑘V𝑘] } .

(48)

Due to∞

∑

𝑘=0

𝐸𝑘

[Δ𝑍 (𝑥𝑘)]

= 𝐸𝑘

[

∞

∑

𝑘=0

Δ𝑍 (𝑥𝑘)] = 𝐸

𝑘[𝑍 (𝑥∞

) − 𝑍 (𝑥0)] = −𝑥

0𝐿1

𝑖𝑥0,

(49)

by (40) and a completing squares technique, (31) can bederived that

𝐽1

(𝑢∗

, V)

=

∞

∑

𝑘=0

𝐸𝑘

[𝑥

𝑘(𝑄1

𝜃𝑘

𝑄1

𝜃𝑘

+ 𝐾1

𝜃𝑘

𝑅1

𝜃𝑘

𝐾1

𝜃𝑘

) 𝑥𝑘

+ V𝑘𝑆1

𝜃𝑘

V𝑘]

+

∞

∑

𝑘=0

𝐸𝑘

[Δ𝑍 (𝑥𝑘)] + 𝑥

0𝐿1

𝑖𝑥0

= 𝑥

0𝐿1

𝑖𝑥0

+

∞

∑

𝑘=0

𝐸𝑘

{𝑥

𝑘[−𝐿1

𝑖+ 𝑝𝑖𝑖

(𝐴𝑖+ 𝐵𝑖𝐾1

𝑖)

× 𝐿1


𝑖) + 𝑝𝑖𝑖


𝑖)

× 𝐿1


𝑖) + 𝑄1

𝑖𝑄1

𝑖

+ 𝐾1

𝑖𝑅1

𝑖𝐾1

𝑖] 𝑥𝑘

+ 𝑥

𝑘𝐾3

𝑖V𝑘

+ V𝑘𝐾3

𝑖𝑥𝑘

+ V𝑘

(𝑆1

𝑖+ 𝑝𝑖𝑖𝐶

𝑖𝐿1

𝑖𝐶𝑖+ 𝑝𝑖𝑖𝐶

𝑖𝐿1

𝑖𝐶𝑖) V𝑘}

= 𝑥

0𝐿1

𝑖𝑥0

+

∞

∑

𝑘=0

𝐸𝑘

[𝑥

𝑘𝐾3

𝑖𝐻1

𝑖(𝐿1

𝑖)−1

𝐾3

𝑖𝑥𝑘

+ 𝑥

𝑘𝐾3

𝑖V𝑘

+ V𝑘𝐾3

𝑖𝑥𝑘

+ V𝑘𝐻1

𝑖(𝐿1

𝑖) V𝑘]

= 𝑥

0𝐿1

𝑖𝑥0

+

∞

∑

𝑘=0

𝐸𝑘

{[V (𝑘) − 𝐾2𝑖𝑥 (𝑘)]

𝐻1

𝑖(𝐿1

𝑖) [V (𝑘) − 𝐾2

𝑖𝑥 (𝑘)]}

≥ 𝑥

0𝐿1

𝑖𝑥0, 𝜏 = 1.

(50)

Then by (32), it follows that V∗(𝑘) = 𝐾2𝑖𝑥(𝑘) and 𝐽1(𝑢∗, V∗) =

𝑥

0𝐿1

𝑖𝑥0. Finally, by substituting V∗(𝑘) into (30), in the same

way as before, we have 𝑢∗(𝑘) = 𝐾1𝑖𝑥(𝑘) and 𝐽2(𝑢∗, V∗) =

𝑥

0𝐿2

𝑖𝑥0.

Theorem 7. If (𝐴𝑖, 𝐴𝑖) is 𝑇

𝑛-stable, and, for system (30),

assume that (40)–(43) admit the solution (𝐿1𝑖, 𝐿2

𝑖; 𝐾1

𝑖, 𝐾2

𝑖) with

(𝐾1

𝑖, 𝐾2

𝑖) ∈ K, then

(i) 𝐿1𝑖

> 0, 𝐿2𝑖

> 0,(ii) the problem of infinite horizon stochastic differential

games admits a pair of solutions (𝑢∗(𝑘), V∗(𝑘)) with𝑢∗

(𝑘) = 𝐾1

𝑖𝑥(𝑘), V∗(𝑘) = 𝐾2

𝑖𝑥(𝑘),

(iii) the optimal cost functions incurred by playing strategies(𝑢∗

(𝑘), V∗(𝑘)) are 𝐽𝜏 = 𝑥0𝐿𝜏

𝑖𝑥0

(𝜏 = 1, 2).

Remark 8. When 𝑤(𝑘) ≡ 0, these results still hold inthe paper. Only for the reason of simplicity, in (1) and(30), we assume the state 𝑥(𝑡) and control inputs (𝑢(𝑡), V(𝑡))depend on the same noise 𝑤(𝑘). If they rely on the different


noises (𝑤1(𝑘), 𝑤

2(𝑘)), then new results will be yielded. The

discussion is omitted.

4. Iterative Algorithm and Simulation

4.1. An Iterative Algorithm. In this section, an iterative algo-rithm is proposed to solve the four coupled GAREs (40)–(43). Infinite horizon Riccati equations are hard to be solved;hence the particular problems can be solved via finite horizonequations. 𝑁 represents the finite number of iterations in thefollowing equations:

𝐿1

𝑖

𝑁

(𝑘) = 𝑝𝑖𝑖


𝑖

𝑁

(𝑘))

𝐿1

𝑖

𝑁

(𝑘 + 1)

× (𝐴𝑖+ 𝐵𝑖𝐾1

𝑖

𝑁

(𝑘)) + 𝑝𝑖𝑖


𝑖

𝑁

(𝑘))

× 𝐿1

𝑖

𝑁

(𝑘 + 1) (𝐴𝑖+ 𝐵𝑖𝐾1

𝑖

𝑁

(𝑘))

+ 𝑄1

𝑖𝑄1

𝑖+ 𝐾1

𝑖

𝑁

(𝑘)

𝑅1

𝑖𝐾1

𝑖

𝑁

(𝑘)

− 𝐾3

𝑖

𝑁

(𝑘)

𝐻1

𝑖(𝐿1

𝑖

𝑁

(𝑘 + 1))

−1

𝐾3

𝑖

𝑁

(𝑘) ,

𝐿1

𝑖

𝑁

(𝑘 + 1) = 0,

𝐻1

𝑖(𝐿1

𝑖

𝑁

(𝑘 + 1)) > 0,

(51)

𝐾1

𝑖

𝑁

(𝑘) = −𝐻2

𝑖(𝐿2

𝑖

𝑁

(𝑘 + 1))

−1

𝐾4

𝑖

𝑁

(𝑘) , (52)

𝐿2

𝑖

𝑁


(𝐴𝑖+ 𝐶𝑖𝐾2

𝑖

𝑁

(𝑘))

𝐿2

𝑖

𝑁

(𝑘 + 1)

× (𝐴𝑖+ 𝐶𝑖𝐾2

𝑖

𝑁

(𝑘)) + 𝑝𝑖𝑖

(𝐴𝑖+ 𝐶𝑖𝐾2

𝑖

𝑁

(𝑘))

× 𝐿2

𝑖

𝑁

(𝑘 + 1) (𝐴𝑖+ 𝐶𝑖𝐾2

𝑖

𝑁

(𝑘))

+ 𝑄2

𝑖𝑄2

𝑖+ 𝐾2

𝑖

𝑁

(𝑘)

𝑆2

𝑖𝐾2

𝑖

𝑁

(𝑘)

− 𝐾4

𝑖

𝑁

(𝑘)

𝐻2

𝑖(𝐿2

𝑖

𝑁

(𝑘 + 1))

−1

𝐾4

𝑖

𝑁

(𝑘) ,

𝐿2

𝑖

𝑁

(𝑘 + 1) = 0,

𝐻2

𝑖(𝐿2

𝑖

𝑁

(𝑘 + 1)) > 0,

(53)

𝐾2

𝑖

𝑁

(𝑘) = −𝐻1

𝑖(𝐿1

𝑖

𝑁

(𝑘 + 1))

−1

𝐾3

𝑖

𝑁

(𝑘) , (54)

where

𝐻1

𝑖(𝐿1

𝑖

𝑁

(𝑘 + 1))

= 𝑆1

𝑖+ 𝑝𝑖𝑖

(𝐶

𝑖𝐿1

𝑖

𝑁

(𝑘 + 1) 𝐶𝑖

+ 𝐶

𝑖𝐿1

𝑖

𝑁

(𝑘 + 1) 𝐶𝑖) ,

𝐻2

𝑖(𝐿2

𝑖

𝑁

(𝑘 + 1))

= 𝑅2

𝑖+ 𝑝𝑖𝑖

(𝐵

𝑖𝐿2

𝑖

𝑁

(𝑘 + 1) 𝐵𝑖

+ 𝐵

𝑖𝐿2

𝑖

𝑁

(𝑘 + 1) 𝐵𝑖) ,

𝐾3

𝑖

𝑁


[𝐶

𝑖𝐿1

𝑖

𝑁

(𝑘 + 1) (𝐴𝑖+ 𝐵𝑖𝐾1

𝑖

𝑁

(𝑘 + 1))

+ 𝐶

𝑖𝐿1

𝑖

𝑁

(𝑘 + 1) (𝐴𝑖+ 𝐵𝑖𝐾1

𝑖

𝑁

(𝑘 + 1))] ,

𝐾4

𝑖

𝑁


[𝐵

𝑖𝐿2

𝑖

𝑁

(𝑘 + 1) (𝐴𝑖+ 𝐶𝑖𝐾2

𝑖

𝑁

(𝑘 + 1))

+𝐵

𝑖𝐿2

𝑖

𝑁

(𝑘 + 1) (𝐴𝑖+ 𝐶𝑖𝐾2

𝑖

𝑁

(𝑘 + 1))] .

(55)

An iterative process for solving (40)–(43) based on the aboverecursions is presented as follows.

(a) Given appropriate natural number 𝑁 and the initialconditions 𝐿1

𝑖

𝑁

(𝑁+1) = 0, 𝐿2𝑖

𝑁

(𝑁+1) = 0, 𝐾1𝑖

𝑁

(𝑁+

1) = 0, and 𝐾2𝑖

𝑁

(𝑁 + 1) = 0.

(b) Through the numerical values of 𝐿1𝑖

𝑁

(𝑁+1), 𝐿2𝑖

𝑁

(𝑁+

1),𝐾1𝑖

𝑁

(𝑁+1), and𝐾2𝑖

𝑁

(𝑁+1), we have𝐻1𝑖(𝐿1

𝑖

𝑁

(𝑁+

1)),𝐻2𝑖(𝐿2

𝑖

𝑁

(𝑁+1)),𝐾3𝑖

𝑁

(𝑁), and𝐾4𝑖

𝑁

(𝑁) accordingto (55).

(c) 𝐾1𝑖

𝑁

(𝑁) and 𝐾2𝑖

𝑁

(𝑁) can be, respectively, computedby (52) and (54). Then, 𝐿1

𝑖

𝑁

(𝑁) and 𝐿2𝑖

𝑁

(𝑁) can alsobe, respectively, obtained by (51) and (53).

(d) Let 𝐿1𝑖

𝑁

(𝑁 + 1) = 𝐿1

𝑖

𝑁

(𝑁), 𝐿2𝑖

𝑁

(𝑁 + 1) = 𝐿2

𝑖

𝑁

(𝑁),𝐾1

𝑖

𝑁

(𝑁 + 1) = 𝐾1

𝑖

𝑁

(𝑁), and 𝐾2𝑖

𝑁

(𝑁 + 1) = 𝐾2

𝑖

𝑁

(𝑁).

(e) Then, 𝑁 := 𝑁 − 1. Repeat steps (b)–(d) until thenumber of iterations is 𝑁 + 1. We can finally obtainthe numerical values of 𝐿1

𝑖

𝑁

(0), 𝐿2𝑖

𝑁

(0), 𝐾1𝑖

𝑁

(0), and𝐾2

𝑖

𝑁

(0).

As in [28], under the assumptions of stabilizability, for any𝑥0

∈ R𝑛,

lim𝑁→∞

𝑥

0𝐿1

𝑖

𝑁

(0) 𝑥0

= lim𝑁→∞

min 𝐽1𝑁

(𝑢∗

𝑁, V) = min 𝐽1

∞

(𝑢∗

, V) = 𝑥0𝐿1

𝑖𝑥0,

lim𝑁→∞

𝑥

0𝐿2

𝑖

𝑁

(0) 𝑥0

= lim𝑁→∞

min 𝐽2𝑁

(𝑢, V∗𝑁

) = min 𝐽2∞

(𝑢, V∗) = 𝑥0𝐿2

𝑖𝑥0,

lim𝑁→∞

𝐾1

𝑖

𝑁

(0) = 𝐾1

𝑖, lim

𝑁→∞

𝐾2

𝑖

𝑁

(0) = 𝐾2

𝑖.

(56)


Therefore,

lim𝑁→∞

(𝐿1

𝑖

𝑁

(0) , 𝐿2

𝑖

𝑁

(0) ; 𝐾1

𝑖

𝑁

(0) , 𝐾2

𝑖

𝑁

(0))

= (𝐿1

𝑖, 𝐿2

𝑖; 𝐾1

𝑖, 𝐾2

𝑖) ,

(57)

where (𝐿1𝑖, 𝐿2

𝑖; 𝐾1

𝑖, 𝐾2

𝑖) are the solutions of (40)–(43).

4.2. A Simulation Example. To verify the efficiency of theabove iterative algorithm, we consider the following 2-Dexample. In the system (30), we set 𝜃

𝑘= 𝑖 ∈ X = {1, 2},

𝑅𝜏

𝑖= 𝑆𝜏

𝑖= 1 (𝜏 = 1, 2),

𝐴1

= [0.65 0

0 0.9] , 𝐴

1= [

0.45 0

0 0.55] ,

𝐵1

= [0.6

0.55] , 𝐵

1= [

0.45

0.85] ,

𝐶1

= [0.75

0.55] , 𝐶

1= [

0.5

0.85] ,

𝑄1

1= [

0.55 0

0 0.65] , 𝑄

2

1= [

0.75 0

0 0.25] ,

𝐴2

= [0.75 0

0 0.7] , 𝐴

2= [

0.35 0

0 0.45] ,

𝐵2

= [0.5

0.45] , 𝐵

2= [

0.55

0.85] ,

𝐶2

= [0.65

0.55] , 𝐶

2= [

0.4

0.85] ,

𝑄1

2= [

0.35 0

0 0.45] , 𝑄

2

2= [

0.55 0

0 0.35] .

(58)

For convenience, let 𝑝11

= 0.4, 𝑝22

= 0.5, and 𝑁 = 50.When 𝜃

𝑘= 1, by applying the above iterative algorithm, we

obtain the solutions of the four coupled equations (51)–(54)as follows:

𝐿1

1

𝑁

(0) = [𝐿1

1(1, 1) 𝐿

1

1(1, 2)

𝐿1

1(2, 1) 𝐿

1

1(2, 2)

] = [0.4023 −0.0588

−0.0588 0.6820] ,

𝐿2

1

𝑁

(0) = [𝐿2

1(1, 1) 𝐿

2

1(1, 2)

𝐿2

1(2, 1) 𝐿

2

1(2, 2)

] = [0.7111 −0.0331

−0.0331 0.1487] ,

𝐾1

1

𝑁

(0) = [𝐾1

1(1, 1) 𝐾

1

1(1, 2)] = [−0.1245 −0.0053] ,

𝐾2

1

𝑁

(0) = [𝐾2

1(1, 1) 𝐾

2

1(1, 2)] = [−0.0390 −0.1739] .

(59)

(𝐿1

1

𝑁

(0), 𝐿2

1

𝑁

(0); 𝐾1

1

𝑁

(0), 𝐾2

1

𝑁

(0)) are also the solutionsof (40)–(43) according to (57). By the solutions, itshows that 𝐿1

1> 0 and 𝐿2

1> 0. The evolution of

(𝐿1

1

𝑁

(𝑘), 𝐿2

1

𝑁

(𝑘); 𝐾1

1

𝑁

(𝑘), 𝐾2

1

𝑁

(𝑘)) is exhibited in Figures 1and 2. And the figures clearly illustrate the convergence andspeediness of the backward iterations. When 𝜃

𝑘= 2, it is easy

0 10 20 30 40 50−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

N

L11(1, 1)

L11(2, 1)

L11(2, 2)

L21(1, 1)

L21(2, 1)

L21(2, 2)

Figure 1: Evolution of 𝐿11

𝑁

(𝑘) and 𝐿21

𝑁

(𝑘).

0 10 20 30 40 50−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

N

K11(1, 1)

K11(1, 2)

K21(1, 1)

K21(1, 2)

Figure 2: Evolution of 𝐾11

𝑁

(𝑘) and 𝐾21

𝑁

(𝑘).

to get (𝐿12

𝑁

(0), 𝐿2

2

𝑁

(0); 𝐾1

2

𝑁

(0), 𝐾2

2

𝑁

(0)) that are also thesolutions of (40)–(43). And 𝐿1

2> 0 and 𝐿2

2> 0. Because it is

the same as the above process (𝜃𝑘

= 1), we do not introduceit again due to space limitations.

5. Conclusions

In this paper we have discussed the 𝑇𝑛-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


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 and𝑊-observability fordiscrete-timeMJSLS are investigated by [29, 30]. On the basisof exact observability and 𝑊-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).

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