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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;
[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
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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)
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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
𝛿{𝑇𝑛≥𝑘}
] = 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
𝑖𝜀𝑖(𝐺) 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
𝑥𝑘𝛿{𝑇1=𝑘}
.
(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
(𝑥𝑘, 𝜃𝑘)]
= 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
𝑥𝑘𝛿{𝑇1=𝑘}
,
(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
(𝑥𝑘, 𝜃𝑘)]
= −𝑥
𝑘(𝑊𝜃𝑘𝛿{𝑇1>𝑘}
+ 𝐺𝜃𝑘𝛿{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)
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4 Mathematical Problems in Engineering
Now, let us observe that𝑁
∑
𝑘=0
𝐸0
[𝑉𝑘+1
(𝑥𝑘+1
, 𝜃𝑘+1
) − 𝑉𝑘
(𝑥𝑘, 𝜃𝑘)]
=
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}
+ 𝐺𝜃𝑘𝛿{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
𝛿{𝑇𝑛≥𝑘}
] < ∞.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
𝑥𝑘𝛿{𝑇𝑛+1≥𝑘}
| 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
𝑥𝑘𝛿{𝑇𝑛+1≥𝑘}
]
< lim sup𝑁→∞
𝑁
∑
𝑘=0
𝐸 [𝑥
𝑘𝑄𝜃𝑘
𝑥𝑘𝛿{𝑇𝑛>𝑘}
+ 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
𝛿{𝑇𝑛≥𝑘}
] < ∞ (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
𝑥𝑘𝛿{𝑇1>𝑘}
+ 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
𝑥𝑘𝛿{𝑇1>𝑘}
+ 𝑥
𝑇1
𝐺𝜃𝑇1
𝑥𝑇1
) 𝛿{𝑇1≥1}
]} .
(24)
In addition,
𝐸𝑇1
[(
∞
∑
𝑘=1
𝑥
𝑘𝑊𝜃𝑘
𝑥𝑘𝛿{𝑇1>𝑘}
+ 𝑥
𝑇1
𝐺𝜃𝑇1
𝑥𝑇1
) 𝛿{𝑇1≥1}
]
= 𝐸𝑇1
[(
∞
∑
𝑘=1
𝑥
𝑘𝑊𝜃𝑘
𝑥𝑘𝛿{𝑇1>𝑘}
+ 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)
-
Mathematical Problems in Engineering 5
Since 𝑥0is arbitrary, and calculating the expected values
above, (26) implies that
𝑝𝑖𝑖
(𝐴
𝑖𝑃𝑖𝐴𝑖+ 𝐴
𝑖𝑃𝑖𝐴𝑖) − 𝑃𝑖+ 𝐴
𝑖𝜀𝑖(𝐺) 𝐴
𝑖+ 𝐴
𝑖𝜀𝑖(𝐺) 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
𝑥𝑘𝛿{𝑇𝑛>𝑘}
+ 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+ 𝐴
𝑖𝑃𝑖𝐴𝑖) − 𝑃𝑖+ 𝑊𝑖+ 𝐴
𝑖𝜀𝑖(𝐺) 𝐴
𝑖
+ 𝐴
𝑖𝜀𝑖(𝐺) 𝐴
𝑖= 0, 𝑖 ∈ X.
(36)
-
6 Mathematical Problems in Engineering
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(𝐴𝑖+ 𝐵𝑖𝐾1
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(𝐴𝑖+ 𝐶𝑖𝐾2
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(𝐴𝑖+ 𝐵𝑖𝐾1
𝑖) + 𝐶
𝑖𝐿1
𝑖(𝐴𝑖+ 𝐵𝑖𝐾1
𝑖)] ,
𝐻2
𝑖(𝐿2
𝑖) = 𝑅2
𝑖+ 𝑝𝑖𝑖
(𝐵
𝑖𝐿2
𝑖𝐵𝑖+ 𝐵
𝑖𝐿2
𝑖𝐵𝑖) ,
𝐾4
𝑖= 𝑝𝑖𝑖
[𝐵
𝑖𝐿2
𝑖(𝐴𝑖+ 𝐶𝑖𝐾2
𝑖) + 𝐵
𝑖𝐿2
𝑖(𝐴𝑖+ 𝐶𝑖𝐾2
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
-
Mathematical Problems in Engineering 7
of matrices 𝑊𝑖in Theorem 4, it is easy to get the following
equations for system (30):
𝑝𝑖𝑖
[(𝐴𝑖+ 𝐵𝑖𝐾1
𝑖+ 𝐶𝑖𝐾2
𝑖)
𝐿1
𝑖(𝐴𝑖+ 𝐵𝑖𝐾1
𝑖+ 𝐶𝑖𝐾2
𝑖)
+ (𝐴𝑖+ 𝐵𝑖𝐾1
𝑖+ 𝐶𝑖𝐾2
𝑖)
𝐿1
𝑖(𝐴𝑖+ 𝐵𝑖𝐾1
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(𝐴𝑖+ 𝐵𝑖𝐾1
𝑖+ 𝐶𝑖𝐾2
𝑖)
+ (𝐴𝑖+ 𝐵𝑖𝐾1
𝑖+ 𝐶𝑖𝐾2
𝑖)
𝐿2
𝑖(𝐴𝑖+ 𝐵𝑖𝐾1
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
+ 𝐶𝜃𝑘V𝑘]
+ 𝑝𝑖𝑖
[(𝐴𝜃𝑘
+ 𝐵𝜃𝑘
𝐾1
𝜃𝑘
) 𝑥𝑘
+ 𝐶𝜃𝑘V𝑘]
× 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(𝐴𝑖+ 𝐵𝑖𝐾1
𝑖) + 𝑝𝑖𝑖
(𝐴𝑖+ 𝐵𝑖𝐾1
𝑖)
× 𝐿1
𝑖(𝐴𝑖+ 𝐵𝑖𝐾1
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
-
8 Mathematical Problems in Engineering
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
(𝐴𝑖+ 𝐵𝑖𝐾1
𝑖
𝑁
(𝑘))
𝐿1
𝑖
𝑁
(𝑘 + 1)
× (𝐴𝑖+ 𝐵𝑖𝐾1
𝑖
𝑁
(𝑘)) + 𝑝𝑖𝑖
(𝐴𝑖+ 𝐵𝑖𝐾1
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
(𝑘) = 𝑝𝑖𝑖
(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
(𝑘) = 𝑝𝑖𝑖
[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
(𝑘) = 𝑝𝑖𝑖
[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)
-
Mathematical Problems in Engineering 9
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
-
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 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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