# Nonlinear StochasticH

Authors: Li Sheng; Meijun Zhu; Weihai Zhang; Yuhong Wang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

This paper is concerned with theH
∞ control problem for nonlinear stochastic Markov jump systems with state, control, and external disturbance-dependent noise. By means of inequality techniques and coupled Hamilton-Jacobi inequalities, both finite and infinite horizon H
∞ control designs of such systems are developed. Two numerical examples are provided to illustrate the effectiveness of the proposed design method.


---

## Body

## 1. Introduction

H
∞ control is one of the most important robust control approaches, which can efficiently eliminate the effect of the exogenous disturbance [1, 2]. Since Hinrichsen and Pritchard introduced H
∞ control to linear stochastic systems [3], the nonlinear stochastic H
∞ control and filtering problems have received considerable attention in both theory and practical applications [4–9]. In [4], the nonlinear stochastic H
∞ designs were first developed by solving a second-order nonlinear Hamilton-Jacobi inequality. The H
∞ filtering problems for general nonlinear continuous-time and discrete-time stochastic systems were discussed in [6] and [7], respectively. In [8], the quantized H
∞ control problem for a class of nonlinear stochastic time-delay network-based systems with probabilistic data missing is studied.On the other hand, Itô stochastic systems with Markov jumps have attracted increasing attention due to their powerful modeling ability in many fields [10, 11]. For linear stochastic systems with Markov jumps, many important issues have been studied, such as stability and stabilization [12, 13], observability and detectability [14], optimal control [15], and H
2
/
H
∞ control [16]. The control issues for nonlinear stochastic Markov jump systems (NSMJSs) have also been widely investigated. In [17], the notion of exponential dissipativity of NSMJSs was introduced, and it was used to estimate the possible variations of the output feedback control. In [18], the stabilization of nonlinear Markov jump systems with partly unknown transition probabilities was studied via fuzzy control. In [19], the H
∞ control problems of NSMJSs were studied, which extended the results of [4] to stochastic systems with Markovian jump parameters.Most of the existing literature was concerned with stochastic Markov jump systems with state-dependent noise or both state and disturbance dependent noise ((
x
,
v
)-dependent noise for short) [16, 19]. However, for most natural phenomena described by Itô stochastic systems, not only state but also control input or external disturbance maybe corrupted by noise. By introducing three coupled Hamilton-Jacobi equations (HJEs), the finite and infinite horizon H
∞ control problems were solved for Itô stochastic systems with all system state, control, and disturbance-dependent noise ((
x
,
u
,
v
)-dependent noise for short) in [20] and [21], respectively. In [22], the finite/infinite horizon H
∞ control of nonlinear stochastic systems with (
x
,
u
,
v
)-dependent noise was solved by means of a Hamilton-Jacobi inequality (HJI) instead of three coupled Hamilton-Jacobi equations (HJEs). However, the control problems of nonlinear stochastic systems with Markov jumps and (
x
,
u
,
v
)-dependent noise have never been tackled and deserved further research.In this paper, the finite and infinite horizonH
∞ control problems are studied for nonlinear stochastic Markov jumps systems with (
x
,
u
,
v
)-dependent noise. Firstly, a very useful elementary identity is proposed. Then, by using the technique of squares completion, a sufficient condition for finite/infinite horizon H
∞ control of NSMJSs is presented based on a set of coupled HJIs. By means of linear matrix inequalities (LMIs), a sufficient condition for infinite horizon H
∞ control of linear stochastic Markov jump systems is derived. Finally, two numerical examples are provided to show the effectiveness of our obtained results.For conveniences, we make use of the following notations throughout this paper.R
n is the n-dimensional Euclidean space. R
n
×
m is the set of all n
×
m real matrices. A
>
0
(
A
≥
0
): A is a positive definite (positive semi-definite) symmetric matrix. A
′ is the transpose of a matrix A. I is the identity matrix. ∥
x
∥ is the Euclidean norm of a vector x. L
F
2
(
[
0
,
T
]
;
R
l
) (resp., L
F
2
(
R
+
;
R
l
)) is the space of nonanticipative stochastic processes y
(
t
)
∈
R
l with respect to increasing σ-algebras F
t
(
t
≥
0
) satisfying ∥
y
(
t
)
∥
L
F
2
(
[
0
,
T
]
;
R
l
)
=
(
E
∫
0
T
‍
∥
y
(
t
)
∥
2
d
t
)
1
/
2
<
∞ (resp., ∥
y
(
t
)
∥
L
F
2
(
R
+
;
R
l
)
=
(
E
∫
0
∞
‍
∥
y
(
t
)
∥
2
d
t
)
1
/
2
<
∞).


## 2. Definitions and Preliminaries

Consider the following time-varying nonlinear stochastic Markov jump systems with(
x
,
u
,
v
)-dependent noise:
(1)
d
x
(
t
)
=
[
f
(
t
,
x
,
r
t
)
+
g
(
t
,
x
,
r
t
)
u
(
t
)
+
h
(
t
,
x
,
r
t
)
v
(
t
)
]
d
t
+
[
l
(
t
,
x
,
r
t
)
+
q
(
t
,
x
,
r
t
)
u
(
t
)
D
S
F
D
+
s
(
t
,
x
,
r
t
)
v
(
t
)
]
d
w
(
t
)
,
z
(
t
)
=
col
(
m
(
t
,
x
,
r
t
)
,
u
(
t
)
)
:
=
[
m
(
t
,
x
,
r
t
)
u
(
t
)
]
,
x
(
0
)
=
x
0
∈
R
n
,
where x
(
t
)
∈
R
n, u
(
t
)
∈
R
n
u, v
(
t
)
∈
R
n
v, and z
(
t
)
∈
R
n
z represent the system state, control input, exogenous input, and regulated output, respectively. w
(
t
) is the one-dimensional standard Wiener process defined on the complete probability space (
Ω
,
F
,
P
), with the natural filter F
t generated by w
(
·
) and r
(
·
) up to time t. The jumping process {
r
t
,
t
≥
0
} is a continuous-time discrete-state Markov process taking values in a finite set T
=
{
1
,
…
,
N
}. The transition probabilities for the process r
t are defined as
(2)
P
(
r
t
+
h
=
j
∣
r
t
=
k
)
=
{
π
k
l
h
+
o
(
h
)
,
if
k
≠
j
,
1
+
π
k
k
h
+
o
(
h
)
,
if
k
=
j
,
where h
>
0, lim
⁡
h
→
0
(
o
(
h
)
/
h
)
=
0, and π
k
l
≥
0
(
k
,
j
∈
T
,
k
≠
j
) denotes the switching rate from mode k at time t to mode j at time t
+
h and π
k
k
=
-
∑
j
=
1
,
j
≠
k
‍
π
k
j for all k
∈
T. In this paper, the processes r
t and w
(
t
) are assumed to be independent. For every r
t
=
k
∈
T, f
(
t
,
x
,
k
), g
(
t
,
x
,
k
), h
(
t
,
x
,
k
), l
(
t
,
x
,
k
), q
(
t
,
x
,
k
), s
(
t
,
x
,
k
), and m
(
t
,
x
,
k
) are Borel measurable functions of suitable dimensions, which guarantee that system (1) has a unique strong solution [23]. The finite horizon H
∞ control for system (1) is defined as follows.Definition 1.
For givenγ
>
0, the state feedback control u
T
*
(
t
)
∈
L
F
2
(
[
0
,
T
]
;
R
n
u
) is called a finite horizon H
∞ control of system (1), if for the zero initial state and any nonzero v
(
t
)
∈
L
F
2
(
[
0
,
T
]
;
R
n
v
), we have ∥
L
T
∥
<
γ with
(3)
∥
L
T
∥
=
sup
⁡
v
(
t
)
∈
L
F
2
(
[
0
,
T
]
;
R
n
v
)
,
v
(
t
)
≠
0
,
u
(
t
)
=
u
T
*
(
t
)
,
x
0
=
0
∥
z
(
t
)
∥
L
F
2
(
[
0
,
T
]
;
R
n
z
)
∥
v
(
t
)
∥
L
F
2
(
[
0
,
T
]
;
R
n
v
)
=
sup
⁡
v
(
t
)
∈
L
F
2
(
[
0
,
T
]
;
R
n
v
)
,
v
(
t
)
≠
0
,
u
(
t
)
=
u
T
*
(
t
)
,
x
0
=
0
E
{
∫
0
T
‍
(
∥
m
(
t
,
x
,
r
t
)
∥
2
f
d
s
f
d
s
f
d
f
d
s
f
h
h
s
d
f
d
d
d
d
d
+
∥
u
T
*
(
t
)
∥
2
)
d
t
∣
r
0
=
k
∫
0
T
‍
}
1
/
2
f
s
d
f
d
s
f
d
s
f
d
s
f
d
s
f
d
f
d
×
(
=
k
∫
0
T
‍
}
1
/
2
E
{
∫
0
T
‍
∥
v
(
t
)
∥
2
d
t
∣
r
0
f
d
s
f
d
f
d
s
f
d
s
f
d
s
f
d
f
d
s
f
s
d
d
d
d
=
k
∫
0
T
‍
}
1
/
2
)
-
1
,
f
d
s
f
d
f
d
s
f
d
s
f
d
f
s
d
f
d
s
d
f
d
f
d
f
d
d
d
f
d
i
∈
T
,
where L
T is an operator associated with system (1) which is defined as
(4)
L
T
:
L
F
2
(
[
0
,
T
]
;
R
n
v
)
⟼
L
F
2
(
[
0
,
T
]
;
R
n
z
)
,
L
T
(
v
(
t
)
)
=
z
(
t
)
|
x
0
=
0
,
t
∈
[
0
,
T
]
.Consider the time-invariant nonlinear stochastic Markov jump systems with(
x
,
u
,
v
)-dependent noise
(5)
d
x
(
t
)
=
[
f
(
x
,
r
t
)
+
g
(
x
,
r
t
)
u
(
t
)
+
h
(
x
,
r
t
)
v
(
t
)
]
d
t
+
[
l
(
x
,
r
t
)
+
q
(
x
,
r
t
)
u
(
t
)
+
s
(
x
,
r
t
)
v
(
t
)
]
d
w
(
t
)
,
z
(
t
)
=
col
(
m
(
x
,
r
t
)
,
u
(
t
)
)
:
=
[
m
(
x
,
r
t
)
u
(
t
)
]
,
x
(
0
)
=
x
0
∈
R
n
.
The infinite horizon H
∞ control for system (5) is defined as follows.Definition 2.
For givenγ
>
0, the control u
∞
*
(
t
)
∈
L
F
2
(
R
+
;
R
n
u
) is called an infinite horizon H
∞ control of system (1), if the following is considered.(i)
For the zero initial state and any nonzerov
(
t
)
∈
L
F
2
(
R
+
;
R
n
v
), we have ∥
L
∞
∥
<
γ with
(6)
∥
L
∞
∥
=
sup
⁡
v
(
t
)
∈
L
F
2
(
R
+
;
R
n
v
)
,
v
(
t
)
≠
0
,
u
(
t
)
=
u
∞
*
(
t
)
,
x
0
=
0
∥
z
(
t
)
∥
L
F
2
(
R
+
;
R
n
z
)
∥
v
(
t
)
∥
L
F
2
(
R
+
;
R
n
v
)
=
sup
⁡
v
(
t
)
∈
L
F
2
(
R
+
;
R
n
v
)
,
v
(
t
)
≠
0
,
u
(
t
)
=
u
∞
*
(
t
)
,
x
0
=
0
E
{
∫
0
∞
‍
(
∥
m
(
x
,
r
t
)
∥
2
G
F
D
G
F
D
G
F
G
F
F
F
F
F
F
F
F
F
F
+
∥
u
∞
*
(
t
)
∥
2
)
d
t
∣
r
0
=
k
∫
0
∞
‍
}
1
/
2
f
s
d
f
d
f
s
d
f
d
f
d
f
d
d
d
d
d
×
(
=
k
∫
0
∞
‍
}
1
/
2
E
{
∫
0
∞
‍
∥
v
(
t
)
∥
2
d
t
∣
r
0
f
s
d
f
s
d
f
d
s
f
s
d
f
d
f
d
d
d
d
d
d
d
d
d
=
k
∫
0
∞
‍
}
1
/
2
)
-
1
,
i
∈
T
,
where L
∞ is an operator associated with system (5) which is defined as
(7)
L
∞
:
L
F
2
(
R
+
;
R
n
v
)
⟼
L
F
2
(
R
+
;
R
n
z
)
,
L
∞
(
v
(
t
)
)
=
z
(
t
)
∣
x
0
=
0
,
t
∈
R
+
.
(ii)
System (5) is internally stable; that is, the following system
(8)
d
x
(
t
)
=
[
f
(
x
,
r
t
)
+
g
(
x
,
r
t
)
u
∞
*
(
x
,
r
t
)
]
d
t
+
[
l
(
x
,
r
t
)
+
q
(
x
,
r
t
)
u
∞
*
(
x
,
r
t
)
]
d
w
(
t
)
,is globally asymptotically stable in probability [10].To give our main results, we need the following lemmas.Lemma 3 (see [10]).
(Generalized Itô formula). Letα
(
t
,
x
,
k
) and β
(
t
,
x
,
k
) be given R
n-valued, F
t-adapted process, k
∈
T, and d
x
(
t
)
=
α
(
t
,
x
(
t
)
,
r
t
)
d
t
+
β
(
t
,
x
(
t
)
,
r
t
)
d
w
(
t
). Then for given V
(
t
,
x
,
k
)
∈
C
1,2
(
[
0
,
T
)
;
R
n
), k
∈
T, we have
(9)
E
{
V
(
T
,
x
(
T
)
,
r
T
)
-
V
(
s
,
x
(
s
)
,
r
s
)
∣
r
s
=
k
}
=
E
{
∫
s
T
‍
L
T
V
(
t
,
x
(
t
)
,
r
t
)
d
t
∣
r
s
=
k
}
,
where
(10)
L
T
V
(
t
,
x
,
k
)
=
∂
V
(
t
,
x
,
k
)
∂
t
+
∂
V
′
(
t
,
x
,
k
)
∂
x
α
(
t
,
x
,
k
)
+
∑
j
=
1
N
‍
π
k
j
V
(
t
,
x
,
k
)
+
1
2
β
′
(
t
,
x
,
k
)
∂
V
2
(
t
,
x
,
k
)
∂
x
2
β
(
t
,
x
,
k
)
.Lemma 4.
Ifx
,
b
∈
R
n, A
∈
R
n
×
n is a symmetric matrix and A
-
1 exists, we have
(11)
x
′
A
x
+
x
′
b
+
b
′
x
=
(
x
+
A
-
1
b
)
′
A
(
x
+
A
-
1
b
)
-
b
′
A
-
1
b
.Proof.
This lemma is very easily proved by using completing squares technique, so the proof is omitted.


## 3. Main Results

### 3.1. Finite Horizon Case

The following sufficient condition is presented for the finite horizonH
∞ control of system (1). For convenience, denote (
·
)
k
:
=
(
·
)
(
t
,
x
,
k
) in this subsection.Theorem 5.
Assume that there exists a set of nonnegative functionsV
(
t
,
x
,
k
)
∈
C
1,2
(
[
0
,
T
]
×
R
n
×
T
;
R
), V
(
0,0
,
k
)
=
0, and ∂
2
V
(
t
,
x
,
k
)
/
∂
x
2
≥
0 for all nonzero x
∈
R
n, k
∈
T. If V
(
t
,
x
,
k
) solves the following coupled HJIs:
(12)
Δ
k
=
∂
V
k
∂
t
+
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
m
k
′
m
k
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
4
(
l
k
′
∂
2
V
k
∂
x
2
s
k
+
∂
V
k
′
∂
x
h
k
)
(
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
)
-
1
×
(
s
k
′
∂
2
V
k
∂
x
2
l
k
+
h
k
′
∂
V
k
∂
x
)
-
1
4
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
<
0
,
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
>
0
,
V
(
T
,
x
,
k
)
=
0
,
then
(13)
u
T
*
(
t
,
x
,
k
)
=
-
1
2
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
is a finite horizon H
∞ control of system (1).Proof.
For anyT
>
0 and the initial state x
0
=
0, r
0
=
k, applying Lemma 3, we have
(14)
E
[
V
(
x
(
T
)
,
r
T
)
-
V
(
0
,
r
0
)
∣
r
0
=
k
]
=
E
{
∫
0
T
‍
[
∑
j
=
1
N
‍
π
k
j
∂
V
r
t
∂
t
+
∂
V
r
t
′
∂
x
(
f
r
t
+
g
r
t
u
+
h
r
t
v
)
fdsfdfdsffdfd
f
+
1
2
(
l
r
t
+
q
r
t
u
+
s
r
t
v
)
′
∂
2
V
r
t
∂
x
2
(
l
r
t
+
q
r
t
u
+
s
r
t
v
)
fdsfdsfdfdfdd
+
∥
m
r
t
∥
2
+
∥
u
∥
2
-
γ
2
∥
v
∥
2
-
∥
z
∥
2
sfdfdfdfdfdff
f
+
γ
2
∥
v
∥
2
+
∑
j
=
1
N
‍
π
k
j
V
j
]
d
t
∣
r
0
=
k
}
=
E
{
∫
0
T
‍
[
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
u
)
∂
V
r
t
∂
t
-
∥
z
∥
2
+
γ
2
∥
v
∥
2
dsadasdsdss
l
f
+
∑
j
=
1
N
‍
π
k
j
V
j
+
Θ
1
(
t
,
v
,
x
,
r
t
)
fsdfdsfdsfsdfdsfsdf
+
Θ
2
(
t
,
x
,
r
t
)
+
Θ
3
(
t
,
u
,
x
,
r
t
)
gdfgdfgfdgfdgfgfgff
+
1
2
(
u
′
q
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
v
fdsfdsfsdfsdfsdfdd
sdsdss
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
u
)
]
d
t
∣
r
0
=
k
}
,
where (
·
)
r
t denotes (
·
)
(
t
,
x
,
r
t
) and
(15)
Θ
1
(
t
,
v
,
x
,
r
t
)
=
v
′
(
-
γ
2
I
+
1
2
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
v
+
1
2
(
l
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
+
∂
V
r
t
′
∂
x
h
r
t
)
v
+
1
2
v
′
(
s
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
h
r
t
′
∂
V
r
t
∂
x
)
,
Θ
2
(
t
,
x
,
r
t
)
=
∂
V
r
t
∂
t
+
∂
V
r
t
′
∂
x
f
r
t
+
1
2
l
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
m
r
t
′
m
r
t
,
Θ
3
(
t
,
u
,
x
,
r
t
)
=
u
′
(
I
+
1
2
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
u
+
1
2
(
l
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
+
∂
V
r
t
′
∂
x
g
r
t
)
u
+
1
2
u
′
(
q
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
g
r
t
′
∂
V
r
t
∂
x
)
.
Since ∂
2
V
k
/
∂
x
2
≥
0, k
∈
T, we have
(16)
1
2
(
-
u
′
q
r
t
′
+
v
′
s
r
t
′
)
∂
2
V
r
t
∂
x
2
(
-
q
r
t
u
+
s
r
t
v
)
≥
0
,
which means that
(17)
1
2
(
u
′
q
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
v
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
u
)
≤
1
2
u
′
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
u
+
1
2
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
v
.
Considering the above inequality and (14), we have
(18)
E
[
V
(
x
(
T
)
,
r
T
)
-
V
(
0
,
r
0
)
∣
r
0
=
k
]
≤
E
{
∫
0
T
‍
[
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
u
)
Θ
1
(
t
,
v
,
x
,
r
t
)
+
Θ
2
(
t
,
x
,
r
t
)
+
Θ
3
(
t
,
u
,
x
,
r
t
)
fdfsdddddfd
fd
+
∑
j
=
1
N
‍
π
k
j
V
(
x
,
j
)
-
∥
z
∥
2
+
γ
2
∥
v
∥
2
gfdgfffffffffff
fd
+
1
2
(
u
′
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
v
g
f
d
g
f
d
g
f
g
f
g
f
f
f
f
d
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
u
)
]
d
t
∣
r
0
=
k
}
=
E
[
∫
0
T
‍
(
∑
j
=
1
N
‍
Θ
-
1
(
t
,
v
,
x
,
r
t
)
+
Θ
2
(
t
,
x
,
r
t
)
+
Θ
-
3
(
t
,
u
,
x
,
r
t
)
f
d
s
f
f
d
f
d
d
d
+
∑
j
=
1
N
‍
π
k
j
V
(
x
,
j
)
-
∥
z
∥
2
+
γ
2
∥
v
∥
2
)
d
t
∣
r
0
=
k
]
,
where
(19)
Θ
-
1
(
v
,
x
,
r
t
)
=
v
′
(
-
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
v
+
1
2
(
l
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
+
∂
V
r
t
′
∂
x
h
r
t
)
v
+
1
2
v
′
(
s
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
h
r
t
′
∂
V
r
t
∂
x
)
.
Θ
-
3
(
t
,
u
,
x
,
r
t
)
=
u
′
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
u
+
1
2
(
l
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
+
∂
V
r
t
′
∂
x
g
r
t
)
u
+
1
2
u
′
(
q
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
g
r
t
′
∂
V
r
t
∂
x
)
.
Applying Lemma4 to Θ
-
1
(
t
,
v
,
x
,
r
t
) and Θ
-
3
(
t
,
u
,
x
,
r
t
), we have
(20)
Θ
-
1
(
t
,
v
,
x
,
r
t
)
=
(
v
+
Γ
1
)
′
(
-
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
(
v
+
Γ
1
)
-
1
4
(
l
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
+
∂
V
r
t
′
∂
x
h
r
t
)
×
(
-
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
-
1
×
(
s
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
h
r
t
′
∂
V
r
t
∂
x
)
,
Θ
-
3
(
t
,
u
,
x
,
r
t
)
=
(
u
+
Γ
2
)
′
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
(
u
+
Γ
2
)
-
1
4
(
l
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
+
∂
V
r
t
′
∂
x
g
r
t
)
×
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
-
1
×
(
q
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
g
r
t
′
∂
V
r
t
∂
x
)
,
where
(21)
Γ
1
=
1
2
(
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
-
1
(
s
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
h
r
t
′
∂
V
r
t
∂
x
)
,
Γ
2
=
1
2
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
-
1
(
q
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
g
r
t
′
∂
V
r
t
∂
x
)
.
Substituting (20) into (18), and considering (12), it yields
(22)
E
[
V
(
x
(
T
)
,
r
T
)
-
V
(
0
,
r
0
)
∣
r
0
=
k
]
<
E
{
∫
0
T
‍
[
(
v
+
Γ
1
)
′
(
-
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
(
v
+
Γ
1
)
dsadsadssss
f
+
(
u
+
Γ
2
)
′
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
(
u
+
Γ
2
)
f
d
s
f
d
f
d
s
f
j
f
+
γ
2
∥
v
∥
2
-
∥
z
∥
2
(
-
γ
2
I
+
s
′
r
t
∂
2
V
r
t
∂
x
2
s
r
t
)
]
d
t
∣
r
0
=
k
}
.
Taking u
=
u
T
*
=
-
Γ
2 and considering the second item of (12), (22) leads to
(23)
E
(
∫
0
T
‍
∥
z
(
t
)
∥
2
d
t
∣
r
0
=
k
)
<
γ
2
E
(
∫
0
T
‍
∥
v
(
t
)
∥
2
d
t
∣
r
0
=
k
)
-
E
[
∫
0
T
‍
(
v
(
t
)
+
Γ
1
)
′
f
s
d
f
d
d
f
f
f
d
×
(
γ
2
I
-
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
d
f
s
d
f
d
f
f
f
d
×
(
v
(
t
)
+
Γ
1
)
d
t
∣
r
0
=
k
∫
0
T
‍
]
,
which means ∥
L
T
∥
<
γ in Definition 1. The theorem is proved.Remark 6.
The proof of Theorem5 is based on an elementary identity (11), which avoids using stochastic dissipative theory as done in [4, 5]. We believe that the identity (11) will have many other applications in system analysis and synthesis.


### 3.2. Infinite Horizon Case

In contrast to the finite horizon case, the infinite horizonH
∞ control exhibits more complexity due to the additional requirement of stabilizing the closed-loop system internally. The following sufficient condition is derived for the infinite horizon H
∞ control of system (5). In this subsection, denote (
·
)
k
:
=
(
·
)
(
x
,
k
).Theorem 7.
Assume that there exists a set of nonnegative functionsV
(
x
,
k
)
∈
C
2
(
R
n
×
T
;
R
), V
(
0
,
k
)
=
0, and ∂
2
V
(
x
,
k
)
/
∂
x
2
≥
0 for all nonzero x
∈
R
n, k
∈
T. If V
(
x
,
k
) solves the following coupled HJIs:
(24)
Λ
k
=
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
m
k
′
m
k
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
4
(
l
k
′
∂
2
V
k
∂
x
2
s
k
+
∂
V
k
′
∂
x
h
k
)
(
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
)
-
1
×
(
s
k
′
∂
2
V
k
∂
x
2
l
k
+
h
k
′
∂
V
k
∂
x
)
-
1
4
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
<
0
,
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
>
0
,
then
(25)
u
∞
*
(
x
,
k
)
=
-
1
2
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
is an infinite horizon H
∞ control of system (5).Proof.
Similar to the proof of Theorem5, it is easy to show ∥
L
∞
∥
<
γ under condition (24). Next, we need to prove system (8) to be globally asymptotically stable in probability. Let u
k
*
=
u
∞
*
(
x
,
k
) and let L
∞ be the infinitesimal generator of the system (8) which is similar to L
T in Lemma 3; then
(26)
L
∞
V
k
=
∂
V
k
′
∂
x
(
f
k
+
g
k
u
k
*
)
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
2
(
l
k
+
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
(
l
k
+
q
k
u
k
*
)
=
∂
V
k
′
∂
x
f
k
+
∂
V
k
′
∂
x
g
k
u
k
*
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
q
k
u
k
*
+
1
2
(
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
l
k
+
1
2
(
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
(
q
k
u
k
*
)
=
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
∑
j
=
1
N
‍
π
k
j
V
j
+
Υ
1
k
+
Υ
2
k
,
where
(27)
Υ
1
k
=
∂
V
k
′
∂
x
g
k
u
k
*
+
1
2
l
k
′
∂
2
V
k
∂
x
2
q
k
u
k
*
+
1
2
(
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
l
k
=
-
1
2
∂
V
k
′
∂
x
g
k
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
-
1
4
l
k
′
∂
2
V
k
∂
x
2
q
k
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
-
1
4
(
∂
V
k
′
∂
x
g
k
+
l
k
′
∂
2
V
k
∂
x
2
q
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
q
k
′
∂
2
V
k
∂
x
2
l
k
,
Υ
2
k
=
1
2
(
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
(
q
k
u
k
*
)
=
1
8
(
∂
V
k
′
∂
x
g
k
+
l
k
′
∂
2
V
k
∂
x
2
q
k
)
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
q
k
′
∂
2
V
k
∂
x
2
q
k
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
.
It can be checked that
(28)
Υ
1
k
=
-
1
2
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
,
(29)
Υ
2
k
≤
1
8
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
.
The following inequality is used during the calculation of (29):
(30)
(
I
+
X
)
-
1
X
(
I
+
X
)
-
1
≤
(
I
+
X
)
-
1
(
I
+
X
)
(
I
+
X
)
-
1
=
(
I
+
X
)
-
1
,
X
>
0
.
Implementing (28) and (29) into (26) and considering (24), it yields
(31)
L
∞
V
k
≤
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
∑
j
=
1
N
‍
π
k
j
V
j
-
3
8
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
≤
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
∑
j
=
1
N
‍
π
k
j
V
j
-
1
4
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
<
-
m
k
′
m
k
-
1
4
(
l
k
′
∂
2
V
k
∂
x
2
s
k
+
∂
V
k
′
∂
x
h
k
)
×
(
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
)
-
1
(
s
k
′
∂
2
V
k
∂
x
2
l
k
+
h
k
′
∂
V
k
∂
x
)
≤
0
which implies that (8) is globally asymptotically stable in probability from [10]. This theorem is completed.Remark 8.
The methods proposed in [19, 24] cannot be applied to study the H
∞ problem of NSMJSs with (
x
,
u
,
v
)-dependent noise, although they are suitable for NSMJSs with (
x
,
v
)-dependent noise. One reason for this is that u and v are no longer separable in the conditions, when they enter the diffusion term simultaneously. In the proofs of Theorems 5 and 7, we resort to Lemma 4 to solve this problem.Remark 9.
When there is no Markov jump parameters, system (1)/(5) will reduce to general nonlinear stochastic systems with (
x
,
u
,
v
)-dependent noise, which was studied in [20, 21]. However, all the conditions of H
∞ control design for nonlinear stochastic systems in [20, 21] were given in terms of three coupled HJEs, which were difficult to be solved. According to this paper, the sufficient condition for H
∞ control of nonlinear stochastic systems can be derived by means of a single set of coupled HJIs, which is easier to be verified than three coupled HJEs in [20, 21].
From Theorem7, the following nonlinear stochastic bounded real lemma will be derived for Markov jump system:
(32)
d
x
(
t
)
=
[
f
(
x
,
r
t
)
+
h
(
x
,
r
t
)
v
]
d
t
+
[
l
(
x
,
r
t
)
+
s
(
x
,
r
t
)
v
]
d
w
,
z
(
t
)
=
m
(
x
,
r
t
)
,
x
(
0
)
=
x
0
∈
R
n
.Lemma 10.
For a prescribedγ
>
0, system (32) is internally stable and ∥
L
∞
∥
u
(
t
)
≡
0
<
γ, if there exists a set of nonnegative functions V
(
x
,
k
)
∈
C
2
(
R
n
×
T
;
R
), V
(
0
,
k
)
=
0, and ∂
2
V
(
x
,
k
)
/
∂
x
2
≥
0 for all nonzero x
∈
R
n, k
∈
T, satisfying the following coupled HJIs:
(33)
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
m
k
′
m
k
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
4
(
l
k
′
∂
2
V
k
∂
x
2
s
k
+
∂
V
k
′
∂
x
h
k
)
×
(
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
)
-
1
(
s
k
′
∂
2
V
k
∂
x
2
l
k
+
h
k
′
∂
V
k
∂
x
)
<
0
,
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
>
0
,
k
∈
T
.Proof.
Lettingg
k
≡
0, q
k
≡
0, and u
≡
0 in Theorem 7, we obtain (33) easily.Next, we present a sufficient condition for the following linear stochastic Markov jump systems with(
x
,
u
,
v
)-dependent noise:
(34)
d
x
(
t
)
=
[
A
(
r
t
)
x
+
B
(
r
t
)
u
+
C
(
r
t
)
v
]
d
t
+
[
A
1
(
r
t
)
x
+
B
1
(
r
t
)
u
+
C
1
(
r
t
)
v
]
d
w
,
z
(
t
)
=
col
(
D
(
r
t
)
x
,
u
)
,
x
(
0
)
=
x
0
∈
R
n
.Corollary 11.
System (34) is internally stable and ∥
L
∞
∥
<
γ for given γ
>
0, if there exist matrices X
k
>
0, Y
k
k
∈
T satisfying the following LMIs:
(35)
[
Σ
k
C
k
X
k
D
k
′
X
k
A
1
k
′
+
Y
k
′
B
1
k
′
0
ψ
k
′
(
X
)
*
-
γ
2
I
0
C
1
k
′
C
1
k
′
0
*
*
-
I
0
0
0
*
*
*
-
X
k
0
0
*
*
*
*
-
X
k
0
*
*
*
*
*
-
ϕ
k
(
X
)
]
<
0
,
where
(36)
Σ
k
=
A
k
X
k
+
B
k
Y
k
+
X
k
A
k
′
+
Y
k
′
B
k
′
+
π
k
k
X
k
,
ψ
k
(
X
)
=
[
π
k
1
X
k
,
…
,
π
k
k
-
1
X
k
,
d
s
s
π
k
k
+
1
X
k
,
…
,
π
k
N
X
k
]
′
,
ϕ
k
(
X
)
=
diag
⁡
{
X
1
,
…
,
X
k
-
1
,
X
k
+
1
,
…
,
X
N
}
.
Moreover, the state feedback gain matrices are given by K
k
=
Y
k
X
k
-
1.Proof.
Firstly, consider the following unforced linear stochastic Markov jump systems:(37)
d
x
(
t
)
=
[
A
(
r
t
)
x
+
C
(
r
t
)
v
]
d
t
+
[
A
1
(
r
t
)
x
+
C
1
(
r
t
)
v
]
d
w
,
z
(
t
)
=
D
(
r
t
)
x
,
x
(
0
)
=
x
0
∈
R
n
.
LetV
k
=
x
′
P
k
x
,
P
k
>
0
,
k
∈
T; (33) in Lemma 10 can be written as follows:
(38)
2
x
′
P
k
A
k
x
+
x
′
A
1
k
′
P
k
A
1
k
x
+
x
′
D
k
′
D
k
x
+
x
′
∑
j
=
1
N
‍
π
k
j
P
j
x
+
(
x
′
A
1
k
′
P
k
C
1
k
+
x
′
P
k
C
k
)
×
(
γ
2
I
-
2
C
1
k
′
P
k
C
1
k
)
-
1
(
C
1
k
′
P
k
A
1
k
x
+
C
k
′
P
k
x
)
=
x
′
[
P
k
A
k
+
A
k
′
P
k
+
A
1
k
′
P
k
A
1
k
+
D
k
′
D
k
+
∑
j
=
1
N
‍
π
k
j
P
j
dsddsds
l
+
(
A
1
k
′
P
k
C
1
k
+
P
k
C
k
)
(
γ
2
I
-
2
C
1
k
′
P
k
C
1
k
)
-
1
fdsfdd
×
(
C
1
k
′
P
k
A
1
k
+
C
k
′
P
k
)
∑
j
=
1
N
‍
]
x
<
0
,
(39)
γ
2
I
-
2
C
1
k
′
P
k
C
1
k
>
0
.
By Schur complement, (38) is equivalent to
(40)
[
{
∑
j
=
1
N
‍
P
k
A
k
+
A
k
′
P
k
+
A
1
k
′
P
k
A
1
k
A
1
k
′
P
k
C
1
k
+
P
k
C
k
D
k
′
+
∑
j
=
1
N
‍
π
k
j
P
j
}
*
-
γ
2
I
+
2
C
1
k
′
P
k
C
1
k
0
*
*
-
I
]
<
0
,
which also implies (39). Moreover, (40) can be rewritten as
(41)
[
P
k
A
k
+
A
k
′
P
k
+
∑
j
=
1
N
‍
π
k
j
P
j
P
k
C
k
D
k
′
*
-
γ
2
I
0
*
*
-
I
]
+
[
A
1
k
′
C
1
k
′
0
]
P
k
[
A
1
k
C
1
k
0
]
+
[
0
C
1
k
′
0
]
P
k
[
0
C
1
k
0
]
<
0
,
which yields
(42)
[
P
k
A
k
+
A
k
′
P
k
+
∑
j
=
1
N
‍
π
k
j
P
j
P
k
C
k
D
k
′
A
1
k
′
0
*
-
γ
2
I
0
C
1
k
′
C
1
k
′
*
*
-
I
0
0
*
*
*
-
P
k
-
1
0
*
*
*
*
-
P
k
-
1
]
<
0
,
according to Schur complement. Pre- and postmultiplying (42) by diag
⁡
{
P
k
-
1
I
I
I
I
} and denoting X
k
=
P
k
-
1, we have
(43)
[
A
k
X
k
+
X
k
A
k
′
+
π
k
k
X
k
C
k
X
k
D
k
′
X
k
A
1
k
′
0
ψ
k
′
(
X
)
*
-
γ
2
I
0
C
1
k
′
C
1
k
′
0
*
*
-
I
0
0
0
*
*
*
-
X
k
0
0
*
*
*
*
-
X
k
0
*
*
*
*
*
-
ϕ
k
(
X
)
]
<
0
,
where ψ
k
(
X
) and ϕ
k
(
X
) are defined as in (35).
Considering closed-loop system (34) with state feedback control u
(
t
)
=
K
k
x
(
t
), k
∈
T. Replacing A
k
,
A
1
k by A
k
+
B
k
K
k, A
1
k
+
B
1
k
K
k, respectively, and setting Y
k
=
K
k
X
k in (43) yield (35). Therefore, the state feedback gain matrices can be obtained by K
k
=
Y
k
X
k
-
1.Remark 12.
Although HJIs (12) in Theorem 5 or (24) in Theorem 7 can be solved by trial and error in some simple cases (see Example 13 in Section 4), they are difficult to be dealt with for high-dimensional systems. In order to avoid solving the HJIs, the Taylor series approach [25] or fuzzy approach based on Takagi-Sugeno model [24, 26] can be considered to design the nonlinear stochastic H
∞ controller.


## 3.1. Finite Horizon Case

The following sufficient condition is presented for the finite horizonH
∞ control of system (1). For convenience, denote (
·
)
k
:
=
(
·
)
(
t
,
x
,
k
) in this subsection.Theorem 5.
Assume that there exists a set of nonnegative functionsV
(
t
,
x
,
k
)
∈
C
1,2
(
[
0
,
T
]
×
R
n
×
T
;
R
), V
(
0,0
,
k
)
=
0, and ∂
2
V
(
t
,
x
,
k
)
/
∂
x
2
≥
0 for all nonzero x
∈
R
n, k
∈
T. If V
(
t
,
x
,
k
) solves the following coupled HJIs:
(12)
Δ
k
=
∂
V
k
∂
t
+
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
m
k
′
m
k
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
4
(
l
k
′
∂
2
V
k
∂
x
2
s
k
+
∂
V
k
′
∂
x
h
k
)
(
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
)
-
1
×
(
s
k
′
∂
2
V
k
∂
x
2
l
k
+
h
k
′
∂
V
k
∂
x
)
-
1
4
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
<
0
,
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
>
0
,
V
(
T
,
x
,
k
)
=
0
,
then
(13)
u
T
*
(
t
,
x
,
k
)
=
-
1
2
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
is a finite horizon H
∞ control of system (1).Proof.
For anyT
>
0 and the initial state x
0
=
0, r
0
=
k, applying Lemma 3, we have
(14)
E
[
V
(
x
(
T
)
,
r
T
)
-
V
(
0
,
r
0
)
∣
r
0
=
k
]
=
E
{
∫
0
T
‍
[
∑
j
=
1
N
‍
π
k
j
∂
V
r
t
∂
t
+
∂
V
r
t
′
∂
x
(
f
r
t
+
g
r
t
u
+
h
r
t
v
)
fdsfdfdsffdfd
f
+
1
2
(
l
r
t
+
q
r
t
u
+
s
r
t
v
)
′
∂
2
V
r
t
∂
x
2
(
l
r
t
+
q
r
t
u
+
s
r
t
v
)
fdsfdsfdfdfdd
+
∥
m
r
t
∥
2
+
∥
u
∥
2
-
γ
2
∥
v
∥
2
-
∥
z
∥
2
sfdfdfdfdfdff
f
+
γ
2
∥
v
∥
2
+
∑
j
=
1
N
‍
π
k
j
V
j
]
d
t
∣
r
0
=
k
}
=
E
{
∫
0
T
‍
[
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
u
)
∂
V
r
t
∂
t
-
∥
z
∥
2
+
γ
2
∥
v
∥
2
dsadasdsdss
l
f
+
∑
j
=
1
N
‍
π
k
j
V
j
+
Θ
1
(
t
,
v
,
x
,
r
t
)
fsdfdsfdsfsdfdsfsdf
+
Θ
2
(
t
,
x
,
r
t
)
+
Θ
3
(
t
,
u
,
x
,
r
t
)
gdfgdfgfdgfdgfgfgff
+
1
2
(
u
′
q
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
v
fdsfdsfsdfsdfsdfdd
sdsdss
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
u
)
]
d
t
∣
r
0
=
k
}
,
where (
·
)
r
t denotes (
·
)
(
t
,
x
,
r
t
) and
(15)
Θ
1
(
t
,
v
,
x
,
r
t
)
=
v
′
(
-
γ
2
I
+
1
2
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
v
+
1
2
(
l
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
+
∂
V
r
t
′
∂
x
h
r
t
)
v
+
1
2
v
′
(
s
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
h
r
t
′
∂
V
r
t
∂
x
)
,
Θ
2
(
t
,
x
,
r
t
)
=
∂
V
r
t
∂
t
+
∂
V
r
t
′
∂
x
f
r
t
+
1
2
l
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
m
r
t
′
m
r
t
,
Θ
3
(
t
,
u
,
x
,
r
t
)
=
u
′
(
I
+
1
2
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
u
+
1
2
(
l
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
+
∂
V
r
t
′
∂
x
g
r
t
)
u
+
1
2
u
′
(
q
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
g
r
t
′
∂
V
r
t
∂
x
)
.
Since ∂
2
V
k
/
∂
x
2
≥
0, k
∈
T, we have
(16)
1
2
(
-
u
′
q
r
t
′
+
v
′
s
r
t
′
)
∂
2
V
r
t
∂
x
2
(
-
q
r
t
u
+
s
r
t
v
)
≥
0
,
which means that
(17)
1
2
(
u
′
q
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
v
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
u
)
≤
1
2
u
′
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
u
+
1
2
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
v
.
Considering the above inequality and (14), we have
(18)
E
[
V
(
x
(
T
)
,
r
T
)
-
V
(
0
,
r
0
)
∣
r
0
=
k
]
≤
E
{
∫
0
T
‍
[
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
u
)
Θ
1
(
t
,
v
,
x
,
r
t
)
+
Θ
2
(
t
,
x
,
r
t
)
+
Θ
3
(
t
,
u
,
x
,
r
t
)
fdfsdddddfd
fd
+
∑
j
=
1
N
‍
π
k
j
V
(
x
,
j
)
-
∥
z
∥
2
+
γ
2
∥
v
∥
2
gfdgfffffffffff
fd
+
1
2
(
u
′
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
v
g
f
d
g
f
d
g
f
g
f
g
f
f
f
f
d
+
v
′
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
u
)
]
d
t
∣
r
0
=
k
}
=
E
[
∫
0
T
‍
(
∑
j
=
1
N
‍
Θ
-
1
(
t
,
v
,
x
,
r
t
)
+
Θ
2
(
t
,
x
,
r
t
)
+
Θ
-
3
(
t
,
u
,
x
,
r
t
)
f
d
s
f
f
d
f
d
d
d
+
∑
j
=
1
N
‍
π
k
j
V
(
x
,
j
)
-
∥
z
∥
2
+
γ
2
∥
v
∥
2
)
d
t
∣
r
0
=
k
]
,
where
(19)
Θ
-
1
(
v
,
x
,
r
t
)
=
v
′
(
-
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
v
+
1
2
(
l
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
+
∂
V
r
t
′
∂
x
h
r
t
)
v
+
1
2
v
′
(
s
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
h
r
t
′
∂
V
r
t
∂
x
)
.
Θ
-
3
(
t
,
u
,
x
,
r
t
)
=
u
′
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
u
+
1
2
(
l
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
+
∂
V
r
t
′
∂
x
g
r
t
)
u
+
1
2
u
′
(
q
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
g
r
t
′
∂
V
r
t
∂
x
)
.
Applying Lemma4 to Θ
-
1
(
t
,
v
,
x
,
r
t
) and Θ
-
3
(
t
,
u
,
x
,
r
t
), we have
(20)
Θ
-
1
(
t
,
v
,
x
,
r
t
)
=
(
v
+
Γ
1
)
′
(
-
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
(
v
+
Γ
1
)
-
1
4
(
l
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
+
∂
V
r
t
′
∂
x
h
r
t
)
×
(
-
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
-
1
×
(
s
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
h
r
t
′
∂
V
r
t
∂
x
)
,
Θ
-
3
(
t
,
u
,
x
,
r
t
)
=
(
u
+
Γ
2
)
′
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
(
u
+
Γ
2
)
-
1
4
(
l
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
+
∂
V
r
t
′
∂
x
g
r
t
)
×
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
-
1
×
(
q
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
g
r
t
′
∂
V
r
t
∂
x
)
,
where
(21)
Γ
1
=
1
2
(
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
-
1
(
s
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
h
r
t
′
∂
V
r
t
∂
x
)
,
Γ
2
=
1
2
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
-
1
(
q
r
t
′
∂
2
V
r
t
∂
x
2
l
r
t
+
g
r
t
′
∂
V
r
t
∂
x
)
.
Substituting (20) into (18), and considering (12), it yields
(22)
E
[
V
(
x
(
T
)
,
r
T
)
-
V
(
0
,
r
0
)
∣
r
0
=
k
]
<
E
{
∫
0
T
‍
[
(
v
+
Γ
1
)
′
(
-
γ
2
I
+
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
(
v
+
Γ
1
)
dsadsadssss
f
+
(
u
+
Γ
2
)
′
(
I
+
q
r
t
′
∂
2
V
r
t
∂
x
2
q
r
t
)
(
u
+
Γ
2
)
f
d
s
f
d
f
d
s
f
j
f
+
γ
2
∥
v
∥
2
-
∥
z
∥
2
(
-
γ
2
I
+
s
′
r
t
∂
2
V
r
t
∂
x
2
s
r
t
)
]
d
t
∣
r
0
=
k
}
.
Taking u
=
u
T
*
=
-
Γ
2 and considering the second item of (12), (22) leads to
(23)
E
(
∫
0
T
‍
∥
z
(
t
)
∥
2
d
t
∣
r
0
=
k
)
<
γ
2
E
(
∫
0
T
‍
∥
v
(
t
)
∥
2
d
t
∣
r
0
=
k
)
-
E
[
∫
0
T
‍
(
v
(
t
)
+
Γ
1
)
′
f
s
d
f
d
d
f
f
f
d
×
(
γ
2
I
-
s
r
t
′
∂
2
V
r
t
∂
x
2
s
r
t
)
d
f
s
d
f
d
f
f
f
d
×
(
v
(
t
)
+
Γ
1
)
d
t
∣
r
0
=
k
∫
0
T
‍
]
,
which means ∥
L
T
∥
<
γ in Definition 1. The theorem is proved.Remark 6.
The proof of Theorem5 is based on an elementary identity (11), which avoids using stochastic dissipative theory as done in [4, 5]. We believe that the identity (11) will have many other applications in system analysis and synthesis.


## 3.2. Infinite Horizon Case

In contrast to the finite horizon case, the infinite horizonH
∞ control exhibits more complexity due to the additional requirement of stabilizing the closed-loop system internally. The following sufficient condition is derived for the infinite horizon H
∞ control of system (5). In this subsection, denote (
·
)
k
:
=
(
·
)
(
x
,
k
).Theorem 7.
Assume that there exists a set of nonnegative functionsV
(
x
,
k
)
∈
C
2
(
R
n
×
T
;
R
), V
(
0
,
k
)
=
0, and ∂
2
V
(
x
,
k
)
/
∂
x
2
≥
0 for all nonzero x
∈
R
n, k
∈
T. If V
(
x
,
k
) solves the following coupled HJIs:
(24)
Λ
k
=
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
m
k
′
m
k
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
4
(
l
k
′
∂
2
V
k
∂
x
2
s
k
+
∂
V
k
′
∂
x
h
k
)
(
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
)
-
1
×
(
s
k
′
∂
2
V
k
∂
x
2
l
k
+
h
k
′
∂
V
k
∂
x
)
-
1
4
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
<
0
,
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
>
0
,
then
(25)
u
∞
*
(
x
,
k
)
=
-
1
2
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
is an infinite horizon H
∞ control of system (5).Proof.
Similar to the proof of Theorem5, it is easy to show ∥
L
∞
∥
<
γ under condition (24). Next, we need to prove system (8) to be globally asymptotically stable in probability. Let u
k
*
=
u
∞
*
(
x
,
k
) and let L
∞ be the infinitesimal generator of the system (8) which is similar to L
T in Lemma 3; then
(26)
L
∞
V
k
=
∂
V
k
′
∂
x
(
f
k
+
g
k
u
k
*
)
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
2
(
l
k
+
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
(
l
k
+
q
k
u
k
*
)
=
∂
V
k
′
∂
x
f
k
+
∂
V
k
′
∂
x
g
k
u
k
*
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
q
k
u
k
*
+
1
2
(
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
l
k
+
1
2
(
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
(
q
k
u
k
*
)
=
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
∑
j
=
1
N
‍
π
k
j
V
j
+
Υ
1
k
+
Υ
2
k
,
where
(27)
Υ
1
k
=
∂
V
k
′
∂
x
g
k
u
k
*
+
1
2
l
k
′
∂
2
V
k
∂
x
2
q
k
u
k
*
+
1
2
(
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
l
k
=
-
1
2
∂
V
k
′
∂
x
g
k
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
-
1
4
l
k
′
∂
2
V
k
∂
x
2
q
k
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
-
1
4
(
∂
V
k
′
∂
x
g
k
+
l
k
′
∂
2
V
k
∂
x
2
q
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
q
k
′
∂
2
V
k
∂
x
2
l
k
,
Υ
2
k
=
1
2
(
q
k
u
k
*
)
′
∂
2
V
k
∂
x
2
(
q
k
u
k
*
)
=
1
8
(
∂
V
k
′
∂
x
g
k
+
l
k
′
∂
2
V
k
∂
x
2
q
k
)
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
q
k
′
∂
2
V
k
∂
x
2
q
k
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
×
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
.
It can be checked that
(28)
Υ
1
k
=
-
1
2
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
,
(29)
Υ
2
k
≤
1
8
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
.
The following inequality is used during the calculation of (29):
(30)
(
I
+
X
)
-
1
X
(
I
+
X
)
-
1
≤
(
I
+
X
)
-
1
(
I
+
X
)
(
I
+
X
)
-
1
=
(
I
+
X
)
-
1
,
X
>
0
.
Implementing (28) and (29) into (26) and considering (24), it yields
(31)
L
∞
V
k
≤
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
∑
j
=
1
N
‍
π
k
j
V
j
-
3
8
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
≤
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
∑
j
=
1
N
‍
π
k
j
V
j
-
1
4
(
l
k
′
∂
2
V
k
∂
x
2
q
k
+
∂
V
k
′
∂
x
g
k
)
×
(
I
+
q
k
′
∂
2
V
k
∂
x
2
q
k
)
-
1
(
q
k
′
∂
2
V
k
∂
x
2
l
k
+
g
k
′
∂
V
k
∂
x
)
<
-
m
k
′
m
k
-
1
4
(
l
k
′
∂
2
V
k
∂
x
2
s
k
+
∂
V
k
′
∂
x
h
k
)
×
(
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
)
-
1
(
s
k
′
∂
2
V
k
∂
x
2
l
k
+
h
k
′
∂
V
k
∂
x
)
≤
0
which implies that (8) is globally asymptotically stable in probability from [10]. This theorem is completed.Remark 8.
The methods proposed in [19, 24] cannot be applied to study the H
∞ problem of NSMJSs with (
x
,
u
,
v
)-dependent noise, although they are suitable for NSMJSs with (
x
,
v
)-dependent noise. One reason for this is that u and v are no longer separable in the conditions, when they enter the diffusion term simultaneously. In the proofs of Theorems 5 and 7, we resort to Lemma 4 to solve this problem.Remark 9.
When there is no Markov jump parameters, system (1)/(5) will reduce to general nonlinear stochastic systems with (
x
,
u
,
v
)-dependent noise, which was studied in [20, 21]. However, all the conditions of H
∞ control design for nonlinear stochastic systems in [20, 21] were given in terms of three coupled HJEs, which were difficult to be solved. According to this paper, the sufficient condition for H
∞ control of nonlinear stochastic systems can be derived by means of a single set of coupled HJIs, which is easier to be verified than three coupled HJEs in [20, 21].
From Theorem7, the following nonlinear stochastic bounded real lemma will be derived for Markov jump system:
(32)
d
x
(
t
)
=
[
f
(
x
,
r
t
)
+
h
(
x
,
r
t
)
v
]
d
t
+
[
l
(
x
,
r
t
)
+
s
(
x
,
r
t
)
v
]
d
w
,
z
(
t
)
=
m
(
x
,
r
t
)
,
x
(
0
)
=
x
0
∈
R
n
.Lemma 10.
For a prescribedγ
>
0, system (32) is internally stable and ∥
L
∞
∥
u
(
t
)
≡
0
<
γ, if there exists a set of nonnegative functions V
(
x
,
k
)
∈
C
2
(
R
n
×
T
;
R
), V
(
0
,
k
)
=
0, and ∂
2
V
(
x
,
k
)
/
∂
x
2
≥
0 for all nonzero x
∈
R
n, k
∈
T, satisfying the following coupled HJIs:
(33)
∂
V
k
′
∂
x
f
k
+
1
2
l
k
′
∂
2
V
k
∂
x
2
l
k
+
m
k
′
m
k
+
∑
j
=
1
N
‍
π
k
j
V
j
+
1
4
(
l
k
′
∂
2
V
k
∂
x
2
s
k
+
∂
V
k
′
∂
x
h
k
)
×
(
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
)
-
1
(
s
k
′
∂
2
V
k
∂
x
2
l
k
+
h
k
′
∂
V
k
∂
x
)
<
0
,
γ
2
I
-
s
k
′
∂
2
V
k
∂
x
2
s
k
>
0
,
k
∈
T
.Proof.
Lettingg
k
≡
0, q
k
≡
0, and u
≡
0 in Theorem 7, we obtain (33) easily.Next, we present a sufficient condition for the following linear stochastic Markov jump systems with(
x
,
u
,
v
)-dependent noise:
(34)
d
x
(
t
)
=
[
A
(
r
t
)
x
+
B
(
r
t
)
u
+
C
(
r
t
)
v
]
d
t
+
[
A
1
(
r
t
)
x
+
B
1
(
r
t
)
u
+
C
1
(
r
t
)
v
]
d
w
,
z
(
t
)
=
col
(
D
(
r
t
)
x
,
u
)
,
x
(
0
)
=
x
0
∈
R
n
.Corollary 11.
System (34) is internally stable and ∥
L
∞
∥
<
γ for given γ
>
0, if there exist matrices X
k
>
0, Y
k
k
∈
T satisfying the following LMIs:
(35)
[
Σ
k
C
k
X
k
D
k
′
X
k
A
1
k
′
+
Y
k
′
B
1
k
′
0
ψ
k
′
(
X
)
*
-
γ
2
I
0
C
1
k
′
C
1
k
′
0
*
*
-
I
0
0
0
*
*
*
-
X
k
0
0
*
*
*
*
-
X
k
0
*
*
*
*
*
-
ϕ
k
(
X
)
]
<
0
,
where
(36)
Σ
k
=
A
k
X
k
+
B
k
Y
k
+
X
k
A
k
′
+
Y
k
′
B
k
′
+
π
k
k
X
k
,
ψ
k
(
X
)
=
[
π
k
1
X
k
,
…
,
π
k
k
-
1
X
k
,
d
s
s
π
k
k
+
1
X
k
,
…
,
π
k
N
X
k
]
′
,
ϕ
k
(
X
)
=
diag
⁡
{
X
1
,
…
,
X
k
-
1
,
X
k
+
1
,
…
,
X
N
}
.
Moreover, the state feedback gain matrices are given by K
k
=
Y
k
X
k
-
1.Proof.
Firstly, consider the following unforced linear stochastic Markov jump systems:(37)
d
x
(
t
)
=
[
A
(
r
t
)
x
+
C
(
r
t
)
v
]
d
t
+
[
A
1
(
r
t
)
x
+
C
1
(
r
t
)
v
]
d
w
,
z
(
t
)
=
D
(
r
t
)
x
,
x
(
0
)
=
x
0
∈
R
n
.
LetV
k
=
x
′
P
k
x
,
P
k
>
0
,
k
∈
T; (33) in Lemma 10 can be written as follows:
(38)
2
x
′
P
k
A
k
x
+
x
′
A
1
k
′
P
k
A
1
k
x
+
x
′
D
k
′
D
k
x
+
x
′
∑
j
=
1
N
‍
π
k
j
P
j
x
+
(
x
′
A
1
k
′
P
k
C
1
k
+
x
′
P
k
C
k
)
×
(
γ
2
I
-
2
C
1
k
′
P
k
C
1
k
)
-
1
(
C
1
k
′
P
k
A
1
k
x
+
C
k
′
P
k
x
)
=
x
′
[
P
k
A
k
+
A
k
′
P
k
+
A
1
k
′
P
k
A
1
k
+
D
k
′
D
k
+
∑
j
=
1
N
‍
π
k
j
P
j
dsddsds
l
+
(
A
1
k
′
P
k
C
1
k
+
P
k
C
k
)
(
γ
2
I
-
2
C
1
k
′
P
k
C
1
k
)
-
1
fdsfdd
×
(
C
1
k
′
P
k
A
1
k
+
C
k
′
P
k
)
∑
j
=
1
N
‍
]
x
<
0
,
(39)
γ
2
I
-
2
C
1
k
′
P
k
C
1
k
>
0
.
By Schur complement, (38) is equivalent to
(40)
[
{
∑
j
=
1
N
‍
P
k
A
k
+
A
k
′
P
k
+
A
1
k
′
P
k
A
1
k
A
1
k
′
P
k
C
1
k
+
P
k
C
k
D
k
′
+
∑
j
=
1
N
‍
π
k
j
P
j
}
*
-
γ
2
I
+
2
C
1
k
′
P
k
C
1
k
0
*
*
-
I
]
<
0
,
which also implies (39). Moreover, (40) can be rewritten as
(41)
[
P
k
A
k
+
A
k
′
P
k
+
∑
j
=
1
N
‍
π
k
j
P
j
P
k
C
k
D
k
′
*
-
γ
2
I
0
*
*
-
I
]
+
[
A
1
k
′
C
1
k
′
0
]
P
k
[
A
1
k
C
1
k
0
]
+
[
0
C
1
k
′
0
]
P
k
[
0
C
1
k
0
]
<
0
,
which yields
(42)
[
P
k
A
k
+
A
k
′
P
k
+
∑
j
=
1
N
‍
π
k
j
P
j
P
k
C
k
D
k
′
A
1
k
′
0
*
-
γ
2
I
0
C
1
k
′
C
1
k
′
*
*
-
I
0
0
*
*
*
-
P
k
-
1
0
*
*
*
*
-
P
k
-
1
]
<
0
,
according to Schur complement. Pre- and postmultiplying (42) by diag
⁡
{
P
k
-
1
I
I
I
I
} and denoting X
k
=
P
k
-
1, we have
(43)
[
A
k
X
k
+
X
k
A
k
′
+
π
k
k
X
k
C
k
X
k
D
k
′
X
k
A
1
k
′
0
ψ
k
′
(
X
)
*
-
γ
2
I
0
C
1
k
′
C
1
k
′
0
*
*
-
I
0
0
0
*
*
*
-
X
k
0
0
*
*
*
*
-
X
k
0
*
*
*
*
*
-
ϕ
k
(
X
)
]
<
0
,
where ψ
k
(
X
) and ϕ
k
(
X
) are defined as in (35).
Considering closed-loop system (34) with state feedback control u
(
t
)
=
K
k
x
(
t
), k
∈
T. Replacing A
k
,
A
1
k by A
k
+
B
k
K
k, A
1
k
+
B
1
k
K
k, respectively, and setting Y
k
=
K
k
X
k in (43) yield (35). Therefore, the state feedback gain matrices can be obtained by K
k
=
Y
k
X
k
-
1.Remark 12.
Although HJIs (12) in Theorem 5 or (24) in Theorem 7 can be solved by trial and error in some simple cases (see Example 13 in Section 4), they are difficult to be dealt with for high-dimensional systems. In order to avoid solving the HJIs, the Taylor series approach [25] or fuzzy approach based on Takagi-Sugeno model [24, 26] can be considered to design the nonlinear stochastic H
∞ controller.


## 4. Numerical Example

In this section, two numerical examples are provided to illustrate the effectiveness of the developed results.Example 13.
Consider one-dimensional two-mode time-invariant nonlinear stochastic Markov jump systems with generator matrixΠ
=
[
-
1
1
1
-
1
], and the two subsystems are as follows:
(44)
I
:
{
d
x
=
(
x
3
+
4
x
u
+
x
v
)
d
t
+
(
x
2
+
u
+
v
)
d
w
,
z
=
col
(
x
2
,
u
)
,
x
(
0
)
=
x
0
∈
R
n
,
II
:
{
d
x
=
(
1
2
x
3
+
3
x
u
+
1
2
x
v
)
d
t
+
(
x
2
+
u
+
v
)
d
w
,
z
=
col
(
1
2
x
2
,
u
)
,
x
(
0
)
=
x
0
∈
R
n
.
Set V
(
x
)
=
p
i
x
2, i
=
1,2, with p
1
>
0 and p
2
>
0 to be determined; then HJIs (24) become
(45)
2
p
1
x
·
x
3
+
1
2
x
2
·
2
p
1
·
x
2
+
x
2
·
x
2
+
1
4
(
x
2
·
2
p
1
+
2
p
1
x
·
x
)
2
(
γ
2
-
2
p
1
)
-
1
-
1
4
(
x
2
·
2
p
1
+
2
p
1
x
·
4
x
)
2
(
1
+
2
p
1
)
-
1
-
p
1
x
2
+
p
2
x
2
<
0
,
γ
2
-
2
p
1
>
0
,
2
p
2
x
·
1
2
x
3
+
1
2
x
2
·
2
p
2
·
x
2
+
1
2
x
2
·
1
2
x
2
+
1
4
(
x
2
·
2
p
2
+
2
p
2
x
·
1
2
x
)
2
(
γ
2
-
2
p
2
)
-
1
-
1
4
(
x
2
·
2
p
2
+
2
p
2
x
·
3
x
)
2
(
1
+
2
p
2
)
-
1
+
p
1
x
2
-
p
2
x
2
<
0
,
γ
2
-
2
p
2
>
0
.
For given γ
=
3, the above inequalities have solutions p
1
=
1 and p
2
=
1. According to Theorem 7, the infinite H
∞ controllers of system (44) are u
∞
*
(
x
,
1
)
=
-
(
5
/
3
)
x
2 and u
∞
*
(
x
,
2
)
=
-
(
4
/
3
)
x
2.Example 14.
Consider two-dimensional two-mode linear stochastic Markov jump systems (34) with the following parameters:
(46)
A
1
=
[
1
1
0
-
2
]
,
B
1
=
C
1
=
[
1
0
]
,
A
11
=
[
1
1
0
1
]
,
B
11
=
C
11
=
[
1
0
]
,
A
2
=
[
2
1
0
-
2
]
,
B
2
=
C
2
=
[
1
1
]
,
A
12
=
[
2
0
1
11
]
,
B
12
=
C
12
=
[
0
1
]
,
D
1
=
[
1
0
]
,
D
2
=
[
1
1
]
,
Π
=
[
-
0.5
0.5
0.5
-
0.5
]
.
With the choice of γ
=
2, a possible solution of LMIs (35) in Corollary 11 can be found by using Matlab LMI control toolbox:
(47)
X
1
=
[
0.8879
-
0.4074
-
0.4074
3.6974
]
,
X
2
=
[
0.5026
0.0464
0.0464
5.4520
]
,
Y
1
=
[
-
3.1690
-
5.3082
]
,
Y
2
=
[
-
4.5025
-
8.5671
]
.
Then, the H
∞ control gain matrices of system (34) are as follows:
(48)
K
1
=
Y
1
X
1
-
1
=
[
-
4.4531
-
1.9263
]
,
K
2
=
Y
2
X
2
-
1
=
[
-
8.8209
-
1.4963
]
.Figure1 shows the result of the changing between modes during the simulation with the initial mode at mode 1. The initial condition is chosen as x
0
=
[
-
0.2
0.3
]
′ and exogenous input v
(
t
)
=
e
-
t. By means of Euler-Maruyama method [27], the state responses of unforced system (37) and controlled system (34) are shown in Figures 2 and 3, respectively. From Figure 3, one can find that the controlled system (34) can achieve stability and attenuation performance in the sense of mean square by using the proposed H
∞ controller.Figure 1
Result of the changing between modes.Figure 2
The state responses of unforced system (37).Figure 3
The state responses of controlled system (34).


## 5. Conclusions

In this paper, we have studied theH
∞ control problem for NSMJSs with (
x
,
u
,
v
)-dependent noise. A sufficient condition for finite/infinite horizon H
∞ control has been derived in terms of a set of coupled HJIs. It can be found that Lemma 4 plays an essential role during the proof of Theorems 5 and 7. The validity of the obtained results has been verified by two examples.



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*Source: 948134-2014-06-18.xml*