Abstract—The paper mainly studies the exponential stability analysis problem for a class of grey neutral stochastic systems with distributed delays. As we know, till now, the stability problem of grey neutral stochastic systems has not been intensively studied except some papers, which motives our research. In this paper, by using an appropriately constructed Lyapunov-Krasovskii functional and some stochastic analysis approaches, especially, by utilizing decomposition technique of continuous matrix-covered sets, sufficient stability criteria are proposed which ensure the mean-square exponential stability and almost surely exponentially robustly stable for the systems. Moreover, an example is provided to illustrate the effectiveness and correctness of the obtained results.
Index Terms—Grey Neutral Stochastic Systems, Distributed Delays, Lyapunov-Krasovskii Functional, Decomposition Technique, Exponential Stability
I. INTRODUCTION
tochastic systems have come to play an important role in many branches of science or engineering applications, and time delays are frequently encountered in many real-word control systems, which is the main causes of instability, oscillation, and poor performance of the systems. Therefore, during the past decades, stochastic systems with time delays have been extensively investigated, many important results have been reported in the literature, see [1-8], and the references therein. On the other hand, in practice, many dynamical systems can be effectively established or described by neutral functional differential equations, such as the distributed networks, population ecology, chemical reactors, water pipes and so on [9]. When the number of summands in a system equation is increased and the differences between neighboring argument values are decreased, another type of time-delays, namely, distributed delays will appear, which can be found in the modeling of feeding systems and combustion chambers in a liquid monopropellant rocket motor with pressure feeding [10,11]. Hence, the study of neutral stochastic systems with distributed delays has come to become a subject of intensive research activity in recent years, much effort has been devoted to the study of this kind of systems [9-20]. For example, in [12], authors investigate the problem of stability analysis of neutral type neural networks with both discrete and unbounded distributed delays. In terms of linear matrix inequalities, delay-dependent conditions are obtained, which The research is supported by the National Natural Science Foundation of China (No. 11301009) and the Natural Science Foundation of Henan Province (No. 0511013800).
Jian Wang is with School of Mathematics and Statistics, Anyang Normal University, Anyang 455002, PR China e-mail: ss2wangxin@sina.com
guarantee the networks to have a unique equilibrium point. It is worth noting that, when the mathematical model of stochastic systems is built, the parameters are difficult to obtain. In addition, as pointed out in [21], if the parameters of the systems are evaluated by grey numbers, the systems should be established indeterminately and become grey systems. Therefore, the study of grey systems has received much attention from many scholars, some significant and innovative results have appeared in the literature [21-24]. For instance, in [22], two easily verified delay-dependent criteria of mean-square exponential robust stability were obtained. However, to the best of our knowledge, till now, the problem of exponential stability for grey neutral stochastic systems with distributed delays has not been full investigated, which is still open and remains challenging. this situation motives our present study.
In this paper, we investigate the exponential stability problem for a class of grey neutral stochastic systems with distributed delays. First, a new type of Lyapunov-Krasovskii functional is constructed. Then, by using the decomposition technique of the continuous matrix-covered sets of grey matrix (see [21-24]), we study the systems model directly with some well-known differential formulas, and sufficient criteria are obtained, which ensure the systems in the mean square exponential stability and almost surely exponentially robustly stable. Finally, an example is given to demonstrate the applicability of the proposed stability criteria.
Notations: The notations are quite standard. Throughout this paper,
R
n andR
nn denote the n-dimensional Euclidean space and the set of all n ×n real matrices. The superscript"
"
T
represents matrix transposition, and the notationY
X
(respectivelyX
Y
) whereX
andY
are symmetric matrices, means thatX
Y
is positive semi-definite (respectively positive definite). The symbol
denotes the Euclidean norm for vector or the spectral norm of matrices. Moreover, Let
,
F
,
F
t t0,
P
be a complete probability space with a filtration
F
t t0satisfying the usual conditions. Let
0
andC
[
,
0
];
R
n
denote the family of all continuousR
n-valued functions
on[
,
0
]
. Let)
];
0
,
([
2
0
n
F
R
L
be the family of allF
0 -measurable boundedC
[
,
0
];
R
n
-valued random Variables
(
)
:
0
.Exponential Stability Analysis for Neutral
Stochastic Systems with Distributed Delays
Jian Wang
II. PRELIMINARIES AND PROBLEM FORMULATION Consider a class of grey neutral stochastic systems with distributed delays:
0
),
];
0
,
([
,
0
),
(
]
)
(
)
(
)
(
)
(
)
(
)
(
[
]
)
(
)
(
)
(
)
(
)
(
)
(
[
)]
(
)
(
)
(
[
2
0
L
0R
t
x
t
t
dW
ds
s
x
F
t
x
E
t
x
D
dt
ds
s
x
C
t
x
B
t
x
A
t
x
G
t
x
d
n F
t
t t t
(2.1) Where
A
(
),
B
(
),
C
(
),
D
(
),
E
(
),
F
(
),
)
(
G
are grey n ×n -matrices, and let)
(
)
(
aij
A
,(
)
(
b)
ijB
,(
)
(
c)
ijC
,)
(
)
(
dij
D
,(
)
(
e)
ijE
,(
)
(
f)
ijF
,)
(
)
(
gij
G
.Here,
aij,
bij,
cij,
ijd ,
eij,
ijf and
ijg are said to be grey elements of),
(
A
B
(
),
C
(
),
D
(
),
E
(
),
F
(
)
andG
(
)
. Now, we define}
,...
2
,1
,
,
:
)
(
)
ˆ
(
{
]
,
[
L
aU
a
A
a
ija
ij
a
ij
a
iji
j
n
}
,...
2
,1
,
,
:
)
(
)
ˆ
(
{
]
,
[
L
bU
b
B
b
ijb
ij
b
ij
b
iji
j
n
}
,...
2
,1
,
,
:
)
(
)
ˆ
(
{
]
,
[
L
cU
c
C
c
ijc
ij
c
ij
c
iji
j
n
}
,...
2
,1
,
,
:
)
(
)
ˆ
(
{
]
,
[
L
dU
d
D
d
ijd
ij
d
ij
d
iji
j
n
}
,...
2
,1
,
,
:
)
(
)
ˆ
(
{
]
,
[
L
eU
e
E
e
ije
ij
e
ij
e
iji
j
n
}
,...
2
,1
,
,
:
)
(
)
ˆ
(
{
]
,
[
L
fU
f
F
f
ijf
ij
f
ij
f
iji
j
n
}
,...
2
,1
,
,
:
)
(
)
ˆ
(
{
]
,
[
L
gU
g
G
g
ijg
ij
g
ij
g
iji
j
n
Which are said to be the continuous matrix-covered sets of
),
(
A
B
(
),
C
(
),
D
(
),
E
(
),
F
(
),
andG
(
)
. Here,),
ˆ
(
A
B
(
ˆ
),
C
(
ˆ
),
D
(
ˆ
),
E
(
ˆ
),
F
(
ˆ
)
andG
(
ˆ
)
are whitened (deterministic) matrices of),
(
A
B
(
),
C
(
),
D
(
),
E
(
),
F
(
)
andG
(
)
. Moreover,]
,
[
a
ija
ij ,[
b
ij,
b
ij]
,[
c
ij,
c
ij]
,[
d
ij,
d
ij]
,[
e
ij,
e
ij]
,]
,
[
f
ijf
ij and[
g
ij,
g
ij]
are said to be the number-covered sets of a
ij
,
bij,
cij,
dij,
ije,
ijf and
ijg.Definition 2.1. System(2.1)is said to be exponentially stable in mean square, if for all
L
2F0([
,
0
];
R
n)
and whitened matrices]
,
[
)
ˆ
(
L
aU
aA
,B
(
ˆ
)
[
L
b,
U
b]
,C
(
ˆ
)
[
L
c,
U
c]
,]
,
[
)
ˆ
(
L
dU
dD
,E
(
ˆ
)
[
L
e,
U
e]
,F
(
ˆ
)
[
L
f,
U
f]
,]
,
[
)
ˆ
(
L
gU
gG
,there exist scalars
r
0
andC
0
, such that0
,
)
(
sup
)
;
(
20
2
E
t
Ce
t
x
E
rt
.
Definition 2.2. System(2.1)is almost surely exponentially robustly stable, if for all
L
2F0([
,
0
];
R
n)
and whitened matrices]
,
[
)
ˆ
(
L
aU
aA
,B
(
ˆ
)
[
L
b,
U
b]
,C
(
ˆ
)
[
L
c,
U
c]
,]
,
[
)
ˆ
(
L
dU
dD
,E
(
ˆ
)
[
L
e,
U
e]
,F
(
ˆ
)
[
L
f,
U
f]
,]
,
[
)
ˆ
(
L
gU
gG
,the following inequality holds:
,
.
.
2
ˆ
)
;
(
ln
1
sup
lim
x
t
r
a
s
t
t
First, let us introduce the following lemmas, in particular, lemma 2.1, which will be important for the proof of our main results.Lemma 2.1. [21] If
A
(
)
(
aij)
mnis a grey m×n -matrix,]
,
[
a
ija
ij is a number-covered sets of grey element
aij, then for whitened matrixA
(
ˆ
)
[
L
a,
U
a]
, it follows thati)
A
(
ˆ
)
L
a
A
ii)0
A
U
a
L
a iii)A
(
ˆ
)
L
a
U
a
L
aWhere
L
a
(
a
ij)
mn,U
a
(
a
ij)
mn,
A
(
ijr
ˆ
ij)
mn,0
ij ijij
a
a
,r
ˆ
ij is a whitened number of
ij, and
ij is said to be a unit grey number.Lemma 2.2. [25] Let
x
,
y
R
n,P
R
nnis a symmetric positive definite matrix,M
,
N
R
nn, constants
0
, then one has the following inequality:PNy
N
y
PMx
M
x
PNy
M
x
T T T T 1 T T2
Lemma 2.3. [26] (Schur complement). Given constant
matrices
22 12
12 11
S
S
S
S
S
T , whereS
11
S
11T,S
22
S
22T , the following conditions are equivalent:i)
S
0
ii)
S
22
0
,S
11
S
12S
221S
12T
0
III. MAIN RESULTS AND PROOFS
Theorem 3.1. Let
k
L
g
U
g
L
g
1
, system (2.1) is exponentially stable in mean square, if there exist symmetric matricesP
0
,
Q
0
,
R
0
, and constants0
1
,
2
0
,
3
0
, such that0
0
0
0
0
6 5 3
5 4 2
3 2 1
J
M
M
T T T T
Where
n T
a
a
L
P
Q
R
k
I
PL
2 11
e T d g T
a
b
L
PL
L
PL
PL
2f T
d
c
L
PL
PL
3n g T b b T
g
PL
L
PL
k
I
L
Q
24
c T
g f T
e
PL
L
PL
L
5n
I
k
R
36
)
(
P
P
P
M
)
(
1I
n 2I
n 3I
ndiag
J
with
2 1
5 2 1
1
U
aL
ak
2 2
6 4 3 1
2
U
bL
bk
2 3
7 3 2
3
U
cL
ck
] )[
(
] )[
(
max max 1
e e d d
d e e e d d
a a g g
g a a a g g
L U L U
L L U L L U P
L U L U
L L U L L U P
] )[
(
max 2
f f d d
d f f f d d
L U L U
L L U L L U P
] )[
(
] )[
(
max max 3
f f e e
e f f f e e
c c g g
g c c c g g
L U L U
L L U L L U P
L U L U
L L U L L U P
] )[
(
max 4
b b g g
g b b b g g
L U L U
L L U L L U P
2 max
5
(
P
)(
L
d
U
d
L
d)
2 max
6
(
P
)(
L
e
U
e
L
e)
2 max
7
(
P
)(
L
f
U
f
L
f)
Then, for all
L
2F0([
,
0
];
R
n)
,rt
r
e
E
ke
k
R
Q
k
P
t
x
E
2
0
max 3 max
2 max
2
)
(
sup
)
1
)(
1
(
)
(
2
)
(
)
2
2
)(
(
)
;
(
Here,
r
satisfies the following inequalities:
0
))
(
2
)
(
(
)
(
)
1
(
)
)(
(
)(
)
(
)
0
(
1
,
0
max 2 max
max 2
max
max 2
max
R
Q
r
rk
r
P
r
rk
P
ke
r
r
with
6 5 3
5 4 2
3 2 3
1 2 1 1
:
T T
T
i i
P
Proof By applying Lemma 2.3, it follows that
0
0
0
0
0
6 5 3
5 4 2
3 2 1
J
M
M
T T T T
is equivalent to
0
:
6 5 3
5 4 2
3 2 3
1 2 1 1
T T
T i i
P
Fix
L
2F0([
,
0
];
R
n)
and whitened matrices]
,
[
)
ˆ
(
L
aU
aA
,B
(
ˆ
)
[
L
b,
U
b]
,C
(
ˆ
)
[
L
c,
U
c]
,]
,
[
)
ˆ
(
L
dU
dD
,E
(
ˆ
)
[
L
e,
U
e]
,F
(
ˆ
)
[
L
f,
U
f]
,]
,
[
)
ˆ
(
L
gU
gG
arbitrarily, and writex
(
t
,
)
=x
(
t
)
. Now, we use the similar methods in[23] to proof the Theorem. First, we choose a Lyapunov-Krasovskii functional candidate for system (2.1) as follows:)
),
(
(
)
),
(
(
)
),
(
(
)
),
(
(
)
),
(
(
4 3
2
1
x
t
t
V
x
t
t
V
x
t
t
V
x
t
t
V
t
t
x
V
Where
)]
(
)
ˆ
(
)
(
[
)]
(
)
ˆ
(
)
(
[
)
),
(
(
1
x
t
G
x
t
P
x
t
G
x
t
t
t
x
V
T
x
Qx
d
t
t
x
V
tt T
(
)
(
)
)
),
(
(
x
d
R
x
d
d
t
t
x
V
t tt t T
]
)
(
[
]
)
(
[
)
),
(
(
3
t
x
R
x
d
d
t
t
x
V
tt
T
(
)
(
)
)
(
[
)
),
(
(
04
By Itô’s formula and the definition of weak infinitesimal generator, we can obtain
)
(
]
)
(
)
ˆ
(
)
(
)
ˆ
(
)
(
)
ˆ
(
[
)]
(
)
ˆ
(
)
(
[
2
)
),
(
(
)
),
(
(
t
dW
ds
s
x
F
t
x
E
t
x
D
P
t
x
G
t
x
dt
t
t
x
LV
t
t
x
dV
t t T
(3.1)with
]
)
(
)
(
)
(
)
(
)
(
)
(
[
)]
(
)
(
)
(
[
2
)
),
(
(
ds
s
x
C
t
x
B
t
x
A
P
t
x
G
t
x
t
t
x
LV
t t T
P
ds
s
x
F
t
x
E
t
x
D
t Tt
(
)
]
)
ˆ
(
)
(
)
ˆ
(
)
(
)
ˆ
(
[
)
),
(
(
)
),
(
(
)
),
(
(
]
)
(
)
ˆ
(
)
(
)
ˆ
(
)
(
)
ˆ
(
[
4 32
x
t
t
V
x
t
t
V
x
t
t
V
ds
s
x
F
t
x
E
t
x
D
tt
(3.2) Here,)
(
)
(
)
(
)
(
)
),
(
(
2
t
Qx
t
x
t
Qx
t
x
t
t
x
V
T T]
)
(
[
]
)
(
[
)
(
]
)
(
[
]
)
(
[
)
(
)
),
(
(
3
d
x
R
d
x
d
t
x
R
d
x
d
d
x
R
t
x
t
t
x
V
t t t t T t t t T t t t T
]
)
(
[
]
)
(
[
)
(
)
(
)
(
)
(
)
(
)
(
d
x
R
d
x
d
t
x
R
x
t
d
x
R
t
x
t
t t t t T t t T t t T
]
)
(
[
]
)
(
[
)
(
)
(
)
(
)
(
)
(
)
(
d
x
R
d
x
d
x
R
x
t
d
t
x
R
t
x
t
t t t t T t t T t t T
and
t t T T t t T Td
Rx
x
t
t
Rx
t
x
d
d
x
R
x
d
t
Rx
t
x
t
t
x
V
)
(
)
(
)
(
)
(
)
(
2
)
(
)
(
[
)
(
)
(
)
),
(
(
2 0 0 4For convenience, let
z
t
tx
d
t
(
)
)
(
, then)
(
)
(
)
(
)
(
)
),
(
(
)
),
(
(
2 43
x
t
t
V
x
t
t
x
t
Rx
t
z
t
Rz
t
V
T
Tand (3.2) formula can be rewritten as
)]
(
)
ˆ
(
)
(
)
ˆ
(
)
(
)
ˆ
(
[
)]
(
)
ˆ
(
)
(
[
2
)
),
(
(
t
z
C
t
x
B
t
x
A
P
t
x
G
t
x
t
t
x
LV
T
P
t
z
F
t
x
E
t
x
D
(
ˆ
)
(
)
(
ˆ
)
(
)
(
ˆ
)
(
)]
T[
)]
(
)
ˆ
(
)
(
)
ˆ
(
)
(
)
ˆ
(
[
D
x
t
E
x
t
F
z
t
)
(
)
(
)
(
)
(
)
(
)
)(
(
2t
Rz
t
z
t
Qx
t
x
t
x
R
Q
t
x
T T T
)
(
)
(
)
(
)
(
)
(
)
)(
(
2t
Rz
t
z
t
Qx
t
x
t
x
R
Q
t
x
T T T
)
(
)
ˆ
(
)
(
2
x
Tt
PA
x
t
2
x
T(
t
)
PB
(
ˆ
)
x
(
t
)
)
(
)
ˆ
(
)
(
2
x
Tt
PC
z
t
)
(
)
ˆ
(
)
ˆ
(
)
(
2
x
Tt
G
T
PA
x
t
)
(
)
ˆ
(
)
ˆ
(
)
(
2
x
Tt
G
TPB
x
t
)
(
)
ˆ
(
)
ˆ
(
)
(
2
x
Tt
G
T
PC
z
t
)
(
)
ˆ
(
)
ˆ
(
)
(
t
D
PD
x
t
x
T T
)
(
)
ˆ
(
)
ˆ
(
)
(
x
Tt
D
TPE
x
t
)
(
)
ˆ
(
)
ˆ
(
)
(
t
D
PF
z
t
x
T T
)
(
)
ˆ
(
)
ˆ
(
)
(
t
E
PD
x
t
x
T
T
)
(
)
ˆ
(
)
ˆ
(
)
(
x
Tt
E
TPE
x
t
)
(
)
ˆ
(
)
ˆ
(
)
(
t
E
PF
z
t
x
T
T
)
(
)
ˆ
(
)
ˆ
(
)
(
t
F
PD
x
t
z
T T
)
(
)
ˆ
(
)
ˆ
(
)
(
z
Tt
F
TPE
x
t
)
(
)
ˆ
(
)
ˆ
(
)
(
t
F
PF
z
t
z
T T
(3.3)By using Lemma 2.1 and Lemma 2.2, we can get
)
(
)
(
2
)
(
)
)(
(
)
(
)
ˆ
(
)
(
2
t
Ax
P
t
x
t
x
P
L
PL
t
x
t
x
PA
t
x
T T a a T T
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2 1 2 11
x
t
P
x
t
U
L
x
t
x
t
t
Px
L
t
x
t
x
PL
t
x
T a a T T a T a T
(3.4)Similar to (3.4), we have
and
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
ˆ
(
)
(
2
2 3
2 1
3
x
t
P
x
t
U
L
z
t
z
t
t
Px
L
t
z
t
z
PL
t
x
t
z
PC
t
x
T c c T
T c T c
T T
(3.6)
Moreover, by Lemma 2.1 and Lemma 2.2, we have
)
(
)
(
2
)
(
)
(
2
)
(
)
(
2
)
(
)
(
)
(
)
(
)
(
)
ˆ
(
)
ˆ
(
)
(
2
t
Ax
P
G
t
x
t
Ax
P
L
t
x
t
x
PL
G
t
x
t
x
PL
L
t
x
t
x
PL
L
t
x
t
x
PA
G
t
x
T T
T g T
a T T
g T
a T
a T
g T
T T
(3.7)
Where
)
(
)
(
2
)
(
)
(
2
t
x
t
x
L
P
G
t
x
PL
G
t
x
T a T
a T T
)]
(
)
(
)
(
)
(
[
)
(
max
P
U
L
L
x
Tt
x
t
x
Tt
x
t
a g g
(3.8)
)]
(
)
(
)
(
)
(
[
)
(
)
(
)
(
2
max
t
x
t
x
t
x
t
x
L
L
U
P
t
Ax
P
L
t
x
T
T g a a
T g T
(3.9)
and
)]
(
)
(
)
(
)
(
[
)
(
)
(
)
(
2
max
t
x
t
x
t
x
t
x
L
U
L
U
P
t
Ax
P
G
t
x
T
T a a g g
T T
(3.10)
Combining with (3.7)-(3.10) and computing them, we see
)]
(
)
(
)
(
)
(
[
]
)[
(
)
(
)
(
)
(
)
(
)
(
)
ˆ
(
)
ˆ
(
)
(
2
max
t
x
t
x
t
x
t
x
L
U
L
U
L
L
U
L
L
U
P
t
x
PL
L
t
x
t
x
PL
L
t
x
t
x
PA
G
t
x
T T
a a g g
g a a
a g g g T a T
a T g T
T T
(3.11)
Using the similar method as in (3.11), the following inequalities hold:
)
(
)
(
]
)[
(
)
(
)
(
)
(
)
(
)
(
)
ˆ
(
)
ˆ
(
)
(
2
max
t
x
t
x
L
U
L
U
L
L
U
L
L
U
P
t
x
PL
L
t
x
t
x
PL
L
t
x
t
x
PB
G
t
x
T b b g g
g b b b g g
g T b T
b T
g T
T T
(3.12)
c g g g T c T
c T
g T
T T
L
L
U
P
t
x
PL
L
t
z
t
z
PL
L
t
x
t
z
PC
G
t
x
)[
(
)
(
)
(
)
(
)
(
)
(
)
ˆ
(
)
ˆ
(
)
(
2
max
)]
(
)
(
)
(
)
(
[
]
t
x
t
x
t
z
t
z
L
U
L
U
L
L
U
T T
c c g g g c c
(3.13)
)]
(
)
(
)
(
)
(
[
]
)[
(
2
1
)
(
2
1
)
(
)
(
2
1
)
(
)
(
)
ˆ
(
)
ˆ
(
)
(
max
t
x
t
x
t
x
t
x
L
U
L
U
L
L
U
L
L
U
P
t
x
PL
L
t
x
t
x
PL
L
t
x
t
x
PE
D
t
x
T T
e e d d d e e
e d d
d T e T
e T d T
T T
(3.14)
)]
(
)
(
)
(
)
(
[
]
)[
(
2
1
)
(
2
1
)
(
)
(
2
1
)
(
)
(
)
ˆ
(
)
ˆ
(
)
(
max
t
z
t
z
t
x
t
x
L
U
L
U
L
L
U
L
L
U
P
t
x
PL
L
t
z
t
z
PL
L
t
x
t
z
PF
D
t
x
T T
f f d d d f f
f d d
d T
f T f
T d T
T T
(3.15))]
(
)
(
)
(
)
(
[
]
)[
(
2
1
)
(
2
1
)
(
)
(
2
1
)
(
)
(
)
ˆ
(
)
ˆ
(
)
(
max
t
x
t
x
t
x
t
x
L
U
L
U
L
L
U
L
L
U
P
t
x
PL
L
t
x
t
x
PL
L
t
x
t
x
PD
E
t
x
T T
d d e e e d d
d e e
e T
d T d
T e T
T T
