Abstract—In this paper, we mainly investigate the stability problem for a class of grey stochastic systems with time delays.
The parameters of system are evaluated by grey numbers.
Firstly, we construct a suitable Lyapunov-Krasovskii functional.
Then, by using Itô’s differential formulation and decomposition technique, some sufficient conditions are derived to ensure the grey system in the mean-square exponential stability and almost surely exponential stability. Finally, an example to illustrative the effectiveness of main results is also given.
Index Terms—Grey Stochastic Systems, Time Delays, Lyapunov-Krasovskii Functional, Decomposition Technique, Mean-Square Exponential Stability, Almost Surely Exponential Stability
I. INTRODUCTION
ow, it has been well recognized that, time delays are unavoidably encountered in many physical systems, which are the main sources of oscillation, bifurcation, or performance degradation [1]. For example, on account of finite switching speed of neurons and amplifiers, time delays may lead to instability and oscillation in a neurons network.
So, the stability problem of time-delay systems has attracted considerable attention over decades, many important results have been presented in the literature [2-11]. In [3], by using a Lyapunov-Krasovskii functional, the authors have provided novel delay-dependent conditions in terms of linear matrix inequalities for a class of neutral BAM neural networks with time-varying delays.
In addition to time-delay effects, stochastic effects are another sources of instability and uncertainties in many systems. For instance, stochastic phenomenon frequently occurs in the electrical circuit design of neural networks, and the effects of stochastic phenomenon should be taken into account [12-13]. In recent years, the stability analysis of stochastic systems has been an attractive topic for many scholars, and a large amount of results have been reported [14-19]. In [17], by using the Razumikhin-type theorem, the authors have considered the stability problem of stochastic interval systems, and several sufficient conditions have been proposed.
Also, due to slowly varying parameters, modeling errors, or unknown uncertainties, it is impossible to obtain some parameters of stochastic systems accurately. In [20], the authors have pointed out that, if the parameters are evaluated by grey numbers, the systems can become grey (uncertain)
Manuscript received January 25, 2016; revised April 27, 2016. The research is supported by the National Natural Science Foundation of China (No.11301009) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province (No.16B110001).
Jian Wang is with School of Mathematics and Statistics, Anyang Normal University, Anyang 455002, PR China e-mail: [email protected]
systems. So, it is significant and important to discuss the stability problem for grey stochastic systems. However, to date, there have been very few works on this problem [20-24].
In [20], the authors have investigated the exponential stability for the grey stochastic systems with distributed delays and interval parameters, and the delay-dependent criteria have been obtained to ensure the systems in p-moment exponential robust stability. In [24], the authors have studied the robust stability problem for the grey stochastic nonlinear systems with distributed delays, and have proposed several novel conditions in terms of linear matrix inequalities.
In this paper, the stability problem for a class of grey stochastic time-delay systems is studied. By constructing a suitable Lyapunov-Krasovskii functional and using Itô’s differential formulation, particularly, using decomposition technique of the continuous matrix-covered sets [20-24], we will obtain several novel exponential stability criteria, which ensure the grey system in mean-square exponential stability and almost surely exponential stability. Moreover, an example is given to illustrate the effectiveness of the stability criteria.
The notations are standard. Rn and Rnn denote, the n-dimensional Euclidean space and the set of real n×n matrices. The superscript"T" denotes the transpose, will refer to the Euclidean norm for vector or the spectral norm of matrices. For real symmetric matrices X andY, the notation
Y
X (respectively, X Y ) means that X Y is positive semi-definite (respectively, positive definite). While,
,F, Ft t0,P is a probability space with a filtration
Ft t0 , and C[,0];Rn denotes the family of all continuous Rn-valued functions on[ ,0]. Denote by
) ];
0 ,
2 ([
0
n
F R
L the family of allF0-measurable bounded
Rn
C[,0]; -valued random variables():0.
II. PRELIMINARIES AND PROBLEM FORMULATION
In this section, we consider the following grey stochastic time-delay system:
On Robust Exponential Stability
for Grey Stochastic Time-Delay Systems
Jian Wang
N
IAENG International Journal of Applied Mathematics, 46:4, IJAM_46_4_09
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0 ),
];
0 , ([
,
0 ), ( ), ( ), (
] ) ( ) (
) ( ) ( ) ( ) ( [ ) (
2 0
0
0 R t
L x
t t dw t t x t x f
dt ds s t x s G
t x B t x A t dx
n F
(1)
whereA()andB()are grey (uncertain) n×n matrices.
LetA()(aij),B()(bij), and aij,bij are grey elements ofA()andB().
} ,...
2 , 1 , , :
) ( ˆ) ( { ] ,
[La Ua A aij aij aij aij i j n } ,...
2 , 1 , , :
) ( ˆ) ( { ] ,
[Lb Ub B bij bij bij bij i j n
are the continuous matrix-covered sets ofA()andB(). where A(ˆ) and B(ˆ) are whitened (deterministic) matrices of A()andB(), [aij,aij] and [bij,bij] are the number-covered sets of aij and bij.
For system (1), the following assumptions are given:
(A1) H:RnRn R Rnn, and satisfies the local Lipschitz condition.
(A2) Assume that there exist constants 0, 0, for arbitrary (x,y,t)H:Rn RnR , the following inequality holds:
, , , , ] 2 2 [f x y t f x y t x y
Trace T
The following definitions and lemmas are introduced:
Definition 2.1. [20] System (1) is said to be mean-square exponentially stability, if for all 2 ([ ,0]; )
0
n
F R
L
and
whitened matrices A(ˆ)[La,Ua], B(ˆ)[Lb,Ub], there exist constants r0andC0, such that
0 , ) ( sup )
;
( 2
0
2
E t
Ce t
x
E rt
Definition 2.2. [20] System (1) is said to be almost surely exponential stability, if for all 2 ([ ,0]; )
0
n
F R
L
and
whitened matricesA(ˆ)[La,Ua], B(ˆ)[Lb,Ub], there exists constant rˆ 0, such that
. . 2, ) ˆ
; ( 1ln sup
lim r as
t t x
t
Lemma 2.1. [20] If A()(ij)mn is a grey matrix,
] ,
[aij aij is a number-covered sets of ij, then for arbitrary whitened matrixA(ˆ)[La,Ua], we have
i) U L A
A a a
ˆ) 2
(
ii) 0 Ua2La
A
iii)
2 ) 2
(ˆ Ua La Ua La
A
where La (aij)mn,Ua(aij)mn, aij aij rij mn
A
ˆ)
( 2 ,
andrˆij is a whitened number ofij,rˆij[1,1], [1,1]is a number-covered sets ofij, andij is a unit grey number.
Lemma 2.2. [25] Letx, yRn,PRnnis a symmetric positive definite matrix, M,NRnn, constant 0, we have
PNy N y PMx M x PNy M
xT T T T 1 T T
2
Lemma 2.3. [26] (Schur complement). Given constant
matrices
22 12
12 11
S S
S
S ST , whereS11 S11T,S22 S22T , the following conditions are equivalent:
i) S 0
ii) S22 0, S11 S12S221S12T 0
III. MAIN RESULTS AND PROOFS
In this section, several sufficient conditions are proposed to guarantee system (1) in mean-square exponential stability and almost surely exponential stability.
Theorem 3.1. System (1) is mean-square exponentially stability, if there exist symmetric matricesP0,Q0,R0 and constants 10,20, such that
2
0 2
min(R) sup G(s)
s
(2)
and
0 0
2 0
2
2 1
J M
L P U
L M PU
T T b T b
b b
(3)
where
L P U L PU R Q
T a T a a a
2
1 2
n
a L I
P U a ]
2 2 [ )
max(
IAENG International Journal of Applied Mathematics, 46:4, IJAM_46_4_09
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n
b L P I
Q U b ( ) ]
[ 2 max
2 1
2
，
) (PP
M ，J diag(1In 2In). Then, for all 2 ([ ,0]; )
0
n
F R
L
, we have
)2
, ( t x
E
rt r
r
P e
R e
r Q e
r
P
) (
) ( ) (
) ( ) (
) (
min
max 2 3 max
2 max
2
0
) ( sup
E ， t0 (4) Here, ris the unique positive solution of the following equation
)]
( )
( [ )
( max max
max P r e Q R
r rt
0 )
max(
(5)
with 0
2 : 2
2 2
1 2 1 1
L P U
L PU P
T b T b
b b i
i
(6)
and 1 r0.
Proof By Lemma 2.3, it follows that
0 0
2 0
2
2 1
J M
L P U
L M PU
T T b T b
b b
is equivalent to
0 2
: 2
2 2
1 2 1 1
L P U
L PU P
T b T b
b b i
i
For convenience, let x( t, )=x(t).
Firstly, we construct a Lyapunov-Krasovskii functional:
) ), ( ( ) ), ( ( ) ), ( ( ) ), (
(x t t V1 x t t V2 x t t V3 x t t
V (7)
where
) ( ) ( ) ), (
1(x t t x t Px t
V T
ds s t Qx s t x t t x
V2( ( ), )0 T( ) ( )
x s Rx s dsd t
t x
V t
t
T( ) ( )
[ ) ), (
( 0
3
By using Itô’s differential formula, we have
) ), ( (x t t LV
) ( ) )(
(t Q R x t xT
) ( )
(
xT t Qx t
0xT(t s)Rx(t s)ds ) ( ˆ) ( ) (
2xT t PA x t
) ( ˆ) ( ) (
2
xT t PB x t
2xT(t)P 0G(s)x(t s)ds
, ( ), , ( ), ]
[f x x t t Pf x x t t
Trace T
(8)
Obviously, we see
0xT(t s)Rx(t s)ds
0
2
min(R) x(t s) ds (9) By Lemma 2.1 and Lemma 2.2, we can get
) ( ˆ) ( ) (
2xT t PA x t
) ( 2 )
)( 2
( U L U L P x t
P t x
T a T a a
T a
) ( ) (
2xT t PAx t
) ( 2 )
)( 2
( U L U L P x t
P t x
T a T a a
T a
) ( ) 2 (
2 )
max( U L x t x t
P a a T
(10)
and
) ( ˆ) ( ) (
2xT t PB x t ) ( 2 ) )(
(
U L x t
P t
xT b b
) ( 2 )
)(
( U L P x t
t x
T b T
T b
) ( ) ( 2
1
1 xT t P x t
) ( ) 2 (
2
1
Ub Lb xT t x t
(11) and
0 ( ) ( ) )
(
2xT t P G s x t s ds )
( ) ( 2
1
2 xT t P x t
) ) ( ) ( ( ) ) ( ) (
( 0 0
2
G s x t s ds T G s x t s ds (12) and
) ) ( ) ( ( ) ) ( ) (
(0G s x ts ds T 0G s x ts ds
IAENG International Journal of Applied Mathematics, 46:4, IJAM_46_4_09
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