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A constructive way to design a switching rule and switching regions to mean square exponential stability of switched stochastic systems with non-differentiable and interval time-varying delay

Abstract

This paper addresses a mean square exponential stability problem for a class of switched stochastic systems with time-varying delay. The time delay is any continuous function belonging to a given interval, but not necessary differentiable. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for the mean square exponential stability of switched stochastic systems with time-varying delay are first established in terms of LMIs. Numerical example is given to show the effectiveness of the obtained result.

MSC:15A09, 52A10, 74M05, 93D05.

1 Introduction

In the past decades, the problem of stability for neutral differential systems, which have delays in both their state and the derivatives of their states, has been widely investigated by many researchers. Such systems are often encountered in engineering, biology, and economics. The existence of time delay is frequently a source of instability or poor performance in the systems. Recently, some stability criteria for a neutral system with time delay have been given [125]. Stability analysis of linear systems with time-varying delays x ˙ (t)=Ax(t)+Dx(th(t)) is fundamental to many practical problems and has received considerable attention [17]. In [817], which are not based on the method of Lyapunov functional, one of them uses the diagonal equations for reducing systems of delay differential equations to ones of integral equations and estimates the norms or spectral radii of corresponding integral operators obtained on the basis of the results in the book. Most of the known results on this problem are derived assuming only that the time-varying delay h(t) is a continuously differentiable function, satisfying some boundedness condition on its derivative: h ˙ (t)δ<1. In delay-dependent stability criteria, the main concern is to enlarge the feasible region of stability criteria in a given time-delay interval. Interval time-varying delay means that a time delay varies in an interval in which the lower bound is not restricted to be zero. By constructing a suitable argument, Lyapunov functional and utilizing free weight matrices, some less conservative conditions for asymptotic stability are derived in [1824] for systems with time delay varying in an interval. However, the shortcoming of the method used in these works is that the delay function is assumed to be differential and its derivative is still bounded: h ˙ (t)δ. To the best of our knowledge, a constructive way to design a switching rule, switching regions, and mean square exponential stability of switched stochastic systems with interval time-varying delay, non-differentiable time-varying delays, which are important in both theory and applications, have not been fully studied yet (see, e.g., [2538] and the references therein). This motivates our research.

This paper gives the improved results for the mean square exponential stability of switched stochastic systems with interval time-varying delay. The time delay is assumed to be a time-varying continuous function belonging to a given interval, but not necessary differentiable. Specifically, our goal is to develop a constructive way to design a switching rule to exponential stability of switched stochastic systems with interval time-varying delay. By constructing a Lyapunov functional combined with the LMI technique, we propose new criteria for the mean square exponential stability of switched stochastic systems with interval time-varying delay. The delay-dependent mean square exponential stability conditions are formulated in terms of LMIs, being thus solvable by utilizing Matlab’s LMI control toolbox available in the literature to date.

The paper is organized as follows. Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results. Delay-dependent mean square exponential stability conditions of switched stochastic systems with interval time-varying delay are presented in Section 3. Numerical example is provided to illustrate the theoretical results in Section 4, and the conclusions are drawn in Section 5.

2 Preliminaries

The following notations will be used in this paper. R + denotes the set of all real non-negative numbers; R n denotes the n-dimensional space with the scalar product , and the vector norm ; M n × r denotes the space of all matrices of (n×r)-dimensions; A T denotes the transpose of matrix A; A is symmetric if A= A T ; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λ min / max (A)=min/max{Reλ;λλ(A)}; x t :={x(t+s):s[h,0]}, x t = sup s [ h , 0 ] x(t+s); C([0,t], R n ) denotes the set of all R n -valued continuous functions on [0,t]; matrix A is called semi-positive definite (A0) if Ax,x0 for all x R n ; A is positive definite (A>0) if Ax,x>0 for all x0; A>B means AB>0. denotes the symmetric term in a matrix.

Consider a switched stochastic system with interval time-varying delay of the form

x ˙ ( t ) = A γ ( x ( t ) ) x ( t ) + D γ ( x ( t ) ) x ( t h ( t ) ) + σ γ ( x ( t ) ) ( x ( t ) , x ( k h ( t ) ) , t ) ω ( t ) , t R + , x ( t ) = ϕ ( t ) , t [ h 2 , 0 ] ,
(2.1)

where x(t) R n is the state; γ(): R n N:={1,2,,N} is the switching rule, which is a function depending on the state at each time and will be designed. A switching function is a rule which determines a switching sequence for a given switching system. Moreover, γ(x(t))=i implies that the system realization is chosen as the i th system, i=1,2,,N. It is seen that system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state x(t) hits predefined boundaries. A i , D i M n × n , i=1,2,,N, are given constant matrices, and ϕ(t)C([ h 2 ,0], R n ) is the initial function with the norm ϕ= sup s [ h 2 , 0 ] ϕ(s).

ω(k) is a scalar Wiener process (Brownian motion) on (Ω,F,P) with

E { ω ( t ) } =0,E { ω 2 ( t ) } =1,E { ω ( i ) ω ( j ) } =0(ij),
(2.2)

and σ i : R n × R n ×R R n , i=1,2,,N, is the continuous function, and it is assumed to satisfy that

σ i T ( x ( t ) , x ( t h ( t ) ) , t ) σ i ( x ( t ) , x ( t h ( t ) ) , t ) ρ i 1 x T ( t ) x ( t ) + ρ i 2 x T ( t h ( t ) ) x ( t h ( t ) ) , x ( t ) , x ( t h ( t ) ) R n ,
(2.3)

where ρ i 1 >0 and ρ i 2 >0, i=1,2,,N, are known constant scalars. For simplicity, we denote σ i (x(t),x(th(t)),t) by σ i , respectively.

The time-varying delay function h(t) satisfies

0 h 1 h(t) h 2 ,t R + .

The mean square stability problem for switched stochastic system (2.1) is to construct a switching rule that makes the system mean square exponentially stable.

Definition 2.1 Given α>0. Switched stochastic system (2.1) is α-exponentially stable in the mean square if there exists a switching rule γ() such that every solution x(t,ϕ) of the system satisfies the following condition:

N>0:E { x ( t , ϕ ) } E { N e α t ϕ } ,t R + .

Definition 2.2 The system of matrices { J i }, i=1,2,,N, is said to be strictly complete if for every x R n {0}, there is i{1,2,,N} such that x T J i x<0.

It is easy to see that the system { J i } is strictly complete if and only if

i = 1 N α i = R n {0},

where

α i = { x R n : x T J i x < 0 } ,i=1,2,,N.

We end this section with the following technical well-known propositions, which will be used in the proof of the main results.

Proposition 2.1 [39]

The system { J i }, i=1,2,,N, is strictly complete if there exist δ i 0, i=1,2,,N, i = 1 N δ i >0 such that

i = 1 N δ i J i <0.

If N=2, then the above condition is also necessary for the strict completeness.

Proposition 2.2 (Cauchy inequality)

For any symmetric positive definite matrix N M n × n and a,b R n , we have

± a T b a T Na+ b T N 1 b.

Proposition 2.3 [40]

For any symmetric positive definite matrix M M n × n , scalar μ>0 and vector function ω:[0,μ] R n such that the integrations concerned are well defined, the following inequality holds:

( 0 μ ω ( s ) d s ) T M ( 0 μ ω ( s ) d s ) μ ( 0 μ ω T ( s ) M ω ( s ) d s ) .

Proposition 2.4 [[41], p.89-90]

Let E, H and F be any constant matrices of appropriate dimensions and F T FI. For any ϵ>0, we have

EFH+ H T F T E T ϵE E T + ϵ 1 H T H.

Proposition 2.5 (Schur complement lemma [42])

Given constant matrices X, Y, Z with appropriate dimensions satisfying X= X T ,Y= Y T >0. Then X+ Z T Y 1 Z<0 if and only if

( X Z T Z Y ) <0or ( Y Z Z T X ) <0.

3 Main results

In this section, we investigate the mean square exponential stability problem for a class of switched stochastic systems (2.1) with time-varying delay. Before introducing the main result, the following notations of several matrix variables are defined for simplicity,

M i = [ M 11 M 12 M 13 M 14 M 15 M 22 0 M 24 M 25 M 33 M 34 M 35 M 44 M 45 M 55 ] , M 11 = A i T P + P A i + 2 α P e 2 α h 1 R M 11 = e 2 α h 2 R + Q + 2 ρ i 1 I , M 12 = e 2 α h 1 R S 2 A i , M 13 = e 2 α h 2 R S 3 A i , M 14 = P D i S 1 D i S 4 A i , M 15 = S 1 S 5 A i , M 22 = e 2 α h 1 Q e 2 α h 1 R e 2 α h 2 U , M 24 = e 2 α h 2 U S 2 D i , M 25 = S 2 , M 33 = e 2 α h 2 Q e 2 α h 2 R e 2 α h 2 U , M 34 = e 2 α h 2 U S 3 D i , M 35 = S 3 , M 44 = 2 S 4 D i 2 e 2 α h 2 U + 2 ρ i 2 I , M 45 = S 4 S 5 D i , M 55 = S 5 + S 5 T + h 1 2 R + h 2 2 R + ( h 2 h 1 ) 2 U , J i = Q S 1 A i A i T S 1 T , α i = { x R n : x T J i x < 0 } , i = 1 , 2 , , N , α ¯ 1 = α 1 , α ¯ i = α i j = 1 i 1 α ¯ j , i = 2 , 3 , , N , λ 1 = λ min ( P ) , λ 2 = λ max ( P ) + 2 h 2 λ max ( Q ) + 2 h 2 2 λ max ( R ) λ 2 = + ( h 2 h 1 ) 2 λ max ( U ) .
(3.1)

The following is the main result of the paper, which gives sufficient conditions for mean square exponential stability problem for a class of switched stochastic systems (2.1) with time-varying delay.

Theorem 3.1 Given α>0. The zero solution of switched stochastic system (2.1) is α-exponentially stable in the mean square if there exist symmetric positive definite matrices P, Q, R, U, and matrices S i , i=1,2,,5, satisfying the following conditions:

  1. (i)

    δ i 0, i=1,2,,N, i = 1 N δ i >0: i = 1 N δ i J i <0,

  2. (ii)

    M i <0, i=1,2,,N.

The switching rule is chosen as γ(x(t))=i, whenever x(t) α ¯ i . Moreover, the solution x(t,ϕ) of the switched stochastic system satisfies

E { x ( t , ϕ ) } E { λ 2 λ 1 e α t ϕ } ,t R + .

Proof We consider the following Lyapunov-Krasovskii functional for system (2.1):

E { V ( t , x t ) } =E { i = 1 6 V i } ,

where

V 1 = x T ( t ) P x ( t ) , V 2 = t h 1 t e 2 α ( s t ) x T ( s ) Q x ( s ) d s , V 3 = t h 2 t e 2 α ( s t ) x T ( s ) Q x ( s ) d s , V 4 = h 1 h 1 0 t + s t e 2 α ( τ t ) x ˙ T ( τ ) R x ˙ ( τ ) d τ d s , V 5 = h 2 h 2 0 t + s t e 2 α ( τ t ) x ˙ T ( τ ) R x ˙ ( τ ) d τ d s , V 6 = ( h 2 h 1 ) t h 2 t h 1 t + s t e 2 α ( τ t ) x ˙ T ( τ ) U x ˙ ( τ ) d τ d s .

It easy to check that

E { λ 1 x ( t ) 2 } E { V ( t , x t ) } E { λ 2 x t 2 } ,t0.
(3.2)

Taking the derivative of V 1 along the solution of system (2.1) and taking the mathematical expectation, we obtain

E { V ˙ 1 } = E { 2 x T ( t ) P x ˙ ( t ) } E { V ˙ 1 } = E { x T ( t ) [ A i T P + A i P ] x ( t ) + 2 x T ( t ) P D i x ( t h ( t ) ) + 2 x T ( t ) P σ i ω ( t ) } ; E { V ˙ 2 } = E { x T ( t ) Q x ( t ) e 2 α h 1 x T ( t h 1 ) Q x ( t h 1 ) 2 α V 2 } ; E { V ˙ 3 } = E { x T ( t ) Q x ( t ) e 2 α h 2 x T ( t h 2 ) Q x ( t h 2 ) 2 α V 3 } ; E { V ˙ 4 } = E { h 1 2 x ˙ T ( t ) R x ˙ ( t ) h 1 t h 1 t e 2 α ( τ t ) x ˙ T ( s ) R x ˙ ( s ) d s 2 α V 4 } E { V ˙ 4 } E { h 1 2 x ˙ T ( t ) R x ˙ ( t ) h 1 e 2 α h 1 t h 1 t x ˙ T ( s ) R x ˙ ( s ) d s 2 α V 4 } ; E { V ˙ 5 } = E { h 2 2 x ˙ T ( t ) R x ˙ ( t ) h 2 t h 2 t e 2 α ( τ t ) x ˙ T ( s ) R x ˙ ( s ) d s 2 α V 5 } E { V ˙ 5 } E { h 2 2 x ˙ T ( t ) R x ˙ ( t ) h 2 e 2 α h 2 t h 2 t x ˙ T ( s ) R x ˙ ( s ) d s 2 α V 5 } ; E { V ˙ 6 } E { ( h 2 h 1 ) 2 x ˙ T ( t ) U x ˙ ( t ) ( h 2 h 1 ) e 2 α h 2 t h 2 t h 1 x ˙ T ( s ) U x ˙ ( s ) d s 2 α V 6 } .

Applying Proposition 2.2 and the Leibniz-Newton formula, we have

E { h i t h i t x ˙ T ( s ) R x ˙ ( s ) d s } E { [ t h i t x ˙ ( s ) d s ] T R [ t h i t x ˙ ( s ) d s ] } E { [ x ( t ) x ( t h i ) ] T R [ x ( t ) x ( t h i ) ] } = E { x T ( t ) R x ( t ) + 2 x T ( t ) R x ( t h i ) x T ( t h i ) R x ( t h i ) } .

Note that

E { t h 2 t h 1 x ˙ T ( s ) U x ˙ ( s ) d s } =E { t h 2 t h ( t ) x ˙ T ( s ) U x ˙ ( s ) d s + t h ( t ) t h 1 x ˙ T ( s ) U x ˙ ( s ) d s } .

Using Proposition 2.2 gives

E { [ h 2 h ( t ) ] t h 2 t h ( t ) x ˙ T ( s ) U x ˙ ( s ) d s } E { [ t h 2 t h ( t ) x ˙ ( s ) d s ] T U [ t h 2 t h ( t ) x ˙ ( s ) d s ] } E { [ x ( t h ( t ) ) x ( t h 2 ) ] T U [ x ( t h ( t ) ) x ( t h 2 ) ] } .

Since h 2 h(t) h 2 h 1 , we have

E { [ h 2 h 1 ] t h 2 t h ( t ) x ˙ T ( s ) U x ˙ ( s ) d s } E { [ x ( t h ( t ) ) x ( t h 2 ) ] T U [ x ( t h ( t ) ) x ( t h 2 ) ] } ,

then

E { ( h 2 h 1 ) t h 2 t h ( t ) x ˙ T ( s ) U x ˙ ( s ) d s } E { [ x ( t h ( t ) ) x ( t h 2 ) ] T U [ x ( t h ( t ) ) x ( t h 2 ) ] } .

Similarly, we have

E { ( h 2 h 1 ) t h ( t ) t h 1 x ˙ T ( s ) U x ˙ ( s ) d s } E { [ x ( t h 1 ) x ( t h ( t ) ) ] T U [ x ( t h 1 ) x ( t h ( t ) ) ] } .

Therefore, we have

E { V ˙ ( ) + 2 α V ( ) } E { x T ( t ) [ A i T P + A i P + 2 α P + 2 Q ] x ( t ) } + E { 2 x T ( t ) P D i x ( t h ( t ) ) + 2 x T ( t ) P σ i ω ( t ) } + E { e 2 α h 1 x T ( t h 1 ) Q x ( t h 1 ) } + E { e 2 α h 2 x T ( t h 2 ) Q x ( t h 2 ) } + E { x ˙ T ( t ) [ ( h 1 2 + h 2 2 ) R + ( h 2 h 1 ) 2 U ] x ˙ ( t ) } + E { e 2 α h 1 [ x ( t ) x ( t h 1 ) ] T R [ x ( t ) x ( t h 1 ) ] } + E { e 2 α h 2 [ x ( t ) x ( t h 2 ) ] T R [ x ( t ) x ( t h 2 ) ] } + E { e 2 α h 2 [ x ( t h ( t ) ) x ( t h 2 ) ] T U [ x ( t h ( t ) ) x ( t h 2 ) ] } + E { e 2 α h 2 [ x ( t h 1 ) x ( t h ( t ) ) ] T U [ x ( t h 1 ) x ( t h ( t ) ) ] } .
(3.3)

By using the following identity relation

x ˙ (t) A i x(t) D i x ( t h ( t ) ) =0,

and multiplying by 2 x T (t) S 1 , 2 x T (t h 1 ) S 2 , 2 x T (t h 2 ) S 3 , 2 x T (th(t)) S 4 , 2 x ˙ T (t) S 5 , 2 ω T (t) σ i T both sides of the identity relation, we have

2 x T ( t ) S 1 x ˙ ( t ) 2 x T ( t ) S 1 A i x ( t ) 2 x T ( t ) S 1 D i x ( t h ( t ) ) 2 x T ( t ) S 1 σ i ω ( t ) = 0 , 2 x T ( t h 1 ) S 2 x ˙ ( t ) 2 x T ( t h 1 ) S 2 A i x ( t ) 2 x T ( t h 1 ) S 2 D i x ( t h ( t ) ) 2 x T ( t h 1 ) S 2 σ i ω ( t ) = 0 , 2 x T ( t h 2 ) S 3 x ˙ ( t ) 2 x T ( t h 2 ) S 3 A i x ( t ) 2 x T ( t h 2 ) S 3 D i x ( t h ( t ) ) 2 x T ( t h 2 ) S 3 σ i ω ( t ) = 0 , 2 x T ( t h ( t ) ) S 4 x ˙ ( t ) 2 x T ( t h ( t ) ) S 4 A i x ( t ) 2 x T ( t h ( t ) ) S 4 D i x ( t h ( t ) ) 2 x T ( t h ( t ) ) S 4 σ i ω ( t ) = 0 , 2 x ˙ T ( t ) S 5 x ˙ ( t ) 2 x ˙ T ( t ) S 5 A i x ( t ) 2 x ˙ T ( t ) S 5 D i x ( t h ( t ) ) 2 x ˙ T ( t ) S 5 σ i ω ( t ) = 0 , 2 ω T ( t ) σ i T x ˙ ( t ) 2 ω T ( t ) σ i T A i x ( t ) 2 ω T ( t ) σ i T D i x ( t h ( t ) ) 2 ω T ( t ) σ i T σ i ω ( t ) = 0 .
(3.4)

Adding all the zero items of (3.4) into (3.3), we obtain

E { V ˙ ( ) + 2 α V ( ) } E { x T ( t ) [ A i T P + P A i + 2 α P e 2 α h 1 R ] x ( t ) } + E { x T ( t ) [ e 2 α h 2 R + S 1 A i + A i T S 1 T + 2 Q ] x ( t ) } + E { 2 x T ( t ) [ e 2 α h 1 R S 2 A i ] x ( t h 1 ) } + E { 2 x T ( t ) [ e 2 α h 2 R S 3 A i ] x ( t h 2 ) } + E { 2 x T ( t ) [ P D i S 1 D i S 4 A i ] x ( t h ( t ) ) } + E { 2 x T ( t ) [ S 1 S 5 A i ] x ˙ ( t ) } + E { 2 x T ( t ) [ P σ i S 1 σ i A i T σ i ] ω ( t ) } + E { x T ( t h 1 ) [ e 2 α h 1 Q e 2 α h 1 R e 2 α h 2 U ] x ( t h 1 ) } + E { 2 x T ( t h 1 ) [ e 2 α h 2 U S 2 D i ] x ( t h ( t ) ) } + E { 2 x T ( t h 1 ) S 2 x ˙ ( t ) } + E { 2 x T ( t h 1 ) [ S 2 σ i ] ω ( t ) } + E { x T ( t h 2 ) [ e 2 α h 2 Q e 2 α h 2 R e 2 α h 2 U ] x ( t h 2 ) } + E { 2 x T ( t h 2 ) [ e 2 α h 2 U S 3 D i ] x ( t h ( t ) ) } + E { 2 x T ( t h 2 ) S 3 x ˙ ( t ) } + E { 2 x T ( t h 2 ) [ S 3 σ i ] ω ( t ) } + E { x T ( t h ( t ) ) [ 2 e 2 α h 2 U 2 S 4 D i ] x ( t h ( t ) ) } + E { 2 x T ( t h ( t ) ) [ S 4 S 5 D i ] x ˙ ( t ) } + E { 2 x T ( t h ( t ) ) [ S 4 σ i σ i T D i ] ω ( t ) } + E { x ˙ T ( t ) [ S 5 + S 5 T + h 1 2 R + h 2 2 R + ( h 2 h 1 ) 2 U ] x ˙ ( t ) } + E { 2 x ˙ T ( t ) [ σ i T S 5 σ i ] ω ( t ) } + E { 2 ω T ( t ) [ σ i T σ i ] ω ( t ) } .

By assumption (2.2), we have

E { V ˙ ( ) + 2 α V ( ) } E { x T ( t ) [ A i T P + P A i + 2 α P e 2 α h 1 R ] } + E { x T ( t ) [ e 2 α h 2 R + S 1 A i + A i T S 1 T + 2 Q ] x ( t ) } + E { 2 x T ( t ) [ e 2 α h 1 R S 2 A i ] x ( t h 1 ) } + E { 2 x T ( t ) [ e 2 α h 2 R S 3 A i ] x ( t h 2 ) } + E { 2 x T ( t ) [ P D i S 1 D i S 4 A i ] x ( t h ( t ) ) } + E { 2 x T ( t ) [ S 1 S 5 A i ] x ˙ ( t ) } + E { x T ( t h 1 ) [ e 2 α h 1 Q e 2 α h 1 R e 2 α h 2 U ] x ( t h 1 ) } + E { 2 x T ( t h 1 ) [ e 2 α h 2 U S 2 D i ] x ( t h ( t ) ) } + E { 2 x T ( t h 1 ) S 2 x ˙ ( t ) } + E { x T ( t h 2 ) [ e 2 α h 2 Q e 2 α h 2 R e 2 α h 2 U ] x ( t h 2 ) } + E { x T ( t h 2 ) [ e 2 α h 2 U S 3 D i ] x ( t h ( t ) ) } + E { 2 x T ( t h 2 ) S 3 x ˙ ( t ) } + E { x T ( t h ( t ) ) [ 2 S 4 D i 2 e 2 α h 2 U ] x ( t h ( t ) ) } + E { 2 x T ( t h ( t ) ) [ S 4 S 5 D i ] x ˙ ( t ) } + E { x ˙ T ( t ) [ S 5 + S 5 T + h 1 2 R + h 2 2 R + ( h 2 h 1 ) 2 U ] x ˙ ( t ) } + E { 2 [ σ i T σ i ] } .

Applying assumption (2.3), the following estimations hold:

E { V ˙ ( ) + 2 α V ( ) } E { x T ( t ) [ A i T P + P A i + 2 α P e 2 α h 1 R ] } + E { x T ( t ) [ e 2 α h 2 R + S 1 A i + A i T S 1 T + 2 Q + 2 ρ i 1 I ] x ( t ) } + E { 2 x T ( t ) [ e 2 α h 1 R S 2 A i ] x ( t h 1 ) } + E { 2 x T ( t ) [ e 2 α h 2 R S 3 A i ] x ( t h 2 ) } + E { 2 x T ( t ) [ P D i S 1 D i S 4 A i ] x ( t h ( t ) ) } + E { 2 x T ( t ) [ S 1 S 5 A i ] x ˙ ( t ) } + E { x T ( t h 1 ) [ e 2 α h 1 Q e 2 α h 1 R e 2 α h 2 U ] x ( t h 1 ) } + E { 2 x T ( t h 1 ) [ e 2 α h 2 U S 2 D i ] x ( t h ( t ) ) } + E { 2 x T ( t h 1 ) S 2 x ˙ ( t ) } + E { x T ( t h 2 ) [ e 2 α h 2 Q e 2 α h 2 R e 2 α h 2 U ] x ( t h 2 ) } + E { x T ( t h 2 ) [ e 2 α h 2 U S 3 D i ] x ( t h ( t ) ) } + E { 2 x T ( t h 2 ) S 3 x ˙ ( t ) } + E { x T ( t h ( t ) ) [ 2 S 4 D i 2 e 2 α h 2 U + 2 ρ i 2 I ] x ( t h ( t ) ) } + E { 2 x T ( t h ( t ) ) [ S 4 S 5 D i ] x ˙ ( t ) } + E { x ˙ T ( t ) [ S 5 + S 5 T + h 1 2 R + h 2 2 R + ( h 2 h 1 ) 2 U ] x ˙ ( t ) } = E { x T ( t ) J i x ( t ) + ζ T ( t ) M i ζ ( t ) } ,
(3.5)

where ζ T (t)=[ x T (t), x T (t h 1 ), x T (t h 2 ), x T (th(t)), x ˙ T (t)].

Therefore, we finally obtain from (3.5) and condition (ii) that

E { V ˙ ( ) + 2 α V ( ) } <E { x T ( t ) J i x ( t ) } ,i=1,2,,N,t R + .

We now apply condition (i) and Proposition 2.1, the system J i is strictly complete, and the sets α i and α ¯ i by (3.1) are well defined such that

i = 1 N α i = R n { 0 } , i = 1 N α ¯ i = R n { 0 } , α ¯ i α ¯ j = , i j .

Therefore, for any x(t) R n , t R + , there exists i{1,2,,N} such that x(t) α ¯ i . By choosing a switching rule as γ(x(t))=i whenever γ(x(t)) α ¯ i , from (3.5) we have

E { V ˙ ( ) + 2 α V ( ) } E { x T ( t ) J i x ( t ) } <0,t R + ,

and hence

E { V ˙ ( t , x t ) } E { 2 α V ( t , x t ) } ,t R + .
(3.6)

Integrating both sides of (3.6) from 0 to t, we obtain

E { V ( t , x t ) } E { V ( ϕ ) e 2 α t } ,t R + .

Furthermore, taking condition (3.2) into account, we have

E { λ 1 x ( t , ϕ ) 2 } E { V ( x t ) } E { V ( ϕ ) e 2 α t } E { λ 2 e 2 α t ϕ 2 } ,

then

E { x ( t , ϕ ) } E { λ 2 λ 1 e α t ϕ } ,t R + .

By Definition 2.1, system (2.1) is exponentially stable in the mean square. The proof is complete. □

To illustrate the obtained result, let us give the following numerical example.

4 Numerical example

Example 4.1 Consider the following switched stochastic systems with interval time-varying delay (2.1), where the delay function h(t) is given by

h(t)=0.2+1.5329 sin 2 t

and

A 1 = ( 2 0.1 0.2 2.5 ) , A 2 = ( 2.5 0.3 0.2 2.9 ) , D 1 = ( 0.3 0.2 0.1 0.39 ) , D 2 = ( 0.5 0.2 0.1 0.4 ) .

It is worth noting that the delay function h(t) is non-differentiable and the exponent α1. Therefore, the methods used in [3, 21, 22, 2428, 3039] are not applicable to this system. By LMI toolbox of Matlab, we find that conditions (i), (ii) of Theorem 3.1 are satisfied with h 1 =0.1, h 2 =1.7329, δ 1 =0.5, δ 2 =0.3, α=1.5, ρ 11 =0.1, ρ 12 =0.2, ρ 21 =0.1, ρ 22 =0.2 and

P = ( 1.2397 0.3984 0.3984 1.3112 ) , Q = ( 1.7931 0.0079 0.0079 0.2397 ) , R = ( 2.3297 0.1121 0.1121 1.3397 ) , U = ( 1.7394 0.0982 0.0982 0.6321 ) , S 1 = ( 0.6210 0.0335 0.0499 0.3576 ) , S 2 = ( 0.3602 0.0170 0.0298 0.3550 ) , S 3 = ( 0.3602 0.0170 0.0298 0.3550 ) , S 4 = ( 0.6968 0.0401 0.0525 0.7040 ) , S 5 = ( 1.4043 0.0265 0.0028 0.9774 ) .

In this case, we have

( J 1 , J 2 )= ( [ 1.5667 0.0031 0.0031 1.9712 ] , [ 1.5511 0.0029 0.0029 1.3297 ] ) .

Moreover, the sum

δ 1 J 1 (R,Q)+ δ 2 J 2 (R,Q)= [ 0.3269 0 0 0.7239 ]

is negative definite; i.e., the first entry in the first row and the first column 0.3269<0 is negative and the determinant of the matrix is positive. The sets α 1 and α 2 are given as

α 1 = { ( x 1 , x 2 ) : 1.5667 x 1 2 0.0062 x 1 x 2 1.9712 x 2 2 < 0 } , α 2 = { ( x 1 , x 2 ) : 1.5511 x 1 2 0.0058 x 1 x 2 + 1.3297 x 2 2 > 0 } .

Obviously, the union of these sets is equal to R 2 {0}. The switching regions are defined as

α ¯ 1 = { ( x 1 , x 2 ) : 1.5667 x 1 2 0.0062 x 1 x 2 1.9712 x 2 2 < 0 } , α ¯ 2 = α 2 α ¯ 1 .

By Theorem 3.1, switched stochastic system (2.1) is 1.5-exponentially stable in the mean square and the switching rule is chosen as γ(x(t))=i whenever x(t) α ¯ i . Moreover, the solution x(t,ϕ) of the system satisfies

E { x ( t , ϕ ) } E { 1.0239 e 1.5 t ϕ } ,t R + .

(The trajectories of solution of switched stochastic systems is shown in Figure 1, respectively.)

Figure 1
figure 1

The simulation of the solutions x 1 (t) and x 2 (t) with the initial condition ϕ(t)= [ 10 5 ] T , t[0.4,0] .

5 Conclusions

In this paper, we have proposed new delay-dependent conditions for the mean square exponential stability of switched stochastic systems with time-varying delay. Based on the improved Lyapunov-Krasovskii functional and the linear matrix inequality technique, a switching rule for the mean square exponential stability of switched stochastic systems with time-varying delay has been established in terms of LMIs.

References

  1. de Oliveira MC, Geromel JC, Hsu L: LMI characterization of structural and robust stability: the discrete-time case. Linear Algebra Appl. 1999, 296: 27–38. 10.1016/S0024-3795(99)00086-5

    MathSciNet  Article  MATH  Google Scholar 

  2. Phat VN, Nam PT: Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function. Int. J. Control 2007, 80: 1333–1341. 10.1080/00207170701338867

    MathSciNet  Article  MATH  Google Scholar 

  3. Rajchakit G: Delay-dependent optimal guaranteed cost control of stochastic neural networks with interval nondifferentiable time-varying delays. Adv. Differ. Equ. 2013., 2013: Article ID 241 10.1186/1687-1847-2013-241

    Google Scholar 

  4. Sun YJ: Global stabilizability of uncertain systems with time-varying delays via dynamic observer-based output feedback. Linear Algebra Appl. 2002, 353: 91–105. 10.1016/S0024-3795(02)00292-6

    MathSciNet  Article  MATH  Google Scholar 

  5. Phat VN, Khongtham Y, Ratchagit K: LMI approach to exponential stability of linear systems with interval time-varying delays. Linear Algebra Appl. 2012, 436: 243–251. 10.1016/j.laa.2011.07.016

    MathSciNet  Article  MATH  Google Scholar 

  6. Phat VN, Ratchagit K: Stability and stabilization of switched linear discrete-time systems with interval time-varying delay. Nonlinear Anal. Hybrid Syst. 2011, 5: 605–612. 10.1016/j.nahs.2011.05.006

    MathSciNet  Article  MATH  Google Scholar 

  7. Ratchagit K, Phat VN: Stability criterion for discrete-time systems. J. Inequal. Appl. 2010., 2010: Article ID 201459

    Google Scholar 

  8. Krasnosel’skii MA, Vainikko GM, Zabreiko PP, Rutitskii JB, Stezenko VJ: Approximate Method for Solving Operator Equations. Nauka, Moscow; 1969.

    Google Scholar 

  9. Azbelev NV, Maksimov VP, Rakhmatullina LF Advanced Series in Mathematical Sciences and Engineering 3. In Introduction to Theory of Linear Functional Differential Equations. World Federation Publishers Company, Atlanta; 1995.

    Google Scholar 

  10. Domoshnitsky A, Sheina MV: Nonnegativity of Cauchy matrix and stability of systems with delay. Differ. Uravn. 1989, 25: 201–208.

    MathSciNet  Google Scholar 

  11. Gyori I: Interaction between oscillation and global asymptotic stability in delay differential equations. Differ. Integral Equ. 1990, 3: 181–200.

    MathSciNet  MATH  Google Scholar 

  12. Gyori I, Hartung F: Fundamental solution and asymptotic stability of linear delay differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2006, 13: 261–287.

    MathSciNet  MATH  Google Scholar 

  13. Hofbauer J, So JW-H: Diagonal dominance and harmless off-diagonal delays. Proc. Am. Math. Soc. 2000, 128: 2675–2682. 10.1090/S0002-9939-00-05564-7

    MathSciNet  Article  MATH  Google Scholar 

  14. Campbell SA: Delay independent stability for additive neural networks. Differ. Equ. Dyn. Syst. 2001, 9(3–4):115–138.

    MathSciNet  MATH  Google Scholar 

  15. Bainov D, Domoshnitsky A: Nonnegativity of the Cauchy matrix and exponential stability of a neutral type system of functional-differential equations. Extr. Math. 1993, 8: 75–82.

    MathSciNet  MATH  Google Scholar 

  16. Domoshnitsky A: About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Discrete Contin. Dyn. Syst. 2011, 2011: 373–380. Supplement 2011, Dedicated to the 8th AIMS Conference, Dresden, Germany, American Institute of Mathematical Sciences

    MathSciNet  MATH  Google Scholar 

  17. Gamliel D, Domoshnitsky A, Shklyar R: Time evolution of spin exchange with a time delay. Funct. Differ. Equ. 2013, 20: 81–114.

    MathSciNet  MATH  Google Scholar 

  18. Shatyrko A, Diblík J, Khusainov D, Ruzickova M: Stabilization of Lur’e-type nonlinear control systems by Lyapunov-Krasovskii functionals. Adv. Differ. Equ. 2012., 2012: Article ID 229 10.1186/1687-1847-2012-229

    Google Scholar 

  19. Diblík J, Dzhalladova I, Ruzickova M: The stability of nonlinear differential systems with random parameters. Abstr. Appl. Anal. 2012., 2012: Article ID 924107 10.1155/2012/924107

    Google Scholar 

  20. Bastinec J, Diblík J, Khusainov DY, Ryvolova A: Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients. Bound. Value Probl. 2010., 2010: Article ID 956121 10.1155/2010/956121

    Google Scholar 

  21. Kwon OM, Park JH: Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays. Appl. Math. Comput. 2009, 208: 58–68. 10.1016/j.amc.2008.11.010

    MathSciNet  Article  MATH  Google Scholar 

  22. Shao H: New delay-dependent stability criteria for systems with interval delay. Automatica 2009, 45: 744–749. 10.1016/j.automatica.2008.09.010

    Article  MathSciNet  MATH  Google Scholar 

  23. Sun J, Liu GP, Chen J, Rees D: Improved delay-range-dependent stability criteria for linear systems with time-varying delays. Automatica 2010, 46: 466–470. 10.1016/j.automatica.2009.11.002

    MathSciNet  Article  MATH  Google Scholar 

  24. Zhang W, Cai X, Han Z: Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations. J. Comput. Appl. Math. 2010, 234: 174–180. 10.1016/j.cam.2009.12.013

    MathSciNet  Article  MATH  Google Scholar 

  25. Rajchakit M, Rajchakit G: LMI approach to robust stability and stabilization of nonlinear uncertain discrete-time systems with convex polytopic uncertainties. Adv. Differ. Equ. 2012., 2012: Article ID 106

    Google Scholar 

  26. Xu S, Shi P, Chu Y, Zou Y:Robust stochastic stabilization and H control of uncertain neutral stochastic time-delay systems. J. Math. Anal. Appl. 2006, 314: 1–16. 10.1016/j.jmaa.2005.03.088

    MathSciNet  Article  MATH  Google Scholar 

  27. Yue D, Won S: Delay-dependent robust stability of stochastic systems with time delay and nonlinear uncertainties. Electron. Lett. 2001, 37: 992–993. 10.1049/el:20010632

    Article  MATH  Google Scholar 

  28. Verriest EI, Florchinger P: Stability of stochastic systems with uncertain time delays. Syst. Control Lett. 1995, 24: 41–47. 10.1016/0167-6911(94)00030-Y

    MathSciNet  Article  MATH  Google Scholar 

  29. Dzhalladova IA, Bastinec J, Diblík J, Khusainov DY: Estimates of exponential stability for solutions of stochastic control systems with delay. Abstr. Appl. Anal. 2011., 2011: Article ID 920412

    Google Scholar 

  30. Tian L, Liang J, Cao J: Robust observer for discrete-time Markovian jumping neural networks with mixed mode-dependent delays. Nonlinear Dyn. 2012, 67: 47–61. 10.1007/s11071-011-9956-y

    MathSciNet  Article  MATH  Google Scholar 

  31. Niamsup P, Rajchakit G: New results on robust stability and stabilization of linear discrete-time stochastic systems with convex polytopic uncertainties. J. Appl. Math. 2013., 2013: Article ID 368259 10.1155/2013/368259

    Google Scholar 

  32. Dong H, Wang Z, Ho DWC, Gao H:Robust H filtering for Markovian jump systems with randomly occurring nonlinearities and sensor saturation: the finite-horizon case. IEEE Trans. Signal Process. 2011, 59: 3048–3057.

    MathSciNet  Article  Google Scholar 

  33. Wang Z, Wei G, Feng G:Reliable H control for discrete-time piecewise linear systems with infinite distributed delays. Automatica 2009, 45: 2991–2994. 10.1016/j.automatica.2009.09.012

    MathSciNet  Article  MATH  Google Scholar 

  34. Rajchakit M, Rajchakit G: Mean square robust stability of stochastic switched discrete-time systems with convex polytopic uncertainties. J. Inequal. Appl. 2012., 2012: Article ID 135 10.1186/1029-242X-2012-135

    Google Scholar 

  35. Rajchakit G: Switching design for the asymptotic stability and stabilization of nonlinear uncertain stochastic discrete-time systems. Int. J. Nonlinear Sci. Numer. Simul. 2013, 14(1):33–44.

    MathSciNet  Google Scholar 

  36. Wang Y, Wang Z, Liang J: A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances. Phys. Lett. A 2008, 372: 6066–6073. 10.1016/j.physleta.2008.08.008

    Article  MATH  Google Scholar 

  37. Wang Z, Wang Y, Liu Y: Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays. IEEE Trans. Neural Netw. 2010, 21: 11–25.

    Article  Google Scholar 

  38. Rajchakit M, Rajchakit G: Mean square exponential stability of stochastic switched system with interval time-varying delays. Abstr. Appl. Anal. 2012., 2012: Article ID 623014 10.1155/2012/623014

    Google Scholar 

  39. Niamsup P, Rajchakit M, Rajchakit G: Guaranteed cost control for switched recurrent neural networks with interval time-varying delay. J. Inequal. Appl. 2013., 2013: Article ID 292 10.1186/1029-242X-2013-292

    Google Scholar 

  40. Wang Y, Xie L, de Souza CE: Robust control of a class of uncertain nonlinear systems. Syst. Control Lett. 1992, 19: 139–149. 10.1016/0167-6911(92)90097-C

    MathSciNet  Article  MATH  Google Scholar 

  41. Boyd S, El Ghaoui L, Feron E, Balakrishnan V: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia; 1994.

    Book  MATH  Google Scholar 

  42. Uhlig F: A recurring theorem about pairs of quadratic forms and extensions. Linear Algebra Appl. 1979, 25: 219–237.

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

This work was supported by the Thailand Research Fund Grant, the Commission for Higher Education and Faculty of Science, Maejo University, Thailand. The second author is supported by the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand. The authors thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the paper.

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Rajchakit, M., Niamsup, P. & Rajchakit, G. A constructive way to design a switching rule and switching regions to mean square exponential stability of switched stochastic systems with non-differentiable and interval time-varying delay. J Inequal Appl 2013, 499 (2013). https://doi.org/10.1186/1029-242X-2013-499

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Keywords

  • switching design
  • mean square exponential stability
  • switched stochastic systems
  • scalar Wiener process
  • Brownian motion
  • interval delay
  • Lyapunov function
  • linear matrix inequalities