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 [1–25]. Stability analysis of linear systems with time-varying delays $\dot{x}(t)=Ax(t)+Dx(t-h(t))$ is fundamental to many practical problems and has received considerable attention [1–7]. In [8–17], 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: $\dot{h}(t)\le \delta <1$. 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 [18–24] 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: $\dot{h}(t)\le \delta$. 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., [25–38] 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 $〈\cdot ,\cdot 〉$ and the vector norm $\parallel \cdot \parallel$; $M^{n\times r}$ denotes the space of all matrices of $(n\times r)$-dimensions; $A^T$ denotes the transpose of matrix A; A is symmetric if $A=A^T$; I denotes the identity matrix; $\lambda (A)$ denotes the set of all eigenvalues of A; ${\lambda }_{\min /\max }(A)=min/max\{Re\lambda ;\lambda \in \lambda (A)\}$; $x_t:=\{x(t+s):s\in [-h,0]\}$, $\parallel x_t\parallel ={sup}_{s\in [-h,0]}\parallel x(t+s)\parallel$; $C([0,t],R^n)$ denotes the set of all $R^n$-valued continuous functions on $[0,t]$; matrix A is called semi-positive definite ($A\ge 0$) if $〈Ax,x〉\ge 0$ for all $x\in R^n$; A is positive definite ($A>0$) if $〈Ax,x〉>0$ for all $x\ne 0$; $A>B$ means $A-B>0$. ∗ denotes the symmetric term in a matrix.

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

$$\begin{array}{r}\dot{x}(t)=A_{\gamma (x(t))}x(t)+D_{\gamma (x(t))}x(t-h(t))+{\sigma }_{\gamma (x(t))}(x(t),x(k-h(t)),t)\omega (t),t\in R^{+},\\ x(t)=\varphi (t),t\in [-h_2,0],\end{array}$$

(2.1)

where $x(t)\in R^n$ is the state; $\gamma (\cdot ):R^n\to \mathcal{N}:=\{1,2,\dots ,N\}$ 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, $\gamma (x(t))=i$ implies that the system realization is chosen as the i th system, $i=1,2,\dots ,N$. 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. $A_i,D_i\in M^{n\times n}$, $i=1,2,\dots ,N$, are given constant matrices, and $\varphi (t)\in C([-h_2,0],R^n)$ is the initial function with the norm $\parallel \varphi \parallel ={sup}_{s\in [-h_2,0]}\parallel \varphi (s)\parallel$.

$\omega (k)$ is a scalar Wiener process (Brownian motion) on $(\mathrm{\Omega },\mathcal{F},\mathcal{P})$ with

$$E\{\omega (t)\}=0,E\{{\omega }^2(t)\}=1,E\{\omega (i)\omega (j)\}=0(i\ne j),$$

(2.2)

and ${\sigma }_i:R^n\times R^n\times R\to R^n$, $i=1,2,\dots ,N$, is the continuous function, and it is assumed to satisfy that

$$\begin{array}{r}{\sigma }_i^T(x(t),x(t-h(t)),t){\sigma }_i(x(t),x(t-h(t)),t)\\ \le {\rho }_{i1}{x}^T(t)x(t)+{\rho }_{i2}{x}^T(t-h(t))x(t-h(t)),\\ x(t),x(t-h(t))\in R^n,\end{array}$$

(2.3)

where ${\rho }_{i1}>0$ and ${\rho }_{i2}>0$, $i=1,2,\dots ,N$, are known constant scalars. For simplicity, we denote ${\sigma }_i(x(t),x(t-h(t)),t)$ by ${\sigma }_i$, respectively.

The time-varying delay function $h(t)$ satisfies

$$0\le h_1\le h(t)\le h_2,t\in R^{+}.$$

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 $\alpha >0$. Switched stochastic system (2.1) is α-exponentially stable in the mean square if there exists a switching rule $\gamma (\cdot )$ such that every solution $x(t,\varphi )$ of the system satisfies the following condition:

$$\mathrm{\exists }N>0:E\left\{\parallel x(t,\varphi )\parallel \right\}\le E\{N{e}^{-\alpha t}\parallel \varphi \parallel \},\mathrm{\forall }t\in R^{+}.$$

Definition 2.2 The system of matrices $\{J_i\}$, $i=1,2,\dots ,N$, is said to be strictly complete if for every $x\in R^n\setminus \{0\}$, there is $i\in \{1,2,\dots ,N\}$ 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

$$\bigcup _{i=1}^N{\alpha }_i=R^n\setminus \{0\},$$

where

$${\alpha }_i=\{x\in R^n:x^T{J}_{i}x<0\},i=1,2,\dots ,N.$$

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 $\{J_i\}$, $i=1,2,\dots ,N$, is strictly complete if there exist ${\delta }_i\ge 0$, $i=1,2,\dots ,N$, ${\sum }_{i=1}^N{\delta }_i>0$ such that

$$\sum _{i=1}^N{\delta }_i{J}_i<0.$$

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 $N\in M^{n\times n}$ and $a,b\in R^n$, we have

$$\pm a^{T}b\le a^{T}Na+b^T{N}^{-1}b.$$

Proposition 2.3 [40]

For any symmetric positive definite matrix $M\in M^{n\times n}$, scalar $\mu >0$ and vector function $\omega :[0,\mu ]\to R^n$ such that the integrations concerned are well defined, the following inequality holds:

$${({\int }_0^{\mu }\omega (s)ds)}^{T}M({\int }_0^{\mu }\omega (s)ds)\le \mu ({\int }_0^{\mu }{\omega }^T(s)M\omega (s)ds).$$

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

Let E, H and F be any constant matrices of appropriate dimensions and $F^{T}F\le I$. For any $ϵ>0$, we have

$$EFH+H^T{F}^T{E}^T\le ϵE{E}^T+{ϵ}^{-1}{H}^{T}H.$$

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

$$\left(\begin{array}{cc}X& Z^T\\ Z& -Y\end{array}\right)<0\mathit{\text{or}}\left(\begin{array}{cc}-Y& Z\\ Z^T& X\end{array}\right)<0.$$

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,

$$\begin{array}{r}{\mathcal{M}}_i=\left[\begin{array}{ccccc}{M}_{11}& M_{12}& M_{13}& M_{14}& M_{15}\\ \ast & M_{22}& 0& M_{24}& M_{25}\\ \ast & \ast & M_{33}& M_{34}& M_{35}\\ \ast & \ast & \ast & M_{44}& M_{45}\\ \ast & \ast & \ast & \ast & M_{55}\end{array}\right],\\ M_{11}=A_i^{T}P+P{A}_i+2\alpha P-e^{-2\alpha h_1}R\\ \phantom{M_{11}=}-e^{-2\alpha h_2}R+Q+2{\rho }_{i1}I,\\ M_{12}=e^{-2\alpha h_1}R-S_2{A}_i,\\ M_{13}=e^{-2\alpha h_2}R-S_3{A}_i,\\ M_{14}=P{D}_i-S_1{D}_i-S_4{A}_i,\\ M_{15}=S_1-S_5{A}_i,\\ M_{22}=-e^{-2\alpha h_1}Q-e^{-2\alpha h_1}R-e^{-2\alpha h_2}U,\\ M_{24}=e^{-2\alpha h_2}U-S_2{D}_i,\\ M_{25}=S_2,\\ M_{33}=-e^{-2\alpha h_2}Q-e^{-2\alpha h_2}R-e^{-2\alpha h_2}U,\\ M_{34}=e^{-2\alpha h_2}U-S_3{D}_i,\\ M_{35}=S_3,\\ M_{44}=-2S_4{D}_i-2e^{-2\alpha h_2}U+2{\rho }_{i2}I,\\ M_{45}=S_4-S_5{D}_i,\\ M_{55}=S_5+S_5^T+h_1^{2}R+h_2^{2}R+{(h_2-h_1)}^{2}U,\\ J_i=Q-S_1{A}_i-A_i^T{S}_1^T,\\ {\alpha }_i=\{x\in R^n:x^T{J}_{i}x<0\},i=1,2,\dots ,N,\\ {\overline{\alpha }}_1={\alpha }_1,{\overline{\alpha }}_i={\alpha }_i\setminus \bigcup _{j=1}^{i-1}{\overline{\alpha }}_j,i=2,3,\dots ,N,\\ {\lambda }_1={\lambda }_{\min }(P),\\ {\lambda }_2={\lambda }_{\max }(P)+2h_2{\lambda }_{\max }(Q)+2h_2^2{\lambda }_{\max }(R)\\ \phantom{{\lambda }_2=}+{(h_2-h_1)}^2{\lambda }_{\max }(U).\end{array}$$

(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 $\alpha >0$. 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 $S_i$, $i=1,2,\dots ,5$, satisfying the following conditions:

• (i) $\mathrm{\exists }{\delta }_i\ge 0$, $i=1,2,\dots ,N$, ${\sum }_{i=1}^N{\delta }_i>0:{\sum }_{i=1}^N{\delta }_i{J}_i<0$,

• (ii) ${\mathcal{M}}_i<0$, $i=1,2,\dots ,N$.

The switching rule is chosen as $\gamma (x(t))=i$, whenever $x(t)\in {\overline{\alpha }}_i$. Moreover, the solution $x(t,\varphi )$ of the switched stochastic system satisfies

$$E\left\{\parallel x(t,\varphi )\parallel \right\}\le E\{\sqrt{\frac{{\lambda }_2}{{\lambda }_1}}{e}^{-\alpha t}\parallel \varphi \parallel \},\mathrm{\forall }t\in R^{+}.$$

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

$$E\{V(t,x_t)\}=E\left\{\sum _{i=1}^6{V}_i\right\},$$

where

$$\begin{array}{c}{V}_1=x^T(t)Px(t),\hfill \\ V_2={\int }_{t-h_1}^t{e}^{2\alpha (s-t)}{x}^T(s)Qx(s)ds,\hfill \\ V_3={\int }_{t-h_2}^t{e}^{2\alpha (s-t)}{x}^T(s)Qx(s)ds,\hfill \\ V_4=h_1{\int }_{-h_1}^0{\int }_{t+s}^t{e}^{2\alpha (\tau -t)}{\dot{x}}^T(\tau )R\dot{x}(\tau )d\tau ds,\hfill \\ V_5=h_2{\int }_{-h_2}^0{\int }_{t+s}^t{e}^{2\alpha (\tau -t)}{\dot{x}}^T(\tau )R\dot{x}(\tau )d\tau ds,\hfill \\ V_6=(h_2-h_1){\int }_{t-h_2}^{t-h_1}{\int }_{t+s}^t{e}^{2\alpha (\tau -t)}{\dot{x}}^T(\tau )U\dot{x}(\tau )d\tau ds.\hfill \end{array}$$

It easy to check that

$$E\left\{{\lambda }_1{\parallel x(t)\parallel }^2\right\}\le E\{V(t,x_t)\}\le E\left\{{\lambda }_2{\parallel x_t\parallel }^2\right\},\mathrm{\forall }t\ge 0.$$

(3.2)

Taking the derivative of $V_1$ along the solution of system (2.1) and taking the mathematical expectation, we obtain

$$\begin{array}{c}E\{{\dot{V}}_1\}=E\{2x^T(t)P\dot{x}(t)\}\hfill \\ \phantom{E\{{\dot{V}}_1\}}=E\{x^T(t)[A_i^{T}P+A_{i}P]x(t)+2x^T(t)P{D}_{i}x(t-h(t))+2x^T(t)P{\sigma }_i\omega (t)\};\hfill \\ E\{{\dot{V}}_2\}=E\{x^T(t)Qx(t)-e^{-2\alpha h_1}{x}^T(t-h_1)Qx(t-h_1)-2\alpha V_2\};\hfill \\ E\{{\dot{V}}_3\}=E\{x^T(t)Qx(t)-e^{-2\alpha h_2}{x}^T(t-h_2)Qx(t-h_2)-2\alpha V_3\};\hfill \\ E\{{\dot{V}}_4\}=E\{h_1^2{\dot{x}}^T(t)R\dot{x}(t)-h_1{\int }_{t-h_1}^t{e}^{2\alpha (\tau -t)}{\dot{x}}^T(s)R\dot{x}(s)ds-2\alpha V_4\}\hfill \\ \phantom{E\{{\dot{V}}_4\}}\le E\{h_1^2{\dot{x}}^T(t)R\dot{x}(t)-h_1{e}^{-2\alpha h_1}{\int }_{t-h_1}^t{\dot{x}}^T(s)R\dot{x}(s)ds-2\alpha V_4\};\hfill \\ E\{{\dot{V}}_5\}=E\{h_2^2{\dot{x}}^T(t)R\dot{x}(t)-h_2{\int }_{t-h_2}^t{e}^{2\alpha (\tau -t)}{\dot{x}}^T(s)R\dot{x}(s)ds-2\alpha V_5\}\hfill \\ \phantom{E\{{\dot{V}}_5\}}\le E\{h_2^2{\dot{x}}^T(t)R\dot{x}(t)-h_2{e}^{-2\alpha h_2}{\int }_{t-h_2}^t{\dot{x}}^T(s)R\dot{x}(s)ds-2\alpha V_5\};\hfill \\ E\{{\dot{V}}_6\}\le E\{{(h_2-h_1)}^2{\dot{x}}^T(t)U\dot{x}(t)-(h_2-h_1)e^{-2\alpha h_2}{\int }_{t-h_2}^{t-h_1}{\dot{x}}^T(s)U\dot{x}(s)ds-2\alpha V_6\}.\hfill \end{array}$$

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

$$\begin{array}{rl}E\{-h_i{\int }_{t-h_i}^t{\dot{x}}^T(s)R\dot{x}(s)ds\}& \le E\{-{[{\int }_{t-h_i}^t\dot{x}(s)ds]}^{T}R[{\int }_{t-h_i}^t\dot{x}(s)ds]\}\\ \le E\{-{[x(t)-x(t-h_i)]}^{T}R[x(t)-x(t-h_i)]\}\\ =E\{-x^T(t)Rx(t)+2x^T(t)Rx(t-h_i)-x^T(t-h_i)Rx(t-h_i)\}.\end{array}$$

Note that

$$E\{{\int }_{t-h_2}^{t-h_1}{\dot{x}}^T(s)U\dot{x}(s)ds\}=E\{{\int }_{t-h_2}^{t-h(t)}{\dot{x}}^T(s)U\dot{x}(s)ds+{\int }_{t-h(t)}^{t-h_1}{\dot{x}}^T(s)U\dot{x}(s)ds\}.$$

Using Proposition 2.2 gives

$$\begin{array}{r}E\{[h_2-h(t)]{\int }_{t-h_2}^{t-h(t)}{\dot{x}}^T(s)U\dot{x}(s)ds\}\\ \ge E\left\{{[{\int }_{t-h_2}^{t-h(t)}\dot{x}(s)ds]}^{T}U[{\int }_{t-h_2}^{t-h(t)}\dot{x}(s)ds]\right\}\\ \ge E\left\{{[x(t-h(t))-x(t-h_2)]}^{T}U[x(t-h(t))-x(t-h_2)]\right\}.\end{array}$$

Since $h_2-h(t)\le h_2-h_1$, we have

$$\begin{array}{r}E\{[h_2-h_1]{\int }_{t-h_2}^{t-h(t)}{\dot{x}}^T(s)U\dot{x}(s)ds\}\\ \ge E\left\{{[x(t-h(t))-x(t-h_2)]}^{T}U[x(t-h(t))-x(t-h_2)]\right\},\end{array}$$

then

$$\begin{array}{r}E\{-(h_2-h_1){\int }_{t-h_2}^{t-h(t)}{\dot{x}}^T(s)U\dot{x}(s)ds\}\\ \le E\{-{[x(t-h(t))-x(t-h_2)]}^{T}U[x(t-h(t))-x(t-h_2)]\}.\end{array}$$

Similarly, we have

$$\begin{array}{r}E\{-(h_2-h_1){\int }_{t-h(t)}^{t-h_1}{\dot{x}}^T(s)U\dot{x}(s)ds\}\\ \le E\{-{[x(t-h_1)-x(t-h(t))]}^{T}U[x(t-h_1)-x(t-h(t))]\}.\end{array}$$

Therefore, we have

$$\begin{array}{r}E\{\dot{V}(\cdot )+2\alpha V(\cdot )\}\\ \le E\{x^T(t)[A_i^{T}P+A_{i}P+2\alpha P+2Q]x(t)\}\\ +E\{2x^T(t)P{D}_{i}x(t-h(t))+2x^T(t)P{\sigma }_i\omega (t)\}\\ +E\{-e^{-2\alpha h_1}{x}^T(t-h_1)Qx(t-h_1)\}\\ +E\{-e^{-2\alpha h_2}{x}^T(t-h_2)Qx(t-h_2)\}\\ +E\{{\dot{x}}^T(t)[(h_1^2+h_2^2)R+{(h_2-h_1)}^{2}U]\dot{x}(t)\}\\ +E\{-e^{-2\alpha h_1}{[x(t)-x(t-h_1)]}^{T}R[x(t)-x(t-h_1)]\}\\ +E\{-e^{-2\alpha h_2}{[x(t)-x(t-h_2)]}^{T}R[x(t)-x(t-h_2)]\}\\ +E\{-e^{-2\alpha h_2}{[x(t-h(t))-x(t-h_2)]}^{T}U[x(t-h(t))-x(t-h_2)]\}\\ +E\{-e^{-2\alpha h_2}{[x(t-h_1)-x(t-h(t))]}^{T}U[x(t-h_1)-x(t-h(t))]\}.\end{array}$$

(3.3)

By using the following identity relation

$$\dot{x}(t)-A_{i}x(t)-D_{i}x(t-h(t))=0,$$

and multiplying by $2x^T(t)S_1$, $2x^T(t-h_1)S_2$, $2x^T(t-h_2)S_3$, $2x^T(t-h(t))S_4$, $2{\dot{x}}^T(t)S_5$, $2{\omega }^T(t){\sigma }_i^T$ both sides of the identity relation, we have

$$\begin{array}{r}2x^T(t)S_1\dot{x}(t)-2x^T(t)S_1{A}_{i}x(t)-2x^T(t)S_1{D}_{i}x(t-h(t))-2x^T(t)S_1{\sigma }_i\omega (t)=0,\\ 2x^T(t-h_1)S_2\dot{x}(t)-2x^T(t-h_1)S_2{A}_{i}x(t)\\ -2x^T(t-h_1)S_2{D}_{i}x(t-h(t))-2x^T(t-h_1)S_2{\sigma }_i\omega (t)=0,\\ 2x^T(t-h_2)S_3\dot{x}(t)-2x^T(t-h_2)S_3{A}_{i}x(t)\\ -2x^T(t-h_2)S_3{D}_{i}x(t-h(t))-2x^T(t-h_2)S_3{\sigma }_i\omega (t)=0,\\ 2x^T(t-h(t))S_4\dot{x}(t)-2x^T(t-h(t))S_4{A}_{i}x(t)\\ -2x^T(t-h(t))S_4{D}_{i}x(t-h(t))-2x^T(t-h(t))S_4{\sigma }_i\omega (t)=0,\\ 2{\dot{x}}^T(t)S_5\dot{x}(t)-2{\dot{x}}^T(t)S_5{A}_{i}x(t)-2{\dot{x}}^T(t)S_5{D}_{i}x(t-h(t))-2{\dot{x}}^T(t)S_5{\sigma }_i\omega (t)=0,\\ 2{\omega }^T(t){\sigma }_i^T\dot{x}(t)-2{\omega }^T(t){\sigma }_i^T{A}_{i}x(t)-2{\omega }^T(t){\sigma }_i^T{D}_{i}x(t-h(t))-2{\omega }^T(t){\sigma }_i^T{\sigma }_i\omega (t)=0.\end{array}$$

(3.4)

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

$$\begin{array}{rl}E\{\dot{V}(\cdot )+2\alpha V(\cdot )\}\le & E\{x^T(t)[A_i^{T}P+P{A}_i+2\alpha P-e^{-2\alpha h_1}R]x(t)\}\\ +E\{x^T(t)[-e^{-2\alpha h_2}R+S_1{A}_i+A_i^T{S}_1^T+2Q]x(t)\}\\ +E\{2x^T(t)[e^{-2\alpha h_1}R-S_2{A}_i]x(t-h_1)\}\\ +E\{2x^T(t)[e^{-2\alpha h_2}R-S_3{A}_i]x(t-h_2)\}\\ +E\{2x^T(t)[P{D}_i-S_1{D}_i-S_4{A}_i]x(t-h(t))\}\\ +E\{2x^T(t)[S_1-S_5{A}_i]\dot{x}(t)\}\\ +E\{2x^T(t)[P{\sigma }_i-S_1{\sigma }_i-A_i^T{\sigma }_i]\omega (t)\}\\ +E\{x^T(t-h_1)[-e^{-2\alpha h_1}Q-e^{-2\alpha h_1}R-e^{-2\alpha h_2}U]x(t-h_1)\}\\ +E\{2x^T(t-h_1)[e^{-2\alpha h_2}U-S_2{D}_i]x(t-h(t))\}\\ +E\{2x^T(t-h_1)S_2\dot{x}(t)\}\\ +E\{2x^T(t-h_1)[-S_2{\sigma }_i]\omega (t)\}\\ +E\{x^T(t-h_2)[-e^{-2\alpha h_2}Q-e^{-2\alpha h_2}R-e^{-2\alpha h_2}U]x(t-h_2)\}\\ +E\{2x^T(t-h_2)[e^{-2\alpha h_2}U-S_3{D}_i]x(t-h(t))\}\\ +E\{2x^T(t-h_2)S_3\dot{x}(t)\}\\ +E\{2x^T(t-h_2)[-S_3{\sigma }_i]\omega (t)\}\\ +E\left\{x^T(t-h(t))[-2e^{-2\alpha h_2}U-2S_4{D}_i]x(t-h(t))\right\}\\ +E\{2x^T(t-h(t))[S_4-S_5{D}_i]\dot{x}(t)\}\\ +E\{2x^T(t-h(t))[-S_4{\sigma }_i-{\sigma }_i^T{D}_i]\omega (t)\}\\ +E\{{\dot{x}}^T(t)[S_5+S_5^T+h_1^{2}R+h_2^{2}R+{(h_2-h_1)}^{2}U]\dot{x}(t)\}\\ +E\{2{\dot{x}}^T(t)[{\sigma }_i^T-S_5{\sigma }_i]\omega (t)\}\\ +E\{2{\omega }^T(t)[-{\sigma }_i^T{\sigma }_i]\omega (t)\}.\end{array}$$

By assumption (2.2), we have

$$\begin{array}{rl}E\{\dot{V}(\cdot )+2\alpha V(\cdot )\}\le & E\{x^T(t)[A_i^{T}P+P{A}_i+2\alpha P-e^{-2\alpha h_1}R]\}\\ +E\{x^T(t)[-e^{-2\alpha h_2}R+S_1{A}_i+A_i^T{S}_1^T+2Q]x(t)\}\\ +E\{2x^T(t)[e^{-2\alpha h_1}R-S_2{A}_i]x(t-h_1)\}\\ +E\{2x^T(t)[e^{-2\alpha h_2}R-S_3{A}_i]x(t-h_2)\}\\ +E\{2x^T(t)[P{D}_i-S_1{D}_i-S_4{A}_i]x(t-h(t))\}\\ +E\{2x^T(t)[S_1-S_5{A}_i]\dot{x}(t)\}\\ +E\{x^T(t-h_1)[-e^{-2\alpha h_1}Q-e^{-2\alpha h_1}R-e^{-2\alpha h_2}U]x(t-h_1)\}\\ +E\{2x^T(t-h_1)[e^{-2\alpha h_2}U-S_2{D}_i]x(t-h(t))\}\\ +E\{2x^T(t-h_1)S_2\dot{x}(t)\}\\ +E\{x^T(t-h_2)[-e^{-2\alpha h_2}Q-e^{-2\alpha h_2}R-e^{-2\alpha h_2}U]x(t-h_2)\}\\ +E\{x^T(t-h_2)[e^{-2\alpha h_2}U-S_3{D}_i]x(t-h(t))\}\\ +E\{2x^T(t-h_2)S_3\dot{x}(t)\}\\ +E\left\{x^T(t-h(t))[-2S_4{D}_i-2e^{-2\alpha h_2}U]x(t-h(t))\right\}\\ +E\{2x^T(t-h(t))[S_4-S_5{D}_i]\dot{x}(t)\}\\ +E\{{\dot{x}}^T(t)[S_5+S_5^T+h_1^{2}R+h_2^{2}R+{(h_2-h_1)}^{2}U]\dot{x}(t)\}\\ +E\left\{2[-{\sigma }_i^T{\sigma }_i]\right\}.\end{array}$$

Applying assumption (2.3), the following estimations hold:

$$\begin{array}{rl}E\{\dot{V}(\cdot )+2\alpha V(\cdot )\}\le & E\{x^T(t)[A_i^{T}P+P{A}_i+2\alpha P-e^{-2\alpha h_1}R]\}\\ +E\{x^T(t)[-e^{-2\alpha h_2}R+S_1{A}_i+A_i^T{S}_1^T+2Q+2{\rho }_{i1}I]x(t)\}\\ +E\{2x^T(t)[e^{-2\alpha h_1}R-S_2{A}_i]x(t-h_1)\}\\ +E\{2x^T(t)[e^{-2\alpha h_2}R-S_3{A}_i]x(t-h_2)\}\\ +E\{2x^T(t)[P{D}_i-S_1{D}_i-S_4{A}_i]x(t-h(t))\}\\ +E\{2x^T(t)[S_1-S_5{A}_i]\dot{x}(t)\}\\ +E\{x^T(t-h_1)[-e^{-2\alpha h_1}Q-e^{-2\alpha h_1}R-e^{-2\alpha h_2}U]x(t-h_1)\}\\ +E\{2x^T(t-h_1)[e^{-2\alpha h_2}U-S_2{D}_i]x(t-h(t))\}\\ +E\{2x^T(t-h_1)S_2\dot{x}(t)\}\\ +E\{x^T(t-h_2)[-e^{-2\alpha h_2}Q-e^{-2\alpha h_2}R-e^{-2\alpha h_2}U]x(t-h_2)\}\\ +E\{x^T(t-h_2)[e^{-2\alpha h_2}U-S_3{D}_i]x(t-h(t))\}\\ +E\{2x^T(t-h_2)S_3\dot{x}(t)\}\\ +E\left\{x^T(t-h(t))[-2S_4{D}_i-2e^{-2\alpha h_2}U+2{\rho }_{i2}I]x(t-h(t))\right\}\\ +E\{2x^T(t-h(t))[S_4-S_5{D}_i]\dot{x}(t)\}\\ +E\{{\dot{x}}^T(t)[S_5+S_5^T+h_1^{2}R+h_2^{2}R+{(h_2-h_1)}^{2}U]\dot{x}(t)\}\\ =E\{x^T(t)J_{i}x(t)+{\zeta }^T(t){\mathcal{M}}_i\zeta (t)\},\end{array}$$

(3.5)

where ${\zeta }^T(t)=[x^T(t),x^T(t-h_1),x^T(t-h_2),x^T(t-h(t)),{\dot{x}}^T(t)]$.

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

$$E\{\dot{V}(\cdot )+2\alpha V(\cdot )\}<E\{x^T(t)J_{i}x(t)\},\mathrm{\forall }i=1,2,\dots ,N,t\in R^{+}.$$

We now apply condition (i) and Proposition 2.1, the system $J_i$ is strictly complete, and the sets ${\alpha }_i$ and ${\overline{\alpha }}_i$ by (3.1) are well defined such that

$$\begin{array}{c}\bigcup _{i=1}^N{\alpha }_i=R^n\setminus \{0\},\hfill \\ \bigcup _{i=1}^N{\overline{\alpha }}_i=R^n\setminus \{0\},{\overline{\alpha }}_i\cap {\overline{\alpha }}_j=\mathrm{\varnothing },i\ne j.\hfill \end{array}$$

Therefore, for any $x(t)\in R^n$, $t\in R^{+}$, there exists $i\in \{1,2,\dots ,N\}$ such that $x(t)\in {\overline{\alpha }}_i$. By choosing a switching rule as $\gamma (x(t))=i$ whenever $\gamma (x(t))\in {\overline{\alpha }}_i$, from (3.5) we have

$$E\{\dot{V}(\cdot )+2\alpha V(\cdot )\}\le E\{x^T(t)J_{i}x(t)\}<0,t\in R^{+},$$

and hence

$$E\{\dot{V}(t,x_t)\}\le E\{-2\alpha V(t,x_t)\},\mathrm{\forall }t\in R^{+}.$$

(3.6)

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

$$E\{V(t,x_t)\}\le E\{V(\varphi )e^{-2\alpha t}\},\mathrm{\forall }t\in R^{+}.$$

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

$$E\left\{{\lambda }_1{\parallel x(t,\varphi )\parallel }^2\right\}\le E\{V(x_t)\}\le E\{V(\varphi )e^{-2\alpha t}\}\le E\left\{{\lambda }_2{e}^{-2\alpha t}{\parallel \varphi \parallel }^2\right\},$$

then

$$E\left\{\parallel x(t,\varphi )\parallel \right\}\le E\{\sqrt{\frac{{\lambda }_2}{{\lambda }_1}}{e}^{-\alpha t}\parallel \varphi \parallel \},t\in R^{+}.$$

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

$$\begin{array}{c}{A}_1=\left(\begin{array}{cc}-2& 0.1\\ 0.2& -2.5\end{array}\right),A_2=\left(\begin{array}{cc}-2.5& 0.3\\ 0.2& -2.9\end{array}\right),\hfill \\ D_1=\left(\begin{array}{cc}-0.3& 0.2\\ 0.1& -0.39\end{array}\right),D_2=\left(\begin{array}{cc}-0.5& 0.2\\ 0.1& -0.4\end{array}\right).\hfill \end{array}$$

It is worth noting that the delay function $h(t)$ is non-differentiable and the exponent $\alpha \ge 1$. Therefore, the methods used in [3, 21, 22, 24–28, 30–39] are not applicable to this system. By LMI toolbox of Matlab, we find that conditions (i), (ii) of Theorem 3.1 are satisfied with $h_1=0.1$, $h_2=1.7329$, ${\delta }_1=0.5$, ${\delta }_2=0.3$, $\alpha =1.5$, ${\rho }_{11}=0.1$, ${\rho }_{12}=0.2$, ${\rho }_{21}=0.1$, ${\rho }_{22}=0.2$ and

$$\begin{array}{c}P=\left(\begin{array}{cc}1.2397& -0.3984\\ -0.3984& 1.3112\end{array}\right),Q=\left(\begin{array}{cc}1.7931& -0.0079\\ -0.0079& 0.2397\end{array}\right),\hfill \\ R=\left(\begin{array}{cc}2.3297& -0.1121\\ -0.1121& 1.3397\end{array}\right),U=\left(\begin{array}{cc}1.7394& -0.0982\\ -0.0982& 0.6321\end{array}\right),\hfill \\ S_1=\left(\begin{array}{cc}-0.6210& -0.0335\\ 0.0499& -0.3576\end{array}\right),S_2=\left(\begin{array}{cc}-0.3602& 0.0170\\ 0.0298& -0.3550\end{array}\right),\hfill \\ S_3=\left(\begin{array}{cc}-0.3602& 0.0170\\ 0.0298& -0.3550\end{array}\right),S_4=\left(\begin{array}{cc}0.6968& -0.0401\\ -0.0525& 0.7040\end{array}\right),\hfill \\ S_5=\left(\begin{array}{cc}-1.4043& 0.0265\\ -0.0028& -0.9774\end{array}\right).\hfill \end{array}$$

In this case, we have

$$(J_1,J_2)=(\left[\begin{array}{cc}-1.5667& -0.0031\\ -0.0031& -1.9712\end{array}\right],\left[\begin{array}{cc}-1.5511& 0.0029\\ 0.0029& -1.3297\end{array}\right]).$$

Moreover, the sum

$${\delta }_1{J}_1(R,Q)+{\delta }_2{J}_2(R,Q)=\left[\begin{array}{cc}-0.3269& 0\\ 0& -0.7239\end{array}\right]$$

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 ${\alpha }_1$ and ${\alpha }_2$ are given as

$$\begin{array}{c}{\alpha }_1=\{(x_1,x_2):-1.5667x_1^2-0.0062x_1{x}_2-1.9712x_2^2<0\},\hfill \\ {\alpha }_2=\{(x_1,x_2):1.5511x_1^2-0.0058x_1{x}_2+1.3297x_2^2>0\}.\hfill \end{array}$$

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

$$\begin{array}{c}{\overline{\alpha }}_1=\{(x_1,x_2):-1.5667x_1^2-0.0062x_1{x}_2-1.9712x_2^2<0\},\hfill \\ {\overline{\alpha }}_2={\alpha }_2\setminus {\overline{\alpha }}_1.\hfill \end{array}$$

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 $\gamma (x(t))=i$ whenever $x(t)\in {\overline{\alpha }}_i$. Moreover, the solution $x(t,\varphi )$ of the system satisfies

$$E\left\{\parallel x(t,\varphi )\parallel \right\}\le E\{1.0239e^{-1.5t}\parallel \varphi \parallel \},\mathrm{\forall }t\in R^{+}.$$

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

Figure 1

The simulation of the solutions ${\mathit{x}}_{\mathbf{1}}\mathbf{(}\mathit{t}\mathbf{)}$ and ${\mathit{x}}_{\mathbf{2}}\mathbf{(}\mathit{t}\mathbf{)}$ with the initial condition $\mathit{\varphi }\mathbf{(}\mathit{t}\mathbf{)}\mathbf{=}{\mathbf{[}\mathbf{10}\mathbf{5}\mathbf{]}}^{\mathit{T}}$ , $\mathit{t}\mathbf{\in }\mathbf{[}\mathbf{-}\mathbf{0.4}\mathbf{,}\mathbf{0}\mathbf{]}$ .

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.