P-th moment and almost sure stability of stochastic switched nonlinear systems

This paper mainly tends to utilize \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}ψ-type function to investigate p-th moment and almost sure stability for a class of stochastic switched nonlinear systems. Based on the multiple Lyapunov functions approach, some sufficient conditions are derived to check the stability criteria of stochastic switched nonlinear systems. One numerical example is provided to demonstrate the effectiveness of the proposed results.

systems is investigated in Zhang et al. (2014) and Wu et al. (2013) and almost sure exponential stability of stochastic switched systems is researched in Cong et al. (2011). Although the stability of stochastic switched systems has stirred some initial research interest, there still leaves much room for further investigations to reduce the possible conservations.
For instance, in Hu et al. (2008), Wu and Hu (2012) and Pavlovic and Jankovic (2012), the researchers introduce ψ-type function and investigate p-th moment and almost surely ψ γ stability for stochastic nonlinear systems. Since ψ γ stability contains exponential stability and polynomial stability, it has a wide applicability. However, there are few research results about p-th moment and almost sure ψ γ stability for stochastic switched nonlinear systems.
In this paper, we attempt to investigate p-th moment and almost sure ψ γ stability of stochastic switched nonlinear systems. Since the switching behavior exists among stochastic switched systems, the stability of subsystems does not guarantee the stability of the whole system. By the aid of the semi-martingale convergence theorem, we obtain the p-th moment ψ γ stability of stochastic switched nonlinear systems. In order to establish the criterion on almost surely ψ γ stable of stochastic switched nonlinear systems, we improve the exponential martingale inequality in this paper.
The paper is organized as follows. Firstly, the problem formulations, definitions of ψ γ stability and some lemmas are given in "Preliminaries" section. In third section, the main results on p-th moment ψ γ stability and almost surely ψ γ stability of stochastic switched nonlinear systems are obtained using multiple Lyapunov functions. An example is presented to illustrate the main results in "Examples" section . In the last section the conclusions are given.

Preliminaries
Throughout this paper, unless otherwise specified, we let R n be the n-dimensional Euclidean space; R + is the set of all non-negative real numbers; R n×m denotes the n × m real matrix space; | · | denotes the standard Euclidean norm for vectors; C 1,2 (R + × R n ) denotes the family of all non-negative functions V(t, x(t)) on R + × R n which are twice continuously differentiable in x and once in t; L p (�, R n ) denotes the family of R n − valued random variables ξ with E|ξ | p < ∞; a b denotes the maximum of a and b; s.; P(·) means the probability of a stochastic process; E[·] means the expectation of a stochastic process; N = 1, 2, . . . , N is a discrete index set, where N is a finite positive integer.
Consider a family of stochastic switched nonlinear systems described by where σ (t) : [t 0 , ∞) → N is the switching signal, let {t 1 < t 2 < · · · < t k < · · · } be a switching sequence and the i k -th subsystem is active at time interval [t k , t k+1 ], where i k is the switching instant, i k ∈ N , k = 0, 1, 2, . . .. System (1) is consisted with many stochastic subsystems dx(t) = f i (t, x(t))dt + g i (t, x(t))dw(t) which are driven by switching signal σ (t). x(t) ∈ R n is the state of the system, w(t) is an m-dimensional Brownian motion defined on the complete probability space (�, F, {F t }, P), with filtration F t satisfying the usual conditions (i.e. it is increasing and right continuous while F 0 contains all P-null sets), functions f : R + × R n → R n , g : R + × R n → R n×m are locally Lipschitz in x(t) ∈ R n and piecewise continuous in t for all t ≥ t 0 and f (t, 0) = 0, g(t, 0) = 0, t ∈ [t 0 , ∞).
For the existence and uniqueness of the solution we impose an assumption (A): Both f i (t, x(t)) and g i (t, x(t)) satisfy the Lipschitz condition and the linear growth condition. That is, there exist a group of constants L i > 0 such that For all t ≥ 0, and x, y ∈ R n , and, moreover, there is a group of constants K i > 0 such that For all t ≥ 0, and x ∈ R n .
The purpose of this paper is to investigate the pth moment and almost sure ψ γ stability of system (1), we first introduce some definitions as follows.
Definition 2 For p > 0, the stochastic switched nonlinear system (1) is said to be p-th moment ψ γ stable, if there exist positive constants β, γ and function ψ(·) defined above, such that when p = 2, we say that it is ψ γ stable in mean square, when ψ(t) = e −t , we say that it is p-th moment exponential stable, when ψ(t) = (1 + t) −1 ,we say that it is p-th moment polynomial stable. (1) is said to be almost surely ψ γ stable, if there exist a positive constant γ and function ψ(·) defined above, such that when ψ(t) = e −t , we say that it is almost surely exponential stable, when ψ(t) = (1 + t) −1 , we say that it is almost surely polynomial stable.

Definition 3 Stochastic switched nonlinear systems
Before giving some efficient lemmas, let us introduce Itô formula. For system (1), give any function V (t, x) ∈ C 1,2 (R + × R n ) and define operators dV(t, x) and LV (t, x) as follows.
where Lemma 4 (Hu et al. 2008; Semi-martingale Convergence Theorem) Let M(t) is a real value continuous local martingale, and M(0) = 0, a.s., ζ is an F 0 measurable non-negative random variable, if X(t) is an F t adapted continuous non-negative process and satisfies with then, EX(t) ≤ Eζ, and X(t) is bounded a.s., that is Lemma 5 (Mao 1997 In order to establish the criterion on almost sure ψ γ stability of stochastic switched nonlinear systems, we need to modify the exponential martingale inequality as follows.
Proof For any positive integer n ≥ 1, define a stopping time: and a process: where [0, τ n ] is a stochastic time interval. When the set is empty, τ n = inf φ = ∞, obviously, τ n ↑ ∞. By Itô formula, we have That is, e y(t) is a non-negative martingale for all t ≥ 0, E[e y(t) ] = 1. By the first part of Lemma 5 with p = 1, we obtain That is Let, n → ∞, we have So, the proof is completed.
Lemma 7 (Mao 1997; Borel-Cantelli's Lemma) If {A k } ⊂ F and ∞ k=1 P(A k ) < ∞, then That is, there exist a set o ∈ F with P(� o ) = 1 and an integer valued random variable k o such that for every ω ∈ � o we have ω / ∈ A k whenever k ≥ k o (ω).

Main results
In this section, we shall tend to investigate a family of stochastic switched nonlinear systems by using multiple Lyapunov functions approach and give some sufficient conditions estimating the p-th moment and almost surely ψ γ stable. Before giving the efficient theorems, we assume that the switching signal σ (t) is right continuous. Let us turn our attention to system (1) and give some sufficient results.

Theorem 1 For stochastic switched nonlinear systems (1), let (A) hold, if there exist a group of Lyapunov functions
x) ∈ C 1,2 (R + × R n ) and positive constants p, b i , c i , γ, and η ≥ 1, such that, for all t ≥ 0, x ∈ R n and at each switching instant t k , (k = 1, 2, . . .), Then, for every x 0 ∈ R n , there exists a solution x(t) = x(t, x 0 ) on [t 0 , ∞) to stochastic switched nonlinear system (1). Moreover, the system (1) is p-th moment ψ γ stable and Proof Let x(t) = x(t, x 0 ) is a solution of stochastic switched nonlinear system (1) and We can give switching signal σ (t) and instant t for arbitrary, and assume that t k is the last switching instant before t, i.e. there is no switching on the interval [t k , t).
By condition (5), we obtain Let t 0 = 0, we obtain By condition (5) By Lemma 4, we have b σ (t k ) h(t) ≤ η k c σ (t k ) |x 0 | p . Then Let b = min{b i { and c = max{c i η k }, we obtain Thus, the system (1) is p-th moment ψ γ stable.
Remark 1 Here we generalize our research to stochastic switched nonlinear systems. The result shows how to derive some useful conditions for stochastic switched systems in terms of the multiple Lyapunov functions method.
In the following, the almost sure ψ γ stability of system (1) is presented.
Theorem 2 For stochastic switched nonlinear system (1), let (A) hold, if there exist positive constants p, b i , γ , η ≥ 1, and a group of Lyapunov functions V i (t, x) ∈ C 1,2 (R + × R n ), such that, for all t ≥ 0 and x � = 0 and at each switching instant t k , (k = 1, 2, . . .), Then, for every x 0 ∈ R n , there exists a solution x(t) = x(t, x 0 ) on [t 0 , ∞) to stochastic switched nonlinear system (1). Moreover, the system (1) is almost surely ψ γ stable and Proof Clearly, (13) holds for x 0 = 0 since x(t, x 0 ) ≡ 0. We therefore only need to show (13) for is a solution of stochastic switched nonlinear system (1). By condition (10), we have ln According to Itô formula, we obtain where We can give switching signal σ (t) and instant t for arbitrary, and assume that t k is the last switching instant before t, i.e. there is no switching on the interval [t k , t).

Examples
In this section, a numerical example is given to illustrate the effectiveness of the main results established in "Main results" section as follows.
For the first subsystem, If x � = 0, we have If x = 0 then, V 1 (t, x) = 0, LV 1 (t, x) = 0. For the second subsystem, If x � = 0, we have If x = 0 then, V 2 (t, x) = 0, LV 2 (t, x) = 0. By Theorem 1, we can choose p = 2, γ = 2, then LV i (t, x) ≤ γ ψ 1 (t)V i (t, x), i = 1, 2 , which means that the conditions of Theorem 1 are satisfied. So the stochastic switched nonlinear systems are p-th moment ψ γ stable. The switching signal and the state trajectory are presented in Figs. 1 and 2 respectively. Remark 3 In this example, a stochastic switched nonlinear system is constructed to show the efficiency of the main results. Figure 1 describes switching signal changes over the time. Figure 2 depicts state trajectory changes over the time and shows the system is p-th moment ψ γ stability.