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

Gu and Gao SpringerPlus (2016)5:2002 DOI 10.1186/s40064-016-3686-z Open Access RESEARCH P‑th moment and almost sure stability of stochastic switched nonlinear systems Haibo Gu and Caixia Gao* *Correspondence: School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, People’s Republic of China Abstract This paper mainly tends to utilize 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. Keywords: Stochastic switched nonlinear system, P-th moment stability, Almost sure stability, Multiple Lyapunov functions approach Mathematics Subject Classification: Primary 60H10, 34D20; Secondary 93C30 Background Switched system is an indispensable class of hybrid dynamical systems, which is composed of a family of subsystems and a rule that orchestrates the switching among them. Yet, there exist stochastic effects in the practical systems such as physics, biology, information science and economic. (for instance, see Xie and Wang 2003; Zhang and Nie 2004; Goetz and Hritonenko 2008). Over the previous few decades, stochastic switched systems have received much attention due to their potential applications in many fields, such as the control of mechanical systems, automotive industry, aircraft and air traffic control, chemical and electrical engineering, etc., see e.g. Krystul (2001), Øksendal (2005). As is well known, stability is one of the major issues in the study of control theory. In particular, the existence, uniqueness and stability of solutions for stochastic systems are investigated in Mao (1997), Hu et al. (2008), Zhang and Chen (2004) and the stability results of switched systems are given in Hu et al. (1999) and Aleksandrov et al. (2011). Some exponential stability, almost sure exponential stability and p-th moment stability criteria are obtained for stochastic systems in Mao (1994), Khasminskii (1980), Shen and Wang (2009). Recently, some efforts have been made to extend the stability results from stochastic systems to stochastic switched systems (Feng and Zhang 2006; Feng et al. 2011; Chatterjee and Liberzon 2004; Filipovic 2009; Ai and Zong 2014). In Branicky (1998), Dimarogonas and Kyriakopoulos (2004) and Chatterjee and Liberzon (2006) the investigators utilize multiple Lyapunov functions approach to study the stability of stochastic switched systems. P-th moment exponential stability of stochastic switched © The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Gu and Gao SpringerPlus (2016)5:2002 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 Rn be the n-dimensional Euclidean space; R+ is the set of all non-negative real numbers; Rn×m denotes the n × m real matrix space; | · | denotes the standard Euclidean norm for vectors; C 1,2 (R+ × Rn ) denotes the family of all non-negative functions V(t, x(t)) on R+ × Rn which are twice continuously differentiable in x and once in t; Lp (�, Rn ) denotes the family of Rn −  valued random variables ξ with E|ξ |p < ∞; a b denotes the maximum of a and b; Lp ([a, b], Rn ) denotes the family of Rn −valued Ft −adapted processes {f (t)}a≤t≤b such b that a |f (t)|p dt < ∞ a.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  dx(t) = fσ (t) (t, x(t))dt + gσ (t) (t, x(t))dw(t) (1) x(t0 ) = x0 , t0 = 0 where σ (t) : [t0 , ∞) → N is the switching signal, let {t1 < t2 < · · · < tk < · · · } be a switching sequence and the ik -th subsystem is active at time interval [tk , tk+1 ], where ik is the switching instant, ik ∈ N , k = 0, 1, 2, . . .. System (1) is consisted with many stochastic subsystems dx(t) = fi (t, x(t))dt + gi (t, x(t))dw(t) which are driven by Page 2 of 13 Gu and Gao SpringerPlus (2016)5:2002 Page 3 of 13 switching signal σ (t). x(t) ∈ Rn is the state of the system, w(t) is an m-dimensional Brownian motion defined on the complete probability space (�, F , {Ft }, P), with filtration Ft satisfying the usual conditions (i.e. it is increasing and right continuous while F0 contains all P-null sets), functions f : R+ × Rn → Rn, g : R+ × Rn → Rn×m are locally Lipschitz in x(t) ∈ Rn and piecewise continuous in t for all t ≥ t0 and f (t, 0) = 0, g(t, 0) = 0, t ∈ [t0 , ∞). For the existence and uniqueness of the solution we impose an assumption (A): Both fi (...truncated)