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Finite Time Stability of Stochastic Hybrid Systems

DOI: 10.1155/2014/867189

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Abstract:

This paper considers the finite time stability of stochastic hybrid systems, which has both Markovian switching and impulsive effect. First, the concept of finite time stability is extended to stochastic hybrid systems. Then, by using common Lyapunov function and multiple Lyapunov functions theory, two sufficient conditions for finite time stability of stochastic hybrid systems are presented. Furthermore, a new notion called stochastic minimum dwell time is proposed and then, combining it with the method of multiple Lyapunov functions, a sufficient condition for finite time stability of stochastic hybrid systems is given. Finally, a numerical example is provided to illustrate the theoretical results. 1. Introduction Nowadays, stochastic modeling, control, and optimization have played a crucial role in many applications especially in the areas of controlling science and communication technology [1, 2]. In practical application, many stochastic systems exhibit impulsive and switching behaviors due to abrupt changes and switches of states at certain instants during the dynamical processes; that is, the systems switch with impulsive effects [3–6]. Moreover, impulsive and switching phenomena can be found in the fields of physics, biology, engineering, and information science. Many sudden and sharp changes occur instantaneously, in the form of impulses and switches, which cannot be well described by using pure continuous or pure discrete models. Therefore, it is important and, in fact, necessary to study hybrid impulsive and switching stochastic systems. In many applications, it is desirable that the stochastic system possesses the property that trajectories which converge to a Lyapunov stable equilibrium state must do so in finite time rather than infinite time. Hence, the concept of finite time stability for stochastic systems arises naturally in stochastic control problems. For the deterministic case, finite time stability for continuous time systems was studied in [7] using H?lder continuous Lyapunov functions. Its improvements and extensions have been given by [8–10] for continuous systems satisfying uniqueness of solutions in forward time, for nonautonomous continuous systems, and for functional differential equations, respectively. In [11], the notion of finite time input-to-state stability is introduced for continuous systems with locally essentially bounded input. The problem of finite time stabilization for deterministic nonlinear systems has been accordingly studied in the literature. Numerous theoretical control design methods, including

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