Asymptotic Stabilizability of a Class of Stochastic Nonlinear Hybrid Systems

Ewelina Seroka

doi:10.1155/2015/231214

Abstract

The problem of the asymptotic stabilizability in probability of a class of stochastic nonlinear control hybrid systems (with a linear dependence of the control) with state dependent, Markovian, and any switching rule is considered in the paper. To solve the issue, the Lyapunov technique, including a common, single, and multiple Lyapunov function, the hybrid control theory, and some results for stochastic nonhybrid systems are used. Sufficient conditions for the asymptotic stabilizability in probability for a considered class of hybrid systems are formulated. Also the stabilizing control in a feedback form is considered. Furthermore, in the case of hybrid systems with the state dependent switching rule, a method for a construction of stabilizing switching rules is proposed. Obtained results are illustrated by examples and numerical simulations.

Highlights

In recent years there has been a growing interest in hybrid systems
By applying the obtained control the trivial solution of hybrid system becomes asymptotically stable in probability
In the second section we present main results regarding the asymptotic stabilizability of stochastic nonlinear hybrid systems with any, state dependent, and Markovian switching rule

Summary

Introduction

In recent years there has been a growing interest in hybrid systems. At the same time one of the basic problems in the control theory is the problem of the stability and the stabilizability of dynamic systems. In the absence of the common Lyapunov function, stability properties of the hybrid system in general depend on the switching signal, and in this case the hybrid system is not stable for any switching rules, but only for the so-called stabilizing switching rules [1]. In the present paper ideas of a feedback control, proposed by Florchinger for nonhybrid stochastic nonlinear control systems [16], are used and combined with the concept of the hybrid control theory [1, 3, 6, 10] to derive results for the asymptotic stabilizability of stochastic nonlinear control hybrid systems (with a linear dependence of the control) described by Itostochastic differential equations with any, state dependent, and Markovian switching rule. In the second section we present main results regarding the asymptotic stabilizability of stochastic nonlinear hybrid systems with any, state dependent, and Markovian switching rule.

Mathematical Preliminaries

Stabilizability of Hybrid Systems

Conclusions