Stability Analysis For Semi-Markov Stochastic Switching Systems

Stochastic switching systems have many applications in circuit systems,industrial manufacturing and electronic communications.The semi-Markov process has received extensive attention from scholars because it breaks through the limitations of the traditional Markov process in the distribution of sojourn time.This paper focuses on the stability of semi-Markov linear and nonlinear stochastic switching systems driven by the L(?)vy process.This paper first discusses the stability of the semi-Markov linear stochastic switching system with Poisson jump,based on the explicit solution of the linear system,generalizes the results of the Markov linear jump diffusion system in Zong et al.(2014)to the case of semiMarkov switching.Based on the ergodic property of embedded Markov chain,the discriminant conditions of almost sure exponential stability and th-moment exponential stability are obtained.Secondly,based on the stochastic integral representation of the semi-Markov process and the generalized It ?o formula,this paper uses the multiple Lyapunov function method to generalize the results of the Markov nonlinear jump diffusion system in Zong et al.(2014)to the case of semi-Markov switching,and establishes sufficient conditions for almost sure exponential stability of the system.Finally,examples are given to illustrate the theoretical results.For the semi-Markov switching system driven by Brownian motion,this paper introduces the -type function,using the multiple Lyapunov function method,obtains the discriminant condition of the general decay rate stability of the system that only depends on the asymptotic properties of the switching signal.In addition,by using the probability analysis method,the stability results of the semi-Markov switching system without noise interference in Wu et al.(2017)are generalized to the situation with Brownian motion,and sufficient conditions for the globally asymptotically stable almost surely of the system are obtained.Finally,examples are given to illustrate the theoretical results.