This paper is concerned with stability of a class of randomly switched systems of ordinary differential equations. The system under consideration can be viewed as a two-component process （X（t）, α（t））, where the...This paper is concerned with stability of a class of randomly switched systems of ordinary differential equations. The system under consideration can be viewed as a two-component process （X（t）, α（t））, where the system is linear in X（t） and α（t） is a continuous-time Markov chain with a finite state space. Conditions for almost surely exponential stability and instability are obtained. The conditions are based on the Lyapunov exponent, which in turn, depends on the associate invaxiant density. Concentrating on the case that the continuous component is two dimensional, using transformation techniques, differential equations satisfied by the invariant density associated with the Lyapunov exponent are derived. Conditions for existence and uniqueness of solutions are derived. Then numerical solutions are developed to solve the associated differential equations.展开更多
基金This research was supported in part by the National Science Foundation under Grant No. DMS-0907753, in part by the Air Force Office of Scientific Research under Grant No. FA9550-10-1-0210, and in part by the National Natural Science Foundation of China under Grant No. 70871055.
文摘This paper is concerned with stability of a class of randomly switched systems of ordinary differential equations. The system under consideration can be viewed as a two-component process （X（t）, α（t））, where the system is linear in X（t） and α（t） is a continuous-time Markov chain with a finite state space. Conditions for almost surely exponential stability and instability are obtained. The conditions are based on the Lyapunov exponent, which in turn, depends on the associate invaxiant density. Concentrating on the case that the continuous component is two dimensional, using transformation techniques, differential equations satisfied by the invariant density associated with the Lyapunov exponent are derived. Conditions for existence and uniqueness of solutions are derived. Then numerical solutions are developed to solve the associated differential equations.
