Stability analysis of switched systems on time scales with all modes unstable

Abstract

Abstract In this paper, global uniform asymptotic stability problems of switched systems on time scales with all modes unstable are considered. First, a more general stability criterion is proposed for nonlinear switched systems on time scales, in which the value of Lyapunov function at some switching instants may not necessarily decrease. Second, based on the discretized Lyapunov function technique, a sufficient condition is derived for stability analysis of linear switched systems with all subsystems unstable on a special kind of time scale\xa0(the graininess function is constant). It is shown that this approach does not require the system matrix of one subsystem must commute with the one of the other subsystem, which is essential for stability analysis of linear switched systems on non-uniform time domains by the eigenvalue based approach in the existing results. Moreover, the idea of this paper not only provides a unified approach to study continuous switched systems and their discrete counterparts simultaneously, but also is effective to investigate stability problems of other systems on uniform or non-uniform time domains. Two examples are given for illustrating the effectiveness of the proposed results.