Stability of Mixed Semi-Markov Systems With All Unstable Modes

Abstract

This paper introduces the concept of mixed semi-Markov switching, whose sojourn time consists of a certain part and an uncertain part, where the uncertain part is determined by a semi-Markov process. Compared with classical semi-Markov switching, the concerned switching process is general and practical since it becomes the classical semi-Markov switching when the certain sojourn time is zero. Under the mixed semi-Markov switching, the stability problem for switched stochastic systems with all unstable modes is studied. By partitioning certain sojourn time, a stability criterion is established, where the conventional restriction that multiple Lyapunov functions are not decreasing at switching instants is relaxed. It is shown that, under the mixed semi-Markov switching, the resultant system achieves global asymptotic stability almost surely (GAS a.s.), even all the modes are unstable. The effectiveness of the theoretical analysis is demonstrated by a numerical example.