Stochastic stability of diffusions with semi-Markovian switching

Abstract

This paper examines the stochastic stability of switched noisy dynamics in continuous time. The systems studied comprise two parts: an indexed finite set of time continuous noisy dynamics given in terms of stochastic differential equations determining the time evolution of the continuous state, and secondly a stochastic time continuous switching process assuming values in a finite index set. The time evolution of the continuous state is at all times, determined by the dynamics indexed by the switching process. The main contribution of this paper appear is stochastic stability results for switched dynamics with semi-Markovian switching. The notion of Moment Stability in the Wide Sense (MSWS) is applied as a generalization of ϵ-moment stability. MSWS is intentionally not based on stochastic convergence, since in most practically appearing systems convergence to any equilibrium is not present. A sufficient criterion for MSWS is presented for the above class of systems comprising a finite set of noisy dynamical systems among which switching is governed by a semi-Markov process, where each subsystem is assumed to be characterized by a Lyapunov function candidate together with an associated growth rate equation. For the set of Lyapunov functions, a compatibility criterion is assumed to be fulfilled, bounding the ratio between pairs of Lyapunov functions.