Stochastic Stability of Markovianly Switched Systems

This technical note examines the stochastic stability of noisy dynamics in discrete and continuous time. The notion of moment stability in the wide sense (MSWS) is presented as a generalization of ϵ-moment stability. MSWS is intentionally not based on stochastic convergence properties, since in most practically appearing systems convergence to any equilibrium is not present. A sufficient criterion for both MSWS and ergodicity is presented for a class of systems comprising a finite set of noisy dynamical systems among which switching is governed by a Markov chain. Stability/instability properties for each separate subsystem are assumed to be quantified 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.