Abstract
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.
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