Lyapunov coupled equations for infinite jump linear systems

Abstract

Deals with Lyapunov equations for continuous-time Markov jump linear systems (MJLS). We focus on the case in which the Markov chain has a countably infinite state space. It is shown that the infinite MJLS is stochastically stabilizable (SS) if and only if the associated countably infinite coupled Lyapunov equations have a unique norm bounded strictly positive solution. This result does not hold for mean square stabilizability (MSS), since SS and MSS are no longer equivalent in our set up. To some extent, tools from operator theory in Banach space and, in particular, from semigroup theory are the technical underpinning of the paper.