On the stability of weakly observable Markov jump linear systems with bounded long run average cost

In this paper we consider Markov jump linear systems with additive noise and the notion of weak observability, and we explore the relation between the long run average cost and the conditional second moment of the state. We show for observable systems that bounded long run average cost is a necessary and sufficient condition for the second moment of the state to be bounded. Under the same hypothesis, we show that the spectral radius of the operator associated with the conditional second moment is lesser than one. Numerical examples are included to illustrate the results.