A Krasovskii-LaSalle function based recurrence principle for a class of stochastic hybrid systems

We characterize the sets to which bounded random solutions generated by a class of stochastic hybrid systems converge under the existence of a Lyapunov-like function that is non-increasing almost surely during flows and on average during jumps. In particular, we establish that we get almost sure convergence to the largest weakly totally recurrent in probability set that is contained in a level set of this function. We also apply this result to establish weak sufficient conditions for uniform global asymptotic stability in probability of compact sets and uniform global recurrence of open, bounded sets for a class of stochastic hybrid systems.