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In the first part of the paper, we consider a discrete-time stochasticcontrol system. We show that, under certain conditions, the set of randomoccupational measures generated by the state-control trajectories of the systemas well as the set of their mathematical expectations converge (as the timehorizon tends to infinity) to a convex and compact (non-random) set, which isshown to coincide with the set of stationary probabilities of the system. Inthe second part, we apply the results obtained in the first part to deal with ahybrid system that evolves in continuous time and is subjected to abruptchanges of certain parameters. We show that the solutions of such a hybridsystem are approximated by the solutions of a differential inclusion, theright-hand side of which is defined by the limit occupational measures set, theexistence and convexity of which is established in the first part of the paper.

On convergence of occupational measures sets of a discrete-time stochastic control system, with applications to averaging of hybrid systems