Stochastic approximation of hybrid systems: Boundedness and asymptotic behavior

Abstract

This paper provides a framework for analyzing numerical simulations of a hybrid system when the system’s flow map is approximated stochastically. In this way, the paper makes one of the first substantive connections between hybrid systems theory and the vast literature on stochastic approximations of non-stochastic dynamical systems. The results include conditions for almost sure boundedness and a characterization of the asymptotic behavior of solutions to the stochastic approximation. Namely, under certain conditions on the expected value and variance of the noisy flow map, and with appropriate step sizes, the sample paths of the solutions of the stochastic approximation are shown to have desired boundedness properties and converge to the chain-recurrent part of the global attractor, when one exists, of the underlying hybrid system. The results are established using martingale methods, Morse decompositions of attractors, and Morse–Lyapunov functions. Since the hybrid system’s flow map and its approximation may be set-valued, stochastic approximations of differential inclusions are covered as a special case. Several examples and simulations are provided to illustrate hybrid systems, their stochastic approximations, and their convergence properties.