Stochastic hybrid systems: a modeling and stability theory tutorial

Abstract — Stochastic hybrid systems are driven by random processes and have states that can both flow continuously and jump instantaneously. Many classes of stochastic hybrid sys-tems, with different modeling strengths, have been considered in the literature. In this tutorial paper we first consider perhaps the simplest class of stochastic hybrid systems: those that admit unique solutions and that do not permit state conditions that force jumps. Several examples are given to illustrate the utility of this simple modeling class and Lyapunov-based sufficient conditions for various stability properties are given. The second half of the tutorial addresses a recent, more general stochastic hybrid systems modeling framework that permits state conditions to trigger jumps and that allows for non-unique solutions, via stochastic differential and difference inclusions and possibly overlapping flow and jump sets. Examples are provided to show the relevance of models that admit non-unique solutions and forced jumps. Lyapunov-based sufficient conditions for various stability properties for this class of stochastic hybrid systems are also provided. I