Stochastic hybrid systems: A modeling and stability theory tutorial

Stochastic hybrid systems are driven by random processes and have states that can both flow continuously and jump instantaneously. Many classes of stochastic hybrid systems, 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.