Averaging in a Class of Stochastic Hybrid Dynamical Systems with Time-Varying Flow Maps

We present stability and recurrence results for a class of stochastic hybrid dynamical systems with oscillating flow maps. These results are developed by introducing averaging tools that parallel those already existing for ordinary differential equations and deterministic hybrid dynamical systems. Such tools can be used to examine the stability properties of the original dynamics based on the properties of a simpler dynamical system constructed from the average of the original oscillating vector field. In this work, we focus on a class of systems for which global stability and recurrence results are achievable under suitable smoothness assumptions on the dynamics. By studying the average stochastic hybrid dynamics using Lyapunov-Foster functions, we derive similar stability and recurrence results for the original stochastic hybrid system.