On Composite Foster Functions for a Class of Singularly Perturbed Stochastic Hybrid Inclusions

Abstract: We study sufficient conditions for stability and recurrence in a class of singularly perturbed stochastic hybrid dynamical systems. The systems considered combine multi-time-scale deterministic continuous-time dynamics, modeled by constrained differential inclusions, with discrete-time dynamics described by constrained difference inclusions subject to random disturbances. Under suitable regularity assumptions on the dynamics and causality of the associated solutions, we develop a family of composite nonsmooth Lagrange-Foster and Lyapunov-Foster functions that certify stability and recurrence properties by leveraging simpler functions related to the slow and fast subsystems. Stability is characterized with respect to compact sets, while recurrence is established for bounded open sets. The proposed framework is illustrated through several examples and applications, including the stability analysis of singularly perturbed switching systems with stochastic spontaneous mode transitions, feedback optimization problems with stochastically switching plants, and momentum-based feedback optimization algorithms with stochastic restarting.