On Lie-Bracket Averaging for a Class of Hybrid Dynamical Systems with Applications to Model-Free Control and Optimization

Abstract: The stability of dynamical systems with oscillatory behaviors and well-defined average vector fields has traditionally been studied using averaging theory. These tools have also been applied to hybrid dynamical systems, which combine continuous and discrete dynamics. However, most averaging results for hybrid systems are limited to first-order methods, hindering their use in systems and algorithms that require high-order averaging techniques, such as hybrid Lie-bracket-based extremum seeking algorithms and hybrid vibrational controllers. To address this limitation, we introduce a novel high-order averaging theorem for analyzing the stability of hybrid dynamical systems with high-frequency periodic flow maps. These systems incorporate set-valued flow maps and jump maps, effectively modeling well-posed differential and difference inclusions. By imposing appropriate regularity conditions, we establish results on $(T,\varepsilon)$-closeness of solutions and semi-global practical asymptotic stability for sets. These theoretical results are then applied to the study of three distinct applications in the context of hybrid model-free control and optimization via Lie-bracket averaging.