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Dynamical systems characterized by oscillatory behaviors and well-definedaverage vector fields have traditionally been subjects of stability analysisthrough methodologies rooted in averaging theory. Such tools have also foundapplication in the stability analysis of systems that combine continuous-timedynamics and discrete-time dynamics, referred to as hybrid dynamical systems.However, in contrast to the existing results available in the literature forcontinuous-time systems, averaging results for hybrid systems have mostly beenlimited to first-order averaging methods. This limitation prevents their directapplication for the analysis and design of systems and algorithms requiringhigh-order averaging techniques. Among other applications, such techniques arenecessary to analyze hybrid Lie-bracket-based extremum seeking algorithms andhybrid vibrational controllers. To address this gap, this paper introduces anovel high-order averaging theorem for the stability analysis of hybriddynamical systems with high-frequency periodic flow maps. The consideredsystems allow for the incorporation of set-valued flow maps and jump maps,effectively modeling well-posed differential and difference inclusions. Byimposing appropriate regularity conditions on the hybrid system's data, resultson $(T,\varepsilon)$-closeness of solutions and semi-global practicalasymptotic stability for sets are established. In this way, our findings yieldhybrid Lie-bracket averaging tools that extend those found in the literature onordinary differential equations. These theoretical results are then applied tothe study of three distinct applications in the context of model-free controland optimization.

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