Stability of hybrid systems: state of the art

This paper collects work on the stability analysis of hybrid systems. The hybrid systems considered are those that combine continuous dynamics (represented by differential or difference equations) with finite dynamics, usually thought of as being a finite automaton. We review multiple Lyapunov functions as a tool for analyzing Lyapunov stability of general hybrid systems. Background results, the author's introductory work, and subsequent extensions are covered. Specializing to hybrid systems with linear dynamics in each constituent mode and linear jump operators, we review some key theorems of Barabanov-Staroshilov (1988), and give corollaries encompassing several recently-derived "stability by first approximation" theorems in the literature. We also comment on the use of computational tests for stability of hybrid systems, and the general complexity. The result is a tutorial on the state of the art in theory and computation of hybrid systems stability.