A direct proof of Conley’s decomposition for well-posed hybrid inclusions

A hybrid inclusion combines a differential inclusion, a difference inclusion, and constraints on the resulting continuous-time and discrete-time dynamics. Under basic assumptions on the data of a hybrid inclusion, the paper proves a version of the Conley’s fundamental theorem, which states that a compact and invariant set for a hybrid inclusion decomposes into a part that is chain-recurrent and the complementary part where the dynamics is gradient-like. This is done by constructing a total Lyapunov function. The proof is inspired by the well-known approaches to the Conley’s fundamental theorem for continuous or for discrete dynamics, and has foundations in the previously established asymptotic stability theory for hybrid inclusions.