Notions and Sufficient Conditions for Pointwise Asymptotic Stability in Hybrid Systems*

Pointwise asymptotic stability is a property of a set of equilibria of a dynamical system, where every equilibrium is Lyapunov stable and every solution converges to some equilibrium. Hybrid systems are dynamical systems which combine continuous-time and discrete-time dynamics. In this paper, they are modeled by a combination of differential equations or inclusions, of difference equations or inclusions, and of constraints on the resulting motions. Sufficient conditions for pointwise asymptotic stability of a closed set are given in terms of set-valued Lyapunov functions: they require that the values of the Lyapunov function shrink along solutions. Cases of strict and weak decrease are considered. Lyapunov functions, not set-valued, which imply that solutions have finite length are used in sufficient conditions and related to the set-valued Lyapunov functions. Partial pointwise asymptotic stability is also addressed.