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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.

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Notions and Sufficient Conditions for Pointwise Asymptotic Stability in Hybrid Systems**The work by the first author was partially supported by the Simons Foundation Grant 315326. The work by the second author was partially supported by NSF Grants no. ECS-1150306 and CNS-1544396, and by AFOSR Grant and FA9550-16-1-0015.