Multiple barrier function certificates for forward invariance in hybrid inclusions

As a continuation of [1] and using multiple barrier functions, this paper studies forward invariance in hybrid systems modeled by hybrid inclusions. After introducing the notion of a multiple barrier function, we propose sufficient conditions to guarantee different forward invariance properties of a closed set for hybrid systems with nonuniqueness of solutions, solutions terminating prematurely, and Zeno solutions. More precisely, we consider forward (pre-)invariance of sets, which guarantees solutions to stay in a set, and (pre-)contractivity, which further requires solutions that stay in the boundary of the set to evolve (continuously or discretely) towards its interior. Our conditions for forward invariance involve infinitesimal conditions in terms of multiple barrier functions while our conditions for pre-contractivity (and contractivity) involve Minkowski functionals. Examples illustrate the results