Characterizations of safety in hybrid inclusions via barrier functions

Abstract

This paper investigates characterizations of safety in terms of barrier functions for hybrid systems modeled by hybrid inclusions. After introducing an adequate definition of safety for hybrid inclusions, sufficient conditions using continuously differentiable as well as lower semicontinuous barrier functions are proposed. Furthermore, the lack of existence of autonomous and continuous barrier functions certifying safety, guides us to propose, inspired by converse Lyapunov theorems for only stability, nonautonomous barrier functions and conditions that are shown to be both necessary as well as sufficient, provided that mild regularity conditions on the system's dynamics holds.