Regularity Properties of Reachability Maps for Hybrid Dynamical Systems with Applications to Safety

In this paper, motivated by the safety problem in hybrid systems, two set-valued reachability maps are introduced. The outer semicontinuity, the continuity, and the local boundedness of the proposed reachability maps with respect to their arguments are analyzed under mild regularity conditions. This study is then used to revisit and improve the existing converse safety statements in terms of barrier functions. In particular, for safe hybrid systems satisfying the aforementioned regularity conditions, we construct time-varying barrier functions that depend on the proposed reachability maps. Consequently, we show that the constructed barrier functions inherit the continuity properties established for the proposed reachability maps.