Local lipschitzness of reachability maps for hybrid systems with applications to safety

Motivated by the safety problem, several definitions of reachability maps, for hybrid dynamical systems, are introduced. It is well established that, under certain conditions, the solutions to continuous-time systems depend continuously with respect to initial conditions. In such setting, the reachability maps considered in this paper are locally Lipschitz (in the Lipschitz sense for set-valued maps) when the right-hand side of the continuous-time system is locally Lipschitz. However, guaranteeing similar properties for reachability maps for hybrid systems is much more challenging. Examples of hybrid systems for which the reachability maps do not depend nicely with respect to their arguments, in the Lipschitz sense, are introduced. With such pathological cases properly identified, sufficient conditions involving the data defining a hybrid system assuring Lipschitzness of the reachability maps are formulated. As an application, the proposed conditions are shown to be useful to significantly improve an existing converse theorem for safety given in terms of barrier functions. Namely, for a class of safe hybrid systems, we show that safety is equivalent to the existence of a locally Lipschitz barrier function. Examples throughout the paper illustrate the results.