How well-posedness of hybrid systems can extend beyond Zeno times

Abstract

The extension of a solution to a hybrid system beyond its Zeno time is a simple exercise in modeling. However, assuring that the extended system is well-posed in a certain sense, in particular, that the extension of a solution depends reasonably on initial, pre-Zeno, conditions, has not been addressed. In this paper it is shown that these results hold for hybrid systems that exhibit Zeno behavior when the set of Zeno equilibria forms a continuum that has certain stability properties. Several scenarios of going past Zeno are presented. Dependence of limits of Zeno solutions, of Zeno times, and of reachable sets on initial conditions is also discussed.