Characterization of Zeno behavior in hybrid systems using homological methods

Abstract

It is possible to associate to a hybrid system a single topological space its underlying topological space. Simultaneously, every hybrid system has a graph as its indexing object its underlying graph. Here we discuss the relationship between the underlying topological space of a hybrid system, its underlying graph and Zeno behavior. When each domain is contractible and the reset maps are homotopic to the identity map, the homology of the underlying topological space is isomorphic to the homology of the underlying graph; the nonexistence of Zeno is implied when the first homology is trivial. Moreover, the first homology is trivial when the null space of the incidence matrix is trivial. The result is an easy way to verify the nonexistence of Zeno behavior.