Regularity of solutions and homotopic equivalence for hybrid systems

We study the trajectories of hybrid systems evolving according to constant, convex inclusions and Lipschitz nonlinear inclusions. Two questions are addressed. First, we investigate the existence of continuous selections of trajectories with respect to the initial conditions. Second, previous work on timed automata and hybrid automata has examined equivalence relations on runs of the automaton that visit the same locations and regions of the state space. Here we examine an equivalence relation defined directly on the trajectories. With suitable conditions on the enabling regions and using a suitable metric, we construct a homotopy on the set of solutions and use the homotopy to form an equivalence relation on the trajectories. We show the relationship between region equivalence introduced in Alur and Dill (1990) and homotopic equivalence. The tools needed for studying homotopic equivalence are the same as for obtaining continuity with respect to initial conditions