Hybrid trajectory spaces

In this paper, we present a general framework for describing and studying hybrid systems. We represent the trajectories of the system as functions on a hybrid time domain, and the system itself by its trajectory space, which is the set of all possible trajectories. The trajectory space is given a natural topology, the compact-open hybrid Skorohod topology, and we prove the existence of limiting trajectories under uniform equicontinuity assumptions. We give a compactness result for the trajectory space of impulse differential inclusions, a class of nondeterministic hybrid system, and discuss how to describe hybrid automata, a widely-used class of hybrid system, as impulse differential inclusions. For systems with compact trajectory space, we obtain results on Zeno properties, symbolic dynamics and invariant measures. We give examples showing the application of the results obtained using the trajectory space approach