Non-uniqueness in Reverse Time of Hybrid System Trajectories

Under standard Lipschitz conditions, trajectories of systems described by ordinary differential equations are well defined in both forward and reverse time. (The flow map is invertible.) However for hybrid systems, uniqueness of trajectories in forward time does not guarantee flow-map invertibility, allowing non-uniqueness in reverse time. The paper establishes a necessary and sufficient condition that governs invertibility through events. It is shown that this condition is equivalent to requiring reverse-time trajectories to transversally encounter event triggering hypersurfaces. This analysis motivates a homotopy algorithm that traces a one-manifold of initial conditions that give rise to trajectories which all reach a common point at the same time.