Differential equations and dynamical systems
Differential equations and dynamical systems
Control System Design
SIAM Journal on Scientific Computing
Uniqueness of solutions of linear relay systems
Automatica (Journal of IFAC)
Brief On solution concepts and well-posedness of linear relay systems
Automatica (Journal of IFAC)
Comparing forward and backward reachability as tools for safety analysis
HSCC'07 Proceedings of the 10th international conference on Hybrid systems: computation and control
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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.