A New Solution Concept and Family of Relaxations for Hybrid Dynamical Systems

Hybrid dynamical systems have proven to be a powerful modeling abstraction, yet fundamental questions regarding their dynamical properties remain. In this paper, we develop a novel solution concept for a class of hybrid systems, which is a generalization of Filippov’s solution concept. In the mathematical theory, these hybrid Filippov solutions eliminate the notion of Zeno executions. Building on previous techniques for relaxing hybrid systems, we then introduce a family of smooth control systems that are used to approximate this solution concept. The trajectories of these relaxations vary differentiably with respect to initial conditions and inputs, may be numerically approximated using existing techniques, and are shown to converge to the hybrid Filippov solution in the limit. Finally, we outline how the results of this paper provide a foundation for future work to control hybrid systems using well-established techniques from Control Theory.