On the Relaxation of Hybrid Dynamical Systems

Hybrid dynamical systems have proven to be a powerful modeling abstraction, yet fundamental questions regarding the dynamical properties of these systems remain. In this paper, we develop a novel class of relaxations which we use to recover a number of classic systems theoretic properties for hybrid systems, such as existence and uniqueness of trajectories, even past the point of Zeno. Our relaxations also naturally give rise to a class of provably convergent numerical approximations, capable of simulating through Zeno. Using our methods, we are also able to perform sensitivity analysis about nominal trajectories undergoing a discrete transition -- a technique with many practical applications, such as assessing the stability of periodic orbits.