  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.
