Hybrid systems exhibit phenomena which do not occur in systems withcontinuous vector fields. One such phenomenon - Zeno executions - ischaracterized by an infinite number of discrete events or transitions occurringover a finite interval of time. This phenomenon is not necessarily undesirableand may indeed be used to capture physical phenomena. In this paper, we examinethe problem of proving the existence and stability of zero executions. Ourapproach is to develop a polynomial-time algorithm - based on thesum-of-squares methodology - for verifying the stability of a Zeno execution.We begin by stating Lyapunov-like theorems for local Zeno stability based onexisting results. Then, for hybrid systems with polynomial vector fields, weuse polynomial Lyapunov functions and semialgebraic geometry(Positivstellensatz results) to reduce the local Lyapunov-like conditions to aconvex feasibility problem in polynomial variables. The feasibility problem isthen tested using an algorithm for sum-of-squares programming - SOSTOOLS. Wealso extend these results to hybrid system with parametric uncertainty, wherethe uncertain parameters lie in a semialgebriac set. We also provide severalexamples illustrating the use of our technique.
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