ANALYSIS OF ZENO STABILITY IN HYBRID SYSTEMS USING SUM-OF-SQUARES PROGRAMMING
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Hybrid dynamical systems are systems that combine continuous dynamics with discrete transitions. Such systems can exhibit many unique phenomena, such as Zeno behavior. Zeno behavior is the occurrence of infinite discrete transitions in finite time. This phenomenon has been likened to a form of finite-time asymptotic stability, wherein trajectories converge asymptotically to compact sets in finite time whilst undergoing infinite transitions. Corresponding Lyapunov theorems have been developed. The main objective of our research was to develop computational techniques to determine whether or not a given hybrid system exhibits this Zeno phenomenon. In this thesis, we propose a method to algorithmically construct Lyapunov functions to prove Zeno stability of compact sets in hybrid systems. We use sum-of-squares programming to construct Lyapunov functions that allow us to prove Zeno stability of compact sets for hybrid systems with polynomial vector fields. Examples illustrating the use of the proposed technique are also provided. Finally, we provide a method using sum-of-squares programming to show Zeno stability of compact sets for systems with parametric uncertainties in the vector field, guard sets and domains, and transition maps. We then discuss potential applications of the proposed methods, along with examples.