Zeno Stability of the Set-Valued Bouncing Ball

Abstract

Hybrid dynamical systems consist of both continuous-time and discrete-time dynamics. A fundamental phenomenon that is unique to hybrid systems is Zeno behavior, where the solution involves an infinite number of discrete transitions occurring in finite time, as best illustrated in the classical example of a bouncing ball. In this note, we study the hybrid system of the set-valued bouncing ball, for which the continuous-time dynamics has a set-valued right-hand side. This system is typically used for deriving bounds on the solution of nonlinear single-valued hybrid systems in a small neighborhood of a Zeno equilibrium point in order to establish its local stability. We utilize methods of Lyapunov analysis and optimal control to derive a necessary and sufficient condition for Zeno stability of the set-valued bouncing ball system and to obtain a tight bound on the Zeno time as a function of initial conditions.