Using the Set-Valued Bouncing Ball for Bounding Zeno Solutions of Lagrangian Hybrid Systems

In this paper, we study Zeno behavior in Lagrangian hybrid systems, which are mechanical systems with unilateral constraints that are undergoing impacts. Zeno solutions involve an infinite number of impacts occurring in a finite amount of time (the Zeno time). In such systems, one is typically not able to explicitly compute the Zeno time and Zeno limit point, and even not to detect a Zeno solution from its initial condition. We address these problems by replacing the nonlinear dynamics with a simple hybrid system whose dynamics is a set-valued version of the bouncing ball. We utilize optimal control analysis to derive conditions for the Zenoness of all solutions and compute bounds on their Zeno time and Zeno limit point, which also apply to solutions of the original Lagrangian hybrid system. Application of the results is demonstrated on a Lagrangian hybrid system with two degrees of freedom.