Optimality of Zeno Executions in Hybrid Systems

TL;DR

This paper investigates the role of Zeno trajectories in hybrid systems, showing that they are sometimes essential for optimal control solutions, contrary to common avoidance practices, using the bouncing ball as an illustrative example.

Contribution

It demonstrates the necessity of Zeno executions in optimal control problems within hybrid systems, challenging the typical view of avoiding them.

Findings

Zeno trajectories can be essential in optimal control solutions.

Avoiding Zeno trajectories may not always be feasible or desirable.

The bouncing ball example illustrates the importance of Zeno control executions.

Abstract

A unique feature of hybrid dynamical systems (systems whose evolution is subject to both continuous- and discrete-time laws) is Zeno trajectories. Usually these trajectories are avoided as they can cause incorrect numerical results as the problem becomes ill-conditioned. However, these are difficult to justifiably avoid as determining when and where they occur is a non-trivial task. It turns out that in optimal control problems, not only can they not be avoided, but are sometimes required in synthesizing the solutions. This work explores the pedagogical example of the bouncing ball to demonstrate the importance of "Zeno control executions."