A Geometric Approach to Optimal Control of Hybrid and Impulsive Systems

TL;DR

This paper introduces a geometric framework for optimal control of hybrid systems, utilizing a hybrid Pontryagin's maximum principle that leverages geometric mechanics to better understand and control system behaviors, including Zeno phenomena.

Contribution

It develops a geometric approach to hybrid optimal control, extending Pontryagin's maximum principle to hybrid systems with applications to mechanical impact systems.

Findings

The geometric maximum principle helps control Zeno behavior.

Application to mechanical impact systems reveals additional structure.

Multiple examples demonstrate the approach's effectiveness.

Abstract

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory is applied to these systems and a hybrid version of Pontryagin's maximum principle is presented. This hybrid maximum principle is presented to emphasize its geometric nature which makes its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach is that Zeno behavior can be strongly controlled for "generic" control problems. Moreover, when the underlying control system is a mechanical impact system, additional structure is present which can be exploited and is thus explored. Multiple examples are presented for both mechanical and non-mechanical systems.