A Study of the Long-Term Behavior of Hybrid Systems with Symmetries via Reduction and the Frobenius-Perron Operator

TL;DR

This paper introduces two novel methods for analyzing the long-term behavior of hybrid systems, including a Frobenius-Perron operator analog and a reduction technique for impact systems on Lie groups, supported by numerical results.

Contribution

It presents a new operator-based approach for ensemble evolution and a reduction method for impact systems on Lie groups, advancing hybrid system analysis.

Findings

The Frobenius-Perron analog encodes asymptotic ensemble behavior.

Reduction simplifies impact systems from 2n to n+1 dimensions.

Numerical results demonstrate applicability across various systems.

Abstract

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. As typical in dynamical analysis, an essential goal is to study the long-term behavior of these systems. In this work, we present two different novel approaches for studying these systems. The first approach is based on constructing an analog of the Frobenius-Perron (transport) operator for hybrid systems. Rather than tracking the evolution of a single trajectory, this operator encodes the asymptotic nature of an ensemble of trajectories. The second approach presented applies to an important subclass of hybrid systems, mechanical impact systems. We develop an analog of Lie-Poisson(-Suslov) reduction for left-invariant impact systems on Lie groups. In addition to the Hamiltonian (and constraints) being left-invariant, the impact surface must also be a right coset of a normal subgroup. This procedure allows a reduction from a $2n$-dimensional system to an $(n+1)$-dimensional one. We conclude the paper by presenting numerical results on a diverse array of applications.