Koopman Operators for Global Analysis of Hybrid Limit-Cycling Systems: Construction and Spectral Properties

TL;DR

This paper develops a Koopman operator framework for hybrid systems with stable limit cycles, enabling global spectral analysis by extending smooth structures and ensuring eigenfunction uniqueness.

Contribution

It introduces a rigorous construction of Koopman operators for hybrid limit-cycling systems, extending spectral theory to hybrid dynamical systems.

Findings

Constructed an observable space preserving hybrid system structures

Derived spectral properties and eigenfunctions of Koopman operators

Facilitated global analysis of hybrid systems using Koopman theory

Abstract

This paper reports a theory of Koopman operators for a class of hybrid dynamical systems with globally asymptotically stable periodic orbits, called hybrid limit-cycling systems. We leverage smooth structures intrinsic to the hybrid dynamical systems, thereby extending the existing theory of Koopman operators for smooth dynamical systems. Rigorous construction of an observable space is carried out to preserve the inherited smooth structures of the hybrid dynamical systems. Complete spectral characterization of the Koopman operators acting on the constructed space is then derived where the existence and uniqueness of their eigenfunctions are ensured. Our results facilitate global analysis of hybrid dynamical systems using the Koopman operator.