An operator-theoretic viewpoint to non-smooth dynamical systems: Koopman analysis of a hybrid pendulum

TL;DR

This paper introduces a novel operator-theoretic approach using Koopman analysis to study non-smooth hybrid dynamical systems, exemplified by a pendulum with state resets, linking spectral properties to geometric features.

Contribution

It develops a framework connecting Koopman operator theory with geometric analysis for hybrid systems, providing new insights into their spectral and geometric properties.

Findings

Established a link between Koopman spectral properties and geometric features.

Analyzed a hybrid pendulum with state resets using the proposed framework.

Demonstrated the applicability of Koopman analysis to non-smooth dynamical systems.

Abstract

We apply an operator-theoretic viewpoint to a class of non-smooth dynamical systems that are exposed to event-triggered state resets. The considered benchmark problem is that of a pendulum which receives a downward kick under certain fixed angles. The pendulum is modeled as a hybrid automaton and is analyzed from both a geometric perspective and the formalism carried out by Koopman operator theory. A connection is drawn between these two interpretations of a dynamical system by means of establishing a link between the spectral properties of the Koopman operator and the geometric properties in the state-space.