Learning Bounded Koopman Observables: Results on Stability, Continuity, and Controllability

TL;DR

This paper analyzes the theoretical properties of Koopman eigenfunctions, such as stability and continuity, and discusses how these properties impact the ability of machine learning methods to accurately compute Koopman representations of dynamical systems.

Contribution

It provides a theoretical analysis of the limitations and properties of Koopman eigenfunctions, linking these to the feasibility of machine learning approaches for their computation.

Findings

Koopman eigenfunctions have inherent stability and continuity properties.

Limitations exist in using machine learning to accurately compute Koopman eigenfunctions.

The analysis clarifies conditions under which Koopman representations can be reliably obtained.

Abstract

The Koopman operator is an useful analytical tool for studying dynamical systems -- both controlled and uncontrolled. For example, Koopman eigenfunctions can provide non-local stability information about the underlying dynamical system. Koopman representations of nonlinear systems are commonly calculated using machine learning methods, which seek to represent the Koopman eigenfunctions as a linear combinations of nonlinear state measurements. As such, it is important to understand whether, in principle, these eigenfunctions can be successfully obtained using machine learning and what eigenfunctions calculated in this way can tell us about the underlying system. To that end, this paper presents an analysis of continuity, stability and control limitations associated with Koopman eigenfunctions under minimal assumptions and provides a discussion that relates these properties to the ability to calculate Koopman representations with machine learning.