A numerical Koopman-based framework to estimate regions of attraction for general vector fields

TL;DR

This paper introduces a Koopman operator-based framework for estimating regions of attraction in general vector fields, combining data-driven eigenfunction approximation with rigorous validation techniques.

Contribution

It presents a novel, general approach that constructs Lyapunov functions from Koopman eigenfunctions and validates them using SOS or adaptive grid methods.

Findings

Framework successfully estimates regions of attraction for various vector fields.

Method extends to non-polynomial vector fields using general basis approximations.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

In this paper, we develop a comprehensive framework to estimate regions of attraction of equilibria for dynamics associated with general vector fields. This framework combines Koopman operator-based methods with rigorous validation techniques. A candidate Lyapunov function is constructed with approximated Koopman eigenfunctions and further validated through polynomial approximation, either with SOS-based techniques or with a worst-case approach using an adaptive grid. The framework is general, not only since it is adapted to non-polynomial vector fields, but also since the Koopman operator can be approximated with general bases yielding non-polynomial Lyapunov functions. The performance of the method is illustrated with several numerical examples.