Error bounds on analytic Koopman-based Lyapunov functions

TL;DR

This paper establishes theoretical error bounds for Koopman eigenfunctions approximated in polynomial spaces, enabling validation of Lyapunov functions and inner region of attraction estimates for nonlinear systems.

Contribution

It provides the first rigorous error bounds for finite-dimensional Koopman eigenfunction approximations of analytic systems.

Findings

Derived explicit error bounds for Koopman eigenfunctions in polynomial subspaces.

Validated the use of these bounds to assess Lyapunov function accuracy.

Developed an optimization-free method for inner region of attraction estimation.

Abstract

The Koopman operator provides an infinite-dimensional linear description of nonlinear dynamical systems that can be leveraged in the context of stability analysis. In particular, Lyapunov functions can be obtained in a systematic way via the eigenfunctions of the Koopman operator. However, these eigenfunctions are computed from finite-dimensional approximations, resulting in approximated Lyapunov functions that must be validated. In this paper, we provide theoretical error bounds on the approximation of the eigenfunctions of the Koopman operator in the case of analytic vector field and finite-dimensional approximation in polynomial subspaces. We leverage these results to assess the validity of Koopman-based Lyapunov functions and obtain an optimization-free inner approximation of the region of attraction of an equilibrium.