Extensions of the Path-integral formula for computation of Koopman eigenfunctions

TL;DR

This paper advances the computation of Koopman eigenfunctions by extending a path-integral approach to finite-time scenarios, relaxing eigenvalue distribution assumptions, and applying to saddle point systems, improving practical utility.

Contribution

It introduces several key developments to the path-integral method for Koopman eigenfunctions, including finite-time computation and broader system applicability.

Findings

Enhanced practical applicability of the method

Finite-time eigenfunction computation achieved

Extended to saddle point systems

Abstract

Representing nonlinear dynamical systems using the Koopman Operator and its spectrum has distinct advantages in terms of linear interpretability of the model as well as in analysis and control synthesis through the use of well-studied techniques from linear systems theory. As such, efficient computation of Koopman eigenfunctions is of paramount importance towards enabling such Koopman-based constructions. To this end, several approaches have been proposed in literature, including data-driven, convex optimization, and Deep Learning-based methods. In our recent work, we proposed a novel approach based on path-integrals that allowed eigenfunction computations using a closed-form formula. In this paper, we present several important developments such as finite-time computations, relaxation of assumptions on the distribution of the principal Koopman eigenvalues, as well as extension towards saddle point systems, which greatly enhance the practical applicability of our method.