Kernel Methods for the Construction of Certified Lyapunov Functions via Approximate Koopman Eigenfunctions

TL;DR

This paper introduces a kernel-based method for constructing Lyapunov functions for nonlinear systems using approximate Koopman eigenfunctions, combining linearization and kernel collocation techniques.

Contribution

It presents a novel approach that decomposes Koopman eigenfunctions into linear and nonlinear parts using RKHS, enabling certified Lyapunov function construction.

Findings

Effective Lyapunov functions for benchmark systems

Error bounds relate approximation quality to collocation points

Numerical experiments validate the method's effectiveness

Abstract

We present a kernel-based methodology for constructing Lyapunov functions for nonlinear dynamical systems using approximate Koopman eigenfunctions. Our approach decomposes principal Koopman eigenfunctions into linear and nonlinear components, where the linear part is obtained from the system's linearization and the nonlinear part is computed by solving a partial differential equation using symmetric kernel collocation in reproducing kernel Hilbert spaces (RKHS). The resulting Lyapunov function is constructed as a quadratic form in the approximate eigenfunctions. We establish error bounds relating the approximation quality to the fill distance of collocation points and provide a certification procedure using continuous piecewise affine (CPA) methods. Numerical experiments on benchmark systems, including a polynomial system and the Duffing oscillator, demonstrate the effectiveness of our approach.