Koopman-Based Neural Lyapunov Functions for General Attractors

TL;DR

This paper introduces a novel method using Koopman spectral theory and neural networks to construct Lyapunov functions for systems with limit-cycle attractors, enabling efficient stability analysis and invariant set computation.

Contribution

It leverages Koopman eigenfunctions to parameterize Lyapunov functions for limit-cycle systems and combines neural networks with polynomial methods for provable stability certificates.

Findings

Efficient Lyapunov function construction for limit-cycle systems.

Reduction in decision variables using Koopman-based polynomial methods.

Integration of neural networks and Sum-of-Squares programming for stability analysis.

Abstract

Koopman spectral theory has grown in the past decade as a powerful tool for dynamical systems analysis and control. In this paper, we show how recent data-driven techniques for estimating Koopman-Invariant subspaces with neural networks can be leveraged to extract Lyapunov certificates for the underlying system. In our work, we specifically focus on systems with a limit-cycle, beyond just an isolated equilibrium point, and use Koopman eigenfunctions to efficiently parameterize candidate Lyapunov functions to construct forward-invariant sets under some (unknown) attractor dynamics. Additionally, when the dynamics are polynomial and when neural networks are replaced by polynomials as a choice of function approximators in our approach, one can further leverage Sum-of-Squares programs and/or nonlinear programs to yield provably correct Lyapunov certificates. In such a polynomial case, our Koopman-based approach for constructing Lyapunov functions uses significantly fewer decision variables compared to directly formulating and solving a Sum-of-Squares optimization problem.