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

P. Giesl, S. Hafstein, B. Hamzi, J. Lee, H. Owhadi, G. Santin, U. Vaidya

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.

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

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.
Paper Structure (15 sections, 4 theorems, 81 equations, 6 figures)

This paper contains 15 sections, 4 theorems, 81 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background on Lyapunov functions
  3. Numerical approaches based on kernels approximation
  4. The Koopman operator and its spectrum
  5. Decomposition of Koopman Eigenfunctions
  6. Existence of Koopman Eigenfunctions
  7. Kernels and kernel-based symmetric collocation
  8. Generalized interpolation and symmetric collocation of linear PDEs
  9. Kernel approximation of Koopman eigenfunctions
  10. Constructing Lyapunov functions by approximated Koopman eigenfunctions
  11. Construction of certified Lyapunov functions
  12. Examples
  13. Example 1
  14. Example 2
  15. Conclusion

Key Result

Theorem 1

Let $\Omega \subset \mathbb{R}^d$ be an open bounded domain containing the origin, and let $f \in C^1(\Omega, \mathbb{R}^d)$ with $f(0) = 0$. Assume the origin is a hyperbolic equilibrium with all eigenvalues of $E = Df(0)$ having strictly negative real parts. Let $\lambda < 0$ be an eigenvalue of $

Figures (6)

  • Figure 1: Schematic overview of the proposed kernel-based methodology for constructing Lyapunov functions using approximate Koopman eigenfunctions. The process involves three main steps: (1) eigenfunction approximation via PDE decomposition, (2) kernel regression in RKHS with error bounds (Theorem 3), and (3) Lyapunov construction and certification (Theorem 4).
  • Figure 2: Example \ref{['eq:ex_one']}: approximated ${V}^*$ (left) and true ${V}$ (right), using 3600 collocation points
  • Figure 3: Example \ref{['eq:ex_one']}: approximated $\dot{V}^*$ (left) and true $\dot{V}$ (right), using 3600 collocation points
  • Figure 4: Lyapunov function for the system \ref{['eq:ex_one']} computed by Koopman regression and certified with the method in Section \ref{['sec:certification']}. The orbital derivative is negative with exception of the red dots close to the origin.
  • Figure 5: Example \ref{['eq:duffing']}: approximated ${V}^*$ (left) and true ${V}$ (right), using 3600 collocation points -- note that $V^*$ is not approximating $V$, in fact, $V^*$ is a strict Lyapunov function while $V$ is not
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 1: Lyapunov function
  • Definition 2: Koopman Operator
  • Definition 3: Eigenvalues and Eigenfunctions of the Koopman operator
  • Definition 4: Open Eigenfunction mezic2020spectrum
  • Theorem 1: Existence of Koopman Eigenfunctions
  • proof
  • Remark 1: Practical implications and path-integral formula
  • Definition 5: Reproducing Kernel Hilbert Spaces (RKHS)
  • Definition 6: Positive definite kernel
  • Theorem 2: Regularity and stability
  • ...and 5 more