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Koopman Operator for Stability Analysis: Theory with a Linear--Radial Product Reproducing Kernel

Wentao Tang, Xiuzhen Ye

TL;DR

This paper addresses the challenge of certifying stability from data-derived Koopman operators by introducing a novel function space defined as a linear–radial product RKHS, which combines a linear kernel with a Wendland kernel to capture both equilibrium sensitivity and global regularity. It proves that the Koopman operator is well-defined on this space under mild smoothness and nondegeneracy assumptions, and establishes a spectrum–stability correspondence: if the linearized system at the equilibrium is stable, the learned Koopman spectrum lies inside the unit disk, with bifurcations causing spectral escape. The authors also provide finite-sample error bounds for kernel EDMD in this RKHS and show how the spectrum of the learned operator yields a probabilistic stability certificate. They illustrate the approach with a Van der Pol example, discuss implications for stability-certified Koopman-based control, and outline future extensions to Lyapunov-type kernel formulations and other invariant sets.

Abstract

Koopman operator, as a fully linear representation of nonlinear dynamical systems, if well-defined on a reproducing kernel Hilbert space (RKHS), can be efficiently learned from data. For stability analysis and control-related problems, it is desired that the defining RKHS of the Koopman operator should account for both the stability of an equilibrium point (as a local property) and the regularity of the dynamics on the state space (as a global property). To this end, we show that by using the product kernel formed by the linear kernel and a Wendland radial kernel, the resulting RKHS is invariant under the action of Koopman operator (under certain smoothness conditions). Furthermore, when the equilibrium is asymptotically stable, the spectrum of Koopman operator is provably confined inside the unit circle, and escapes therefrom upon bifurcation. Thus, the learned Koopman operator with provable probabilistic error bound provides a stability certificate. In addition to numerical verification, we further discuss how such a fundamental spectrum--stability relation would be useful for Koopman-based control.

Koopman Operator for Stability Analysis: Theory with a Linear--Radial Product Reproducing Kernel

TL;DR

This paper addresses the challenge of certifying stability from data-derived Koopman operators by introducing a novel function space defined as a linear–radial product RKHS, which combines a linear kernel with a Wendland kernel to capture both equilibrium sensitivity and global regularity. It proves that the Koopman operator is well-defined on this space under mild smoothness and nondegeneracy assumptions, and establishes a spectrum–stability correspondence: if the linearized system at the equilibrium is stable, the learned Koopman spectrum lies inside the unit disk, with bifurcations causing spectral escape. The authors also provide finite-sample error bounds for kernel EDMD in this RKHS and show how the spectrum of the learned operator yields a probabilistic stability certificate. They illustrate the approach with a Van der Pol example, discuss implications for stability-certified Koopman-based control, and outline future extensions to Lyapunov-type kernel formulations and other invariant sets.

Abstract

Koopman operator, as a fully linear representation of nonlinear dynamical systems, if well-defined on a reproducing kernel Hilbert space (RKHS), can be efficiently learned from data. For stability analysis and control-related problems, it is desired that the defining RKHS of the Koopman operator should account for both the stability of an equilibrium point (as a local property) and the regularity of the dynamics on the state space (as a global property). To this end, we show that by using the product kernel formed by the linear kernel and a Wendland radial kernel, the resulting RKHS is invariant under the action of Koopman operator (under certain smoothness conditions). Furthermore, when the equilibrium is asymptotically stable, the spectrum of Koopman operator is provably confined inside the unit circle, and escapes therefrom upon bifurcation. Thus, the learned Koopman operator with provable probabilistic error bound provides a stability certificate. In addition to numerical verification, we further discuss how such a fundamental spectrum--stability relation would be useful for Koopman-based control.

Paper Structure

This paper contains 29 sections, 13 theorems, 21 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Function space defining the Koopman operator.
  3. Stability in Koopman operator modeling.
  4. Koopman-based control.
  5. Contributions of this work.
  6. Preliminaries
  7. Koopman operator on Sobolev--Hilbert space
  8. Sobolev--Hilbert space as an RKHS
  9. Koopman operator learning by kernel EDMD
  10. Koopman Operator using Linear--Radial Product Kernel
  11. Linear--radial product kernel
  12. Well-definedness of Koopman operator on the RKHS
  13. Koopman Spectrum and Stability of Equilibrium Point
  14. Spectrum under stability
  15. Spectrum upon bifurcation
  16. ...and 14 more sections

Key Result

lemma 1

Suppose that $\mathbb{X}\subset \mathbb{R}^d$ is compact. If $f\in C^s(\mathbb{X})$ and is non-degenerate in the sense of $\inf_{x\in \mathbb{X}} \lvert \mathrm{det}~\mathrm{D}f(x) \rvert > 0$, then $A$ is a bounded linear operator on $W^{s,2}(\mathbb{X})$.

Figures (10)

  • Figure 1: Trajectory prediction based on estimated Koopman operator when $\mu=-1$.
  • Figure 2: Spectrum of the estimated Koopman operator when $\mu=-1$.
  • Figure 3: Trajectory prediction based on estimated Koopman operator when $\mu=+1$.
  • Figure 4: Spectrum of the estimated Koopman operator when $\mu=+1$.
  • Figure 5: Trajectory prediction based on estimated Koopman operator when $\mu=+1$ with initial states in the domain $[-2.5, 2.5]$.
  • ...and 5 more figures

Theorems & Definitions (19)

  • lemma 1
  • lemma 2
  • lemma 3
  • corollary 1
  • lemma 4
  • theorem 1
  • lemma 5
  • theorem 2
  • theorem 3
  • lemma 6
  • ...and 9 more