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

TL;DR

This paper introduces a novel RKHS using a linear-radial product kernel to analyze the stability of nonlinear dynamical systems via the Koopman operator, providing theoretical guarantees and practical insights.

Contribution

It develops a new kernel-based RKHS framework that ensures invariance under the Koopman operator and links the spectrum to system stability, with provable error bounds.

Findings

RKHS with linear-radial product kernel is invariant under Koopman operator

Spectrum of Koopman operator confined inside the unit circle for stable systems

Provides stability certificates and insights for Koopman-based control

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.