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

Wentao Tang, Xiuzhen Ye
Proceedings of The 8th Annual Learning for Dynamics and Control Conference, PMLR 331:282-298, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v331-tang26a, title = {Koopman Operator for Stability Analysis: Theory with a Linear–Radial Product Reproducing Kernel}, author = {Tang, Wentao and Ye, Xiuzhen}, booktitle = {Proceedings of The 8th Annual Learning for Dynamics and Control Conference}, pages = {282--298}, year = {2026}, editor = {Sukhatme, Gaurav and Lindemann, Lars and Tu, Stephen and Wierman, Adam and Atanasov, Nikolay}, volume = {331}, series = {Proceedings of Machine Learning Research}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v331/main/assets/tang26a/tang26a.pdf}, url = {https://proceedings.mlr.press/v331/tang26a.html}, 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.} }
Endnote
%0 Conference Paper %T Koopman Operator for Stability Analysis: Theory with a Linear–Radial Product Reproducing Kernel %A Wentao Tang %A Xiuzhen Ye %B Proceedings of The 8th Annual Learning for Dynamics and Control Conference %C Proceedings of Machine Learning Research %D 2026 %E Gaurav Sukhatme %E Lars Lindemann %E Stephen Tu %E Adam Wierman %E Nikolay Atanasov %F pmlr-v331-tang26a %I PMLR %P 282--298 %U https://proceedings.mlr.press/v331/tang26a.html %V 331 %X 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.
APA
Tang, W. & Ye, X.. (2026). Koopman Operator for Stability Analysis: Theory with a Linear–Radial Product Reproducing Kernel. Proceedings of The 8th Annual Learning for Dynamics and Control Conference, in Proceedings of Machine Learning Research 331:282-298 Available from https://proceedings.mlr.press/v331/tang26a.html.

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