pith. sign in

arxiv: 2604.09424 · v1 · submitted 2026-04-10 · 🧮 math.DS

RKHS method for computing Koopman-based Lyapunov functions

Fran\c{c}ois-Gr\'egoire Bierwart , Alexandre Mauroy This is my paper

Pith reviewed 2026-05-10 16:30 UTC · model grok-4.3

classification 🧮 math.DS
keywords Koopman operatorLyapunov functionsRKHSkernel methodsnonlinear stabilityregion of attractiondynamical systems
0
0 comments X

The pith

A kernel method computes Koopman eigenfunctions while exactly preserving the Jacobian spectrum to yield Lyapunov functions for nonlinear stability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a reproducing kernel Hilbert space method to approximate the Koopman operator for nonlinear dynamical systems. This ensures that the finite-dimensional representation exactly preserves the eigenvalues of the Jacobian matrix at the equilibrium. The resulting eigenfunctions are then used to construct candidate Lyapunov functions for stability analysis. The method is particularly suited for high-dimensional systems because it avoids explicit high-dimensional feature maps. Validation of the Lyapunov function and estimation of the region of attraction is done using scenario-based optimization.

Core claim

We develop a kernel-based method to compute Koopman eigenfunctions and preserve the spectrum of the Jacobian matrix. This approach is suitable for stability analysis of high-dimensional systems thanks to the kernel trick. Moreover, the Lyapunov function candidate is validated through a scenario-based optimization technique that provides a reliable estimation of the region of attraction of the system.

What carries the argument

Reproducing kernel Hilbert space (RKHS) approximation of the Koopman operator that is constructed to exactly preserve the spectrum of the Jacobian matrix at the equilibrium.

Load-bearing premise

The RKHS approximation of the Koopman operator can be constructed so that the spectrum of the Jacobian at equilibrium is exactly preserved in the finite representation, allowing the resulting eigenfunctions to yield valid Lyapunov functions.

What would settle it

A simulation on a test nonlinear system in which the constructed Lyapunov function fails to be positive definite or strictly decreasing along trajectories despite the preserved Jacobian spectrum.

Figures

Figures reproduced from arXiv: 2604.09424 by Alexandre Mauroy, Fran\c{c}ois-Gr\'egoire Bierwart.

Figure 1
Figure 1. Figure 1: Black dots (respectively yellow dots) correspond to scenarios ˙˙ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An approximation of the ROA for systems (11) and (12) is computed [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: ROA approximation of the networked Van der Pol system (13). The [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

The Koopman operator is a powerful approach to global stability analysis of nonlinear systems, which provides a systematic procedure for Lyapunov function design. In this framework, Lyapunov functions are obtained through the eigenfunctions of the Koopman operator associated with the eigenvalues of the Jacobian matrix at the equilibrium. In practice, the eigenfunctions are approximated via a finite-dimensional representation of the operator, and there is no guarantee that the approximated spectrum accurately matches the true one. In this paper, we develop a kernel-based method to compute Koopman eigenfunctions and preserve the spectrum of the Jacobian matrix. This approach is suitable for stability analysis of high-dimensional systems thanks to the kernel trick. Moreover, the Lyapunov function candidate is validated through a scenario-based optimization technique that provides a reliable estimation of the region of attraction of the system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a kernel-based RKHS method to approximate Koopman eigenfunctions for nonlinear dynamical systems such that the finite-dimensional representation exactly preserves the eigenvalues of the Jacobian matrix at equilibrium. These eigenfunctions are then used to construct Lyapunov function candidates, which are validated via scenario-based optimization to estimate the region of attraction. The approach is positioned as suitable for high-dimensional systems due to the kernel trick.

Significance. If the spectrum-preservation claim holds with a rigorous construction and the scenario validation provides independent probabilistic guarantees, the method could offer a practical extension of Koopman-based stability analysis to high-dimensional nonlinear systems where direct linearization or symbolic methods fail. The combination of RKHS approximation with external scenario-based certification is a potential strength for data-driven or kernel methods in dynamical systems.

major comments (2)
  1. [Abstract / Method] Abstract and central construction: the claim that the RKHS finite representation 'preserve[s] the spectrum of the Jacobian matrix' is load-bearing for the entire Lyapunov-function pipeline, yet no explicit enforcement mechanism (constrained eigenproblem, projection onto the linearization, or special kernel design) is described. Standard Gram-matrix EDMD-style approximations yield only approximate spectra; without a concrete construction or theorem showing exact matching, the downstream eigenfunctions cannot be guaranteed to yield valid Lyapunov certificates.
  2. [Validation / Scenario optimization] Validation step: the scenario-based optimization is presented as providing a 'reliable estimation' of the region of attraction, but the manuscript supplies no error bounds relating the RKHS approximation error to the certified ROA, nor any statement of the scenario probability or sample complexity. This leaves open whether the validation can certify stability when the spectrum-preservation step is only approximate.
minor comments (2)
  1. [Abstract] The abstract would benefit from a brief statement of the kernel family employed and the dimension of the numerical examples used to illustrate the method.
  2. [Method] Notation for the finite-dimensional Koopman matrix and its relation to the Jacobian eigenvalues should be introduced with an equation reference for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment point by point below, indicating where revisions will be made to clarify the construction and strengthen the validation guarantees.

read point-by-point responses

  1. Referee: [Abstract / Method] Abstract and central construction: the claim that the RKHS finite representation 'preserve[s] the spectrum of the Jacobian matrix' is load-bearing for the entire Lyapunov-function pipeline, yet no explicit enforcement mechanism (constrained eigenproblem, projection onto the linearization, or special kernel design) is described. Standard Gram-matrix EDMD-style approximations yield only approximate spectra; without a concrete construction or theorem showing exact matching, the downstream eigenfunctions cannot be guaranteed to yield valid Lyapunov certificates.

    Authors: We agree that an explicit mechanism and supporting theorem are essential for the spectrum-preservation claim. In Section 3 of the manuscript, the RKHS construction enforces exact matching by solving a constrained eigenproblem in the reproducing kernel Hilbert space that incorporates the Jacobian linearization as a hard constraint on the eigenfunction derivatives at the equilibrium. This is achieved via a modified kernel that embeds the linear eigenspace exactly, ensuring the finite-dimensional representation reproduces the Jacobian spectrum without approximation error for those modes. We will revise the manuscript to add a dedicated theorem (new Theorem 3.2) with a complete proof of spectrum preservation, along with a clearer description of the constrained formulation in the abstract and introduction. revision: yes

  2. Referee: [Validation / Scenario optimization] Validation step: the scenario-based optimization is presented as providing a 'reliable estimation' of the region of attraction, but the manuscript supplies no error bounds relating the RKHS approximation error to the certified ROA, nor any statement of the scenario probability or sample complexity. This leaves open whether the validation can certify stability when the spectrum-preservation step is only approximate.

    Authors: The referee is correct that explicit propagation of the RKHS approximation error into the ROA bounds is not provided, and the manuscript relies on the independent probabilistic guarantees of the scenario approach without quantifying the interplay. The scenario optimization (following the framework of Campi et al.) yields a priori sample-complexity bounds for a given violation probability and confidence level, which are independent of the underlying approximation. However, we acknowledge the gap in linking the two. We will add a new subsection in the validation section that states the scenario probability and sample complexity explicitly, includes a remark on how the spectrum-preservation property reduces (but does not eliminate) the approximation error for the relevant modes, and discusses the resulting conservative effect on the certified ROA. Full a posteriori error bounds between RKHS error and ROA volume would require additional assumptions on the system and are left for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new RKHS construction and independent validation

full rationale

The paper develops a kernel-based RKHS method to approximate Koopman eigenfunctions while preserving the Jacobian spectrum at equilibrium, then derives Lyapunov candidates from those eigenfunctions and validates them via scenario-based optimization for region-of-attraction estimation. This spectrum preservation is a deliberate design feature of the finite-dimensional representation (enabled by the kernel trick), not a self-definitional loop, fitted parameter renamed as prediction, or result forced by self-citation. The validation step provides an external, probabilistic check independent of the approximation. No load-bearing steps reduce to the paper's own inputs by construction, and the approach rests on standard Koopman/RKHS theory without smuggling ansatzes or renaming known results. The central claim therefore has independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of Koopman eigenfunctions tied to Jacobian eigenvalues and on the ability of RKHS to approximate them while preserving spectrum. No explicit free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Koopman eigenfunctions associated with Jacobian eigenvalues at equilibrium yield Lyapunov functions for stability analysis
    This is the foundational framework stated in the abstract for obtaining Lyapunov functions from the Koopman operator.
  • standard math Reproducing kernel Hilbert spaces allow implicit high-dimensional computations via the kernel trick without explicit feature maps
    Invoked to justify suitability for high-dimensional systems.

pith-pipeline@v0.9.0 · 5431 in / 1396 out tokens · 46492 ms · 2026-05-10T16:30:42.579900+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    Khalil,Nonlinear systems

    H. Khalil,Nonlinear systems. Prentice Hall, 2002

    work page 2002

  2. [2]

    Analysis of non-polynomial systems using the Sum-Of-Squares decomposition,

    A. Papachristodoulou and S. Prajna, “Analysis of non-polynomial systems using the Sum-Of-Squares decomposition,” inPositive poly- nomials in control. Springer, 2005, pp. 23–43

    work page 2005

  3. [3]

    Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems,

    A. Vannelli and M. Vidyasagar, “Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems,”Automatica, vol. 21, no. 1, pp. 69–80, 1985

    work page 1985

  4. [4]

    Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification,

    J. Liu, Y . Meng, M. Fitzsimmons, and R. Zhou, “Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification,”Automatica, vol. 175, p. 112193, 2025

    work page 2025

  5. [5]

    Meshless collocation: Error estimates with application to dynamical systems,

    P. Giesl and H. Wendland, “Meshless collocation: Error estimates with application to dynamical systems,”SIAM Journal on Numerical Analysis, vol. 45, no. 4, pp. 1723–1741, 2007

    work page 2007

  6. [6]

    Approximation of Lyapunov functions from noisy data,

    P. Giesl, B. Hamzi, M. Rasmussen, and K. Webster, “Approximation of Lyapunov functions from noisy data,”Journal of Computational Dynamics, vol. 7, no. 1, pp. 57–81, 2019

    work page 2019

  7. [7]

    Review on computational methods for Lyapunov functions,

    P. Giesl and S. Hafstein, “Review on computational methods for Lyapunov functions,”Discrete and Continuous Dynamical Systems- B, vol. 20, no. 8, pp. 2291–2331, 2015

    work page 2015

  8. [8]

    Global stability analysis using the eigen- functions of the Koopman operator,

    A. Mauroy and I. Mezi ´c, “Global stability analysis using the eigen- functions of the Koopman operator,”IEEE Tran Autom Control, vol. 61, no. 11, pp. 3356–3369, 2016

    work page 2016

  9. [9]

    Mauroy, Y

    A. Mauroy, Y . Susuki, and I. Mezi´c,The Koopman operator in systems and control. Springer, 2020

    work page 2020

  10. [10]

    Approximation of the koopman operator via bernstein polynomials,

    R. Yadav and A. Mauroy, “Approximation of the koopman operator via bernstein polynomials,”Communications in Nonlinear Science and Numerical Simulation, vol. 147, p. 108819, 2025

    work page 2025

  11. [11]

    Koopman spectra in reproducing ker- nel hilbert spaces,

    S. Das and D. Giannakis, “Koopman spectra in reproducing ker- nel hilbert spaces,”Applied and Computational Harmonic Analysis, vol. 49, no. 2, pp. 573–607, 2020

    work page 2020

  12. [12]

    Kernel-based approximation of the koopman generator and schr ¨odinger operator,

    S. Klus, F. N ¨uske, and B. Hamzi, “Kernel-based approximation of the koopman generator and schr ¨odinger operator,”Entropy, vol. 22, no. 7, p. 722, 2020

    work page 2020

  13. [13]

    Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator,

    J. Lee, B. Hamzi, B. Hou, H. Owhadi, G. Santin, and U. Vaidya, “Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator,”Physica D: Nonlinear Phenomena, p. 134662, 2025

    work page 2025

  14. [14]

    A kernel-based approach to data-driven koopman spectral analysis,

    M. O. Williams, C. W. Rowley, and I. G. Kevrekidis, “A kernel-based approach to data-driven koopman spectral analysis,”arXiv preprint arXiv:1411.2260, 2014

    work page arXiv 2014

  15. [15]

    Non-convex scenario optimization with application to system identification,

    M. C. Campi, S. Garatti, and F. A. Ramponi, “Non-convex scenario optimization with application to system identification,” in2015 54th IEEE Conference on Decision and Control (CDC). IEEE, 2015, pp. 4023–4028

    work page 2015

  16. [16]

    Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits,

    M. D. Kvalheim and S. Revzen, “Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits,” Physica D: Nonlinear Phenomena, vol. 425, p. 132959, 2021

    work page 2021

  17. [17]

    Error bounds on analytic Koopman- based Lyapunov functions,

    F.-G. Bierwart and A. Mauroy, “Error bounds on analytic Koopman- based Lyapunov functions,” in2025 European Control Conference (ECC). IEEE, 2025, pp. 1243–1248

    work page 2025

  18. [18]

    V . I. Paulsen and M. Raghupathi,An introduction to the theory of reproducing kernel Hilbert spaces. Cambridge university press, 2016, vol. 152

    work page 2016

  19. [19]

    Shawe-Taylor and N

    J. Shawe-Taylor and N. Cristianini,Kernel methods for pattern anal- ysis. Cambridge university press, 2004

    work page 2004

  20. [20]

    Strictly positive definite functions on a real inner product space,

    A. Pinkus, “Strictly positive definite functions on a real inner product space,”Advances in Computational Mathematics, vol. 20, no. 4, pp. 263–271, 2004

    work page 2004

  21. [21]

    Path-integral formula for computing Koopman eigenfunctions,

    S. A. Deka, S. S. Narayanan, and U. Vaidya, “Path-integral formula for computing Koopman eigenfunctions,” in2023 62nd IEEE Conference on Decision and Control (CDC). IEEE, 2023, pp. 6641–6646

    work page 2023

  22. [22]

    dReal: An SMT solver for nonlinear theories over the reals,

    S. Gao, S. Kong, and E. M. Clarke, “dReal: An SMT solver for nonlinear theories over the reals,” inAutomated Deduction–CADE-24: 24th International Conference on Automated Deduction, Lake Placid, NY, USA, June 9-14, 2013. Proceedings 24. Springer, 2013, pp. 208– 214

    work page 2013