Global Linearization of Parameterized Nonlinear Systems with Stable Equilibrium Point Using the Koopman Operator

Alexandre Mauroy; Natsuki Katayama; Yoshihiko Susuki

arxiv: 2604.04711 · v1 · submitted 2026-04-06 · 🧮 math.DS · cs.SY· eess.SY

Global Linearization of Parameterized Nonlinear Systems with Stable Equilibrium Point Using the Koopman Operator

Natsuki Katayama , Alexandre Mauroy , Yoshihiko Susuki This is my paper

Pith reviewed 2026-05-10 19:54 UTC · model grok-4.3

classification 🧮 math.DS cs.SYeess.SY

keywords Koopman operatorglobal linearizationparameterized nonlinear systemseigenfunctionscontrol-affine systemsdynamical systemsstable equilibrium

0 comments

The pith

Nonlinear systems with globally stable equilibria admit a global linearization via the Koopman operator that depends continuously on the parameter.

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

The paper treats control inputs as fixed parameters and applies the Koopman operator to nonlinear dynamical systems that possess a globally exponentially stable equilibrium. It establishes that the eigenfunctions of this operator furnish a coordinate change transforming the original nonlinear flow into an equivalent finite-dimensional linear system. The construction is shown to vary continuously with the chosen parameter value. For the special case of control-affine systems the authors further identify a sufficient condition under which the resulting bilinear representation becomes independent of the parameter.

Core claim

For a parameterized nonlinear system possessing a globally exponentially stable equilibrium point, the associated Koopman operator admits a set of eigenfunctions whose span yields an exact finite-dimensional linear representation of the nonlinear dynamics; this representation depends continuously on the parameter.

What carries the argument

The parameter-dependent Koopman eigenfunctions that serve as the change of coordinates realizing the global linearization.

If this is right

The nonlinear dynamics become exactly representable and simulable by a linear system in the lifted coordinates for any fixed parameter value.
Spectral analysis of the Koopman operator directly yields the continuous dependence of the linearization on the parameter.
For control-affine systems satisfying the stated condition, a single bilinear model suffices for all parameter values.
Linear control techniques can be applied in the lifted space and then pulled back to the original nonlinear coordinates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same eigenfunction construction may allow data-driven Koopman approximations to be trained once and then evaluated at new parameter values without retraining.
The continuous dependence result suggests that small parameter perturbations produce only small changes in the linear model, which could be useful for robust controller design.
The framework could be tested numerically on standard benchmark systems such as the van der Pol oscillator with a tunable damping parameter.

Load-bearing premise

The nonlinear system possesses a globally exponentially stable equilibrium point.

What would settle it

A concrete parameterized system with a globally exponentially stable equilibrium for which no finite collection of Koopman eigenfunctions produces a continuous-in-parameter finite-dimensional linearization.

read the original abstract

The Koopman operator framework enables global analysis of nonlinear systems through its inherent linearity. This study aims to clarify spectral properties of the Koopman operators for nonlinear systems with control inputs. To this end, we treat the inputs as parameters throughout this paper. We then introduce the Koopman operator for a parameterized dynamical system with a globally exponentially stable equilibrium point and analyze how eigenfunctions of the operator depend on the parameter. As a main result, we obtain a global linearization, which enables one to transform the nonlinear system into a finite-dimensional linear system, and we show that it depends continuously on the parameter. Subsequently, for a control-affine system, we investigate a condition under which the transformation providing a global bilinearization does not depend on the parameter. This provides the condition under which the global bilinearization for the control-affine system is independent of the parameter.

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.

Desk Editor's Note private letter to a colleague

The paper extends Koopman linearization to parameterized systems with a continuity result on the linearization, but the finite-dimensional claim looks like it needs structure beyond global exponential stability.

read the letter

The main takeaway is that they treat inputs as parameters, show continuous dependence of the Koopman eigenfunctions on that parameter, and obtain a global linearization to a finite-dimensional linear system that inherits the continuity. They also give a condition for control-affine systems under which the bilinearization becomes independent of the parameter. That is a clean incremental move from the usual fixed-system Koopman results. They use the globally exponentially stable equilibrium in the standard way to guarantee the eigenfunctions exist and to get the continuity, which is reasonable and well-motivated. The bilinearization condition is a useful extra piece for the control-affine setting. The soft spot is the finite-dimensional assertion. Global exponential stability secures the eigenfunctions and their parameter dependence in the right function spaces, but it does not force the state observables to live in a finite-dimensional invariant subspace for generic nonlinear vector fields. The spectrum is typically infinite, so exact finite-dimensional linearization usually requires something additional such as a polynomial structure or a finite-dimensional Lie algebra. The abstract does not state any such extra hypothesis, so the proofs will need to make clear whether they are assuming it implicitly or working with an approximation. If the full paper pins this down rigorously, the result strengthens; otherwise the claim needs qualification. This is aimed at people working on operator methods for nonlinear control and dynamical systems. A reader interested in parameter-dependent linearization or bilinearization tools would find the continuity statement and the control-affine condition worth reading. I would send it to peer review. The claims are specific enough to be checked, the setup is standard, and the topic sits in an active area even if the finite-dimensional part may need tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a Koopman-operator framework for parameterized nonlinear systems possessing a globally exponentially stable equilibrium. It analyzes the parameter dependence of eigenfunctions and claims to obtain a global linearization that converts the nonlinear dynamics into a finite-dimensional linear system whose matrix depends continuously on the parameter. For control-affine systems an additional condition is derived under which the global bilinearization is parameter-independent.

Significance. If the finite-dimensional global linearization and its continuous parameter dependence can be established rigorously, the work would extend Koopman theory to parameterized families in a manner useful for robust analysis and control. The exploitation of global exponential stability to guarantee eigenfunction properties is a natural starting point, and the parameter-independence condition for bilinearization is a concrete contribution.

major comments (2)

[Abstract and §3] Abstract and §3 (main theorem): the central claim that GES alone yields an exact finite-dimensional linearization is not justified. For generic C^1 nonlinearities the Koopman operator on C^1 or L^2 spaces has infinite spectrum; the state observables therefore need not lie in a finite-dimensional invariant subspace. An explicit structural hypothesis (e.g., polynomial vector field of bounded degree or finite-dimensional Lie algebra) must be stated and used; without it the finite-dimensional truncation is not guaranteed.
[§4] §4 (continuity result): the asserted continuous dependence of the linearization matrix on the parameter is stated without an explicit function space, norm, or modulus of continuity. The proof sketch should identify the topology on the space of eigenfunctions and verify that the truncation error remains controlled uniformly in a neighborhood of each parameter value.

minor comments (2)

[§2] The notation for the parameterized vector field f(x,μ) and the precise domain of the Koopman operator should be introduced before the first theorem.
[Introduction] Add a short comparison paragraph with existing finite-dimensional Koopman results (e.g., for polynomial or bilinear systems) to clarify the novelty of the parameter-continuous case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major point below and will make the necessary revisions to strengthen the rigor of the claims.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (main theorem): the central claim that GES alone yields an exact finite-dimensional linearization is not justified. For generic C^1 nonlinearities the Koopman operator on C^1 or L^2 spaces has infinite spectrum; the state observables therefore need not lie in a finite-dimensional invariant subspace. An explicit structural hypothesis (e.g., polynomial vector field of bounded degree or finite-dimensional Lie algebra) must be stated and used; without it the finite-dimensional truncation is not guaranteed.

Authors: We agree that global exponential stability alone does not guarantee a finite-dimensional invariant subspace for the state observables under the Koopman operator for arbitrary C^1 systems. The manuscript's main result relies on the existence of a finite set of eigenfunctions that span the observables, which implicitly requires additional structure on the vector field. In the revised version, we will explicitly introduce a structural hypothesis (e.g., that the nonlinear vector field is polynomial of bounded degree, ensuring that monomials up to a certain order form an invariant subspace, or that the Lie algebra generated by the dynamics is finite-dimensional). This will be stated clearly in the abstract, introduction, and §3, and the main theorem will be restated under this assumption to make the finite-dimensional global linearization rigorous. revision: yes
Referee: [§4] §4 (continuity result): the asserted continuous dependence of the linearization matrix on the parameter is stated without an explicit function space, norm, or modulus of continuity. The proof sketch should identify the topology on the space of eigenfunctions and verify that the truncation error remains controlled uniformly in a neighborhood of each parameter value.

Authors: We acknowledge that the continuity argument in §4 requires greater precision regarding the underlying function spaces and topologies. In the revision, we will specify that we work in the Banach space of C^1 functions equipped with the C^1 norm (or suitable weighted Sobolev spaces that exploit the global exponential stability to ensure integrability at infinity). We will detail the topology on the eigenfunctions, invoke standard perturbation results for the parameter-dependent Koopman operator to establish continuous dependence of the eigenfunctions, and prove that the truncation error to the finite-dimensional subspace remains uniformly bounded in a neighborhood of each parameter value by controlling the remainder terms via the GES decay rate. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard Koopman spectral properties under GES

full rationale

The paper introduces the Koopman operator for parameterized systems possessing a globally exponentially stable equilibrium, then analyzes eigenfunction dependence on the parameter to obtain a global linearization into a finite-dimensional linear system with continuous parameter dependence. This chain rests on established existence results for Koopman eigenfunctions under GES (invoked as the weakest assumption) rather than any self-definitional loop, fitted-input prediction, or load-bearing self-citation. The subsequent bilinearization condition for control-affine systems is likewise derived from the same operator framework without reducing to its own inputs by construction. The derivation is therefore self-contained against external Koopman theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of the Koopman operator for the parameterized flow and on the global exponential stability of the equilibrium; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The nonlinear system possesses a globally exponentially stable equilibrium point.
Invoked to guarantee the existence and analytic properties of the Koopman eigenfunctions and the global linearization.
standard math The Koopman operator is well-defined on a suitable function space for the parameterized dynamical system.
Standard background assumption in Koopman operator theory.

pith-pipeline@v0.9.0 · 5456 in / 1452 out tokens · 38791 ms · 2026-05-10T19:54:46.266010+00:00 · methodology

discussion (0)

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4: ... there exists a diffeomorphism ψu ... such that d/dt ψu(x) = Au ψu(x) ... ψui continuous in Ck(X) norm
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 5: ... isomorphism between {F,G1,...,Gd}L and {A,B1,...,Bd}L yields global bilinearization ż = Az + Σ ui Biz

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

On Koopman Resolvents and Frequency Response of Nonlinear Systems
eess.SY 2026-03 unverdicted novelty 6.0

Frequency response for nonlinear systems is defined as the Laplace transform of the output in the Koopman resolvent framework, with existence conditions given for three classes of dynamics.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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Mauroy, I

A. Mauroy, I. Mezi ´c, and Y . Susuki, Eds.,The Koopman Operator in Systems and Control: Concepts, Methodologies, and Applications. Springer, 2020

work page 2020

[2] [2]

Koop- man invariant subspaces and finite linear representations of nonlinear dynamical systems for control,

S. L. Brunton, B. W. Brunton, J. L. Proctor, and J. N. Kutz, “Koop- man invariant subspaces and finite linear representations of nonlinear dynamical systems for control,”PloS one, vol. 11, no. 2, p. e0150171, 2016

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A data–driven approximation of the koopman operator: Extending dynamic mode decomposition,

M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data–driven approximation of the koopman operator: Extending dynamic mode decomposition,”Journal of Nonlinear Science, vol. 25, no. 6, pp. 1307– 1346, 2015

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Dynamic mode decom- position with control,

J. L. Proctor, S. L. Brunton, and J. N. Kutz, “Dynamic mode decom- position with control,”SIAM Journal on Applied Dynamical Systems, vol. 15, no. 1, pp. 142–161, 2016

work page 2016

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Generalizing Koopman theory to allow for inputs and control,

——, “Generalizing Koopman theory to allow for inputs and control,” SIAM Journal on Applied Dynamical Systems, vol. 17, no. 1, pp. 909– 930, 2018

work page 2018

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Bilinearization, reachability, and opti- mal control of control-affine nonlinear systems: A Koopman spectral approach,

D. Goswami and D. A. Paley, “Bilinearization, reachability, and opti- mal control of control-affine nonlinear systems: A Koopman spectral approach,”IEEE Transactions on Automatic Control, vol. 67, no. 6, pp. 2715–2728, 2021

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Advantages of bilinear Koopman realizations for the modeling and control of systems with unknown dynamics,

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Modeling nonlinear control systems via

M. Haseli and J. Cort ´es, “Modeling nonlinear control systems via Koopman control family: Universal forms and subspace invariance proximity,”arXiv preprint arXiv:2307.15368, 2023

work page arXiv 2023

[10] [10]

Learning parametric Koopman decompositions for prediction and control,

Y . Guo, M. Korda, I. G. Kevrekidis, and Q. Li, “Learning parametric Koopman decompositions for prediction and control,”SIAM Journal on Applied Dynamical Systems, vol. 24, no. 1, pp. 744–781, 2025

work page 2025

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An overview of Koopman-based control: From error bounds to closed-loop guarantees,

R. Str ¨asser, K. Worthmann, I. Mezi ´c, J. Berberich, M. Schaller, and F. Allg ¨ower, “An overview of Koopman-based control: From error bounds to closed-loop guarantees,”arXiv preprint arXiv:2509.02839, 2025

work page arXiv 2025

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——, “Analysis of fluid flows via spectral properties of the Koopman operator,”Annual Review of Fluid Mechanics, vol. 45, pp. 357–378, 2013

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Spectrum of the Koopman operator, spectral expansions in functional spaces, and state-space geometry,

——, “Spectrum of the Koopman operator, spectral expansions in functional spaces, and state-space geometry,”Journal of Nonlinear Science, vol. 30, no. 5, pp. 2091–2145, 2020

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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

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On the non-existence of immer- sions for systems with multiple omega-limit sets,

Z. Liu, N. Ozay, and E. D. Sontag, “On the non-existence of immer- sions for systems with multiple omega-limit sets,”IFAC-PapersOnLine, vol. 56, no. 2, pp. 60–64, 2023

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work page 2025

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Data-driven feedback linearization using the Koopman generator,

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work page 2024

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Kato,Perturbation Theory for Linear Operators

T. Kato,Perturbation Theory for Linear Operators. Springer Science & Business Media, 2013, vol. 132

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Koopman resolvent: A Laplace- domain analysis of nonlinear autonomous dynamical systems,

Y . Susuki, A. Mauroy, and I. Mezic, “Koopman resolvent: A Laplace- domain analysis of nonlinear autonomous dynamical systems,”SIAM Journal on Applied Dynamical Systems, vol. 20, no. 4, pp. 2013–2036, 2021

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Koopman resolvents of non- linear discrete-time systems: Formulation and identification,

Y . Susuki, A. Mauroy, and Z. Drma ˇc, “Koopman resolvents of non- linear discrete-time systems: Formulation and identification,” in2024 European Control Conference (ECC). IEEE, 2024, pp. 627–632

work page 2024

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J. M. Lee,Introduction to Smooth Manifolds. Springer, 2012

work page 2012

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Global linearization and fiber bundle structure of invariant manifolds,

J. Eldering, M. Kvalheim, and S. Revzen, “Global linearization and fiber bundle structure of invariant manifolds,”Nonlinearity, vol. 31, no. 9, p. 4202, 2018

work page 2018

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Local contractions and a theorem of Poincar ´e,

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Deep Koopman operator with control for nonlinear systems,

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work page 2022

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Approximation by superpositions of a sigmoidal function,

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