Koopman Representations of Dynamic Systems with Control

Craig Bakker; Kathleen E. Nowak; Steven Rosenthal

REVIEW 3 major objections 6 minor 1 cited by

Koopman Representations of Dynamic Systems with Control

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper introduces a consistency criterion showing that a hybrid Koopman representation with joint state-control observables can represent controlled nonlinear systems with state-dependent control while keeping the Koopman operator…

desk verdict The negative results on separable and affine Koopman control formulations are the real contribution; the hybrid claim is under-supported but easily fixed, and the fix reveals the hybrid representation is universal. read the letter →

arxiv 1908.02233 v1 pith:OQ6FEJEX submitted 2019-08-06 math.DS cs.LG

Craig Bakker , Steven Rosenthal , Kathleen E. Nowak This is my paper · ORCID

classification math.DScs.LG MSC 37N3593C10

keywords Koopmanoperatorcontrolleddynamicalsystemsconsistencyobservablesstate-dependentcontrolhybridformulationmodelpredictivedata-drivenapproximation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Koopman representations turn nonlinear dynamics into linear evolution on a space of observables, but for controlled systems the standard simplifications—state-only observables with affine control, or a control-dependent operator—cannot represent every nonlinear system. This paper introduces a consistency condition: a representation is consistent when its observable evolution matches the chain-rule time derivative of the true flow. Applying the condition shows that separable and affine-control forms force the dynamics to have no state-control interaction term whenever the observables include the state itself. The paper's proposed hybrid form, which combines state-only observables for the drift with observables that jointly depend on state and control for the control response, keeps the Koopman operator constant yet remains consistent with state-dependent control. A sympathetic reader would care because the result tells control designers, before fitting data, which Koopman ansatz can in principle represent their system.

What carries the argument

The load-bearing mechanism is the paper's definition of dynamical consistency: a Koopman representation is consistent with $\dot{x}=f(x,u)$ when the propagated observables satisfy the chain-rule identity $\partial_t\psi=(\partial\psi/\partial x)f(x,u)$, and for discrete maps the analogous partial-derivative identities with respect to $x_k$ and $u_k$. Evaluating those identities at $u=0$ and $x=0$ splits the dynamics into an uncontrolled drift, a state-independent control term, and an interaction term $f_{xu}(x,u)$. That split exposes the restrictions: in separable and affine representations the interaction term must vanish, while in the hybrid representation it is absorbed by a joint observable $\psi_{xu}(x,u)$ with $\psi_{xu}(x,0)=0$, producing separate algebraic conditions that can be satisfied sequentially.

What would settle it

Take a scalar system with state-dependent control, such as $\dot{x}=x u$, and search for all finite-dimensional candidates $\psi_x,\psi_{xu}$ that satisfy the hybrid consistency equations (49)-(50), then test whether the span of the resulting observables is invariant under the Koopman operator. If every such candidate provably lacks a finite invariant subspace while still meeting the equations, the conditions are not sufficient and the claimed wider admissibility of the hybrid form fails; if a finite invariant solution exists, the claim is supported.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a classification of Koopman representations of controlled systems by their consistency with the underlying dynamics. For a discrete system $x_{k+1}=f(x_k,u_k)$, consistency requires $\partial \psi_{k+1}/\partial x_k = (\partial \psi_{k+1}/\partial x_{k+1})(\partial f_k/\partial x_k)$ and the analogous identity in $u$. From these, the paper proves that a state-inclusive representation (one whose observables include $\mathrm{Id}[x]=x$) of the separable form $\psi_x(x_{k+1})=K_x\psi_x(x_k)+K_u\psi_u(u_k)$, or the affine form $K\psi(x_k)+B u_k$, necessarily has $f_{xu}(x,u)=0$: those forms can only represent systems whose control action does not interact with the state. The hybrid formulation $\psi_x(x_{k+1})=K_x\psi_x(x_k)+K_{xu}\psi_{xu}(x_k,u_k)$, with $\psi_{xu}(x,0)=0$, removes that obstruction because the joint observable carries the $f_{xu}$ term while both operators remain constant. In continuous time the same split reads $\dot{x}=f_x(x)+f_{xu}(x,u)$ with conditions $(\partial\psi_x/\partial x)f_x(x)=L_x\psi_x(x)$ and $(\partial\psi_x/\partial x)f_{xu}(x,u)=L_{xu}\psi_{xu}(x,u)$.

Load-bearing premise

The comparison assumes that satisfying the paper's necessary consistency conditions is enough for a finite Koopman representation of that form to exist; the paper proves only that failing the conditions rules a form out, not that passing them guarantees a representation.

Editorial extensions

If this is right

For any system whose observables include the state itself, the separable representation $K_x\psi_x+K_u\psi_u$ and the affine-control representation $K\psi+Bu$ are consistent only if $f_{xu}(x,u)=0$.
Even without state-inclusive observables, the separable form forces the Jacobian of the state observables to be constant along the control flow, which in practice pushes the observables toward linearity in the affected directions.
The hybrid form separates the fitting problem: the drift operator is fixed by the uncontrolled dynamics, and the joint observable is then determined from the controlled dynamics, so the two pieces can be solved sequentially.
The earlier control-dependent-operator formulation is a special case of the hybrid form through $\psi_{xu}(x,u)=(K(u)-K(0))\psi_x(x)$.
The consistency conditions provide a pre-computation check for data-driven Koopman control: if the chosen ansatz fails them for the target dynamics, any finite approximation will carry unavoidable model error.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the consistency conditions are necessary but not proven sufficient, the 'relatively large space' claim is best read as an exclusion result: it says which systems are not ruled out, not that every system passing the conditions has a finite Koopman representation.
A natural algorithmic consequence the paper does not develop is a two-stage fitting procedure: estimate the drift operator from control-off data, then fit the joint observable and its operator from control-on data.
The same consistency framework can rank other proposed Koopman control constructions—for instance different dictionaries for the joint observable—by the class of dynamics they exclude.
Constructing an explicit finite-dimensional hybrid representation for a concrete system with $f_{xu}\neq 0$ would upgrade the paper's comparison from exclusion to existence; the paper provides no such example.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The negative results in this paper are the part worth taking seriously. The consistency definitions are sensible, and the necessary conditions showing that the Korda-Mezic, Arbabi, Kaiser, and affine DMD-type formulations cannot represent systems with state-control coupling when the observables are state-inclusive (Corollaries 2.1, 4.1, 2.3, 4.3) are genuine contributions. As far as I know, these constraints are not stated in the prior literature. The chain-rule derivations check out; the math is sound. People who apply these control Koopman methods should know these limitations.

The soft spot is the paper’s central positive claim for the hybrid formulation (45)/(84). The theorems give only necessary conditions, and the paper treats the absence of a restrictive necessary condition as evidence that the hybrid class is 'relatively large.' That is not a proof. The stress-test note says the gap is closable by a simple construction, and on reading the paper I think the note is right. For any discrete-time system x_{k+1}=f(x_k,u_k), take ψx=x, Kx=0, Kxu=I, and ψxu=f. Then ψx(x_{k+1})=x_{k+1}=f(x_k,u_k)=Kxψx+Kxuψxu. The same construction works in continuous time with Lx=0, Lxu=I. So the hybrid representation is actually universal under the paper’s own consistency definition. That makes the 'relatively large space' claim true but trivial, and the Section IV remark that hybrid 'can model a nonlinear system with state-dependent control' becomes a conspicuous understatement. The authors should state this construction, or at least acknowledge that the hybrid formulation imposes no restriction when ψxu is free. Without that, the comparison leans entirely on the negative results, which are solid but less exciting than the abstract suggests.

There are also a few typos a referee will want cleaned: Corollary 2.1 says f(x,u)=0 but the proof and surrounding text clearly mean fxu(x,u)=0; Theorem 3.5 has missing ∂ symbols in the chain-rule products. Minor.

Who should read this: anyone using Koopman-based control with state-inclusive dictionaries. The negative conditions are practical warnings. The hybrid section needs revision before acceptance. I would send this to peer review rather than desk reject; it earns referee time. Expect minor-to-moderate revision.

Referee Report

3 major / 6 minor

Summary. The paper studies Koopman representations of controlled dynamical systems. It introduces a notion of dynamical consistency based on the chain rule: a representation is consistent if the time evolution of the observables obeys the same partial differential identities that the underlying system's chain rule imposes. The authors then derive necessary conditions for several existing and proposed Koopman control formulations: a separable form ψ_x(x_{k+1}) = K_x ψ_x(x_k) + K_u ψ_u(u_k) (and its continuous-time analogue), an affine form ψ(x_{k+1}) = Kψ(x_k) + Bu_k, and a new 'hybrid' form ψ_x(x_{k+1}) = K_x ψ_x(x_k) + K_xu ψ_xu(x_k,u_k) in which the control response is driven by a joint observable of state and control. The main claim is that the hybrid formulation admits a relatively large space of dynamical systems, in particular systems with state-dependent control (f_xu ≠ 0), while keeping the Koopman operators independent of state and control. The paper analyzes discrete- and continuous-time settings and discusses the relation of the hybrid form to the Williams et al. formulation.

Significance. If the claims are fully substantiated, the paper provides a useful framework for comparing Koopman control formulations and for deciding, from necessary conditions, which formulations cannot represent a given controlled system. The negative results for separable and affine state-inclusive formulations, showing that they force f_xu = 0, are clearly derived and are likely to be of practical value. The consistency framework is self-contained and does not rely on fitted parameters or external benchmarks. The main positive claim about the hybrid formulation, however, is currently supported only by necessary conditions, not by a sufficiency proof or a nontrivial example, which leaves the headline claim under-supported. The paper is otherwise well organized and the derivations are mostly transparent, apart from specific proof errors noted below.

major comments (3)

[Section IV, Theorems 3.3 and 3.5] The central claim that the hybrid formulation (45)/(84) 'can model a nonlinear system with state-dependent control (i.e. fxu(x,u) ≠ 0) while still having state-inclusive observables' is not established by the theorems as written. Theorems 3.3 and 3.5 and Corollary 5.2 provide only necessary conditions, and the paper itself states these are 'necessary (though possibly not sufficient)' in Deﬁnitions 3.1 and 3.2. No explicit example of any system with fxu ≠ 0 is constructed, and no sufficiency argument is given. I request either a theorem giving sufficient conditions for existence of a hybrid representation under the stated hypotheses, or at least a worked example such as the discrete-time system x_{k+1} = x_k + x_k u_k with ψx(x) = x, Kx = 1, Kxu = 1, ψxu(x,u) = x u, which satisfies the conditions of Corollary 5.2. Without such an addition, the positive claim in Section IV is a conjecture rather than a proved result.
[Section III-C, Theorem 3.4 proof] The proof of Theorem 3.4 is flawed. After deriving equation (68), the text states that 'evaluating at uk=0 yields' equation (69), but (69) is identical to (67) (both are evaluated at xk=0 and involve ∂fu,k/∂uk), whereas the needed condition (59) involves ∂fx,k/∂xk and should come from evaluating the ∂/∂xk equation at uk=0. A correct derivation of (59) and (62) is missing. Additionally, equation (66) has 'Kx ∂ψu,k/∂uk' on the right-hand side, which should be 'Ku ∂ψu,k/∂uk'. These errors need to be fixed for the theorem's proof to support the stated necessary conditions.
[Section II, equations (5)-(6) and Section IV] The hybrid formulation without any normalization is trivially universal: for any discrete-time system, the choice ψx(x) = x, Kx = 0, Kxu = I, and ψxu(x,u) = f(x,u) satisfies (84), and the analogous continuous-time choice satisfies (45). Since the paper's comparison of formulations and its claim of a 'relatively large space' are based on the necessary conditions derived under the normalization ψxu(x,0) = 0, the discussion should explicitly acknowledge this trivial universality and state that the comparison is relative to that normalization. Without this caveat, the reader cannot tell what 'relatively large' means; with the normalization imposed, the positive claim becomes substantive but still requires an example or sufficiency proof as noted above.

minor comments (6)

[Throughout] There are frequent typos: 'inﬁnte' for 'infinite', 'observerables' for 'observables', 'simpiﬁes' for 'simplifies', 'approxinatly' for 'approximately', 'advantagous' for 'advantageous', and 'and and' in Section III-A. A careful proofreading pass is needed.
[Section III-B] The corollaries are numbered 'Corollary 2.1', 'Corollary 2.2', and 'Corollary 2.3' even though they appear in Section III; they should be numbered as Corollary 3.1, 3.2, and 3.3 to avoid confusion.
[Section III-C, proof of Corollary 4.1] The proof says 'evaluating (61) and (61) for the subset of observables Id[x] = x', but the second reference should be to equation (62).
[Equations (85), (91), (93), (99)] Several equations in Theorem 3.5 and its corollaries have 'Kx ψx,k / ∂xk' where the partial derivative symbol is missing; these should read 'Kx ∂ψx,k/∂xk' (and similarly for Kxu and ψxu).
[Corollary 4.2, equations (74)-(75)] The notation with simultaneous subscripts 'xk=x1 uk=u1' is hard to parse; consider writing the conditions in a cleaner form, for example as separate equations for each fixed u1 or x1.
[Deﬁnition 3.2] The consistency conditions (14)-(18) are chain-rule identities that hold for any differentiable ψ and f, so they are automatically satisfied; the real content comes from substituting the specific Koopman representation. A brief note clarifying this would prevent the reader from misinterpreting the definitions as substantive constraints.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the hybrid-generality claim is under-supported by necessary conditions, but this is a rigor gap, not a derivation that reduces to its inputs.

full rationale

The derivation chain is self-contained. Definition 3.1 defines dynamical consistency via the chain rule, and Theorem 3.1 correctly proves that any actual Koopman representation is consistent. The subsequent theorems and corollaries are algebraic consequences of applying these necessary conditions to specific representation forms; no fitted parameter is relabeled as a prediction, and no load-bearing self-citation or imported uniqueness theorem appears. The only substantive weakness is in Section IV, where the paper concludes that the hybrid formulation (45)/(84) 'can model a nonlinear system with state-dependent control (i.e. fxu(x,u) ≠ 0)' even though the preceding consistency conditions are explicitly only 'necessary (though possibly not sufficient).' This is a logical gap between necessary conditions and the existence of a finite-dimensional representation, not a circular identification of the target claim with an input. The reader's take and skeptic note correctly identify this gap, but a proof gap is not circularity under the specified standards: no equation is shown to be equivalent to its own input by construction, and the paper does not use the unproved claim as a premise. Score 0 reflects the absence of circularity; the rigor gap should be closed by an explicit example or sufficiency theorem, but it does not make the argument circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The paper introduces no fitted parameters. The central claim rests on smoothness and invariance assumptions standard in Koopman theory, plus the implicit sufficiency assumption that the hybrid representation exists whenever the necessary conditions hold. The joint observable ψxu is a new mathematical object whose existence is not demonstrated.

assumptions (6)

domain assumption The observables ψ are continuously differentiable with respect to state and control.
Consistency definitions and proofs rely on partial derivatives of ψ with respect to x and u (e.g., equations (11)-(13), (14)-(18)). The paper states this for continuous-time observables in Section II but does not state it as a global assumption.
domain assumption The dynamical system f is differentiable and defined on a neighborhood of x=0 and u=0.
Theorems 3.2 and 3.4 evaluate conditions at x=0 and u=0 and use partial derivatives of f (e.g., equations (27)-(29), (59)-(62)).
domain assumption The exact Koopman representation (K,ψ) or (L,ψ) exists, i.e., the span of the observables is invariant under the Koopman operator.
The consistency analysis assumes a Koopman representation of the given form is given; existence of finite-dimensional invariant subspaces is not proven. This underlies all theorems.
standard math The decomposition f(x,u) = fx(x) + fu(u) + fxu(x,u) with fu(0) = fxu(x,0) = fxu(0,u) = 0 is always possible.
This follows by setting fx(x) = f(x,0), fu(u) = f(0,u) - f(0,0), and fxu = f - fx - fu; used in Theorems 3.2 and 3.4.
standard math The Koopman operator is linear.
Linearity of the Koopman operator is part of its definition (Section II); all representations assume linearity.
ad hoc to paper Sufficiency of necessary consistency conditions for existence of the hybrid representation.
The paper's conclusion that the hybrid formulation admits a larger class of systems relies on the unstated assumption that satisfying the necessary conditions is enough for a representation to exist. The paper only proves necessary conditions; no sufficiency or construction is given.

invented entities (1)

Joint control-response observable ψxu(x,u)
purpose: Used in the hybrid formulation (45)/(84) to allow the Koopman operator to remain independent of state and control while capturing state-control coupling fxu.
This is a new mathematical object introduced by the paper. It has no falsifiable handle outside the paper; its existence is not proven for any concrete system. It is a functional component of the proposed representation, not a physical entity.

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Cite this review

Pith. "Pith review of Koopman Representations of Dynamic Systems with Control." pith.science (2026). https://pith.science/paper/OQ6FEJEX

@misc{pith2026190802233,
  author       = {Pith},
  title        = {Pith review of: Koopman Representations of Dynamic Systems with Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQ6FEJEX}},
  note         = {Machine review of arXiv:1908.02233}
}

read the original abstract

The design and analysis of optimal control policies for dynamical systems can be complicated by nonlinear dependence in the state variables. Koopman operators have been used to simplify the analysis of dynamical systems by mapping the flow of the system onto a space of observables where the dynamics are linear (and possibly infinte). This paper focuses on the development of consistent Koopman representations for controlled dynamical system. We introduce the concept of dynamical consistency for Koopman representations and analyze several existing and proposed representations deriving necessary constraints on the dynamical system, observables, and Koopman operators. Our main result is a hybrid formulation which independently and jointly observes the state and control inputs. This formulation admits a relatively large space of dynamical systems compared to earlier formulations while keeping the Koopman operator independent of the state and control inputs. More generally, this work provides an analysis framework to evaluate and rank proposed simplifications to the general Koopman representation for controlled dynamical systems.

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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. Full citation record

Operator-Theoretic Methods for Differential Games
math.OC 2025-07 conditional novelty 7.0 of 10

Two Koopman-based solution approaches, resolvent feedback and EDMD-MCP, reproduce the analytical turret-defense equilibrium for most initial states but visibly fail on singular surfaces and lack error metrics.

Reference graph

Works this paper leans on

15 extracted references · 9 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Applied koopmanis m,

M. Budiˇ si´ c, R. Mohr, and I. Mezi´ c, “Applied koopmanis m,” Chaos: An Interdisciplinary Journal of Nonlinear Science , vol. 22, no. 4, p. 047510, 2012

work page 2012

[2] [2]

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J. Page and R. R. Kerswell, “Koopman analysis of burgers equation,” Physical Review Fluids, vol. 3, no. 7, p. 071901, 2018

work page 2018

[3] [3]

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J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz, “On dynamic mode decomposition: Theory and applications,” Journal of Computational Dynamics, vol. 1, no. 2, pp. 391–421, 2014

2014

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

work page 2015

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arXiv 2016

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J. L. Proctor, S. L. Brunton, and J. N. Kutz, “Generalizin g 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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M. O. Williams, M. S. Hemati, S. T. Dawson, I. G. Kevrekidi s, and C. W. Rowley, “Extending data-driven koopman analysis t o actuated systems,” in IFAC Symposium on Nonlinear Control Systems (NOLCOS), 2016

work page 2016

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Linear predictors for nonlinear dynami- cal systems: Koopman operator meets model predictive contr ol,

M. Korda and I. Mezi´ c, “Linear predictors for nonlinear dynami- cal systems: Koopman operator meets model predictive contr ol,” Automatica, vol. 93, pp. 149–160, 2018

work page 2018