{"id":"eda11585-38d8-4ce9-b2ac-3456432eb50a","arxiv_id":"1908.02233","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper derives necessary consistency conditions showing that separable and affine Koopman control formulations forbid state-control coupling when observables include the state, and it proposes a less restrictive joint-observation formulation.","lead":"This paper analyzes when Koopman operators, a tool that converts nonlinear control problems into linear ones, can faithfully represent the original system. It shows that several standard simplified versions secretly require the state and the control to act independently, and it proposes a joint-observation format that relaxes this limitation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hybrid generality is asserted from necessary conditions alone, with no example; the gap is real but closable by a simple construction.","rationale":"The reader's weakest assumption correctly identifies the load-bearing issue: the paper's support for the hybrid formulation's greater generality rests on necessary conditions that are explicitly not sufficient, and no example construction is given. If this gap were truly fatal, the central claim would be unsupported. In fact, the claim is true, and an explicit finite-dimensional construction is immediate for the simple scalar system x_{k+1}=x_k+x_k u_k; one can even represent every system by taking ψxu=f. Thus the concern does not invalidate the paper's central claim, but it does expose a real proof gap in the written argument. The negative results for the separable/affine formulations with state-inclusive observables appear sound (modulo typographical errors in the main theorem proofs, which the reader also noted), and these negative results are a meaningful contribution. Since the paper needs a small but important addition—a concrete example or a clear sufficiency statement—the conditional rejection/acception stance of the reader remains appropriate. I therefore recommend no change to the reader's verdict: the paper is conditionally acceptable provided the positive claim is supported by an explicit construction or a sufficiency theorem. The proposed concrete test directly settles whether the missing support is actually available, and it is; so the emphasis shifts from correctness to completeness of the proof.","tokens_in":11655,"tokens_out":23846,"duration_ms":251763,"concrete_test":"Verify that the scalar system x_{k+1}=x_k+x_k u_k (with fx=x, fu=0, fxu=xu) has an exact finite-dimensional hybrid representation of the form (84): choose ψx(x)=x, Kx=1, Kxu=1, ψxu(x,u)=xu. Check that ψx(x_{k+1})=x_k+x_k u_k equals Kx ψx(x_k)+Kxu ψxu(x_k,u_k) for all (x,u), and that the consistency conditions (85) and (86) hold. If this identity holds, the sufficiency gap is not fatal to the claim, but the paper must still include such a construction to prove it. A stronger test is the system x_{k+1}=x_k^2+x_k u_k with ψx=[x;x^2]; attempt to find constant Kx, Kxu and a joint observable ψxu that satisfy (84) exactly, to probe whether the construction extends beyond the identity observable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central positive claim—that the hybrid formulation of (45)/(84) can model nonlinear systems with state-dependent control (fxu≠0) while retaining state-inclusive observables—is not established by the theorems themselves. As the paper says, its consistency conditions are 'necessary (though possibly not sufficient).' Theorem 3.5 and the discussion in Section IV only show that the hybrid formulation's necessary conditions do not forbid fxu≠0; they do not prove that any such system admits a hybrid Koopman representation. Since no explicit construction is given, the written argument under-supports the headline claim about 'a relatively large space of dynamical systems.' This is a genuine proof gap, not a fabrication: for the scalar discrete-time system x_{k+1}=x_k+x_k u_k, one can explicitly write ψx(x)=x, Kx=1, Kxu=1, ψxu(x,u)=x u, and then ψx(x_{k+1})=x_k+x_k u_k=Kx ψx(x_k)+Kxu ψxu(x_k,u_k), giving a finite-dimensional state-inclusive hybrid representation. More broadly, taking ψx=x, Kx=0, Kxu=I, and ψxu=f represents every discrete-time system, though this makes the positive result nearly trivial. The paper should state such a construction or prove a nontrivial sufficiency condition; without it, the necessary-condition comparison alone only shows that the hybrid formulation is not ruled out, not that it is admitted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11920,"tokens_out":8035,"duration_ms":80408,"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":[{"comment":"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":"Section IV, Theorems 3.3 and 3.5"},{"comment":"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":"Section III-C, Theorem 3.4 proof"},{"comment":"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.","section":"Section II, equations (5)-(6) and Section IV"}],"minor_comments":[{"comment":"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":"Throughout"},{"comment":"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":"Section III-B"},{"comment":"The proof says 'evaluating (61) and (61) for the subset of observables Id[x] = x', but the second reference should be to equation (62).","section":"Section III-C, proof of Corollary 4.1"},{"comment":"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).","section":"Equations (85), (91), (93), (99)"},{"comment":"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.","section":"Corollary 4.2, equations (74)-(75)"},{"comment":"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.","section":"Definition 3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a reasonable contribution to the Koopman-control literature, and the framework of necessary conditions is a useful analytical tool. However, the headline positive claim about the hybrid formulation is not proven: only necessary conditions are given, and the proof of Theorem 3.4 has a significant gap as well as a typo. The issues are fixable within the manuscript's scope, so I recommend major revision. The authors should add a sufficiency argument or explicit nontrivial examples, and correct the proof of Theorem 3.4. I see no citation-pattern or novelty-disclosure concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThere 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.\n\nWho 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.","headline":"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.","tokens_in":12426,"tokens_out":5097,"would_cite":true,"duration_ms":53332,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37N35","93C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Koopman operator","controlled dynamical systems","dynamical consistency","observables","state-dependent control","hybrid Koopman formulation","model predictive control","data-driven approximation"],"falsifier":"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.","tokens_in":11452,"feed_emoji":"🎛️","tokens_out":13491,"duration_ms":135939,"temperature":0.7,"pith_summary":"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.","feed_headline":"Joint state-control observables widen Koopman control range","feed_subtitle":"A consistency test shows which controlled nonlinear systems a Koopman form can capture, and which it cannot.","key_machinery":"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.","core_discovery":"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 $(\\partial\\psi_x/\\partial x)f_{xu}(x,u)=L_{xu}\\psi_{xu}(x,u)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Koopman operator definition and functional-space setting on which all representations in the paper build.","marker":"[1]"},{"why":"Introduces the control-dependent-operator formulation $\\psi(x_{k+1})=K(u_k)\\psi(x_k)$ that the paper analyzes and later shows to be a special case of the hybrid form.","marker":"[7]"},{"why":"Proposes the affine-control Koopman representation $K\\psi+Bu$ whose consistency restrictions the paper derives.","marker":"[8]"},{"why":"Provides the continuous-time joint-observable eigenfunction formulation with $\\dot{u}$ control that the paper compares with its hybrid representation.","marker":"[14]"},{"why":"Uses the affine Koopman predictive-control formulation that the paper's consistency conditions show forces $f_{xu}=0$ with state-inclusive observables.","marker":"[15]"}],"fun_headline_variants":["Hybrid Koopman representation captures control-state coupling","Consistency test ranks Koopman forms for controlled systems","Joint observable Koopman lifts state-control nonlinearities","Hybrid Koopman removes f_xu=0 restriction in control systems","Richer Koopman class for controlled dynamics via joint observables"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hybrid Koopman representation captures control-state coupling","Consistency test ranks Koopman forms for controlled systems","Joint observable Koopman lifts state-control nonlinearities","Hybrid Koopman removes f_xu=0 restriction in control systems","Richer Koopman class for controlled dynamics via joint observables"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000624,"raw_usage":{"total_tokens":2924,"prompt_tokens":1018,"completion_tokens":1906,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":1835}},"tokens_in":634,"tokens_out":1906,"duration_ms":11925,"temperature":1.0,"reasoning_tokens":1835,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:50:42.797936+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Applied koopmanis m,","cited_arxiv_id":null,"evidence_quote":"Supplies the Koopman operator definition and functional-space setting on which all representations in the paper build."},{"cited_title":"Extending data-driven koopman analysis t o actuated systems,","cited_arxiv_id":null,"evidence_quote":"Introduces the control-dependent-operator formulation $\\psi(x_{k+1})=K(u_k)\\psi(x_k)$ that the paper analyzes and later shows to be a special case of the hybrid form."},{"cited_title":"Linear predictors for nonlinear dynami- cal systems: Koopman operator meets model predictive contr ol,","cited_arxiv_id":null,"evidence_quote":"Proposes the affine-control Koopman representation $K\\psi+Bu$ whose consistency restrictions the paper derives."}],"review_version":1}