{"id":"05ca7750-3f80-466e-a8aa-7e927a6030e0","arxiv_id":"2411.16640","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Any right-invariant optimal control problem on a Lie groupoid is claimed to reduce to one on its co-adjoint Lie algebroid, but the key step is cited, not proven.","lead":"This paper claims that symmetric optimal control problems on Lie groupoids can be moved to a simpler geometric setting, the co-adjoint Lie algebroid. The proof is incomplete and rests on earlier work by the author, and the illustrative example contains algebraic mistakes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The co-adjoint algebroid is not well-defined: ad*_Xξ is not a section of A*IG, so the central reduction and Hamiltonian system (6) lack a legitimate target.","rationale":"The deepest requirement for the paper's main theorem is that the co-adjoint Lie algebroid exists and is equivalent to the original. The formulas given for the infinitesimal co-adjoint action do not satisfy the defining C∞-linearity of a section; this is a direct, checkable algebraic fact, not a reliance on unpublished references. The reader's weakest point (the kernel of ad* must be an ideal) is a downstream consequence: if ad*_Xξ were a section, well-definedness would still require the kernel to be an ideal for the bracket to descend, but here the action is not even a section. The trivial-groupoid computation above isolates the failure: with an abelian structure group and nonzero anchor, the pairing with fY differs from f times the pairing with Y. This means AG cannot carry the structure functions claimed by [2], so Corollary 13 overreaches. The paper may still contain a correct statement under additional assumptions, but the abstract and Corollary 13 claim all invariant problems on arbitrary Lie groupoids, and that claim is not supported.","tokens_in":18399,"tokens_out":20560,"duration_ms":209554,"concrete_test":"Work with the trivial Lie groupoid G = M×R×M over M=R, so AG = TM⊕(M×R) and AIG = M×R. Choose a nonzero constant covector ξ with ξ(e)=1, X=∂_x∈Γ(AG), and Y=e∈Γ(AIG). For any f∈C∞(M), the paper's formula gives ⟨ad*_Xξ, f e⟩ = -ξ([∂_x, f e]) = -ξ(∂_x f · e) = -∂_x f, while f⟨ad*_Xξ, e⟩ = -f ξ([∂_x,e]) = 0. Since -∂_x f ≠ f·0 for generic f, ad*_Xξ is not C∞-linear in Y and is not a section of A^*IG. Consequently AG={ad*_Xξ} is not a vector bundle, and the structure-function equality from [2] used before Eq. (6) is not meaningful. This settles that the central reduction in Corollary 13 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Corollary 13 and system (6)) requires the co-adjoint Lie algebroid AG = {ad*_X ξ | X∈AG} to be a genuine Lie algebroid whose sections have the same structure functions as AG. The paper defines (Section 2.3) ⟨ad*_X ξ, Y⟩ = -⟨ξ, [X,Y]⟩ for sections X∈Γ(AG), Y∈Γ(AIG). For any f∈C∞(M), the Lie algebroid Leibniz rule gives [X,fY]=f[X,Y]+(ρ(X)f)Y, so ⟨ad*_X ξ, fY⟩ = -f⟨ξ,[X,Y]⟩ - (ρ(X)f)⟨ξ,Y⟩. This is not f times ⟨ad*_Xξ,Y⟩ unless (ρ(X)f)⟨ξ,Y⟩=0 for all f,Y, which fails whenever the anchor ρ(X) is nonzero and ξ is nontrivial. Hence ad*_Xξ is not C∞-linear in Y and is not a section of A^*IG. The paper's own trivial-groupoid example exhibits this immediately. Thus AG is not a well-defined vector bundle, let alone a Lie algebroid with the claimed structure functions; the appeal to [2] cannot repair this. Since system (6) is built on this target algebroid, the reduction of every invariant optimal control problem to the co-adjoint algebroid is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that any right-invariant control system and any right-invariant optimal control problem on a Lie groupoid can be reduced to the co-adjoint Lie algebroid associated with the groupoid. The authors first recall their construction of a co-adjoint Lie groupoid whose points are co-adjoint orbits, define control systems and cost functions on it, and then apply the standard Lie-algebroid optimal control formalism of E. Martinez to write a Hamiltonian system (equation (6)) for the critical trajectories. A worked example for the trivial Lie groupoid M×G×M is presented in Section 6.","tokens_in":18769,"tokens_out":10241,"duration_ms":89218,"significance":"If valid, the reduction would constitute a useful extension of Jurdjevic's reduction of invariant optimal control problems on Lie groups to the groupoid setting, and would connect groupoid symmetry with Lie-algebroid Hamiltonian equations. The paper is ambitious and clearly written in intent, and equation (6) is an explicit falsifiable prediction of the form the reduced dynamics should take. However, the central object of the paper, the co-adjoint Lie algebroid, is not established as a well-defined Lie algebroid, and the main reduction result depends on an equality of structure functions cited without proof from the author's unpublished preprint [2]. The paper does not provide machine-checked proofs or reproducible code; its main value would lie in the reduction principle if that principle were rigorously proved.","major_comments":[{"comment":"The co-adjoint Lie algebroid AG = {ad*_X ξ | X ∈ AG} is not well-defined as a vector bundle over M. For X ∈ Γ(AG), Y ∈ Γ(AIG), and f ∈ C∞(M), the Leibniz rule gives ⟨ad*_X ξ, fY⟩ = ⟨ξ, [fY, X]⟩ = f⟨ξ, [Y, X]⟩ - ρ(X)(f)⟨ξ, Y⟩ = f⟨ad*_X ξ, Y⟩ - ρ(X)(f)⟨ξ, Y⟩. Thus the assignment Y ↦ ⟨ad*_X ξ, Y⟩ is not C∞(M)-linear in Y unless ρ(X)(f)⟨ξ, Y⟩ = 0 for all f, Y, which is generically false. Consequently ad*_X ξ is not a section of A*IG, and the alleged vector bundle AG with sections {ad*_X ξ} does not exist in the usual sense. Since the Hamiltonian system (6) and the reduction in Corollary 13 are formulated on the dual A*G of this object, the central construction fails.","section":"Section 2.3 and Section 5 (definition of co-adjoint algebroid; paragraph 'As it is shown in [2]')"},{"comment":"Corollary 13 states that every right-invariant control system and every optimal control problem on a Lie groupoid reduce to its co-adjoint Lie algebroid, but this statement is not supported by the preceding results. The co-adjoint Lie groupoid in Section 2.3 is defined only for regular Lie groupoids and only for ξ whose stabilizer G_ξ is a normal Lie subgroupoid; these hypotheses are absent from Corollary 13. Moreover, the reduction to the co-adjoint Lie algebroid relies on the equality of structure functions of AG and AG, which is cited from the author's preprint [2] without proof. Without that equality, equations (6) are not the Hamiltonian equations of the co-adjoint algebroid and the reduction is not established.","section":"Corollary 13"},{"comment":"In the converse part of Theorem 8, the proof infers dR_g ∘ F(h,u) = F(R_g(h),u) from the equality ad*_{dR_g∘F(h,u)} ξ = ad*_{F(R_g(h),u)} ξ by saying that 'ad* is linear.' Linearity alone does not imply injectivity; the map X ↦ ad*_X ξ has a kernel given by the stabilizer of ξ under the co-adjoint action, which is generically nontrivial. The proof therefore needs an additional injectivity statement, which is neither stated nor proved, so the claimed equivalence of right-invariance of the original and reduced control systems is not established.","section":"Theorem 8, converse direction"},{"comment":"The illustrative example contains technical errors that affect its validity. The text states 'ρ^i_α = 1 for the Lie algebroid AG and as well for AG.' For the trivial Lie algebroid AG = TM ⊕ (M×g), with a local basis {∂/∂x^i} of TM and {e_α} of g, the anchor sends ∂/∂x^i to itself and e_α to 0, so the anchor matrix is δ^i_α or 0, not 1. Therefore the simplified Hamiltonian equations (15) and the associated bivector expression are not justified. In addition, equation (20) lists '0 = ∂H/∂η_α' as one of the critical trajectory equations; the correct control stationarity condition is '0 = ∂H/∂u_c', which already appears in the same display. As written, the condition 0 = ∂H/∂η_α contradicts the Hamiltonian system (6) and renders the example inconsistent.","section":"Section 6, equations (15) and (20)"}],"minor_comments":[{"comment":"There are repeated typographical errors, e.g., 'extermal' for 'extremal' in the abstract and introduction, and inconsistent notation between the original Lie algebroid and the co-adjoint Lie algebroid, both called AG in Section 5.","section":"Throughout"},{"comment":"The statement that the linear Poisson structure on A*G 'is easy to check' is not demonstrated; in particular, the verification of the Jacobi identity and the compatibility with the anchor are not provided, and these are not immediate in the presence of the equality of structure functions cited from [2].","section":"Section 5.1"},{"comment":"The formula for the anchor of the co-adjoint Lie algebroid in the example is unclear: the expression ρ tilde(x, ad*_V ξ')(p) = X(p) mixes a point p in M, a point x in the fiber, and a vector field X without a precise definition of the domain and the evaluation point.","section":"Section 6, equation (7)"},{"comment":"Lemma 7 is stated without proof; it is then used as the basis for Theorem 8, so a proof or a reference would be needed.","section":"Section 3, Lemma 7"}],"recommendation":"reject","confidential_remarks":"The manuscript relies almost entirely on the author's own previous work [1,2] for the definition of the co-adjoint Lie groupoid and for the key equality of structure functions; reference [2] is an unpublished ResearchGate preprint. The central construction is not shown to be well-defined, and the main reduction claim is not proved. This is a scope issue as well as a correctness issue: the paper overstates the generality of its results relative to the restrictive hypotheses under which the co-adjoint groupoid is defined. In my view, the errors are load-bearing and cannot be repaired by local edits within the manuscript's current scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the paper claims that every right-invariant optimal control problem on a regular Lie groupoid reduces to its co-adjoint Lie algebroid, with critical trajectories given by the Hamiltonian system (6). The idea is new — Martinez and Jozwikowski reduce to the Lie algebroid, not to a co-adjoint algebroid — and the author's earlier co-adjoint groupoid construction is a reasonable starting point. The paper also cleanly restates the standard Lie algebroid optimal control formalism.\n\nThe central problem is that the co-adjoint Lie algebroid is not a well-defined object. The paper defines AG = {ad*_X ξ : X ∈ Γ(AG)} with ⟨ad*_X ξ, Y⟩ = -⟨ξ, [X,Y]⟩ for Y ∈ Γ(AIG). For f ∈ C∞(M), the Leibniz rule gives [X, fY] = f[X,Y] + (ρ(X)f)Y, so ⟨ad*_X ξ, fY⟩ = f⟨ad*_X ξ, Y⟩ - (ρ(X)f)⟨ξ, Y⟩. The extra term is generically nonzero. The correct co-adjoint representation of a Lie algebroid on its dual must include the anchor term ρ(X)⟨ξ,Y⟩; without it, ad*_X ξ is not C∞-linear in Y and is not a section of A*IG. The image AG is therefore not a vector bundle over M, let alone a Lie algebroid with the claimed structure functions. The stress-test note is correct, and the trivial-groupoid example works only because it effectively restricts to the isotropy part, ignoring the base component.\n\nThis is a load-bearing flaw. Corollary 13 and system (6) are built on AG, so the main reduction is not established. The paper also relies on the author's own unpublished preprint [2] for the equality of structure functions between AG and AG; that equality is exactly what the stress-test shows to be false in general, and the citation cannot be checked. The example contains index errors in (15) and (20), and the claim ρ_i^α = 1 for a trivial algebroid is simply wrong.\n\nWhat is worth keeping: the strategy of reducing to co-adjoint objects is interesting, and the paper's restatement of Martinez's Hamiltonian equations is competent. If the author were to use the correct dual representation and restrict the construction so that AG is actually a vector bundle, there might be a salvageable paper. As presented, the central theorem is unsupported.\n\nI recommend reject, and I would accept a desk rejection. The paper deserves a serious referee only if the main object is well-defined; this version needs fundamental repairs first.","headline":"The co-adjoint Lie algebroid that underpins the main reduction is not a well-defined vector bundle, so the central theorem and Hamiltonian system (6) do not hold as stated.","tokens_in":19250,"tokens_out":8963,"would_cite":false,"duration_ms":79109,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J15","53D17","22A22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that every right-invariant control system and optimal control problem on a Lie groupoid can be reduced to a Hamiltonian system on the co-adjoint Lie algebroid, whose structure functions coincide with those of the…","keywords":["optimal control problem","invariant control system","Hamiltonian system","co-adjoint Lie groupoid","co-adjoint Lie algebroid","reduction","critical trajectories"],"falsifier":"Compute the co-adjoint algebroid for a concrete regular Lie groupoid where the stabilizer of some $\\xi$ is not normal—for example an action or frame groupoid—and check whether $[\\mathrm{ad}^*_X\\xi, \\mathrm{ad}^*_Y\\xi] = \\mathrm{ad}^*_{[X,Y]}\\xi$ and whether the anchor of $\\mathrm{ad}^*_X\\xi$ equals $\\rho(X)$; a single discrepancy shows that Hamiltonian system (6) does not describe the original problem's extremals.","tokens_in":18205,"feed_emoji":"🎛️","tokens_out":9754,"duration_ms":80299,"temperature":0.7,"pith_summary":"This paper claims that any right-invariant control system on a regular Lie groupoid—and any optimal control problem built from it—can be reduced to a smaller object, the co-adjoint Lie algebroid, whose fibers are spanned by the co-adjoint action $\\mathrm{ad}^*_X \\xi$. The reduction matters because it turns a search over curves in the whole groupoid into a Hamiltonian system on the dual of this algebroid, where the critical trajectories are computed from structure functions $\\rho^i_\\alpha$ and $C^\\gamma_{\\alpha\\beta}$ of the original Lie algebroid. The central result is Corollary 13 together with the Hamiltonian equations (6), which give the candidate extremals of the reduced optimal control problem. In the worked example of the trivial Lie groupoid, the reduction lands in the cotangent bundle of a co-adjoint orbit of the underlying Lie group, simplifying the equations further.","feed_headline":"Control problems on Lie groupoids reduce to a co-adjoint system","feed_subtitle":"Extremal trajectories come from a Hamiltonian system on the co-adjoint algebroid, using the original structure functions.","key_machinery":"The load-bearing object is the co-adjoint Lie algebroid $\\mathcal{A}_G = \\mathrm{Im}(\\mathrm{ad}^*_\\bullet \\xi)$, defined as the image of the map $X \\mapsto \\mathrm{ad}^*_X \\xi$ on the original Lie algebroid, together with the identity that its anchor and bracket structure functions equal those of $AG$. This identity lets every local basis section of the co-adjoint algebroid be written $\\tilde e_\\alpha = \\mathrm{ad}^*_{e_\\alpha}\\xi$, so the prolongation $T\\mathcal{A}_G^*$ carries a canonical symplectic form $\\omega = \\tilde X^\\alpha \\wedge \\tilde V_\\alpha + \\tfrac12 C^\\gamma_{\\alpha\\beta} \\tilde X^\\alpha \\wedge \\tilde X^\\beta$. Solving the symplectic equation $i_{f_H}\\omega = dH$ for the Hamiltonian $H(\\eta,c) = \\langle \\eta, f(c)\\rangle - L(c)$ produces the reduced Hamiltonian vector field and the critical-trajectory equations (6).","core_discovery":"The paper's central claim is that the right-invariant geometry of a Lie groupoid is retained by its co-adjoint Lie algebroid $\\mathcal{A}_G = \\{\\mathrm{ad}^*_X \\xi : X \\in AG\\}$, the image of the infinitesimal co-adjoint action. Right-invariant control systems and right-invariant cost functions on the groupoid descend to right-invariant objects on the co-adjoint Lie groupoid, and hence on the co-adjoint Lie algebroid, and the structure functions of the co-adjoint algebroid coincide with those of $AG$: $\\tilde\\rho^i_\\alpha = \\rho^i_\\alpha$ and $\\tilde C^\\gamma_{\\alpha\\beta} = C^\\gamma_{\\alpha\\beta}$. Because of this equality, the canonical prolongation and symplectic form of the co-adjoint algebroid yield the Hamiltonian system (6), whose integral curves are the critical trajectories of the original problem. The paper's strongest formulation is Corollary 13: every right-invariant control system and every optimal control problem on a Lie groupoid reduces to its co-adjoint Lie algebroid.","pith_inferences":["Because the reduction is a chain of equalities of structure functions, it could be applied iteratively: successive co-adjoint reductions would keep collapsing the algebroid whenever the relevant stabilizer remains normal.","A practical test of the main theorem is numerical: solve one right-invariant optimal control problem directly on a nontrivial groupoid and again through system (6), then compare extremal trajectories.","The paper assumes the stabilizer $G_\\xi$ is a normal Lie subgroupoid; whether the reduction survives without this normality condition is left open, and it is the most natural place to probe the generality of the result."],"forward_implications":["Right-invariant control systems and cost functions on any regular Lie groupoid descend without loss to the co-adjoint Lie algebroid, so extremal solutions can be sought there.","Critical trajectories of the reduced problem are exactly the integral curves of Hamiltonian system (6).","In the trivial groupoid case, the reduction sends the problem to the cotangent bundle of the co-adjoint orbit $O(\\xi')$, where the Hamiltonian equations simplify to (21), a purely orbit-level system.","The equality of structure functions means the reduced computation uses only the original anchor and bracket coefficients, so no new algebroid structure needs to be computed.","Composing this reduction with the existing reduction of groupoid problems to Lie algebroids gives a direct route from a right-invariant groupoid problem to Hamiltonian equations on the co-adjoint algebroid."],"supporting_citations":[{"why":"Constructs the co-adjoint Lie groupoid and its Lie algebroid, the objects to which the paper reduces invariant control problems.","marker":"[1]"},{"why":"Supplies the claim that the structure functions of the co-adjoint Lie algebroid equal those of the original algebroid, the identity that makes the Hamiltonian reduction work.","marker":"[2]"},{"why":"Provides the prolongation and Hamiltonian-section method that converts the reduced problem into the Hamiltonian system (6).","marker":"[7]"},{"why":"Shows control systems and optimal control problems on a Lie groupoid reduce to its Lie algebroid, the step the paper composes with the co-adjoint reduction.","marker":"[4]"},{"why":"Generalizes the Pontryagin maximum principle to the algebroid setting, justifying that critical trajectories are candidates for optimal solutions.","marker":"[5]"},{"why":"Sets up the classical Lie-group optimal control theory that the present reduction extends to groupoids.","marker":"[10]"},{"why":"Develops optimal control on Lie groups and co-adjoint orbits, the model for the trivial-groupoid example.","marker":"[11]"}],"fun_headline_variants":["Groupoid control reduces to co-adjoint dynamics","Optimal groupoid control collapses to co-adjoint algebroid","Co-adjoint algebroid encodes groupoid optimal control","Groupoid optimal control lives on co-adjoint algebroid"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reduction assumes that the image of the co-adjoint map, sending $X$ to $\\mathrm{ad}^*_X \\xi$, is a Lie algebroid with the same bracket and anchor as the original; this requires those elements mapped to zero to form an ideal of the section space, which fails in some cases when the stabilizer of $\\xi$ is not normal.","fun_headline_variants_meta":{"raw":{"variants":["Groupoid control reduces to co-adjoint dynamics","Optimal groupoid control collapses to co-adjoint algebroid","Co-adjoint algebroid encodes groupoid optimal control","Groupoid optimal control lives on co-adjoint algebroid"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000327,"raw_usage":{"total_tokens":1761,"prompt_tokens":812,"completion_tokens":949,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":877}},"tokens_in":428,"tokens_out":949,"duration_ms":6993,"temperature":1.0,"reasoning_tokens":877,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:53:44.286502+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the co-adjoint algebroid for a concrete regular Lie groupoid where the stabilizer of some $\\xi$ is not normal—for example an action or frame groupoid—and check whether $[\\mathrm{ad}^*_X\\xi, \\mathrm{ad}^*_Y\\xi] = \\mathrm{ad}^*_{[X,Y]}\\xi$ and whether the anchor of $\\mathrm{ad}^*_X\\xi$ equals $\\rho(X)$; a single discrepancy shows that Hamiltonian system (6) does not describe the original problem's extremals.","supporting_citations":[{"cited_title":"Haghighatdoost and R","cited_arxiv_id":null,"evidence_quote":"Constructs the co-adjoint Lie groupoid and its Lie algebroid, the objects to which the paper reduces invariant control problems."},{"cited_title":"Haghighatdoost and R","cited_arxiv_id":null,"evidence_quote":"Supplies the claim that the structure functions of the co-adjoint Lie algebroid equal those of the original algebroid, the identity that makes the Hamiltonian reduction work."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the prolongation and Hamiltonian-section method that converts the reduced problem into the Hamiltonian system (6)."},{"cited_title":"Optimal Control Theory on almost-Lie Algebroids","cited_arxiv_id":"1111.1549","evidence_quote":"Shows control systems and optimal control problems on a Lie groupoid reduce to its Lie algebroid, the step the paper composes with the co-adjoint reduction."},{"cited_title":"Pontryagin Maximum Principle - a generalization","cited_arxiv_id":"0905.2767","evidence_quote":"Generalizes the Pontryagin maximum principle to the algebroid setting, justifying that critical trajectories are candidates for optimal solutions."},{"cited_title":"Jurdjevic, Geometric control theory, Cambridge Uni versity Press 1997","cited_arxiv_id":null,"evidence_quote":"Sets up the classical Lie-group optimal control theory that the present reduction extends to groupoids."},{"cited_title":"Jurdjevic: Optimal control problems on Lie groups: C rossroads between geometry and mechanics, In Geometry of Feedback and Optimal Control, B","cited_arxiv_id":null,"evidence_quote":"Develops optimal control on Lie groups and co-adjoint orbits, the model for the trivial-groupoid example."}],"review_version":1}