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REVIEW 4 major objections 4 minor 14 references

Optimal control problems on the co-adjoint Lie groupoids

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.16640 v1 pith:6TGPBXJB submitted 2024-11-25 math.OC math-phmath.MP

Ghorbanali Haghighatdoost This is my paper · ORCID
classification math.OCmath-phmath.MP MSC 49J1553D1722A22
keywords optimalcontrolprobleminvariantsystemHamiltonianco-adjointLiegroupoidalgebroidreductioncriticaltrajectories
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

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.

What carries the argument

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).

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

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.

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 (4)
  1. [Section 2.3 and Section 5 (definition of co-adjoint algebroid; paragraph 'As it is shown in [2]')] 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.
  2. [Corollary 13] 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.
  3. [Theorem 8, converse direction] 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.
  4. [Section 6, equations (15) and (20)] 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.
minor comments (4)
  1. [Throughout] 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.
  2. [Section 5.1] 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].
  3. [Section 6, equation (7)] 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.
  4. [Section 3, Lemma 7] 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.

Circularity Check

2 steps flagged · score 7.0 of 10

The reduction to the co-adjoint Lie algebroid is carried by a self-cited equality of structure functions that is built into the definition of AG; system (6) is the standard Lie-algebroid Hamiltonian system on a relabeled bundle.

  1. self definitional [Section 5, paragraph introducing the structure functions of AG and AG (before eq. (6))]
    "Also, it is proven in [2] that the structure functions of the Lie algebroids AG and AG are equal, i.e. if ρi α,C γ αβ are structure functions of the Lie algebroid AG and ˜ρi α, ˜C γ αβ are structure functions of Lie algebroid AG, then ˜ρi α =ρi α, ˜C γ αβ =C γ αβ ."

    The co-adjoint Lie algebroid AG was introduced in Section 2.3/5 as AG={ad∗_Xξ} and its bracket is given earlier in the same section by ~[X̃,Ỹ]=ad∗_{[X,Y]}ξ. Therefore the equality of structure functions with AG is an immediate consequence of how AG is defined, not an independently derived fact. This equality is the only input needed to write the Hamiltonian system (6) with the original ρ,C. Corollary 13 and system (6) thus reduce to the standard Martinez optimal-control equations for the original Lie algebroid AG, with the bundle merely renamed AG; the citation to the author's own preprint [2] supplies no independent evidence for the claimed reduction.

  2. renaming known result [Section 6, final paragraph before the Conclusion (equations (18)-(21))]
    "as we proved in [1], for Hamiltonian H = (p,h) : M × T ∗ ξ′O(ξ′) − →R, we have {F,H}A∗G = {f,h}K.K , (18) ... according to the equations (15), (16) and (19), the equations for critical trajectories will be ˙ηα = −ηγC γ αβ ∂h/∂ηβ, 0 = ∂h/∂uc. (21)"

    In the example, the co-adjoint algebroid is M×T_{ξ′}O(ξ′) with bracket [ad∗_Vξ′, ad∗_Wξ′]=ad∗_{[V,W]}ξ′, and its dual Poisson structure is shown, via (18) imported from [1], to be the Kirillov-Kostant bracket on T∗O(ξ′). The resulting 'reduced' critical equations (21) are exactly the standard Lie-Poisson/Kirillov-Kostant Hamiltonian equations on a co-adjoint orbit. The paper presents this classical system as the outcome of a new reduction; it is a relabeling of a known result in co-adjoint-orbit coordinates, not an independent prediction.

full rationale

The first link in the chain (right-invariant control systems and costs descend to the co-adjoint groupoid) is a genuine, if elementary, projection of invariant data. The problematic link is the identification of AG as a Lie algebroid with structure functions equal to those of AG and the consequent Hamiltonian system (6). That equality is not proved in this paper; it is imported from the author's own preprint [2], and the bracket on AG was already set, in [1] and in Section 5, to be ad∗_{[X,Y]}ξ, so the equality is definitional rather than a theorem. As a result, Corollary 13 and the central equations (6) are the standard Lie-algebroid optimality equations for (AG,ρ,C) with the bundle renamed AG; the announced reduction carries no new content beyond the ansatz. The example similarly reproduces the known Kirillov-Kostant/Lie-Poisson system on a co-adjoint orbit as the 'reduced' system. Separately, the co-adjoint action as defined is not C∞-linear in the test section, so AG is not shown to be a well-defined Lie algebroid; that is a serious correctness defect, but the circularity is already present in the definitional equality of structure functions that drives the main result.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The reduction rests on two external facts: the normal-stabilizer condition and the equality of structure functions. The former is an explicit hypothesis, the latter is a self-cited result. Neither is established in this paper, so the central claim carries a large unstated burden.

assumptions (4)
  • domain assumption G is a regular Lie groupoid, meaning the map (beta, alpha) has constant rank.
    Regularity is invoked in Section 2.3 to ensure the isotropy groupoid is a Lie groupoid; the co-adjoint construction depends on it.
  • domain assumption The stabilizer G_xi = {g : Ad*_g xi = xi} is a normal Lie subgroupoid of G.
    This assumption is explicit in Section 2.3 and is required for the multiplication on the co-adjoint orbit O(xi) to be well-defined.
  • ad hoc to paper The kernel of the map X -> ad*_X xi is an ideal in Gamma(AG).
    Without this, the bracket [|ad*_X xi, ad*_Y xi|] = ad*_{[X,Y]} xi is not well-defined and AG is not a Lie algebroid. The paper does not state or prove this.
  • ad hoc to paper The structure functions of AG and AG are equal (rho tilde = rho, C tilde = C).
    This is cited from the authors' own [2] in Section 5.2 and is the key identification used to write the Hamiltonian equations (6). It is not derived in this paper.
invented entities (1)
  • Co-adjoint Lie algebroid AG = {ad*_X xi : X in AG}
    purpose: The reduced state space on which invariant optimal control problems are claimed to simplify; it underlies the Hamiltonian equations (6).
    The object is introduced by the authors and its Lie algebroid structure is not independently verified in this paper; well-definedness depends on the kernel of ad* being an ideal, which is not shown.

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Pith. "Pith review of Optimal control problems on the co-adjoint Lie groupoids." pith.science (2026). https://pith.science/paper/6TGPBXJB

@misc{pith2026241116640,
  author       = {Pith},
  title        = {Pith review of: Optimal control problems on the co-adjoint Lie groupoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6TGPBXJB}},
  note         = {Machine review of arXiv:2411.16640}
}
read the original abstract

In this work we study the invariant optimal control problem on Lie groupoids. We show that any invariant optimal control problem on a Lie groupoid reduces to its co-adjoint Lie algebroid. In the final section of the paper, we present an illustrative example.

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