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arxiv: 2504.19554 · v2 · pith:XJBVVW33new · submitted 2025-04-28 · 🧮 math.AP · math.OC

Approximation of an optimal control problem on a network with a perturbed problem in the whole space

Mohamed Camar-Eddine (IRMAR) , M\'eriadec Chuberre (IRMAR) , Mounir Haddou (IRMAR) , Olivier Ley (IRMAR) This is my paper

Pith reviewed 2026-05-25 08:22 UTC · model grok-4.3

classification 🧮 math.AP math.OC
keywords optimal controlsingular perturbationnetworksEikonal equationvalue functionstrajectorieslimit problem
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The pith

A singular penalty of size 1/ε forces optimal-control trajectories in the plane to converge onto a fixed network Γ as ε approaches zero.

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

The authors start from a standard optimal-control problem posed throughout R^2 and add a large singular term whose size grows like 1/ε. The term is designed to penalize any path that leaves a prescribed network Γ. They prove that any sequence of minimizing trajectories for the perturbed problems has a subsequential limit whose image lies entirely on Γ. When the running cost is the Eikonal Hamiltonian, the value functions of the perturbed problems converge to the value function of the natural optimal-control problem stated directly on the network. A reader cares because the construction supplies a concrete way to recover network problems as limits of ordinary whole-space problems.

Core claim

We prove that the sequence of trajectories admits a subsequential limit evolving on Γ. Moreover, in the case of the Eikonal equation, we show that the sequence of value functions associated with the perturbed optimal control problems converges to a limit which, in particular, coincides with the value function of the expected optimal control problem set on the network Γ.

What carries the argument

The singular perturbation term of magnitude ε^{-1} that penalizes distance from the prescribed network Γ.

If this is right

  • Any sequence of perturbed minimizers has at least one subsequential limit whose support lies on Γ.
  • In the Eikonal case the value functions converge pointwise to the value function of the network problem.
  • The whole-space perturbed problems therefore serve as an approximating family for the network optimal-control problem.
  • The same limit procedure recovers the network problem studied in the existing literature on optimal control on graphs.

Where Pith is reading between the lines

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

  • Numerical schemes for the perturbed problem in R^2 could be used to compute approximate solutions on Γ without having to discretize the network geometry directly.
  • The same penalization device may produce convergence for other Hamilton-Jacobi equations whose Hamiltonians satisfy the same structural assumptions used for the Eikonal case.

Load-bearing premise

The singular term of size 1/ε is strong enough by itself to force every minimizing trajectory onto Γ in the limit, with no further regularity or geometric conditions imposed on Γ.

What would settle it

An explicit network Γ together with initial data and running cost for which some sequence of perturbed minimizers stays bounded away from Γ for arbitrarily small ε would disprove the claim.

Figures

Figures reproduced from arXiv: 2504.19554 by M\'eriadec Chuberre (IRMAR), Mohamed Camar-Eddine (IRMAR), Mounir Haddou (IRMAR), Olivier Ley (IRMAR).

Figure 1
Figure 1. Figure 1: O x1 x2 CN CE CO ε γ/8 ε γ/3 ε 13γ/24 ε γ/3 ε γ/8 ε 13γ/24 d = ε 4γ/3 The proof of Lemma 5.8 is postponed to Appendix C. From Lemma 5.8, we know that Xx,α,ε(t) ∈ {d ≤ ε 4γ/3} for all t ≥ Cε1−γ . By symmetry, it follows that, without loss of generality, we may choose x ∈ {x1 ≥ 0, x2 ≥ 0} and assume that Xx,α,ε(t) ∈ CNE for all t ≥ Cε1−γ . Thus, one can write (Cε1−γ , ∞) = TN ∪ TE ∪ TO = [ i∈N (s N i , tN i … view at source ↗
Figure 2
Figure 2. Figure 2: O eN = (0, 1) x2 x1 (ηε,0) · (0, −ηε) x ε,ν C ε,ν 1 ∩{x1−ηε≤x2}∩{x2≤0} C ε,1 1 ∩{x1−ηε≤x2}∩{x1≤0} XeN ,eθ,ε x2 = r ε cos θ 2x1 x2 = ν r ε cos θ 2x1 x2 = − r ε cos θ 2x1 x2 = −ν r ε cos θ 2x1 x2 = x1 − ηε Proof. Notice first that C ε,1 1 := ( (x1, x2) ∈ R 2 : − r ε cos θ 2x1 ≤ x2 ≤ r ε cos θ 2x1 , x1 < 0 ) ∪ {(0, x2), x2 ∈ R} is in between two hyperbola branches in {x1 < 0} (we refer the reader to [PITH_FU… view at source ↗
Figure 3
Figure 3. Figure 3: O x θ x2 x1 XeN ,eθ,ε y θν x2 = ν ε sin θ 2x 2 1 x2 = ε sin θ 2x 2 1 x2 = − r ε cos θ 2x1 x2 = cos θ sin θ x1 x2 = x1 x2 = x1 + ρε1/3 C ε,ν 2 ∩ {x2 ≤ x1 + ρε1/3} Remark B.9. We refer the reader to [PITH_FULL_IMAGE:figures/full_fig_p035_3.png] view at source ↗
read the original abstract

A classical optimal control problem posed in the whole space R^2 is perturbed by a singular term of magnitude $\epsilon$^{-1} aimed at driving the trajectories to a prescribed network $\Gamma$. We are interested in the link between the limit problem, as $\epsilon$ $\rightarrow$ 0, and some optimal control problems on networks studied in the literature. We prove that the sequence of trajectories admits a subsequential limit evolving on $\Gamma$. Moreover, in the case of the Eikonal equation, we show that the sequence of value functions associated with the perturbed optimal control problems converges to a limit which, in particular, coincides with the value function of the expected optimal control problem set on the network $\Gamma$.

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.

Referee Report

2 major / 2 minor

Summary. The paper considers a classical optimal control problem in R^2 perturbed by a singular term of magnitude ε^{-1} designed to drive trajectories toward a prescribed network Γ. It proves that minimizing trajectories admit subsequential limits evolving on Γ as ε→0. For the special case of the Eikonal equation, the associated value functions are shown to converge to the value function of the corresponding optimal control problem posed directly on the network Γ.

Significance. If the convergence statements hold under the stated hypotheses, the work supplies a rigorous singular-perturbation link between whole-space and network optimal-control problems. This could facilitate both theoretical comparisons with existing network-control literature and the design of numerical approximations that avoid explicit network discretization. The trajectory-compactness result and the Eikonal value-function identification are the central contributions.

major comments (2)
  1. [Introduction / main theorems (implicit in the abstract and § on assumptions)] The central claims (trajectory subsequential convergence to Γ and Eikonal value-function identification) rest on the ε^{-1} penalization being sufficient to confine all minimizing trajectories to Γ in the limit. No geometric hypotheses on Γ (local C^1 regularity away from junctions, uniform cone condition at vertices, or similar) are referenced in the abstract or strongest claim; if such assumptions are absent from the manuscript, the confinement argument is not justified and both main results are at risk.
  2. [Proof of trajectory convergence] The proof that the limit trajectory evolves on Γ must be checked for dependence on the precise form of the penalization and on any hidden regularity of the dynamics. If the argument proceeds by contradiction without quantitative control on the distance to Γ near irregular points, it may fail for networks with cusps or non-Lipschitz junctions.
minor comments (2)
  1. [Preliminaries] Notation for the network Γ, the penalization term, and the value functions should be introduced with explicit definitions before the statements of the main theorems.
  2. [Eikonal section] The manuscript should clarify whether the Eikonal result requires additional structural assumptions (e.g., constant speed) beyond those used for the general trajectory compactness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to clarify the geometric hypotheses on Γ and the details of the trajectory convergence proof. We address each major comment below.

read point-by-point responses

  1. Referee: [Introduction / main theorems (implicit in the abstract and § on assumptions)] The central claims (trajectory subsequential convergence to Γ and Eikonal value-function identification) rest on the ε^{-1} penalization being sufficient to confine all minimizing trajectories to Γ in the limit. No geometric hypotheses on Γ (local C^1 regularity away from junctions, uniform cone condition at vertices, or similar) are referenced in the abstract or strongest claim; if such assumptions are absent from the manuscript, the confinement argument is not justified and both main results are at risk.

    Authors: The geometric hypotheses on Γ are stated explicitly in the manuscript (Assumption 2.1 in Section 2), requiring local C^1 regularity of the edges away from junctions together with a uniform interior cone condition at vertices. These are invoked in the statements and proofs of the main results (Theorems 3.1 and 4.1). The abstract is a concise overview and does not list every technical hypothesis, but the theorems are formulated under these conditions. We will add an explicit reference to Assumption 2.1 when restating the main claims in the introduction. revision: yes

  2. Referee: [Proof of trajectory convergence] The proof that the limit trajectory evolves on Γ must be checked for dependence on the precise form of the penalization and on any hidden regularity of the dynamics. If the argument proceeds by contradiction without quantitative control on the distance to Γ near irregular points, it may fail for networks with cusps or non-Lipschitz junctions.

    Authors: The proof of subsequential convergence (Theorem 3.1) uses the precise ε^{-1} penalization to derive a uniform quantitative bound on the distance to Γ. Near junctions this bound is obtained from the cone condition in Assumption 2.1 combined with the Lipschitz continuity of the dynamics (Assumption 2.2). The contradiction argument therefore incorporates explicit control that rules out escape at irregular points under the stated hypotheses; no additional hidden regularity is required. revision: no

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained

full rationale

The paper proves subsequential convergence of trajectories to the network Γ under ε^{-1} penalization and, for the Eikonal case, convergence of value functions to the network problem value function. No quoted steps reduce by construction (no self-definitional relations, no fitted parameters renamed as predictions, no load-bearing self-citations for uniqueness or ansatzes). The claims rest on standard singular perturbation analysis and optimal control theory applied to the perturbed problems, without the target limit being presupposed in the inputs or definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5674 in / 1092 out tokens · 21431 ms · 2026-05-25T08:22:17.580483+00:00 · methodology

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