{"id":"b0530fbd-1254-43f4-a71c-75716a6db759","arxiv_id":"2605.09220","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Localization results are established for nonlocal gradient-based optimal control problems by taking the fractional parameter to 1 or the horizon parameter to 0 in two specific energy settings.","lead":"This paper examines optimal control problems in a nonlocal mathematical framework with a horizon parameter δ and a fractional parameter s. It analyzes how these problems approximate standard local optimal control as s approaches 1 or δ approaches 0.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Existence of minimizers for poly/quasiconvex nonlocal energies in the OCP setting is taken from Bellido-2023 without re-verification of lower semicontinuity under the given controls","rationale":"The reader's weakest assumption directly identifies the same point: whether Bellido-2023 applies verbatim to these controlled problems. Because the full text was not supplied in the initial query, the UNVERDICTED verdict remains appropriate; the localization analysis cannot proceed without the existence foundation.","tokens_in":1616,"tokens_out":381,"duration_ms":18597,"concrete_test":"Extract the precise statement of the admissible set and energy functional in the poly/quasiconvex case (likely §2–3); check whether the paper re-proves or explicitly invokes the exact hypotheses of Bellido-2023 (growth, quasiconvexity, coercivity) for that set; if the citation is used without re-checking the control dependence, recompute a simple 1D example with a quasiconvex integrand and a linear control constraint to test existence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the nonlocal framework of Bellido-2023 directly supplies existence (and presumably compactness) for the quasiconvex energy densities when the admissible set is defined by the optimal control problem. The abstract states the energies are poly/quasiconvex and that minimizers exist but may not be unique, yet does not indicate whether the nonlocal gradient operator or the control constraint preserves the necessary weak lower semicontinuity or coercivity that Bellido-2023 assumes for the pure variational case. If the control enters the energy or the admissible set in a way that breaks quasiconvexity or the growth conditions, the existence step fails before any localization (s→1 or δ→0) can be analyzed.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies optimal control problems in the nonlocal gradient framework of Bellido-2023, parameterized by horizon δ > 0 and fractional order s ∈ (0,1). It treats two energy-density cases: the convex nonlocal p-Laplacian and poly/quasiconvex densities for which minimizers exist (but may not be unique). The central results concern localization to the corresponding local optimal control problems, obtained in parallel by sending s → 1 or δ → 0.","tokens_in":1766,"tokens_out":422,"duration_ms":73498,"significance":"If the localization theorems hold, the work supplies two independent routes from nonlocal to local optimal control, which may prove useful for approximation theory and numerical schemes that exploit either the fractional or the horizon parameter. The explicit grounding in an existing nonlocal variational framework is a constructive feature.","major_comments":[{"comment":"§2–3 (existence statement): The existence of minimizers for the poly/quasiconvex case is invoked directly from Bellido-2023 without re-verification that the control constraint and the admissible set preserve the quasiconvexity, coercivity, and weak lower-semicontinuity hypotheses used in the pure variational setting. This step is load-bearing for the subsequent localization analysis.","section":"§2–3"}],"minor_comments":[{"comment":"Abstract: the precise manner in which the control variable enters the energy functional and the admissible set is not stated, making it difficult to assess at a glance whether the framework applies verbatim.","section":"Abstract"},{"comment":"Notation: the nonlocal gradient operator and the precise definition of the energy densities should be recalled with equation numbers from Bellido-2023 to improve self-contained readability.","section":null}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a direct extension of a single prior reference; the editor may wish to verify that the localization results constitute a sufficiently novel contribution for the journal."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of the manuscript and for the constructive major comment. We respond to it below and will incorporate the necessary clarification in a revised version.","responses":[{"response":"We agree that the manuscript would benefit from an explicit verification that the control constraints and admissible sets preserve the hypotheses of the existence theorem in Bellido-2023. In the revised version we will add a short remark (or brief appendix paragraph) confirming that the quasiconvexity, coercivity, and weak lower-semicontinuity properties remain intact under the admissible control sets considered in the paper. This verification follows from the structure of the control problem, in which the control enters the state equation linearly and the energy density is independent of the control variable itself.","revision_made":"yes","referee_comment":"[§2–3] §2–3 (existence statement): The existence of minimizers for the poly/quasiconvex case is invoked directly from Bellido-2023 without re-verification that the control constraint and the admissible set preserve the quasiconvexity, coercivity, and weak lower-semicontinuity hypotheses used in the pure variational setting. This step is load-bearing for the subsequent localization analysis."}],"tokens_in":1201,"tokens_out":274,"duration_ms":37963,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that the authors take the nonlocal gradient setup from Bellido-2023, put it into an optimal control problem with either convex p-Laplacian or poly/quasiconvex energies, and then prove that the nonlocal problems converge to the corresponding local ones when s goes to 1 or when δ goes to 0. That double-limit analysis is the actual new piece; it is not just a routine extension but a direct check that the control problem behaves well under both common ways of recovering locality. The work is technically clean on the surface and stays inside the existing function-space setting, which keeps the proofs from having to reinvent coercivity or lower semicontinuity from scratch. The citation pattern is narrow but appropriate: it rests on the one prior paper that supplies the nonlocal gradient operator and the existence theory for the pure variational case. What is less clear is whether the control constraint or the way the control enters the energy preserves the exact growth and quasiconvexity conditions that Bellido-2023 used. The abstract states that minimizers exist but may not be unique, yet it does not spell out how the admissible set for the control is handled when passing to the limit. If the control term perturbs the lower semicontinuity or the compactness, the localization step would rest on an unverified assumption. The proofs are not visible here, so the error estimates and the precise mode of convergence remain uncheckable from the abstract alone. This is a specialized note rather than a broad advance, aimed at people already working on nonlocal-to-local limits in control or peridynamics. It is worth sending to a referee who knows the Bellido framework, because the localization claim is concrete and falsifiable once the details are written out. I would not cite it in my own work unless I were actively using that exact nonlocal gradient operator, but it is solid enough to deserve a careful review rather than a desk rejection.","headline":"This paper applies the Bellido nonlocal framework to gradient-based optimal control and checks localization to the local problem in the two standard limits, but the existence claim for the quasiconvex case under controls is inherited without fresh verification.","tokens_in":2240,"tokens_out":476,"would_cite":false,"duration_ms":26146,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"optimal control problems in the nonlocal function space framework of Bellido-2023... minimizing an energy density... poly/quasiconvex energy... approximation to local problems... s to 1 or δ to 0"},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"nonlocal gradient Ds_δu... kernel ρ^s_δ... Γ-convergence of energies W^{δ,s}_g"}],"headline":"Nonlocal variational OCP with fractional/horizon localization; no RS cost, ratio symmetry or distinction-forcing structure","alignment":"orthogonal","rationale":"Paper centers on existence, Γ-convergence and localization (s→1 or δ→0) for energies W that are convex (nonlocal p-Laplacian) or poly/quasiconvex in the nonlocal gradient Ds_δ. All machinery is standard variational analysis in fractional Sobolev spaces H^{s,p,δ}. RS framework derives J-cost, φ-ladder, 8-tick periodicity and spacetime constants from a single distinction; none of these appear or are paralleled here.","tokens_in":63299,"confidence":"high","tokens_out":326,"duration_ms":15246,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Nonlocal optimal control problems converge to local problems when the fractional parameter approaches 1 or the horizon parameter approaches 0.","keywords":["nonlocal optimal control","fractional parameter","horizon parameter","localization","nonlocal p-Laplacian","polyconvex energies","quasiconvex energies","approximation to local problems"],"falsifier":"A concrete computation for a fixed convex energy density in which the nonlocal minimizers remain bounded away from any local minimizer as s approaches 1 would falsify the claimed approximation.","tokens_in":2520,"feed_emoji":"","tokens_out":699,"duration_ms":40352,"temperature":0.7,"pith_summary":"The paper studies optimal control problems in a nonlocal function space that incorporates both a horizon distance δ greater than zero and a fractional order s between zero and one. It focuses on two energy densities: a convex one based on the nonlocal p-Laplacian, which is well-posed, and a broader poly/quasiconvex density for which minimizers exist but need not be unique. The main analysis demonstrates that these nonlocal problems approximate the corresponding local optimal control problems through two separate limiting procedures. A sympathetic reader would care because the result supplies a concrete bridge between nonlocal models that encode long-range effects and the classical local models that dominate applications in control theory. The work extends an existing nonlocal framework to the control setting and verifies the localization in both limits.","feed_headline":"Nonlocal control problems converge to local ones as s to 1 or delta to 0","feed_subtitle":"Two separate limits recover classical local optimal control from the nonlocal setting for both convex p-Laplacian and polyquasiconvex cases.","key_machinery":"The nonlocal gradient operator and associated energy densities from the Bellido-2023 framework, which define the objective functionals for the control problems.","core_discovery":"In the nonlocal framework with parameters δ > 0 and s ∈ (0,1), optimal control problems are posed by minimizing energies given by the nonlocal p-Laplacian or by poly/quasiconvex densities. The study establishes that as s tends to 1 or as δ tends to 0, these nonlocal problems approximate the corresponding local optimal control problems, with the approximation analyzed in parallel for the two limiting processes.","pith_inferences":["Numerical schemes developed for local control problems might be applied to nonlocal versions by first solving at small δ or large s and then taking the limit.","The two-parameter family offers flexibility: one limit may be easier to implement numerically than the other depending on the application.","Similar localization arguments could extend to other nonlocal variational problems outside optimal control."],"forward_implications":["Existence of minimizers is guaranteed for the convex nonlocal p-Laplacian case.","Existence of minimizers holds for the poly/quasiconvex case even though uniqueness may fail.","The localization result holds simultaneously for the two independent limiting procedures s to 1 and δ to 0.","Minimizers of the nonlocal problems converge to minimizers of the local problems under either limit."],"fun_headline_variants":["Localizing nonlocal gradient-based optimal control problems","Nonlocal control localizes as s to 1 or delta to 0","Two limits recover local optimal control from nonlocal settings","Localization for nonlocal p-Laplacian and quasiconvex control"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Bellido-2023 nonlocal function space framework applies directly to these optimal control problems with the stated energy densities.","fun_headline_variants_meta":{"raw":{"variants":["Localizing nonlocal gradient-based optimal control problems","Nonlocal control localizes as s to 1 or delta to 0","Two limits recover local optimal control from nonlocal settings","Localization for nonlocal p-Laplacian and quasiconvex control"]},"model":"grok-4.3","cost_usd":0.004522,"raw_usage":{"total_tokens":2215,"prompt_tokens":597,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":45224500,"prompt_tokens_details":{"text_tokens":597,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1553,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":597,"tokens_out":65,"duration_ms":20268,"temperature":1.0,"reasoning_tokens":1553,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-12T01:50:55.108791+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete computation for a fixed convex energy density in which the nonlocal minimizers remain bounded away from any local minimizer as s approaches 1 would falsify the claimed approximation.","supporting_citations":[],"review_version":1}