{"id":"94f94fd9-43d0-4f3b-b52b-14e85f1a2b78","arxiv_id":"2607.21822","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Nonlocal optimal control problems with ε-quasi-minimization constraints have solutions that converge, as s→1⁻ or δ→0⁺, to solutions of a local optimal-control problem.","lead":"This paper proves existence and convergence results for nonlocal optimal-control problems in which the state is only required to be a near-minimizer (up to tolerance ε) of a quasiconvex energy, rather than an exact minimizer. Using this relaxation, the author shows that as the nonlocal/fractional parameters vanish, solutions converge to a local PDE-constrained optimal-control limit—a convergence that was unattainable for the exact-minimizer problems studied earlier.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Missing boundedness of the admissible control set: Assumption 2.13 does not bound Z_ad, yet Lemmas 3.1/4.8 and Theorem 4.10 rely on uniform boundedness of controls to extract weakly convergent subsequences; without it the weak-limit control g need not exist and (4.19) collapses.","rationale":"The reader identified (4.19) as the weakest assumption. I agree that (4.19) is asserted without proof, but the more load-bearing gap is the missing boundedness of controls. Theorem 4.10 first needs a weak limit g of g_{δ,s}; this is not supplied by Assumption 2.13. Moreover, equicoercivity of W^{δ,s}_{g_{δ,s}} needed for the cited [22, Cor. 7.20] also requires uniform L^{p'}-boundedness of the controls. Once that boundedness is added, the moving-control Γ-convergence is obtainable by combining Theorem 4.7 with the convergences g_s⇀g and u_s→u in the linear term, so (4.19) is a corollary-level repair. Thus the central idea appears sound, but the written proof has a genuine structural omission in its compactness argument. This matches the reader's CONDITIONAL verdict: not rejectable, but requiring revision. I set agreement to 'partial' because the reader's rationale noted the false boundedness assertion in Theorem 3.4 but did not make it the primary weakness; I believe it is more fundamental than (4.19).","tokens_in":19757,"tokens_out":21765,"duration_ms":207609,"concrete_test":"Test Lemma 3.1 with an unbounded Z_ad: take p=2, Ω=(0,1), W(x,z,A)=|A|^2, s=1/2, δ=1, Z_ad=L^2(Ω), and g_k=k on Ω. Let u_k be an exact minimizer of W^{1/2,1}_{g_k} in H^{1/2,2}_0(Ω_{−1}). Compute or estimate ||D^{1/2}_1 u_k||_{L^2}. In the local analogue the minimizer is u_k=(k/2)φ with −φ''=1, giving ||∇u_k||_2∼k; if the nonlocal norm grows similarly, Lemma 3.1 is false as stated. Then confirm that adding boundedness of Z_ad, or deriving sup_s ||g_{δ,s}||_{p'}<∞ from the Λ|g|^{p'} cost term, is enough to make Lemma 4.8 and (4.19) valid; if so, the theorem is salvageable and the correct verdict remains conditional.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Assumption 2.13 only requires Z_ad to be closed, nonempty, and convex in L^{p'}(Ω;R^n); it does not require boundedness. Lemma 3.1 nevertheless claims V^{δ,s}_ε is bounded for all g∈Z_ad. This is false in general: with p=2, W(x,z,A)=|A|^2, Z_ad=L^2(Ω), and g_k=k, exact minimizers u_k of W^{δ,s}_{g_k} satisfy ||D^s_δ u_k||_{L^p}→∞, so the image of T^{δ,s}_ε is unbounded. The proof of Lemma 3.1 omits the linear term −⟨g,u⟩; the bound W_g(u)≤C_w|Ω|+ε does not control ||D^s_δ u||_p unless ||g||_{p'} is known to be bounded. Lemma 4.8 repeats this defect. Consequently, in Theorem 4.10 the passage to a weak limit g∈Z_ad by 'reflexivity and weak closedness' is unjustified: without sup_s ||g_{δ,s}||_{p'}<∞, there may be no convergent subsequence. This is load-bearing because (4.18)–(4.20) require a limiting control g; if g cannot be extracted, the admissibility and optimality of the limit pair collapse. The gap is repairable by adding boundedness of Z_ad (e.g., box constraints as in Example 2.14) or by proving from the Λ|g|^{p'} term that solutions have uniformly bounded controls, but the present text asserts rather than proves either.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies nonlocal optimal control problems with constraints given by epsilon-quasi-minimizers of quasiconvex nonlocal energies. It proves existence of solutions for fixed s and delta (Problem 2.18), and then establishes Gamma-convergence-based localization results as s -> 1^- and delta -> 0^+: solutions converge along subsequences to solutions of a local PDE-constrained problem with epsilon-quasi-minimization constraint (Theorems 4.10 and 4.20). The key novelty claimed is the use of epsilon-relaxed minimality to overcome the obstacle identified in prior work [20] for global-minimizer constraints, together with a recovery sequence constructed via convex combinations of the candidate limit and approximate minimizers.","tokens_in":20134,"tokens_out":2838,"duration_ms":30513,"significance":"If the convergence theorems are correct, the paper makes a substantive advance: it provides a general framework for localizing a class of ill-posed nonlocal control problems with non-unique constraints, extending the earlier partial results of [20] to a broader class of cost functionals and to the quasi-minimization setting. The use of Gamma-convergence with a carefully designed recovery sequence is a natural and potentially reusable technique. The paper also states all assumptions explicitly and gives concrete examples (box constraints, desired-state and compliance costs), which aids reproducibility. However, the proof of the central convergence claim relies on two unproved compactness and commutation steps; these are load-bearing and need to be addressed before the theorems can be considered established.","major_comments":[{"comment":"Assumption 2.13 only postulates closedness, nonemptiness, and convexity of Z_ad in L^{p'}(Omega;R^n); no boundedness is imposed. Lemma 3.1 nevertheless asserts that the image V^{delta,s}_epsilon is bounded. The proof omits the linear term -<g,u> in (2.8). From (2.7) and the epsilon-quasi-minimality one gets only c_w ||D^s_delta u||_p^p <= c0|Omega| + C_w|Omega| + epsilon + <g,u>. Without a uniform bound on ||g||_{p'} this does not yield boundedness; with p=2, Z_ad = L^2, and g_k = k, exact minimizers have ||D^s_delta u_k||_2 -> infinity. The same defect propagates to Lemma 3.2 and to Theorem 3.4, where the sentence 'Z_ad, which is itself a bounded subset of L^{p'}' is asserted without support. It also invalidates Lemma 4.8 and the subsequence extraction of g in Theorem 4.10. This is repairable by adding an explicit boundedness assumption on Z_ad (e.g., box constraints as in Example 2.14)","section":"Assumption 2.13, Lemma 3.1, Theorem 3.4"},{"comment":"Equation (4.19) asserts that lim_s min_v W^{delta,s}_{g_{delta,s}}(v) = min_v W^{loc}_g(v) for the weakly convergent family of controls g_{delta,s} -> g. The paper only proves Gamma-convergence for a fixed control g (Theorem 4.7). Applying [22, Corollary 7.20] to functionals with moving linear terms requires a joint Gamma-convergence or an equi-coercivity plus convergence of the linear perturbations in the appropriate topology. That point is not established. When g_{delta,s} is not strongly convergent, the linear term -<g,u> need not pass to the limit along arbitrary recovery sequences, and the advertised equality of minima is exactly what needs proof. This step is load-bearing for admissibility of the limit pair (4.20) and for the optimality argument.","section":"Theorem 4.10, Eq. (4.19)"},{"comment":"The local limit space is defined as W^{1,p}_0(tilde(Omega);R^n), with tilde(Omega)=Omega_{-delta} in Subsection 4.2 and tilde(Omega)=Omega in Subsection 4.3. However, in the proof of Theorem 4.10, lines (4.20) and (4.22) write W^{1,p}_0(Omega;R^n) while the statement and the local problem use W^{1,p}_0(tilde(Omega);R^n). In the s->1^- case these do not agree, since Omega_{-delta} is strictly smaller than Omega. This inconsistency leaves ambiguous which boundary conditions are meant and whether the compactness result [18, Lemma 9] actually yields a limit in W^{1,p}_0(Omega_{-delta}). The same issue appears in the delta->0^+ analogue. The notation should be cleaned up and every occurrence matched to the domain of the local problem.","section":"Theorem 4.10 and Section 4.1 (domain inconsistency)"},{"comment":"The lim-inf inequalities for the nonlocal cost integrand are asserted with reference to [21, Theorem 8.11] and the translation argument of [18]. This is plausible, but the proof as written is only a sketch: it states that the argument of [18] can be 'repeated' with u-dependence, without verifying that the translation map used in [18] is compatible with the u-dependence of F under the assumed strong L^p convergence of u_s. In particular, the translation changes the spatial argument of u, and the Caratheodory regularity plus growth (2.14) must be shown to allow that interchange. Since Lemma 4.9 is used in the energy-convergence part of Theorem 4.10, a complete proof (or a precise citation of a theorem that covers u-dependent integrands) is needed.","section":"Lemma 4.9 and Lemma 4.19"}],"minor_comments":[{"comment":"Minor wording: 'we get stronger convergence results' and 'main convergence results establish a novel analytic technique' would benefit from concrete quantification of what 'strong' means (state convergence, control convergence, cost convergence). Also, the abstract says 'a fractional parameters' (typo, 'parameter').","section":"Abstract and Introduction"},{"comment":"The statements say the family {W^{delta,s}_g}_{s<1} Gamma-converges to W^{loc}_g, but the notation 'Wloc_g' is inconsistent with the earlier definition W^{loc}_g. Also, Theorem 4.17 states the Gamma-convergence as delta->0^+ but writes '{W^{delta,s}_g}_{s<1}' in the notation; the index should be delta.","section":"Theorem 4.7 and Theorem 4.17"},{"comment":"Remark 4.14 discusses the epsilon->0^+ limit but appears to use 'F^{loc}' where the nonlocal cost is intended at the end of the bullet list; also the claim that a direct verification proves optimality of (u_s,g_s) for the epsilon=0 problem is not detailed. This is a remark, not a theorem, so it is not blocking, but it is currently too cryptic.","section":"Remark 4.14"},{"comment":"The concluding remarks mention enforcing admissible pairs to be equilibrium points as future work, but do not mention the boundedness-of-controls issue or the moving-target Gamma-convergence step that the paper leaves open. Adding a sentence on the limitations and possible fixes would improve the manuscript.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is clearly a serious contribution in an active area and the core strategy is promising. However, the missing boundedness of Z_ad is not a cosmetic issue: it affects Lemma 3.1, Theorem 3.4, Lemma 4.8, and Theorem 4.10, and the proof as written asserts the needed compactness. The moving-target Gamma-convergence in (4.19) is likewise a real gap. I believe both are fixable within the manuscript's scope (e.g., by adding bounded box constraints or deriving control bounds from the cost, and by proving a joint Gamma-convergence lemma for sequences with weakly converging linear perturbations), so I lean major revision rather than rejection. The author should also carefully reconcile the two different domains tilde(Omega)=Omega_{-delta} vs Omega in the limit statements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nThe headline: this paper has a real new idea — replacing the global-minimizer constraint in the predecessor [20] with an ε-quasi-minimizer constraint — and it pays off. The relaxation lets the author prove convergence of minimizers for nonlocal optimal control problems to a local limit, with a general cost depending on the nonlocal gradient, which is exactly what [20] couldn't do. The convex-combination recovery sequence in Theorem 4.10 is a genuine technical innovation, and the paper is honest about where [20] left off.\n\nThe good parts deserve credit: the problem class is meaningful for peridynamics and nonlocal variational methods, the Γ-convergence blueprint is recognizable, and the self-citation to [20] is fair. No circularity.\n\nThe soft spots are real but likely repairable. Most important: Lemma 3.1 asserts that the set of admissible states V^{δ,s}_ε is bounded, but the proof drops the linear term −⟨g,u⟩. Under Assumption 2.13, Z_ad is only closed and convex — not bounded — so the claim is false as stated (example: p=2, Z_ad=L^2, g_k=k). This matters because Theorem 3.4 and Theorem 4.10 extract weakly convergent subsequences of controls. In Theorem 3.4 the Λ||g||^{p'} term in the cost can bound a minimizing sequence, so that theorem may be salvageable; in Theorem 4.10 you need a uniform control bound along the family, which neither the assumptions nor the proof provide. The stress-test note is right about this. Fix: add box constraints or prove the bound from the cost.\n\nSecond, equation (4.19) asserts convergence of the minimal energies for weakly varying controls g_{δ,s} ⇀ g. The paper only proves Γ-convergence for a fixed g. The result is plausible with bounded controls, but it needs proof, not a bare citation.\n\nThird, Theorem 4.10's optimality step is garbled: the same symbols (u,g) are reused for the limit pair and an arbitrary competitor, and the conclusion (4.29) is a tautology as written. Repairable, but currently confusing.\n\nMinor: the domain eΩ = Ω_{−δ} in the statement vs W^{1,p}_0(Ω) in the proof needs reconciling.\n\nWho this is for: people working on Γ-convergence for nonlocal variational problems and optimal control under nonunique constraints. It deserves a serious referee — the gaps are substantial but not fatal. I'd send it to review with a request for major revision.","headline":"A genuine new relaxation with a real payoff, but the proofs as written have load-bearing gaps — most importantly the missing boundedness of Z_ad — that need patching before the main theorems are solid.","tokens_in":20641,"tokens_out":6217,"would_cite":false,"duration_ms":58888,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J20","49J45","35R11","49K20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Nonlocal optimal control converges to the local problem once minimizer constraints are relaxed by a tolerance.","keywords":["nonlocal optimal control","epsilon-quasi-minimizers","Gamma-convergence","fractional gradient","vanishing horizon","quasiconvex energy","PDE-constrained optimization"],"falsifier":"Compute, for a one-dimensional double-well quasiconvex energy with controls g_delta oscillating between two forces that select different wells and converging weakly to a limit g, the difference between min W^{delta,s}_{g_delta} and min W^loc_g. If this difference does not tend to zero, or if the associated minimizers do not converge, then equation (4.19) fails and Theorem 4.10 collapses; if it tends to zero, the moving-target step is confirmed.","tokens_in":19562,"feed_emoji":"🎯","tokens_out":8060,"duration_ms":78405,"temperature":0.7,"pith_summary":"This paper tries to close a gap in the asymptotic theory of nonlocal optimal control. Earlier work constrained the state to be a global minimizer of a quasiconvex energy; because such energies can have many minimizers, no proof of convergence to the corresponding local, PDE-constrained problem was available. The paper instead imposes an epsilon-quasi-minimizer constraint — the state only needs to come within epsilon of the minimal energy — and shows that, for every fixed tolerance, solutions of the nonlocal problems converge, as the fractional order tends to 1 or the interaction horizon shrinks to 0, to solutions of the local problem. A sympathetic reader would care because this supplies a general Gamma-convergence route for parameterized control problems with nonunique constraints, and because the tolerance mirrors what a numerical solver actually enforces.","feed_headline":"Tolerance-based constraints make nonlocal controls converge","feed_subtitle":"Replacing hard minimizer constraints by epsilon-quasi-minimizer constraints closes the gap to local PDE-constrained problems.","key_machinery":"The central object is the admissible set T^{delta,s}_epsilon = {(u,g): W^{delta,s}_g(u) - m_{g,delta,s} <= epsilon}, where W^{delta,s}_g(u) = integral over Omega of W(x,u,D^s_delta u) dx - <g,u> is a quasiconvex nonlocal energy with a linear control term, and m_{g,delta,s} is its minimal value. The epsilon-quasi-minimizer inequality acts as a relaxed constraint that keeps admissible states bounded, weakly closed, and compatible with Gamma-convergence of W^{delta,s}_g to the local functional W^loc_g. Two auxiliary tools carry the argument: compactness lemmas that convert bounded nonlocal gradients into weak convergence of D^s_delta u to nabla u as s -> 1^- or delta -> 0^+, and a recovery sequ","core_discovery":"On its own terms, the paper establishes two convergence theorems. For fixed horizon delta > 0, every solution (u^{delta,s}, g^{delta,s}) of the nonlocal epsilon-quasi-minimizer problem has a subsequence with u^{delta,s} -> u strongly in L^p and g^{delta,s} -> g weakly in L^{p'}, where (u,g) solves the local optimal control problem with the same tolerance epsilon, and the values of the cost functionals converge. The identical statement holds as the horizon shrinks to 0 with s fixed. The proof works through Gamma-convergence of the nonlocal energy functionals to the local one; the central step is proving that the limit pair is admissible, which is handled by constructing a recovery sequence fr","pith_inferences":["Editorial extension: the interpolation step (u_t = t u + (1-t) v_s) looks like a general regularization device — any family of constraints defined by Gamma-convergent, equicoercive energies with nonunique minimizers can likely be handled by adding an epsilon-band around the minimizer set.","Editorial extension: a decisive test is to compute, in a one-dimensional double-well energy with controls g_delta oscillating between the two wells and converging weakly to g, whether min W^{delta,s}_{g_delta} - min W^loc_g tends to zero; this would confirm or refute the moving-target Gamma-convergence step behind equation (4.19).","Editorial extension: the stated limit domain shifts between Omega_{-delta} and Omega in the s -> 1^- theorem, and reconciling this inconsistency is necessary before the result can be used in numerical schemes on shrinking domains.","Editorial extension: if the moving-target energy convergence fails for some controls, a natural repair is to impose a compactness or equicontinuity condition on the family of controls, or to upgrade weak convergence of controls to a stronger mode in the admissible-set characterization."],"forward_implications":["For every fixed tolerance epsilon > 0, the nonlocal optimal control problem is asymptotically compatible: any limit point of solutions solves the local problem, so fractional or long-range models can be replaced by classical PDE constraints in the limit.","The result applies to cost functionals that depend explicitly on the nonlocal gradient, not just to compliance-type costs, widening the class of tracking and design problems that can be localized.","The proof gives an alternative existence route for the local problem: existence in the nonlocal family plus compactness produces a local solution.","The controls converge strongly in every L^r space, not merely weakly, so the recovered control is a genuine limit object rather than an abstract equivalence class.","The tolerance is essential: the paper states explicitly that the argument cannot be pushed to epsilon = 0, so the global-minimizer problem remains open."],"fun_headline_variants":["Epsilon-quasi-minimizers drive nonlocal controls to local limit","Relaxed minimizer constraints fix nonlocal control convergence","Quasi-minimizers close gap between nonlocal and local controls","Tolerance parameter enables convergence in nonlocal optimal control","Gamma-convergence: nonlocal controls converge with quasi-minimizers"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the minimal values of the nonlocal energies converge to the local minimal value even while the control g in the linear term is itself varying and only weakly convergent — the moving-target version of Gamma-convergence — and the paper proves only the fixed-control version, while the stated limit domain shifts between Omega_{-delta} and Omega.","fun_headline_variants_meta":{"raw":{"variants":["Epsilon-quasi-minimizers drive nonlocal controls to local limit","Relaxed minimizer constraints fix nonlocal control convergence","Quasi-minimizers close gap between nonlocal and local controls","Tolerance parameter enables convergence in nonlocal optimal control","Gamma-convergence: nonlocal controls converge with quasi-minimizers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000724,"raw_usage":{"total_tokens":3087,"prompt_tokens":749,"completion_tokens":2338,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":2250}},"tokens_in":493,"tokens_out":2338,"duration_ms":17589,"temperature":1.0,"reasoning_tokens":2250,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T06:34:41.904408+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a one-dimensional double-well quasiconvex energy with controls g_delta oscillating between two forces that select different wells and converging weakly to a limit g, the difference between min W^{delta,s}_{g_delta} and min W^loc_g. If this difference does not tend to zero, or if the associated minimizers do not converge, then equation (4.19) fails and Theorem 4.10 collapses; if it tends to zero, the moving-target step is confirmed.","supporting_citations":[],"review_version":1}