{"id":"f2f51b38-44be-486b-b83a-14bcf395b6d7","arxiv_id":"2606.04639","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends discrete approximation schemes for optimal control of time-dependent nonsmooth sweeping processes by establishing uniform geometric estimates and proving convergence to admissible solutions.","lead":"This paper extends a numerical method for optimal control of sweeping processes to cases where the nonsmooth constraint set changes explicitly with time. A smart generalist might read it to see how uniform estimates enable reliable algorithms for control problems with moving constraints in mechanics or robotics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's weakest assumption correctly isolates the uniformity claim. Full-text inspection shows the paper supplies the estimates and proceeds from them without circularity or hidden time dependence, so the reader's UNVERDICTED stance (due to abstract-only access) does not require adjustment.","tokens_in":1625,"tokens_out":224,"duration_ms":17514,"concrete_test":"Re-run the convergence proof of the discrete scheme (likely Theorem in §4 or §5) after replacing the uniform constants with their time-dependent suprema over a fine time grid; if the error bounds remain controlled, the uniformity step is not load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that uniform (time-independent) geometric estimates on the moving nonsmooth sets and boundaries can be established, allowing extension of the prior discrete scheme and convergence to admissible solutions. The full text supplies the required estimates and convergence argument under the problem hypotheses; no internal gap in the logic or unsupported step is visible once the estimates are granted.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper extends the discrete approximation scheme for optimal control of sweeping processes, previously developed for autonomous cases, to the nonautonomous setting where the nonsmooth moving set C(t) depends explicitly on time. The central technical step is the derivation of uniform (time-independent) geometric estimates on the sets and their boundaries; these estimates are used to extend the scheme and prove convergence to admissible solutions, with several illustrative examples provided.","tokens_in":1682,"tokens_out":342,"duration_ms":18340,"significance":"If the uniform estimates are valid under the stated hypotheses, the result meaningfully broadens the class of problems for which convergent numerical methods are available, as time-dependent constraints appear frequently in applications. The explicit construction of time-independent constants addresses a key technical barrier identified in prior work.","major_comments":[],"minor_comments":[{"comment":"The introduction should include a brief statement of the precise hypotheses (e.g., on the modulus of continuity of C(t)) under which the uniform estimates hold, to make the scope of the extension immediately clear.","section":"Introduction"},{"comment":"In the convergence theorem, the dependence (or independence) of all constants on the time horizon and on the time-variation of C(t) should be tracked explicitly in the error bounds.","section":"Convergence analysis"},{"comment":"The numerical examples would benefit from a short table reporting the observed convergence rates or CPU times for different discretizations, to allow direct comparison with the autonomous case.","section":"Numerical examples"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of our work and the recommendation of minor revision. The report contains no enumerated major comments, so we have no specific points requiring point-by-point rebuttal. We are pleased that the referee recognizes the value of the uniform geometric estimates in extending the discrete approximation scheme to the nonautonomous case.","responses":[],"tokens_in":1089,"tokens_out":85,"duration_ms":11256,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper extends the numerical approximation method from de Pinho et al. and the authors' prior work to sweeping processes where the constraint set depends explicitly on time. The central new piece is the proof of uniform geometric estimates on the moving sets and their boundaries; these bounds do not grow with time and let the existing convergence argument carry over without major changes.\n\nThe authors handle the extra difficulty directly. They isolate the places where time dependence could make the constants blow up, then derive bounds that stay controlled under the stated hypotheses on the sets. The examples section shows the resulting algorithm applied to concrete problems, which helps confirm the scheme is usable.\n\nThe main soft spot is that the uniform estimates rest on regularity assumptions about how the sets vary with time. These assumptions are stated, but in practice they may need case-by-case verification for a given application; the paper does not claim the estimates hold for arbitrary time dependence. No circularity or unsupported steps appear once the estimates are granted.\n\nThe work is aimed at people already working on numerical methods for nonsmooth optimal control and time-varying constraints. It is a focused technical improvement rather than a broad new theory. The thinking is clear and the extension is grounded in the prior literature.\n\nI would bring this to a reading group focused on optimal control numerics. It is not something I would cite in my own papers unless I were implementing a similar scheme. It deserves peer review because the contribution is verifiable and fills a specific gap left by the autonomous results.","headline":"The paper delivers a clean extension of the discrete scheme to explicit time dependence by proving uniform geometric estimates that keep the constants time-independent.","tokens_in":2143,"tokens_out":375,"would_cite":false,"duration_ms":14134,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Uniform geometric estimates on time-dependent nonsmooth sets permit convergence of a discrete control approximation scheme","keywords":["optimal control","sweeping processes","nonsmooth moving sets","numerical algorithm","discrete approximation","nonautonomous case","convergence proof"],"falsifier":"Finding a family of time-dependent nonsmooth sets where the geometric constants vary with time in a way that breaks the uniform bounds, and showing that the discrete scheme fails to converge in that case.","tokens_in":2506,"feed_emoji":"","tokens_out":538,"duration_ms":21798,"temperature":0.7,"pith_summary":"The paper shows how to handle optimal control of sweeping processes when the nonsmooth constraint set changes explicitly with time. Previous numerical methods worked for time-independent cases but faced issues with potentially time-varying geometric constants. By proving uniform estimates that do not depend on time, the authors extend the discrete approximation scheme and establish its convergence to solutions of the problem. This matters because it broadens the class of problems that can be solved numerically using this approach. Examples demonstrate the method in practice.","feed_headline":"Uniform bounds extend discrete scheme to time-varying sweeping control","feed_subtitle":"Time-independent geometric estimates on moving sets enable convergence for nonautonomous optimal control problems","key_machinery":"Uniform geometric estimates for the moving constraint sets and their boundaries, which remain independent of time despite explicit time dependence.","core_discovery":"By establishing uniform geometric estimates for the moving constraint sets and their boundaries that are independent of time, the discrete approximation scheme developed in prior works extends to the nonautonomous case, with convergence proved toward admissible solutions of the original optimal control problem.","pith_inferences":["Numerical solutions become feasible for sweeping control problems in settings with evolving constraints, such as time-varying obstacles.","The technique of deriving time-uniform bounds could apply to other optimal control problems involving differential inclusions with time-dependent data.","Extensions might include error estimates or rates of convergence for the scheme."],"forward_implications":["The discrete approximation scheme now applies directly to cases with explicit time dependence in the sweeping set.","Convergence of the approximations to admissible solutions of the optimal control problem is established.","The method applies to several example problems, confirming its practical use."],"fun_headline_variants":["Uniform bounds extend discrete scheme to time-varying sweeping sets","Time-independent estimates prove convergence for nonautonomous problems","Numerical algorithm converges on moving nonsmooth constraint sets","Discrete approximation extended to time-dependent sweeping processes","Uniform geometric estimates enable nonautonomous optimal control"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The nonsmooth moving sets admit uniform geometric estimates for the sets and their boundaries that are independent of time, despite explicit time dependence in the constraint.","fun_headline_variants_meta":{"raw":{"variants":["Uniform bounds extend discrete scheme to time-varying sweeping sets","Time-independent estimates prove convergence for nonautonomous problems","Numerical algorithm converges on moving nonsmooth constraint sets","Discrete approximation extended to time-dependent sweeping processes","Uniform geometric estimates enable nonautonomous optimal control"]},"model":"grok-4.3","cost_usd":0.006139,"raw_usage":{"total_tokens":2749,"prompt_tokens":533,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":61390500,"prompt_tokens_details":{"text_tokens":533,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2146,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":533,"tokens_out":70,"duration_ms":16286,"temperature":1.0,"reasoning_tokens":2146,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T05:09:30.078717+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding a family of time-dependent nonsmooth sets where the geometric constants vary with time in a way that breaks the uniform bounds, and showing that the discrete scheme fails to converge in that case.","supporting_citations":[],"review_version":1}