{"id":"39ea9e84-6f1c-4772-a598-4e07745e455f","arxiv_id":"2412.11320","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new equivalence between Moreau sweeping processes and extended projected dynamical systems enables accurate switch-detection discretization and efficient optimal control of constrained nonsmooth systems.","lead":"The paper shows that Moreau sweeping processes with moving constraints can be rewritten as extended projected dynamical systems, and uses this equivalence to build a high-accuracy discretization for optimal control. The practical upshot is a numerical method for optimizing systems that must stay inside time-varying sets, such as robots pushing objects through moving doorways.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's proof contains an internal sign inconsistency: the normal cone displayed in Section IV.A and the final equation of Theorem 4 have opposite signs to Lemma 3's DCS, so the claimed FOSwP-ePDS correspondence is not established as written.","rationale":"I agree with the reader's identification of the weakest point. The paper's central equivalence depends on Lemma 3, which is unproved and cited out of house, and the surrounding equations have an internal sign conflict. The sign error is not merely cosmetic: it changes the direction of the normal force and, if present in the implementation, changes the numerical results. The ePDS construction itself is plausible, and the direct projection computation in Example 1 confirms which sign is correct, so this is a correctable but real flaw rather than a false central claim. Because the reader already rendered a CONDITIONAL verdict and the necessary corrections are localized, my stress-test does not move that verdict; hence UNCHANGED.","tokens_in":9860,"tokens_out":17172,"duration_ms":166474,"concrete_test":"Analytic check on Example 1: with y = (x,tau), c(y) = x - tau, E = span{(1,0)} and f = (-1,1), compute P_{T_C,E}(f) at the boundary y = (1,1). Since T_C = {d : d_x - d_tau >= 0} and the projection forces d_tau = 1, the unique projected derivative is dot y = (1,1). Substitute this into DCS (7): dot x = -1 + lambda = 1 gives lambda = 2 >= 0, with 0 <= lambda perp c(y) = 0 satisfied. The paper's final minus equation would require -1 - lambda = 1, which no lambda >= 0 can satisfy. Separately, re-derive Lemma 3 from [20, Section 4] using the standard polar normal cone for C(t) = {x : c(x,t) >= 0}; if the DCS sign is plus, the central claim is recoverable after fixing the sign errors and adding a full proof of Lemma 3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing spot is the chain FOSwP (3) -> Lemma 3 DCS (7) -> Theorem 4 ePDS (8). Lemma 3 is not proved; it is deferred to [20, Section 4], and the paper's own definitions do not align with the claimed DCS. For C(t) = {x : c(x,t) >= 0}, the polar normal cone is N_{C(t)}(x) = {-sum_i lambda_i grad_x c_i(x,t), lambda_i >= 0}, not the positive span printed in Section IV.A. If the printed positive span is used literally, sweeping process (3) becomes dot x in f - sum lambda grad c, which has the opposite sign to DCS (7) dot x = f + grad_x c(x,t) lambda. Theorem 4's proof compounds this by writing dot x = f(x,t) - grad_x c(x,t) lambda', flipping the sign a second time relative to both Theorem 2 and Lemma 3. Since Theorem 4 is the bridge that justifies applying FESD to time-varying sweeping processes, the numerical claims inherit whichever sign is actually implemented. Example 1 is decisive: at the boundary c(y) = x - tau, the correct ePDS projection along E gives dot y = (1,1), which forces lambda = 2 in the plus-sign DCS and is impossible in the minus-sign equation. The intended theorem is plausible after a sign correction and a proper proof of Lemma 3, but the paper as written is internally inconsistent and cannot be verified as is.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript establishes (or aims to establish) equivalences between three classes of nonsmooth dynamical systems: perturbed first-order Moreau sweeping processes with time-varying sets, extended projected dynamical systems (ePDS), and dynamic complementarity systems (DCS). It then uses these equivalences to adapt the Finite Elements with Switch Detection (FESD) discretization to the ePDS/DCS form, enabling direct optimal control of sweeping processes with moving sets. Two numerical examples, a wave-rider optimal control problem and a moving-obstacle path planning problem, demonstrate the accuracy of the FESD approach.","tokens_in":10143,"tokens_out":12209,"duration_ms":105275,"significance":"If the equivalences are correct, the paper gives a practical route to high-order discretization of sweeping processes with time-varying sets, which is a genuine extension of the authors' previous FESD method for PDS. The numerical results are promising and the implementation is open source. However, the central proof chain contains an internal sign inconsistency in the normal cone definition and in Theorem 4, and Lemma 3 is not proved in the paper; these issues must be resolved before the equivalences can be accepted. The paper's value is therefore conditional on a careful correction of signs and a complete proof of the FOSwP–DCS equivalence.","major_comments":[{"comment":"The normal cone definition in Section IV.A is stated as the positive span of the active constraint gradients: N_{C(t)}(x) = { Σ_{i∈A(x)} λ_i ∇c_i(x,t), λ_i ≥ 0 }. For a set C(t) = {c(x,t) ≥ 0}, the polar normal cone is the negative span, { −Σ_{i∈A(x)} λ_i ∇c_i(x,t), λ_i ≥ 0 }. With the printed definition, Eq. (3) becomes ˙x ∈ f − Σλ∇c, which has the opposite sign to Lemma 3's DCS (Eq. (7), ˙x = f + ∇c λ). Theorem 4's proof then writes ˙x = f − ∇c λ′, reversing the sign a second time. Consequently, the equivalence chain FOSwP (3) → DCS (7) → ePDS (8) is not established as written; the sign errors must be corrected consistently throughout, including in Example 1, where the displayed normal cone has the same incorrect sign.","section":"Section IV.A, Eq. (3), Lemma 3 (Eq. (7)), Theorem 4"},{"comment":"Lemma 3 is the load-bearing bridge from the sweeping process to a dynamic complementarity system, but it is not proved in the paper; the proof is delegated to 'the transformation in [20, Section 4]'. The paper should either give a self-contained proof or state precisely which theorem of [20] applies, and verify that the hypotheses (uniform prox-regularity, bounded variation, and the sign convention for the normal cone) are satisfied in the setting of Eq. (3). Without this, the subsequent equivalence in Theorem 4 rests on an unverified premise.","section":"Lemma 3, Section IV.A"},{"comment":"In the proof of Theorem 2, the KKT conditions yield the velocity complementarity 0 ≤ λ_j ⊥ ∇c_j^T ˙x ≥ 0 for active constraints, and the proof asserts without argument that this is equivalent to the position complementarity 0 ≤ λ ⊥ c(x) ≥ 0. This equivalence relies on absolute continuity of the trajectory and on the fact that a feasible trajectory cannot have \\dot c < 0 at a point where c = 0; a short proof or a precise reference should be supplied, because the DCS (6) with position complementarity is the object used by the FESD discretization.","section":"Theorem 2 proof"}],"minor_comments":[{"comment":"The notation \"E = span E\" is confusing because E is used for both a matrix and the subspace it spans; please use distinct symbols, e.g., E for the matrix and E for the subspace.","section":"Section II.A"},{"comment":"The indexing of the complementarity constraints in Eqs. (9c)-(9f) is unclear: λ_{n,i} is used without being defined, and Eqs. (9e)-(9f) appear to mix elements n and n-1 in a way that is not explained. Please align these equations with the formulation in [13].","section":"Section V, Eqs. (9c)-(9f)"},{"comment":"The assumptions on C(t) are introduced as \"forward Lipschitz\" and later as \"uniformly prox-regular and of bounded variation\"; the relationship between these conditions, and which one is used in Lemma 3 versus Theorem 4, should be clarified.","section":"Section IV.A"},{"comment":"The proof of Lemma 1 is hard to follow; for instance, the inequality \"∥ϕ−v∥2^2 < ∥ϕE−vE∥2^2\" compares quantities in different spaces. Please rewrite the proof with explicit definitions of the norms and a clearer chain of inequalities.","section":"Lemma 1 proof"}],"recommendation":"major_revision","confidential_remarks":"The sign inconsistency is the kind of error that a careful reviewer expects the authors to fix; it is not a fundamental flaw, since the correct signs are standard and the numerical code likely uses the plus-sign convention. Please ask the authors to verify the implementation against the corrected equations and to add a proof of Lemma 3 or a precise reference. Also, check that the published equations match the implemented discretization in nosnoc."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper has a genuinely new result: the clock-state lifting of time-varying sweeping processes into an extended projected dynamical system, plus a FESD-based discretization for the resulting DCS. Theorem 2 (ePDS to DCS) is proved carefully and is useful on its own. The numerical examples (wave-rider, moving doorway) are well chosen, and the observed high-order convergence suggests the implemented sign is the correct one. The code is available in nosnoc, and the experiments look reproducible.\n\nThe soft spot is exactly where the stress-test puts its finger. Section IV.A defines the normal cone to C(t) as the positive span of gradients, {Σλ∇c}, while the paper's own general definition in the notation section is the polar cone, which for c≥0 is {−Σλ∇c}. Lemma 3's DCS (7) uses ˙x = f + ∇c λ, matching the polar cone. Theorem 4's proof then writes ˙y = (f,1) + EE^T∇_y c λ' and later ˙x = f − ∇_x c λ', a sign flip with no justification. So, as written, the central equivalence FOSwP→ePDS is not established. The intended statement is almost certainly correct: Example 1 works, and a consistent sign convention fixes the proof. Lemma 3 is cited rather than proved; that's acceptable if [20] really covers it, but the statements should agree in sign so a reader can verify the chain.\n\nI also want to give credit where due: the paper builds on prior FESD work rather than fitting parameters, the equivalences are derived from first principles, and the citation pattern is reasonable. The heavy reliance on the authors' own [13] and nosnoc is legitimate prior work, not circularity.\n\nBottom line: a solid, useful paper that currently has a load-bearing sign inconsistency. A referee should send it for major revision with a request to fix the signs in IV.A, Lemma 3, and Theorem 4, and to re-run Example 1 against the corrected definitions. After that, it would be a nice addition to the nonsmooth optimal control literature. If you want a paper for a student to practice careful verification, this one is a great candidate because the error is instructive.\n\nRecommendation: accept for peer review with the expectation of major revision.","headline":"Good ideas, fixable sign error in the main proof – worth a referee but not as-is.","tokens_in":10725,"tokens_out":7166,"would_cite":false,"duration_ms":60017,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34A60","49J15","49M25","90C33"],"pacs":[],"model":"deepseek-v4-flash","headline":"Time-varying Moreau sweeping processes, previously outside the scope of accurate direct discretization, are shown to be equivalent to extended projected dynamical systems, and this equivalence yields a high-accuracy FESD scheme for their…","keywords":["sweeping process","extended projected dynamical system","dynamic complementarity system","finite elements with switch detection","optimal control","nonsmooth dynamics","time-varying constraints","prox-regular sets"],"falsifier":"Take a sweeping process with a known closed-form solution, such as Example 1's $x(t)=2-t$ for $t<1$ and $x(t)=t$ for $t\\ge1$ with $C(t)=\\{x:x\\ge t\\}$ and $f=-1$, and integrate the ePDS (8) and the DCS (7) numerically. If the lifted trajectory does not reproduce the closed-form solution at the contact time $t=1$, or if the complementarity formulation (7) with the sign convention used in the paper gives a different boundary reaction than the normal-cone definition of Section IV.A, the claimed equivalence fails.","tokens_in":9625,"feed_emoji":"🕐","tokens_out":12564,"duration_ms":103366,"temperature":0.7,"pith_summary":"Moreau's first-order sweeping process describes a state confined to a moving set and pushed by the set's boundary whenever it is reached. The paper targets a gap: when the set moves with time, the known equivalence between sweeping processes and ordinary projected dynamical systems breaks, and standard Runge-Kutta discretizations of the resulting nonsmooth systems suffer low accuracy. The paper's central claim is that, under prox-regularity and mild regularity of the set-valued map, a time-varying sweeping process can be rewritten exactly as an extended projected dynamical system (ePDS), a projection dynamics in which the projection is allowed only along a chosen subspace, by adding a clock state whose own evolution is not projected. It further proves that such ePDS are equivalent to dynamic complementarity systems, which the finite-elements-with-switch-detection (FESD) discretization can handle at high order. If the claim is right, optimal control of moving-boundary sweeping processes—marine vehicles, crowd motion, soft robots, moving obstacles—becomes numerically tractable with much coarser time grids.","feed_headline":"Moving-boundary sweeping processes get high-order discretization","feed_subtitle":"A clock state makes time-varying sweeping processes equivalent to projected systems, enabling accurate optimal control.","key_machinery":"The central object is the extended projection operator $P_{K,E}(v)$: the solution of a convex quadratic program that minimizes $\\tfrac{1}{2}\\|w-v\\|^2$ subject to $w\\in K$ and $w-v\\in E$. It is the mechanism that replaces the unprojected dynamics by the closest admissible velocity reachable along the subspace $E$. In this paper $K$ is the tangent cone of the moving constraint set and $E$ is the subspace of the $x$-coordinates only, so the clock component is never projected. Lemma 1 gives a subspace decomposition of this operator, and Theorem 2 converts its KKT conditions into complementarity form; along $\\tau=t$ the identity $EE^\\top\\nabla_x c(x,t)\\lambda=\\nabla_x c(x,t)\\lambda$ collapses the lifted system back to the original sweeping dynamics.","core_discovery":"The paper establishes an equivalence chain linking three system classes. Theorem 2 shows that, when the constraint set is finitely defined (given by finitely many inequalities $c(x)\\ge 0$), satisfies LICQ at active constraints, and the tangent cone always meets the projected dynamics, the extended projected dynamical system $\\dot{x} = P_{T_C(x),E}(f(x,u))$ is equivalent to the dynamic complementarity system $\\dot{x} = f(x,u) + EE^\\top \\nabla c(x)\\lambda$, $0 \\le \\lambda \\perp c(x)\\ge 0$. Theorem 4 extends this to time-varying sets: the perturbed sweeping process $\\dot{x}\\in f(x,u) - N_{C(t)}(x)$ with uniformly prox-regular $C(t)$ (the standard regularity class for which sweeping-process solutions exist) corresponds to an ePDS in the lifted state $y=(x,\\tau)$, where the added clock state satisfies $\\dot{\\tau}=1$ and the projection subspace $E$ spans only the $x$-directions. Because the clock is not projected, the boundary moves with physical time, and the complementarity form collapses back to $\\dot{x}=f(x,u)+\\nabla_x c(x,t)\\lambda$ with $0\\le \\lambda\\perp c(x,t)\\ge 0$. This chain is what lets the FESD method for projected dynamical systems be applied, with only the term $\\nabla c(x)\\lambda$ replaced by $EE^\\top\\nabla c(x)\\lambda$.","pith_inferences":["A step the paper does not take but which follows directly: the same clock-lifting construction should apply to any time-varying constrained dynamics, not just sweeping processes, since the clock coordinate is the only place where time enters the lifted constraint set.","The subspace freedom in the extended projection suggests a recipe for designing partially projected dynamics—where some directions are free to react and others are not—which could be used to model actuators that cannot react instantly to boundary contacts.","A natural next test is to deploy the FESD-discretized complementarity system in a real-time iteration or moving-horizon setting; the coarse-grid accuracy demonstrated here would be the enabling property."],"forward_implications":["Time-varying sweeping processes can be discretized with FESD, recovering the nominal order of the Runge-Kutta scheme: the paper's experiments show $O(h)$, $O(h^3)$, $O(h^5)$, and $O(h^7)$ errors for one- to four-stage schemes, instead of first-order behavior for fixed-step discretization.","Optimal control problems with moving constraint sets, such as the wave-rider and moving-doorway examples, can be solved with far fewer control stages while keeping accurate terminal states.","The reformulation turns sweeping-process optimal control into a mathematical program with complementarity constraints, so existing relaxation and homotopy solution techniques apply directly.","Because the equivalence is bidirectional, any ePDS with the same structure can also be treated as a sweeping process or a dynamic complementarity system, unifying the three formulations under one discretization."],"supporting_citations":[{"why":"Introduces the extended projected dynamical system and the associated projection operator that the equivalence theorems build on.","marker":"[1]"},{"why":"Supplies the finite-elements-with-switch-detection discretization for projected dynamical systems that the paper extends to ePDS and sweeping processes.","marker":"[13]"},{"why":"Provides the transformation from which Lemma 3 concludes that the sweeping process is equivalent to a dynamic complementarity system.","marker":"[20]"},{"why":"Gives existence and uniqueness of solutions for nonconvex sweeping processes with perturbation, used as the regularity setting for the time-varying set.","marker":"[19]"},{"why":"Provides the representation of the normal cone of a finitely defined set used to rewrite the sweeping process.","marker":"[16]"},{"why":"Justifies convexity of the tangent cone under a constraint qualification such as LICQ, needed in the KKT argument of Theorem 2.","marker":"[17]"},{"why":"Defines forward-Lipschitz time-varying sets, the regularity assumption on the moving constraint set.","marker":"[18]"}],"fun_headline_variants":["Sweeping processes meet projected dynamics for optimal control","Unified view of sweeping and projected systems enables fast solvers","Equivalence unlocks high-order methods for moving-boundary systems","From sweeping to projected systems: a bridge for optimal control","Projected dynamics reformulate sweeping processes for discretization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 3: the sweeping process with time-varying prox-regular set $C(t)$ really is equivalent to the complementarity system (7), with the correct sign of the boundary-reaction term; the paper imports this equivalence from a cited result instead of proving it here.","fun_headline_variants_meta":{"raw":{"variants":["Sweeping processes meet projected dynamics for optimal control","Unified view of sweeping and projected systems enables fast solvers","Equivalence unlocks high-order methods for moving-boundary systems","From sweeping to projected systems: a bridge for optimal control","Projected dynamics reformulate sweeping processes for discretization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000872,"raw_usage":{"total_tokens":3804,"prompt_tokens":1005,"completion_tokens":2799,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":2719}},"tokens_in":621,"tokens_out":2799,"duration_ms":18074,"temperature":1.0,"reasoning_tokens":2719,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:04:58.280790+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a sweeping process with a known closed-form solution, such as Example 1's $x(t)=2-t$ for $t<1$ and $x(t)=t$ for $t\\ge1$ with $C(t)=\\{x:x\\ge t\\}$ and $f=-1$, and integrate the ePDS (8) and the DCS (7) numerically. If the lifted trajectory does not reproduce the closed-form solution at the contact time $t=1$, or if the complementarity formulation (7) with the sign convention used in the paper gives a different boundary reaction than the normal-cone definition of Section IV.A, the claimed equivalence fails.","supporting_citations":[{"cited_title":"Extended pro- jected dynamical systems with applications to hybrid integrator-gain systems,","cited_arxiv_id":null,"evidence_quote":"Introduces the extended projected dynamical system and the associated projection operator that the equivalence theorems build on."},{"cited_title":"Existence and uniqueness of solutions for non-autonomous complementarity dynamical systems,","cited_arxiv_id":null,"evidence_quote":"Provides the transformation from which Lemma 3 concludes that the sweeping process is equivalent to a dynamic complementarity system."},{"cited_title":"Bv solutions of nonconvex sweeping process differential inclusion with perturbation,","cited_arxiv_id":null,"evidence_quote":"Gives existence and uniqueness of solutions for nonconvex sweeping processes with perturbation, used as the regularity setting for the time-varying set."},{"cited_title":"Acary and B","cited_arxiv_id":null,"evidence_quote":"Provides the representation of the normal cone of a finitely defined set used to rewrite the sweeping process."},{"cited_title":"Rockafellar and R","cited_arxiv_id":null,"evidence_quote":"Justifies convexity of the tangent cone under a constraint qualification such as LICQ, needed in the KKT argument of Theorem 2."},{"cited_title":"Time- varying projected dynamical systems with applications to feedback optimization of power systems,","cited_arxiv_id":null,"evidence_quote":"Defines forward-Lipschitz time-varying sets, the regularity assumption on the moving constraint set."}],"review_version":1}