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REVIEW 3 major objections 4 minor 28 references

First-Order Sweeping Processes and Extended Projected Dynamical Systems: Equivalence, Time-Discretization and Numerical Optimal Control

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Pith's one-line read 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…

desk verdict Good ideas, fixable sign error in the main proof – worth a referee but not as-is. read the letter →

arxiv 2412.11320 v1 pith:ZJY2WNXN submitted 2024-12-15 math.OC

Anton Pozharskiy , Armin Nurkanović , Moritz Diehl This is my paper · ORCID
classification math.OC MSC 34A6049J1549M2590C33
keywords sweepingprocessextendedprojecteddynamicalsystemdynamiccomplementarityfiniteelementswithswitchdetectionoptimalcontrolnonsmoothdynamicstime-varyingconstraintsprox-regularsets
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $EE^\top\nabla c(x)\lambda$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section IV.A, Eq. (3), Lemma 3 (Eq. (7)), Theorem 4] 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.
  2. [Lemma 3, Section IV.A] 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.
  3. [Theorem 2 proof] 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.
minor comments (4)
  1. [Section II.A] 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.
  2. [Section V, Eqs. (9c)-(9f)] 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].
  3. [Section IV.A] 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.
  4. [Lemma 1 proof] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equivalence theorems are derived in-paper or imported from external references, and the FESD extension is built on prior work without circular reduction.

full rationale

The derivation chain is FOSwP (3) -> DCS (7) via Lemma 3 -> ePDS (8) via Theorem 4, and ePDS -> DCS (6) via Theorem 2. Theorem 2 is proved from first principles using the KKT conditions of the extended projection QP; it does not assume the DCS it derives. Lemma 3 is not proved in the paper but is explicitly delegated to Brogliato and Thibault [20, Section 4], an external reference, not a self-citation. Theorem 4 combines Lemma 3 with Theorem 2, and the key algebraic step EE^T grad_y c-hat(y) = grad_x c(x,t) is stated explicitly and is not a restatement of the theorem's conclusion. The FESD discretization is a modification of the authors' own prior FESD method [13], but this is normal building on previous work rather than a circular argument: the paper's contribution is the equivalence reformulation, and the numerical experiments validate the discretization against accurate simulation rather than fitting constants. The sign inconsistency between the normal cone definition in Section IV.A, Lemma 3's DCS (7), and the final equation in Theorem 4 is a genuine correctness or consistency issue, but it is not a circularity: the theorem is not forced by its inputs by construction; it is potentially internally inconsistent as written. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors' own work is invoked to forbid alternatives, and no ansatz is smuggled in via self-citation. Accordingly, no circular step is identified and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The method introduces no fitted constants or new physical entities. The clock state τ is a mathematical lifting device, not an invented entity. The main axiomatic burden is the regularity of C(t) and the cited DCS equivalence.

assumptions (5)
  • domain assumption C(t) is uniformly prox-regular and of bounded variation
    Invoked in Lemma 3 and Theorem 4 to guarantee existence, uniqueness and the DCS representation of the sweeping process; not proved in the paper.
  • domain assumption LICQ holds for active constraints along the trajectory
    Required in Theorem 2 to ensure the tangent cone is closed and convex and the KKT conditions are necessary and sufficient.
  • domain assumption TC(x) ∩ (f(x,u)+E) is nonempty for all time
    Assumed in Theorem 2 so that the extended projection QP is feasible; needed for equivalence to the DCS.
  • domain assumption f is continuously differentiable in x and u
    Stated in Section II.B and used throughout for the sweeping process and its DCS representation.
  • domain assumption The sweeping process is equivalent to the DCS (7) (Lemma 3)
    This lemma is the bridge between FOSwP and ePDS, but it is only cited to [20, Section 4] and not proved. It also carries the sign convention that conflicts with the paper's own normal cone definition.

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Pith. "Pith review of First-Order Sweeping Processes and Extended Projected Dynamical Systems: Equivalence, Time-Discretization and Numerical Optimal Control." pith.science (2026). https://pith.science/paper/ZJY2WNXN

@misc{pith2026241211320,
  author       = {Pith},
  title        = {Pith review of: First-Order Sweeping Processes and Extended Projected Dynamical Systems: Equivalence, Time-Discretization and Numerical Optimal Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJY2WNXN}},
  note         = {Machine review of arXiv:2412.11320}
}
read the original abstract

Constrained dynamical systems are systems such that, by some means, the state stays within a given set. Two such systems are the (perturbed) Moreau sweeping process and the recently proposed extended Projected Dynamical System (ePDS). We show that under certain conditions solutions to the ePDS correspond to the solutions of a dynamic complementarity system, similar to the one equivalent to ordinary PDS. We then show that the perturbed sweeping process with time varying set can, under similar conditions, be reformulated as an ePDS. In this paper, we leverage these equivalences to develop an accurate discretization method for perturbed first-order Moreau sweeping processes via the finite elements with switch detection method. This allows the efficient optimal control of systems governed by ePDS and perturbed first-order sweeping processes.

Figures

Figures reproduced from arXiv: 2412.11320 by the authors.

Figure 1
Figure 1. Relationships between systems in this paper with edge [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. Schematic of vectors in eq. (5) applied to [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Example 1 ePDS trajectory. subspace E = { s ∈ R n | EE⊤x = s, x ∈ R n }, with E = (e1, . . . , en) ∈ R (nx+1)×nx where ei is the ith basis vector of R nx+1 . Proof. As a consequence of Lemma 3 we have an equivalent DCS for Equation (3). By Theorem 2, for y(0) = (x(0), 0) ∈ C the solution to Equation (8) is the solution to the DCS: y˙ = (f(x, u), 1) + EE⊤∇ycˆ(y)λ ′ 0 ≤ λ ′ ⊥ cˆ(y) ≥ 0 From the structure of E, τ˙ = 1 … view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: Plots of terminal error vs step size. with n = 1, . . . , Nfe, i = 1, . . . , ns, and j = 1, . . . , ns, and a fixed control u. Note that in this case hn is allowed to vary which allows the complementarity conditions to be satisfied. The complementarity constraints her…
Figure 6
Figure 6. Figure 6: Plot of wave-rider example trajectory. with x ∈ R 2 , u ∈ R, xˆ = (0, 2), x¯ = (1, 4), and time-varying set C(t) = { z ∈ R 2 | z1 + cos(z2 − t) ≥ 0 }. The system is under-actuated and the controller must use the moving boundary defined by C(t) to translate in the unact…
Figure 7
Figure 7. Figure 7: Several frames of the solution for the manipulation problem with autonomous discs. [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

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