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Optimal Control of Sweeping Processes with Nonsmooth Moving Sets: A Numerical Algorithm

T0 review · 0 major / 3 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Uniform geometric estimates on time-dependent nonsmooth sets permit convergence of a discrete control approximation scheme

desk verdict 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. read the letter →

arxiv 2606.04639 v1 pith:HUYXEYQ3 submitted 2026-06-03 math.OC

Chadi Nour This is my paper · ORCID
classification math.OC
keywords optimalcontrolsweepingprocessesnonsmoothmovingsetsnumericalalgorithmdiscreteapproximationnonautonomouscaseconvergenceproof
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

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.

What carries the argument

Uniform geometric estimates for the moving constraint sets and their boundaries, which remain independent of time despite explicit time dependence.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

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.

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.

minor comments (3)
  1. [Introduction] 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.
  2. [Convergence analysis] 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.
  3. [Numerical examples] 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.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper's core step is establishing new uniform (time-independent) geometric estimates on the moving nonsmooth sets and boundaries from the problem hypotheses; these estimates are then used to extend the discrete scheme from prior references and prove convergence. No equation reduces a claimed prediction or result to a fitted parameter or self-citation by construction, and the central estimates are presented as derived content rather than renamed inputs. Self-citations to [22,33,37] support the base scheme but are not load-bearing for the new estimates themselves.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the existence of time-independent geometric estimates for the moving sets; this is treated as a domain assumption rather than derived from first principles in the abstract.

assumptions (1)
  • domain assumption Moving nonsmooth sets admit uniform geometric estimates independent of time
    Invoked explicitly to overcome difficulties arising from explicit time dependence in the constraint sets.

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Cite this review

Pith. "Pith review of Optimal Control of Sweeping Processes with Nonsmooth Moving Sets: A Numerical Algorithm." pith.science (2026). https://pith.science/paper/HUYXEYQ3

@misc{pith2026260604639,
  author       = {Pith},
  title        = {Pith review of: Optimal Control of Sweeping Processes with Nonsmooth Moving Sets: A Numerical Algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUYXEYQ3}},
  note         = {Machine review of arXiv:2606.04639}
}
read the original abstract

This paper extends the numerical method introduced by de Pinho et al. [22] and later developed in Nour and Zeidan [33, 37] to the nonautonomous case where the nonsmooth sweeping set depends explicitly on time. The moving character of the sweeping set creates substantial additional difficulties, since several geometric constants involved in the analysis may a priori depend on time. To overcome this issue, we establish uniform geometric estimates for the moving constraint sets and their boundaries. These estimates allow us to extend the discrete approximation scheme developed in [22, 33, 37] and prove its convergence toward admissible solutions of the original problem. Several examples illustrating the applicability of the proposed method are also presented.

Figures

Figures reproduced from arXiv: 2606.04639 by the authors.

Figure 1
Figure 1. Numerical vs exact optimal trajectory reached the prescribed tolerance ε after 187 iterations, corresponding to the value γ187 = 2170. The obtained numerical cost is g187 = g(ˆx N γ187 (3)) = −2.747601272285, which is in excellent agreement with the exact minimum value of (P) computed above. The total running time of the algorithm was 50.394 seconds.3 Moreover, [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Convergence with respect to γk Hence, ψ1, ψ2 : [0, T] × R 3 −→ R are defined by: • ψ1(t, x1, x2, x3) := (x1 − 4)2 + x 2 2 + x 2 3 − ρ(t) 2 . • ψ2(t, x1, x2, x3) := (x1 + 4)2 + x 2 2 + x 2 3 − ρ(t) 2 . Note that C(t) is a nonsmooth and convex set. • The objective function g : R 3 −→ R is defined by g(x1, x2, x3) = x 2 1 +  x2 − 1 2 2 + [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. C(0), Γ(0), and x0 where θ(t) = 2 arctan(e t/3 ) − π 2 . Thus, the trajectory first evolves along the nonsmooth part Γ(t) of the boundary of the moving set, and then enters the interior of C(t) and evolves freely until the terminal time. In particular, x¯(T) =  0, 1 2 , 3 √ 2 2  , and therefore min(P) = 0. For this example, we choose N = 50, ε = 10−5 , γ = 20,4 and δ = 10, and apply Algorithm 1 in order to numeric… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Numerical vs exact optimal trajectory . 20 40 60 80 100 120 140 Minim u m v alu e 0 0.1 0.2 0.3 0.4 0.5 0.6 Convergence with respect to . Exact value [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Convergence with respect to γk [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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