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arxiv: 2604.10774 · v1 · submitted 2026-04-12 · 🧮 math.OC

Convergence and Stability of a Catching-Up Algorithm for Differential Inclusions with Maximal Monotone Operators

Tan H. Cao , Hassan Saoud This is my paper

Pith reviewed 2026-05-10 15:31 UTC · model grok-4.3

classification 🧮 math.OC
keywords catching-up algorithmdifferential inclusionsmaximal monotone operatorsconvergence of discretizationsenergy inequalitystability estimateserror bounds
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The pith

A catching-up algorithm for differential inclusions with maximal monotone operators converges on finite horizons with explicit error bounds.

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

The paper proves existence and uniqueness for solutions to differential inclusions driven by maximal monotone operators under a mild tangent dissipativity condition and local Lipschitz perturbations. It then introduces a time-discretized catching-up scheme with variable steps and approximate projections that approximates these solutions. The analysis shows that discrete trajectories converge to continuous ones, with uniform boundedness of iterates, stability estimates, and explicit error bounds coming from a discrete velocity decomposition and energy inequality. Additional results establish that constraint violations from the free step vanish in an averaged sense as the mesh is refined.

Core claim

The catching-up algorithm, built on a decomposition of the maximal monotone operator into the closed convex hull of its single-valued part and the normal cone to a closed convex set, yields global energy bounds and uniqueness when the perturbation is locally Lipschitz. The corresponding time-discretized scheme with variable step sizes and approximate projections produces trajectories that converge to continuous solutions on any finite horizon; a discrete velocity decomposition plus discrete energy inequality then supplies uniform boundedness, quantitative stability with respect to initial data, and explicit error estimates.

What carries the argument

The catching-up scheme with variable step sizes and approximate projections, driven by a discrete velocity decomposition together with a discrete energy inequality that controls the iterates.

If this is right

  • Discrete trajectories converge to continuous solutions as the maximum step size tends to zero on any fixed time interval.
  • The iterates remain uniformly bounded on finite horizons.
  • Stability estimates quantify how changes in initial data affect the solutions.
  • Explicit error bounds between discrete and continuous trajectories are available.
  • The predictor step becomes asymptotically feasible in the L2 sense and satisfies a Cesaro averaged feasibility property.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The energy-based bounding technique could be adapted to other discretization methods for monotone inclusions, such as proximal-point schemes.
  • Explicit error bounds may enable adaptive step-size control in numerical implementations without a priori mesh refinement.
  • The feasibility results suggest the scheme could be combined with projection-free methods when the constraint set is hard to project onto exactly.

Load-bearing premise

The monotone operator must satisfy a mild tangent dissipativity condition and the perturbation must be locally Lipschitz.

What would settle it

A concrete maximal monotone operator violating the tangent dissipativity condition for which the discrete trajectories fail to remain bounded or diverge from the continuous solution as the step size goes to zero.

read the original abstract

We study a catching-up algorithm for a class of differential inclusions driven by maximal monotone operators with continuous perturbations. Using a decomposition of the monotone operator into the closed convex hull of its single-valued part and the normal cone to a closed convex set, we establish existence of solutions and derive global energy bounds under a mild tangent dissipativity assumption. Under an additional local Lipschitz assumption on the perturbation, we also obtain uniqueness and stability with respect to the initial data. We then analyze a time-discretized catching-up scheme with variable step sizes and approximate projections. On every finite horizon, we prove convergence of the discrete trajectories to solutions of the continuous problem. A discrete velocity decomposition together with a discrete energy inequality yields uniform boundedness of the iterates, quantitative stability estimates, and explicit error bounds. We also establish asymptotic feasibility of the predictor step in an $L^2$ sense, as well as a Ces\`aro-type averaged feasibility property, showing that the constraint violations generated by the free step vanish as the discretization is refined. Finally, we illustrate the theory on explicit examples, including a fully explicit one--dimensional test case and a multidimensional constrained dry-friction system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies a catching-up algorithm for differential inclusions driven by maximal monotone operators with continuous perturbations. Using the decomposition of the operator into the closed convex hull of its single-valued part plus the normal cone to a closed convex set, it proves existence of solutions and derives global energy bounds under a mild tangent dissipativity assumption on the operator. Under an additional local Lipschitz condition on the perturbation, uniqueness and stability with respect to initial data are obtained. The paper then analyzes a time-discretized catching-up scheme with variable step sizes and approximate projections, proving convergence of discrete trajectories to continuous solutions on every finite horizon, uniform boundedness, quantitative stability estimates, explicit error bounds, asymptotic L2 feasibility of the predictor step, and a Cesàro-type averaged feasibility property. The theory is illustrated on a one-dimensional test case and a multidimensional constrained dry-friction system.

Significance. If the derivations hold, the work supplies a convergent discretization framework for a class of differential inclusions with explicit quantitative bounds and stability results. This is useful in optimization and control applications involving constraints and nonsmooth dynamics, extending standard monotone-operator techniques with energy estimates and feasibility analysis for the discrete scheme.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction invoke the 'mild tangent dissipativity assumption' for global energy bounds without a dedicated subsection clarifying its relation to standard monotonicity or providing verifiable conditions under which it holds for common maximal monotone operators (e.g., subdifferentials of convex functions).
  2. [§3 and §4] Notation for the decomposition A = cl conv(A_s) + N_C and the discrete velocity splitting is introduced without an explicit comparison table or diagram showing how the continuous and discrete versions align term-by-term.
  3. [§6] The examples in the final section are presented without tabulated error values or plots of the explicit error bounds derived earlier, making it harder to verify the quantitative claims numerically.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive summary of our work on the catching-up algorithm for differential inclusions with maximal monotone operators. The recommendation for minor revision is noted. No specific major comments were raised in the report, so we have no points requiring rebuttal or revision at this stage. We will proceed with any final polishing of the manuscript prior to publication.

Circularity Check

0 steps flagged

No significant circularity; results conditional on explicit assumptions with independent analysis

full rationale

The paper derives convergence of the discrete catching-up scheme to continuous solutions on finite horizons via a discrete velocity decomposition and energy inequality, using the operator splitting A = cl conv(A_s) + N_C. These steps are presented as consequences of the stated mild tangent dissipativity assumption on A and local Lipschitz continuity on f, which are introduced upfront rather than derived from the target results. No equation reduces the convergence claim to a fitted parameter, self-definition, or load-bearing self-citation chain; the quantitative bounds and error estimates follow from the discrete energy inequality under those assumptions. Foundational monotone-operator techniques are referenced from prior literature but do not render the discretization analysis tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard properties of maximal monotone operators and two additional assumptions (tangent dissipativity and local Lipschitz continuity of the perturbation) that are typical in the literature but not derived inside the paper.

axioms (3)
  • standard math Maximal monotone operators admit a decomposition into the closed convex hull of their single-valued part and the normal cone to a closed convex set.
    Invoked to establish existence of solutions and energy bounds.
  • domain assumption Mild tangent dissipativity holds for the operator.
    Required to obtain global energy bounds on solutions.
  • domain assumption The perturbation is locally Lipschitz continuous.
    Used to obtain uniqueness and stability with respect to initial data.

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Reference graph

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