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arxiv: 1907.03412 · v1 · pith:YFKWUCCNnew · submitted 2019-07-08 · 🧮 math.AP

Stochastic optimal control of a evolutionary p-Laplace equation with multiplicative L\'{e}vy noise

Ananta K. Majee This is my paper

Pith reviewed 2026-05-25 01:25 UTC · model grok-4.3

classification 🧮 math.AP
keywords stochastic optimal controlp-Laplace equationLévy noisevariational methodexistence of optimal solutionweak solutionsmultiplicative noiseevolutionary equations
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The pith

An optimal control exists for the evolutionary p-Laplace equation driven by multiplicative Lévy noise.

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

The paper first establishes well-posedness of weak solutions to the evolutionary p-Laplace equation with multiplicative Lévy noise through implicit time discretization and the Jakubowski-Skorokhod theorem on non-metric spaces. It then sets up the associated optimal control problem and proves existence of an optimal solution by applying the variational method to a convex cost functional. A sympathetic reader would care because this supplies a rigorous existence result for steering nonlinear parabolic systems subject to jump noise, extending classical control theory into stochastic regimes where deterministic methods fail.

Core claim

We first present wellposedness of a weak solution by using an implicit time discretization of the problem, along with the Jakubowski version of the Skorokhod theorem for a non-metric space. We then formulate associated control problem, and establish existence of an optimal solution by using variational method and exploiting the convexity property of the cost functional.

What carries the argument

The variational method applied to the control problem, which relies on convexity of the cost functional to guarantee existence of a minimizer.

If this is right

  • Well-posedness of the state equation holds via the implicit discretization scheme.
  • Existence of an optimal control follows directly once the cost functional satisfies convexity.
  • The control problem admits at least one solution in the stochastic setting with multiplicative Lévy noise.
  • The result applies to initial-value problems for the evolutionary p-Laplace operator.

Where Pith is reading between the lines

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

  • The implicit discretization used for well-posedness might serve as the basis for convergent numerical schemes.
  • Relaxing convexity could require different arguments such as relaxation or Gamma-convergence.
  • The same variational approach may transfer to other nonlinear SPDEs driven by Lévy processes.

Load-bearing premise

The cost functional is convex.

What would settle it

A concrete example of a non-convex cost functional for which no optimal control exists for the given p-Laplace equation with Lévy noise.

read the original abstract

In this article, we are interested in an initial value optimal control problem for a evolutionary $p$-Laplace equation driven by multiplicative L\'{e}vy noise. We first present wellposedness of a weak solution by using an implicit time discretization of the problem, along with the Jakubowski version of the Skorokhod theorem for a non-metric space. We then formulate associated control problem, and establish existence of an optimal solution by using variational method and exploiting the convexity property of the cost functional.

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

1 major / 0 minor

Summary. The paper establishes well-posedness of weak solutions to the evolutionary p-Laplace equation with multiplicative Lévy noise via implicit time discretization combined with the Jakubowski version of the Skorokhod theorem on a non-metric space. It then formulates the associated optimal control problem and proves existence of an optimal control by the direct variational method, relying on convexity of the cost functional.

Significance. If the central claims hold, the work contributes to stochastic optimal control for nonlinear parabolic PDEs driven by jump noise. The use of Jakubowski's Skorokhod theorem to obtain compactness in a non-metric setting is a technical strength that supports the well-posedness argument.

major comments (1)
  1. [Abstract (final paragraph)] Abstract (final paragraph) and the control-problem section: existence of an optimal solution is obtained by the variational method exploiting convexity of the cost functional. The state equation is the nonlinear evolutionary p-Laplace PDE (p>2) with multiplicative Lévy noise, so the control-to-state map is nonlinear. A quadratic tracking-type cost composed with this map is not convex in general. The manuscript provides no explicit verification or proof that the reduced functional remains convex under the given nonlinearity and noise.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting this point concerning the convexity of the reduced cost functional. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract (final paragraph)] Abstract (final paragraph) and the control-problem section: existence of an optimal solution is obtained by the variational method exploiting convexity of the cost functional. The state equation is the nonlinear evolutionary p-Laplace PDE (p>2) with multiplicative Lévy noise, so the control-to-state map is nonlinear. A quadratic tracking-type cost composed with this map is not convex in general. The manuscript provides no explicit verification or proof that the reduced functional remains convex under the given nonlinearity and noise.

    Authors: We agree that the reduced functional obtained by composing a quadratic tracking-type cost with the nonlinear control-to-state map is not convex in general, and that the manuscript does not contain an explicit verification or proof of convexity for this reduced functional. The original cost is convex jointly in the state and control, but this property does not automatically transfer. In the revised manuscript we will remove the phrasing 'exploiting the convexity property of the cost functional' from the abstract and the control-problem section. We will instead state that existence follows from the direct method by establishing coercivity and weak lower semicontinuity of the reduced functional, which holds because the control-to-state map is continuous from the control space into the state space (in the topologies used for the compactness argument). A short remark will be added explaining why convexity of the reduced functional is not required and is not claimed. revision: yes

Circularity Check

0 steps flagged

No circularity; existence via external variational methods under stated convexity assumption

full rationale

The derivation chain rests on well-posedness via implicit discretization + Jakubowski-Skorokhod (external theorems) followed by a direct-method existence argument that explicitly requires convexity of the cost functional as an input assumption. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation loop, or definitional tautology. Convexity is invoked rather than derived inside the paper, so any failure of convexity is a correctness issue, not a circularity issue. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper rests on standard background results for monotone operators, Lévy processes, and weak convergence in non-metric spaces. No free parameters or new postulated entities appear in the abstract.

axioms (3)
  • standard math The p-Laplace operator satisfies the standard monotonicity and coercivity properties needed for implicit discretization to be well-defined
    Invoked implicitly in the wellposedness step.
  • domain assumption The cost functional is convex
    Explicitly used to obtain existence of an optimal control via variational methods.
  • standard math The Jakubowski version of the Skorokhod theorem applies in the non-metric space setting of the problem
    Cited for passage to the limit after time discretization.

pith-pipeline@v0.9.0 · 5610 in / 1400 out tokens · 36801 ms · 2026-05-25T01:25:39.034642+00:00 · methodology

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Reference graph

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