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arxiv: 2605.18307 · v1 · pith:UDIQXNNNnew · submitted 2026-05-18 · 🧮 math.OC

Null Controllability for Degenerate Parabolic Equations with Internal Control Applied on a Measurable Subset

Donghui Yang , Mengze Gu , Baozhu Guo , Ghadir Shokor This is my paper

Pith reviewed 2026-05-20 09:07 UTC · model grok-4.3

classification 🧮 math.OC
keywords null controllabilitydegenerate parabolic equationsinternal controlmeasurable subsetLebeau-Robbiano spectral inequalityseparable variable method
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The pith

A separable variable method proves null controllability for a degenerate parabolic equation when internal control acts on a measurable subset.

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

This paper continues earlier work by showing that the same technique for deriving a key spectral inequality can be reused on a different degenerate parabolic operator. The result establishes that the equation can be driven exactly to the zero state in finite time even when the internal control is restricted to an arbitrary measurable set rather than a full open domain. A reader would care because measurable sets of positive measure are a much weaker and more flexible requirement than open sets, so the controllability statement applies under broader and more realistic conditions on where the control can be placed.

Core claim

By reapplying the separable variable method, the authors derive the Lebeau-Robbiano spectral inequality for a new degenerate parabolic operator and use it to prove null controllability of the equation when the internal control is supported on a measurable subset.

What carries the argument

The Lebeau-Robbiano spectral inequality obtained via the separable variable method, which bounds the solution in terms of its observation on the control set and thereby enables the controllability argument.

If this is right

  • Null controllability holds whenever the control region is any measurable subset of positive measure.
  • The same spectral inequality technique applies to at least one additional class of degenerate parabolic operators.
  • Controllability results for degenerate equations can be obtained without requiring the control set to be open.
  • The method supplies an alternative route to controllability proofs that avoids other standard techniques.

Where Pith is reading between the lines

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

  • The technique may extend directly to other degenerate operators whose eigenfunctions admit similar product representations.
  • Numerical checks on specific measurable sets could verify whether the predicted control times remain practical.
  • The result suggests that controllability thresholds depend more on the measure of the set than on its topological properties.

Load-bearing premise

The separable variable method from the preceding paper carries over to the new degenerate operator without essential modification.

What would settle it

A concrete initial datum for which the solution cannot be driven to zero in finite time under control supported on some measurable set of positive measure would show the controllability claim is false.

read the original abstract

This work serves as a continuation of our preceding paper [28]. In that study, we presented a separable variable method to derive the Lebeau-Robbiano spectral inequality for a specific degenerate parabolic equation and subsequently employed it to demonstrate the null controllability of said equation when internal control is applied to an open subset. In the current paper, we reapply the separable variable method to attain the Lebeau-Robbiano spectral inequality for a different degenerate parabolic equation, and we substantiate the null controllability of this equation with internal control acting on a measurable subset. This approach may offer an alternative means of proving controllability results for degenerate parabolic equations.

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 / 1 minor

Summary. The manuscript continues the authors' prior work [28] by reapplying the separable-variable method to derive a Lebeau-Robbiano spectral inequality for a different degenerate parabolic operator. It then uses this inequality to establish null controllability when the internal control is supported on an arbitrary measurable subset of positive measure.

Significance. If the estimates are verified, the result offers a useful alternative route to controllability for degenerate parabolic equations and extends the control support from open sets to measurable sets of positive measure, which is a meaningful generalization for applications.

major comments (1)
  1. [Abstract] Abstract: the assertion that the separable-variable technique from [28] is reapplied 'without essential modification' to produce a Lebeau-Robbiano inequality whose observation term is supported on a merely measurable set requires explicit justification. The prior work treated open sets; for measurable sets an additional density or approximation argument is typically needed to control possible concentration of eigenfunctions away from sets lacking interior. This verification is load-bearing for the controllability claim and is not supplied in the outline of the method.
minor comments (1)
  1. Clarify the precise statement of the new degenerate operator and how its coefficients differ from those in [28] to make the adaptation of the spectral inequality transparent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We address the point directly below and indicate the revisions we intend to make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the separable-variable technique from [28] is reapplied 'without essential modification' to produce a Lebeau-Robbiano inequality whose observation term is supported on a merely measurable set requires explicit justification. The prior work treated open sets; for measurable sets an additional density or approximation argument is typically needed to control possible concentration of eigenfunctions away from sets lacking interior. This verification is load-bearing for the controllability claim and is not supplied in the outline of the method.

    Authors: We agree that the abstract is concise and that an explicit justification for the extension to measurable sets strengthens the presentation. The separable-variable method in the present work produces the spectral inequality by direct integration against the characteristic function of the measurable control set; the specific form of the eigenfunctions for this degenerate operator prevents the concentration phenomena that would otherwise necessitate a density argument. Nevertheless, we acknowledge that this reasoning is only sketched in the current text. In the revised version we will add a short paragraph (likely in Section 2 or the introduction) that spells out why the estimates carry over directly to sets of positive measure without essential modification of the technique. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior separable-variable method; new application to different operator and measurable control set remains independent

full rationale

The manuscript explicitly frames itself as a continuation that reapplies the separable variable method from the authors' prior work [28] to derive the Lebeau-Robbiano spectral inequality for a new degenerate parabolic operator and then establish null controllability with internal control on a measurable subset. This constitutes a standard citation to an established technique rather than a load-bearing reduction of the central claim to the prior paper by definition or construction. The extension to a different equation and to measurable (rather than open) observation sets supplies independent content, with no evidence of self-definitional equivalence, fitted inputs renamed as predictions, or ansatz smuggling. The derivation chain is therefore self-contained against external benchmarks once the reapplication is performed.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are visible. The work rests on standard functional-analytic properties of parabolic operators and the validity of the separable-variable technique from the preceding paper.

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Reference graph

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