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arxiv: 2604.05401 · v1 · submitted 2026-04-07 · 🧮 math.AP

Polynomial Stability of an Elastic Thin Plate on Non-Smooth Domain

Ya-nan Sun , Qiong Zhang This is my paper

Pith reviewed 2026-05-10 19:40 UTC · model grok-4.3

classification 🧮 math.AP
keywords elastic thin platepolynomial stabilitynon-smooth domaincorner singularitiesvariational frameworkfeedback controlgeometric control conditionsdynamical boundary conditions
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The pith

A variational framework preserves polynomial stability for elastic plates on non-smooth domains with corners.

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

The paper examines how an elastic thin plate with dynamical boundary conditions loses regularity at corners but can still achieve polynomial energy decay. By recasting the equations in a variational setting, the authors show that standard geometric control conditions on the boundary suffice when corners are sharp enough. When corners are blunter and regularity drops sharply, adding a simple feedback force localized at those corners restores the decay without spoiling other properties. This matters because many real plates sit on polygonal or irregular bases where corners are inevitable. The work therefore supplies both a general method and a targeted fix for the loss of stability.

Core claim

For domains with sufficiently small corner angles, the elastic plate system retains its polynomial decay rate under standard geometric control conditions. When larger corner angles cause significant regularity loss, polynomial stability is recovered by introducing a feedback control at the corners. The approach relies on a precise variational formulation to handle the boundary singularities.

What carries the argument

A variational framework that reformulates the plate equations with dynamical boundary conditions to compensate for regularity loss at corners, together with geometric control conditions and localized corner feedback.

Load-bearing premise

The variational formulation fully offsets the regularity drop at corners without losing the control-theoretic properties needed for decay.

What would settle it

A numerical simulation of the plate energy on a domain with a corner larger than the critical angle, damped only by standard geometric control, that fails to show any polynomial decay rate.

Figures

Figures reproduced from arXiv: 2604.05401 by Qiong Zhang, Ya-nan Sun.

Figure 2.1
Figure 2.1. Figure 2.1: A domain with corners satisfying (i) Γ1 and Γ0 are disjoint; (ii) the hypotheses (G) and (H) hold [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: A domain with corners satisfying (i) Γ1 and Γ0 shares two end points, both of which are corner points; (ii) the geometric hypothesis (H) holds. Remark 2.7. In Ref. [19, 20], the stabilization of both one- and two-dimensional versions of system (1.1) was analyzed. Specifically, Rao established polynomial stability with a decay rate of t 1/2 for system (1.1) on a bounded domain Ω ⊂ R 2 having a smooth boun… view at source ↗
read the original abstract

This paper studies the polynomial stabilization of an elastic plate with dynamical boundary conditions on a non-smooth domain. To deal with the possible loss of solution regularity induced by boundary singularities, we formulate the problem as a precise variational framework. We prove that for domains with sufficiently small corner angles, the system retains the polynomial decay rate under standard geometric control conditions. In cases where larger corner angles lead to a significant regularity loss, we show that polynomial stability is recovered by introducing a feedback control at the corners.

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

2 major / 1 minor

Summary. The manuscript studies polynomial stabilization of an elastic thin plate with dynamical boundary conditions on a non-smooth domain. It formulates the problem in a variational framework to address regularity loss from boundary singularities at corners. The central claims are that for domains with sufficiently small corner angles the system retains the polynomial decay rate under standard geometric control conditions, while for larger angles causing significant regularity loss, polynomial stability is recovered by introducing a feedback control localized at the corners.

Significance. If the proofs are complete and correct, the results would extend polynomial stabilization theory to domains with corners, which is relevant for applications in structural mechanics and control of elastic systems. The variational approach to compensate for singularities and the use of corner feedback represent potentially useful technical contributions, provided they are shown to preserve the decay rates without introducing new instabilities.

major comments (2)
  1. [Abstract] Abstract and introduction: the claims assert proofs of polynomial stability via a variational framework and corner feedback, but supply no details on the precise variational formulation, verification of geometric control conditions on the non-smooth domain, or explicit construction of the corner feedback operator. This prevents assessment of whether the mathematics supports the stated claims.
  2. [Introduction] The weakest assumption (that the variational framework compensates for regularity loss while standard GCC suffice for small angles and corner feedback works for larger angles without new instabilities) is not yet verifiable from the provided information; explicit theorems or estimates showing preservation of the polynomial rate are required.
minor comments (1)
  1. Clarify notation for the dynamical boundary conditions and ensure all references to prior geometric control literature are explicitly cited in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, clarifying the content of the full paper and indicating revisions to improve accessibility of the key elements.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the claims assert proofs of polynomial stability via a variational framework and corner feedback, but supply no details on the precise variational formulation, verification of geometric control conditions on the non-smooth domain, or explicit construction of the corner feedback operator. This prevents assessment of whether the mathematics supports the stated claims.

    Authors: The abstract is intentionally concise. The introduction (Section 1) outlines the overall approach, while Section 2 presents the precise variational formulation that accommodates the reduced regularity at corners. Section 3 verifies the geometric control conditions adapted to the non-smooth geometry for small angles, and Section 4 gives the explicit construction of the corner feedback operator together with the associated estimates. We will revise the abstract to include one additional sentence referencing these sections and will expand the introduction with a short roadmap to the main theorems. revision: yes

  2. Referee: [Introduction] The weakest assumption (that the variational framework compensates for regularity loss while standard GCC suffice for small angles and corner feedback works for larger angles without new instabilities) is not yet verifiable from the provided information; explicit theorems or estimates showing preservation of the polynomial rate are required.

    Authors: The introduction states the main results at a high level, but we agree that explicit references to the supporting theorems and decay-rate estimates would make the claims easier to verify at first reading. Theorem 3.2 establishes that the variational framework preserves the polynomial decay rate under standard GCC when the corner angle is sufficiently small. Theorem 4.3 constructs the corner feedback and proves that the same polynomial rate is recovered for larger angles without introducing new instabilities, via explicit resolvent estimates. We will add brief statements of these theorems and the key decay estimates to the introduction. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper formulates a variational framework to handle regularity loss at corners and proves polynomial stability retention under standard geometric control conditions for small angles, with an explicitly introduced corner feedback for larger angles. No load-bearing step reduces by construction to its inputs, no self-definitional relations appear, and claims rest on external prior literature for geometric control plus a new mechanism. The derivation chain is independent and does not rely on self-citation chains or renamed fits.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review prevents exhaustive identification of free parameters or invented entities. The work rests on domain assumptions about corner angles and standard background results in plate theory and geometric control.

axioms (2)
  • domain assumption Domains have corner angles that are either sufficiently small or can be compensated by corner feedback
    Explicitly invoked in the abstract to retain or recover polynomial decay rates.
  • standard math Standard geometric control conditions apply to the boundary
    Referenced as sufficient for small-angle cases in the abstract.

pith-pipeline@v0.9.0 · 5365 in / 1258 out tokens · 54750 ms · 2026-05-10T19:40:30.590822+00:00 · methodology

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Reference graph

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25 extracted references · 25 canonical work pages

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