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arxiv: 2604.25909 · v1 · submitted 2026-04-28 · 🧮 math.OC

H² Stabilization of the 2-D and 3-D Heat Equation via Modal Decomposition

Mohamed Amine Ouchdiri , Mohamed-Camil Belhadjoudja , Mohamed Maghenem , Saad Benjelloun , Adnane Saoud This is my paper

Pith reviewed 2026-05-07 15:15 UTC · model grok-4.3

classification 🧮 math.OC
keywords modal decompositionH2 stabilizationheat equationboundary controlexponential stabilityparabolic PDEmaximum norm
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The pith

A modal-decomposition boundary controller achieves H² exponential stability for the 2-D and 3-D heat equation.

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

The paper proves that the modal-decomposition controller previously shown to deliver H¹ exponential stability for the heat equation in two and three dimensions also guarantees H² exponential stability. This matters because H¹ stability alone does not guarantee boundedness or convergence of the state in the maximum norm in dimensions greater than one, while H² stability does. The argument rewrites the Laplacian term appearing in the H² norm as a linear combination of the state and its time derivative. Since the H¹ norm already controls the L² norm of the state and the controller bounds the L² norm of the time derivative, the H² norm follows directly.

Core claim

The modal-decomposition based controller guarantees, not only H¹ exponential stability, but also H² exponential stability. This implies, in particular, boundedness and asymptotic convergence of the state to zero in the sense of the max norm.

What carries the argument

Rewriting the Laplacian of the state as a linear combination of the state and its time derivative to transfer H¹ and time-derivative bounds onto the H² norm.

If this is right

  • The closed-loop state remains bounded and converges to zero in the maximum norm.
  • The same full-state feedback law works for the stronger H² norm without modification.
  • Exponential decay holds simultaneously in H¹ and H² for both two- and three-dimensional domains.
  • Maximum-norm convergence follows as a direct corollary of H² stability.

Where Pith is reading between the lines

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

  • The same rewriting technique could be tested on other linear parabolic equations to obtain higher-norm stability from lower-norm controllers.
  • If the time-derivative estimate can be obtained from output measurements, the result would extend to output-feedback designs.
  • The approach supplies a concrete route to norm-equivalence arguments needed when physical quantities such as pointwise temperature must be bounded.

Load-bearing premise

The Laplacian of the state can be rewritten as a linear combination of the state and its time derivative in a manner that lets the H² norm be bounded once the H¹ norm and the L² norm of the time derivative are controlled.

What would settle it

A closed-loop simulation of the 2-D or 3-D heat equation in which the H¹ norm and time-derivative L² norm decay exponentially but the H² norm does not.

Figures

Figures reproduced from arXiv: 2604.25909 by Adnane Saoud, Mohamed Amine Ouchdiri, Mohamed-Camil Belhadjoudja, Mohamed Maghenem, Saad Benjelloun.

Figure 1
Figure 1. Figure 1: The state u of Σ in closed loop, at times t = 0, 0.5, 1.5, 4. The domain is the disk Ω = {(x, y) ∈ R 2 : |(x, y)| < 2}. Combining (43), (44), and (47), there exist Cs > 0 and 0 < σ ∗ ≤ σ such that X n≥N+1 |w˙ n(t)| 2 ≤ C 2 s ∥uo∥ 2 H2(Ω) e −2σ ∗ t ∀ t ≥ 0. (48) Combining (36) and (48) and using Parseval’s identity yields (32) with Cw := p C2 u + C2 s , which concludes the proof. V. SIMULATIONS RESULTS In t… view at source ↗
Figure 3
Figure 3. Figure 3: The function t 7→ ∥u(·, t)∥L∞(Ω) in closed loop (blue) vs in open loop (red), in the 2-D case view at source ↗
Figure 2
Figure 2. Figure 2: The functions t 7→ ∥u(·, t)∥H2(Ω) (top) and t 7→ ∥u(·, t)∥L∞(Ω) (bottom) in closed loop, in the 2-D case. (γ1, . . . , γ5) = (6.17, 7.17, 8.17, 9.17, 10.17). In view at source ↗
Figure 4
Figure 4. Figure 4: shows the state of the closed-loop system at times t = 0, 0.5, 1.5, 4. The state converges to zero at every spatial location view at source ↗
Figure 5
Figure 5. Figure 5: The functions t 7→ ∥u(·, t)∥H2(Ω) (top) and t 7→ ∥u(·, t)∥L∞(Ω) (bottom) for Σ in closed loop, and in the 3- D case. [3] T. L. Bergman, A. S. Lavine, F. P. Incropera, and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 8th ed. Wiley, 2020. [4] U.S. Environmental Protection Agency (EPA), “2024 Appendix W Final Rule,” Support Center for Regulatory Atmospheric Modeling (SCRAM), 2024. [Online]. Available… view at source ↗
read the original abstract

Boundary controllers have been recently proposed in the literature, via modal decomposition, to achieve $H^1$ stabilization of linear parabolic equations in two and three dimensions. In one dimension ($1$-D), $H^1$ exponential stability is known to imply boundedness and asymptotic convergence of the state to zero in the sense of the max norm. However, in two ($2$-D) and three dimensions ($3$-D), this implication does not systematically hold. In this paper, focusing on the full-state feedback case, our objective is to prove that the modal-decomposition based controller in \cite{Munteanu2017IJC} guarantees, not only $H^1$ exponential stability, but also $H^2$ exponential stability. This implies, in particular, boundedness and asymptotic convergence of the state to zero in the sense of the max norm. Our approach consists in rewriting the Laplacian of the state, required in the $H^2$ norm, as a linear combination of the state and its time derivative. The $L^2$ norm of the state being bounded by the $H^1$ norm, we only analyze the $L^2$ norm of the time derivative of the state.

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 paper claims that the modal-decomposition boundary feedback controller from Munteanu2017IJC, already known to yield H^1 exponential stability for the 2-D and 3-D heat equation, also yields H^2 exponential stability. The argument proceeds by rewriting the Laplacian appearing in the H^2 norm as a linear combination of the state u and its time derivative u_t (via the closed-loop PDE), bounding the L^2 norm of u by the H^1 norm, and then establishing decay of ||u_t||_L2; the resulting H^2 bound implies convergence in the maximum norm.

Significance. If the central estimates close, the result supplies the missing link between H^1 and uniform-norm stability for modal controllers in dimensions greater than one, extending the 1-D situation and strengthening the practical utility of the existing feedback law without redesign. The approach is economical, relying only on the prior controller and standard embeddings.

major comments (2)
  1. [Abstract and main stability theorem] Abstract and the section deriving the H^2 estimate: the rewriting of Δu as a linear combination of u and u_t is asserted to suffice for ||u||_H2 ≤ C(||u||_H1 + ||u_t||_L2), yet the manuscript supplies no explicit constant or verification that the time-derivative term remains bounded under the closed-loop modal dynamics in 2-D/3-D; without this, the implication from H^1 to H^2 stability does not follow.
  2. [Proof of H^2 stability] The section treating the closed-loop operator: modal boundary feedback renders the domain of the Laplacian non-homogeneous; the elliptic regularity estimate therefore contains trace terms on the boundary whose norms must be absorbed into the H^1 term. The paper does not show that these traces are controlled independently of the modal gains or that the constant C remains uniform in dimension.
minor comments (1)
  1. [Introduction] The citation to Munteanu2017IJC should be accompanied by a brief recall of the precise form of the modal feedback law and the H^1 decay rate obtained there, to make the present argument self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions that will be incorporated to clarify the H^2 stability argument.

read point-by-point responses

  1. Referee: [Abstract and main stability theorem] Abstract and the section deriving the H^2 estimate: the rewriting of Δu as a linear combination of u and u_t is asserted to suffice for ||u||_H2 ≤ C(||u||_H1 + ||u_t||_L2), yet the manuscript supplies no explicit constant or verification that the time-derivative term remains bounded under the closed-loop modal dynamics in 2-D/3-D; without this, the implication from H^1 to H^2 stability does not follow.

    Authors: The inequality ||u||_H2 ≤ C(||u||_H1 + ||u_t||_L2) follows directly from the definition of the Sobolev norm and the closed-loop PDE u_t = Δu (in the interior), so that the Laplacian term is replaced by the time derivative. The constant C is the standard one arising from norm equivalence on bounded domains with smooth boundary and does not need to be computed explicitly for the purpose of establishing exponential decay. Regarding verification that ||u_t||_L2 decays, the manuscript already indicates that the L^2 norm of the time derivative is analyzed via the same modal decomposition; however, to make this fully explicit, we will add a dedicated lemma in the revision deriving the dynamics satisfied by u_t and showing its exponential decay in L^2 using an analogous Lyapunov functional to the one employed for the H^1 estimate. This will confirm that the H^2 norm decays exponentially as claimed. revision: yes

  2. Referee: [Proof of H^2 stability] The section treating the closed-loop operator: modal boundary feedback renders the domain of the Laplacian non-homogeneous; the elliptic regularity estimate therefore contains trace terms on the boundary whose norms must be absorbed into the H^1 term. The paper does not show that these traces are controlled independently of the modal gains or that the constant C remains uniform in dimension.

    Authors: We agree that the modal boundary feedback renders the domain of the Laplacian non-homogeneous, so that a direct elliptic regularity estimate on u would involve boundary trace terms. Our approach circumvents a direct application of elliptic regularity to u by substituting Δu = u_t from the PDE, thereby shifting the estimate to the decay of u_t (already controlled in L^2). Nevertheless, to address the trace terms rigorously, we will add a clarifying paragraph in the revision explaining that any residual boundary traces can be absorbed into the H^1 norm of u via the standard trace theorem. The modal gains are fixed by the controller design (independent of the state norm) and the absorption constants depend only on the domain geometry, not on the specific gains or the spatial dimension; uniformity in dimension follows from the fact that the modal decomposition and Lyapunov analysis hold uniformly for the 2-D and 3-D cases considered. This clarification will be inserted in the section on the closed-loop operator. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation; H2 bound obtained via external controller and standard elliptic estimate

full rationale

The paper cites an external reference (Munteanu2017IJC) for the modal controller that already achieves H1 exponential stability, then applies the closed-loop PDE to rewrite the Laplacian term appearing in the H2 norm as a linear combination of the state and its time derivative. This is followed by bounding the L2 norm of the time derivative separately and invoking standard Sobolev embeddings. No step reduces the target H2 result to a fitted parameter, a self-referential definition, or a load-bearing self-citation chain; the derivation remains independent of the claimed conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard functional-analytic properties of the Laplacian eigenbasis and Sobolev embeddings rather than new postulates.

axioms (2)
  • standard math The eigenfunctions of the Laplacian form a complete orthogonal basis allowing modal decomposition of the state and its derivatives.
    Invoked when the controller is constructed via modal decomposition and when the Laplacian is expressed in the same basis.
  • standard math Sobolev embedding theorems relate the H2 norm to the maximum norm on the given domain.
    Used to conclude max-norm convergence from H2 exponential stability.

pith-pipeline@v0.9.0 · 5549 in / 1361 out tokens · 61761 ms · 2026-05-07T15:15:58.431134+00:00 · methodology

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