Quantitative Fredholm backstepping and rapid stabilization

Amaury Hayat; Christophe Zhang; Ludovick Gagnon; Shengquan Xiang; Swann Marx

arxiv: 2605.17941 · v1 · pith:O3Z2M45Nnew · submitted 2026-05-18 · 🧮 math.OC · math.AP· math.FA

Quantitative Fredholm backstepping and rapid stabilization

Ludovick Gagnon , Amaury Hayat , Swann Marx , Shengquan Xiang , Christophe Zhang This is my paper

Pith reviewed 2026-05-20 09:51 UTC · model grok-4.3

classification 🧮 math.OC math.APmath.FA

keywords Fredholm backsteppingrapid stabilizationquantitative estimatesself-adjoint operatorsskew-adjoint operatorsnull controllabilityinfinite-dimensional systems

0 comments

The pith

An explicit isomorphism constructs Fredholm backstepping transformations for self-adjoint operators of order greater than 1.

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

The paper proves existence of a Fredholm backstepping transformation T for self-adjoint and skew-adjoint operators A of order strictly greater than 1, under suitable assumptions on A and the control operator B. The construction identifies the underlying isomorphism explicitly, which removes the need for compactness arguments or Riesz-basis methods. This identification produces sharp, quantitative bounds on the operator norms of T and its inverse that depend directly on the target decay rate λ. The bounds immediately yield quantitative rapid stabilization and small-time null controllability results that apply to a broad class of infinite-dimensional linear systems. A sympathetic reader cares because the estimates turn abstract stabilization claims into concrete, usable rates for systems governed by PDEs.

Core claim

Under suitable assumptions on the operator A and the possibly unbounded control operator B, we prove the existence of a Fredholm backstepping transformation for operators of order strictly greater than 1. This work overcomes two major limitations of the classical Fredholm backstepping framework. One of the main contributions is the explicit identification of the underlying isomorphism used in the construction of the transformation T, thereby bypassing the compactness arguments and Riesz basis mechanisms traditionally used in the literature. This explicit structure enables us to derive quantitative and sharp estimates for ||T|| and ||T^{-1}|| with respect to the decay rate λ. As a consequence

What carries the argument

The explicit underlying isomorphism that defines the Fredholm backstepping transformation T and supplies the norm estimates with respect to the decay rate λ.

Load-bearing premise

The operator A must be self-adjoint or skew-adjoint and satisfy the specific assumptions that allow the explicit isomorphism to be constructed without compactness or Riesz-basis arguments.

What would settle it

Compute the transformation explicitly on a concrete example such as the one-dimensional heat equation with boundary control and check whether the observed operator norm of T grows exactly as predicted by the paper's bound when the target decay rate λ is increased.

read the original abstract

In this paper, we address the existence of Fredholm backstepping transformations for self-adjoint and skew-adjoint operators $A$. Under suitable assumptions on the operator $A$ and the possibly unbounded control operator $B$, we prove the existence of a Fredholm backstepping transformation for operators of order strictly greater than $1$. This work overcomes two major limitations of the classical Fredholm backstepping framework. One of the main contributions is the explicit identification of the underlying isomorphism used in the construction of the transformation $T$, thereby bypassing the compactness arguments and Riesz basis mechanisms traditionally used in the literature. This explicit structure enables us to derive quantitative and sharp estimates for $\|T\|_{\mathcal{L}(H;H)}$ and $\|T^{-1}\|_{\mathcal{L}(H;H)}$ with respect to the decay rate $\lambda$. As a consequence, we obtain quantitative rapid stabilization and small-time null controllability results for a broad class of operators.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit construction of the Fredholm isomorphism for backstepping on self-adjoint operators, yielding sharp quantitative norm bounds in terms of the decay rate λ.

read the letter

The main point is that this paper constructs an explicit Fredholm isomorphism for backstepping transformations on self-adjoint and skew-adjoint operators, leading to sharp quantitative bounds on the norms of the transformation and its inverse in terms of the target decay rate λ. This is new compared to the usual compactness-based approaches. They do a good job showing how this explicit structure works for operators of order greater than 1 and how it gives quantitative rapid stabilization and small-time null controllability results. The derivation avoids Riesz basis arguments and instead uses a direct definition that yields the estimates directly. The assumptions on the operator A and the control B are laid out clearly, and the stress test confirms the central claims hold without inconsistencies. The potential soft spot is that the class of operators is restricted to self-adjoint or skew-adjoint ones with suitable properties, so it does not apply to arbitrary generators. But within that setting the results look solid, and the quantitative aspect is a real improvement over existence-only results. No signs of circular reasoning or unfalsifiable claims here. This kind of work is for people in infinite-dimensional systems theory and PDE control who care about explicit constants for stabilization rates. A reader looking for practical bounds in applications would get something out of the estimates. I think it deserves peer review because the quantitative estimates address a genuine limitation in the literature and the math appears carefully done.

Referee Report

0 major / 2 minor

Summary. The manuscript proves the existence of a Fredholm backstepping transformation for self-adjoint and skew-adjoint operators A of order strictly greater than 1. Under suitable assumptions on A and the possibly unbounded control operator B, the authors construct an explicit isomorphism T that bypasses compactness and Riesz-basis arguments. This explicit structure yields quantitative and sharp estimates for ||T|| and ||T^{-1}|| in terms of the target decay rate λ, from which they derive quantitative rapid stabilization and small-time null controllability results.

Significance. If the central claims hold, the work is significant for mathematical control theory. The explicit construction of the isomorphism and the resulting quantitative norm bounds with respect to λ constitute a clear advance over prior qualitative Fredholm backstepping results. The paper ships an explicit, non-compactness-based derivation together with sharp estimates that directly support the stabilization and controllability conclusions.

minor comments (2)

[§2.2] §2.2: the precise meaning of 'order strictly greater than 1' for the operator A is invoked repeatedly but defined only implicitly through the spectral assumptions; an explicit definition or reference to the precise functional setting would improve readability.
[Theorem 3.1] Theorem 3.1: the statement of the quantitative bound on ||T^{-1}|| is given in terms of λ, but the dependence on the constants appearing in the assumptions on B is not tracked explicitly; adding this dependence would make the sharpness claim easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report does not list any specific major comments under the MAJOR COMMENTS section, so we have no individual points to address point-by-point. We will incorporate any minor editorial or presentational improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a pure existence proof that constructs an explicit Fredholm isomorphism T for self-adjoint or skew-adjoint operators A of order >1 under stated assumptions on A and the control operator B. Quantitative bounds on ||T|| and ||T^{-1}|| are obtained directly from the construction as functions of the target decay rate λ; no data-fitting, self-referential predictions, or load-bearing self-citations appear in the derivation chain. The argument bypasses compactness and Riesz-basis arguments by direct definition rather than reducing to prior results by the same authors. The derivation is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted beyond standard functional-analysis background.

axioms (1)

domain assumption A is self-adjoint or skew-adjoint with suitable domain and spectrum properties
Invoked to guarantee the Fredholm transformation exists for order >1

pith-pipeline@v0.9.0 · 5704 in / 1233 out tokens · 24811 ms · 2026-05-20T09:51:52.813986+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We observe that these operators can be represented through an infinite-dimensional Cauchy matrix... ˜T^{-1}_{i,j} = λ²/(λ_j - λ_i - λ) ∏_{m≠i} (1 + λ/(λ_i - λ_m)) ∏_{n≠j} (1 - λ/(λ_j - λ_n))
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.3 (Quantitative e^{c λ^{1/α}}-type cost) ... ||T_λ|| + ||T_λ^{-1}|| ≤ C e^{c λ^{1/α}}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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[1] [1]

Balogh and M

A. Balogh and M. Krsti´ c. Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability.Eur. J. Control, 8(3):165–175, 2002

work page 2002

[2] [2]

Barbu.Controllability and stabilization of parabolic equations, volume 90 ofProgress in Nonlinear Differential Equations and their Applications

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work page 2018

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work page 2003

[4] [4]

Breiten, K

T. Breiten, K. Kunisch, and S. Rodrigues. Feedback stabilization to nonstationary solutions of a class of reaction diffusion equations of FitzHugh-Nagumo type.SIAM J. Control Optim., 55(4):2684–2713, 2017. 43

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J.-M. Coron, L. Hu, G. Olive, and P. Shang. Boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space.J. Differential Equations, 271:1109–1170, 2021

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J.-M. Coron and Q. L¨ u. Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right.J. Math. Pures Appl. (9), 102(6):1080–1120, 2014

work page 2014

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Coron and Q

J.-M. Coron and Q. L¨ u. Fredholm transform and local rapid stabilization for a Kuramoto–Sivashinsky equation. J. Differential Equations, 259(8):3683–3729, 2015

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Coron and E

J.-M. Coron and E. Tr´ elat. Global steady-state controllability of one-dimensional semilinear heat equations. SIAM J. Control Optim., 43(2):549–569, 2004

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Stabilization of the linearized water tank system.Arch

Jean-Michel Coron, Amaury Hayat, Shengquan Xiang, and Christophe Zhang. Stabilization of the linearized water tank system.Arch. Ration. Mech. Anal., 244(3):1019–1097, 2022

work page 2022

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Jean-Michel Coron and Hoai-Minh Nguyen. Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach.Archive for Rational Mechanics and Analysis, 225:993–1023, 2017

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Small-time global stabilization of the viscous burgers equation with three scalar controls.Journal de Math´ ematiques Pures et Appliqu´ ees, 151:212–256, 2021

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Fredholm backstepping control of coupled linear parabolic pdes with input and output delays.IEEE Transactions on Automatic Control, 65(7):3128–3135, 2019

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work page 2021

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Fredholm transformation on Laplacian and rapid stabilization for the heat equation.J

Ludovick Gagnon, Amaury Hayat, Shengquan Xiang, and Christophe Zhang. Fredholm transformation on Laplacian and rapid stabilization for the heat equation.J. Funct. Anal., 283(12):Paper No. 109664, 67, 2022

work page 2022

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Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves.Ann

Ludovick Gagnon, Amaury Hayat, Shengquan Xiang, and Christophe Zhang. Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves.Ann. Inst. Fourier, 75(6):2423– 2500, 2025

work page 2025

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BZ Guo and Heiko J Zwart. Riesz spectral systems. 2001

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Hayat and P

A. Hayat and P. Shang. Exponential stability of density-velocity systems with boundary conditions and source term for theH 2 norm.J. Math. Pures Appl. (9), 153:187–212, 2021

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Fredholm backstepping and rapid stabilization of general linear systems

Amaury Hayat and Epiphane Loko. Fredholm backstepping and rapid stabilization of general linear systems. 2024

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M. Krsti´ c and A. Smyshlyaev.Boundary control of PDEs, volume 16 ofAdvances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. A course on backstepping designs

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K. K. Kunisch, G. Wang, and H. Yu. Frequency-domain criterion on the stabilizability for infinite-dimensional linear control systems.J. Math. Pures Appl., 196:Paper No. 103690, 35, 2025

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Rapid stabilization of a degenerate parabolic equation using a backstepping approach: the case of a boundary control acting at the degeneracy.Math

Pierre Lissy and Claudia Moreno. Rapid stabilization of a degenerate parabolic equation using a backstepping approach: the case of a boundary control acting at the degeneracy.Math. Control Relat. Fields, 14(3):1007– 1032, 2024

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H. Liu, G. Wang, Y. Xu, and H. Yu. Characterizations of complete stabilizability.SIAM J. Control Optim., 44 60:2040–2069, 2022

work page 2040

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Stabilization of control systems associated with a strongly continuous group.ESAIM Control Optim

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