pith. sign in

arxiv: 2508.21605 · v3 · submitted 2025-08-29 · 🧮 math.AP

F-equivalence for parabolic systems and applications to the stabilization of nonlinear PDE

Vincent Boulard , Amaury Hayat This is my paper

Pith reviewed 2026-05-18 20:40 UTC · model grok-4.3

classification 🧮 math.AP MSC 93B5235K5593C20
keywords F-equivalenceparabolic systemsrapid stabilizationfeedback controlapproximate controllabilitysemilinear PDENavier-Stokes equationsheat equation
0
0 comments X

The pith

Parabolic control systems admit an F-equivalence pair that transforms them into exponentially stable systems with arbitrarily large decay rates under optimal conditions.

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

The paper develops an extended framework for F-equivalence of parabolic operators to overcome limitations of prior approaches in infinite-dimensional settings. It proves existence of a pair (T, K) that renders a given system (A, B) equivalent to one that decays exponentially at any prescribed rate. This construction directly supplies feedback operators that achieve rapid stabilization for semilinear parabolic PDEs, including those in spatial dimensions greater than one. Uniqueness of the pair holds exactly when the original system is approximately controllable. Concrete applications are given to the heat equation, the Kuramoto-Sivashinsky equation, the Navier-Stokes equations, and the quasilinear heat equation.

Core claim

Our main result establishes optimal conditions for the existence of an F-equivalence pair (T,K) for a given parabolic control system (A,B). We introduce an extended framework for F-equivalence of parabolic operators, addressing key limitations of existing approaches, and we prove that the pair (T,K) is unique if and only if (A,B) is approximately controllable. As a consequence, this provides a method to construct feedback operators for the rapid stabilization of semilinear parabolic systems, possibly multi-dimensional in space.

What carries the argument

The F-equivalence pair (T,K), where T is a transformation operator and K is a feedback operator that together convert the original parabolic system into an exponentially stable target system.

If this is right

  • Feedback operators can be constructed for rapid stabilization of semilinear parabolic systems in any spatial dimension.
  • The method applies directly to the heat equation, Kuramoto-Sivashinsky equation, Navier-Stokes equations, and quasilinear heat equation.
  • Uniqueness of the stabilizing pair is equivalent to approximate controllability of the original system.
  • The framework extends previous finite-dimensional and one-dimensional results to multi-dimensional parabolic operators.

Where Pith is reading between the lines

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

  • Similar structural conditions might allow the same equivalence approach for other classes of evolution equations beyond the parabolic case.
  • The link between uniqueness and approximate controllability could be tested numerically on discretized multi-dimensional domains.
  • Explicit construction of T and K may yield new numerical schemes for computing stabilizing feedbacks in practice.

Load-bearing premise

The parabolic operator A and control operator B must satisfy the structural conditions required by the extended F-equivalence framework, especially for systems with spatial dimension larger than one.

What would settle it

A parabolic system (A,B) that meets the structural conditions but for which no F-equivalence pair (T,K) exists, or a system that is not approximately controllable yet admits a unique F-equivalence pair.

read the original abstract

We consider the $F$-equivalence problem for parabolic systems: under which conditions a control system, governed by a parabolic operator $A$ and a control operator $B$, can be made equivalent to an exponentially stable system with arbitrarily large decay rate through an appropriate control feedback law? While this problem has been resolved for finite-dimensional systems fifty years ago, good conditions for infinite-dimensional systems remain a challenge, especially for systems in spatial dimension larger than one. Our main result establishes optimal conditions for the existence of an $F$-equivalence pair $(T,K)$ for a given parabolic control system $(A,B)$. We introduce an extended framework for $F$-equivalence of parabolic operators, addressing key limitations of existing approaches, and we prove that the pair $(T,K)$ is unique if and only if $(A,B)$ is approximately controllable. As a consequence, this provides a method to construct feedback operators for the rapid stabilization of semilinear parabolic systems, possibly multi-dimensional in space. We provide several illustrative examples, including the rapid stabilization of the heat equation, the Kuramoto-Sivashinsky equation, the Navier-Stokes equations and the quasilinear heat equation.

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 / 2 minor

Summary. The paper considers the F-equivalence problem for parabolic control systems (A,B): it seeks conditions under which there exists a pair (T,K) transforming the system into an exponentially stable one with arbitrarily large decay rate. The main result claims optimal conditions for existence of such a pair via an extended framework for F-equivalence of parabolic operators that overcomes limitations of prior approaches. It further proves that (T,K) is unique if and only if (A,B) is approximately controllable. As applications, the framework yields feedback operators for rapid stabilization of semilinear parabolic systems, including multi-dimensional examples such as the heat equation, Kuramoto-Sivashinsky equation, Navier-Stokes equations, and quasilinear heat equation.

Significance. If the central claims hold under the stated hypotheses, the work would advance the theory of stabilization for infinite-dimensional parabolic systems by providing an extended F-equivalence framework applicable in spatial dimensions d>1, where existing methods face limitations. The uniqueness characterization tied to approximate controllability supplies a clean criterion, and the constructive aspect for feedback laws on nonlinear systems (Navier-Stokes, quasilinear heat) could have concrete implications for control design. The manuscript supplies machine-checked elements only in the finite-dimensional reduction; the infinite-dimensional proofs rely on standard functional-analytic tools.

major comments (2)
  1. [§3 (main existence theorem)] §3 (main existence theorem): the structural conditions on the parabolic operator A and control operator B required by the extended F-equivalence framework are invoked to guarantee existence of (T,K) but are not explicitly verified for the multi-dimensional examples (Navier-Stokes and quasilinear heat equation). If these conditions encode extra spectral or boundary-regularity requirements that fail for standard boundary-controlled systems in d>1, both the existence claim and the subsequent uniqueness statement become conditional rather than optimal.
  2. [Uniqueness direction (Theorem on iff with approximate controllability)] Uniqueness direction (Theorem on iff with approximate controllability): the argument that (T,K) is unique precisely when (A,B) is approximately controllable rests on the same structural hypotheses as the existence result. A direct check or counter-example showing whether the hypotheses hold for at least one d>1 boundary-controlled system would be needed to confirm that the iff statement is not an artifact of the framework's restrictions.
minor comments (2)
  1. [Notation] Notation for the feedback operator K and the transformation T is introduced without a consolidated table of symbols; a short notation summary would improve readability for readers outside the immediate subfield.
  2. [Abstract] The abstract states that the pair (T,K) is unique iff (A,B) is approximately controllable, yet the precise definition of 'approximate controllability' used in the infinite-dimensional setting is only referenced rather than restated; a one-sentence reminder would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The suggestions help clarify the applicability of our framework to multi-dimensional systems. We address each major comment below and will incorporate explicit verifications in the revised version to strengthen the presentation.

read point-by-point responses

  1. Referee: §3 (main existence theorem): the structural conditions on the parabolic operator A and control operator B required by the extended F-equivalence framework are invoked to guarantee existence of (T,K) but are not explicitly verified for the multi-dimensional examples (Navier-Stokes and quasilinear heat equation). If these conditions encode extra spectral or boundary-regularity requirements that fail for standard boundary-controlled systems in d>1, both the existence claim and the subsequent uniqueness statement become conditional rather than optimal.

    Authors: We appreciate this observation. Our structural conditions are formulated precisely to apply to standard parabolic boundary control systems in dimensions d>1 without introducing extraneous spectral or boundary-regularity assumptions beyond those required for well-posedness of the linear and semilinear problems. For the Navier-Stokes equations (Section 5.3) and quasilinear heat equation (Section 5.4), the examples are chosen so that the conditions hold under the usual Dirichlet or Neumann boundary controls. In the revised manuscript we will add an explicit verification subsection (new Section 3.4 or Appendix) that checks each structural hypothesis item-by-item for these two systems, confirming that no additional restrictions are imposed and that the existence result remains optimal as stated. revision: yes

  2. Referee: Uniqueness direction (Theorem on iff with approximate controllability): the argument that (T,K) is unique precisely when (A,B) is approximately controllable rests on the same structural hypotheses as the existence result. A direct check or counter-example showing whether the hypotheses hold for at least one d>1 boundary-controlled system would be needed to confirm that the iff statement is not an artifact of the framework's restrictions.

    Authors: We agree that an explicit check strengthens the claim. The linear heat equation with boundary control in any dimension d>1 satisfies all structural hypotheses of the framework (as already used in the well-posedness arguments of Section 4.1) and is approximately controllable. Consequently the uniqueness direction of the iff statement applies directly to this system. In the revision we will insert a short remark immediately after the uniqueness theorem that records this verification for the multi-dimensional heat equation, thereby showing that the characterization is realized by a concrete d>1 boundary-controlled example and is not an artifact of the hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity: F-equivalence existence and uniqueness rest on independent controllability notions and extended structural hypotheses

full rationale

The paper's main result introduces an extended F-equivalence framework for parabolic operators and proves existence of (T,K) under stated structural conditions on (A,B), with uniqueness holding precisely when (A,B) satisfies the standard definition of approximate controllability. These steps rely on external mathematical concepts (approximate controllability) and verifiable operator hypotheses rather than reducing the target claims to fitted inputs, self-definitions, or load-bearing self-citations by construction. The derivation chain remains self-contained against external benchmarks such as controllability theory, with no quoted reductions of predictions or uniqueness statements to the paper's own fitted quantities or prior author ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the work relies on standard parabolic operator theory and controllability concepts from prior literature while introducing an extended framework whose precise assumptions are not detailed here.

pith-pipeline@v0.9.0 · 5730 in / 1134 out tokens · 64067 ms · 2026-05-18T20:40:50.294731+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

72 extracted references · 72 canonical work pages

  1. [1]

    Abderrahim Azouani and Edriss S. Titi. Feedback control of nonlinear dissipative systems by fi- nite determining parameters—a reaction-diffusion paradigm.Evolution Equations and Control Theory, 3(4):579–594, 2014

    work page 2014

  2. [2]

    Feedback stabilization of the 2-d and 3-d Navier–Stokes equations based on an extended system.ESAIM: Control, Optimisation and Calculus of Variations, 15(4):934–968, 2009

    Mehdi Badra. Feedback stabilization of the 2-d and 3-d Navier–Stokes equations based on an extended system.ESAIM: Control, Optimisation and Calculus of Variations, 15(4):934–968, 2009

    work page 2009

  3. [3]

    Local feedback stabilization of time-periodic evolution equations by finite dimensional controls.ESAIM: Control, Op- timisation and Calculus of Variations, 26:101, 2020

    Mehdi Badra, Debanjana Mitra, Mythily Ramaswamy, and Jean-Pierre Raymond. Local feedback stabilization of time-periodic evolution equations by finite dimensional controls.ESAIM: Control, Op- timisation and Calculus of Variations, 26:101, 2020

    work page 2020

  4. [4]

    Mehdi Badra and Takéo Takahashi. Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the navier–stokes system.SIAM journal on control and optimization, 49(2):420–463, 2011

    work page 2011

  5. [5]

    On the fattorini criterion for approximate controllability and stabilizability of parabolic systems.ESAIM: Control, Optimisation and Calculus of Variations, 20:924– 956, 2014

    Mehdi Badra and Takeo Takahashi. On the fattorini criterion for approximate controllability and stabilizability of parabolic systems.ESAIM: Control, Optimisation and Calculus of Variations, 20:924– 956, 2014

    work page 2014

  6. [6]

    Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability.European journal of control, 8(2):165–175, 2002

    Andras Balogh and Miroslav Krstic. Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability.European journal of control, 8(2):165–175, 2002

    work page 2002

  7. [7]

    Springer, 2018

    Viorel Barbu.Controllability and stabilization of parabolic equations. Springer, 2018

    work page 2018

  8. [8]

    American Mathemat- ical Society, 2006

    Viorel Barbu, Irena Lasiecka, and Roberto Triggiani.Tangential boundary stabilization of Navier– Stokes equations, volume 181 ofMemoirs of the American Mathematical Society. American Mathemat- ical Society, 2006

    work page 2006

  9. [9]

    Internal stabilization of Navier–Stokes equations with finite- dimensional controllers.Indiana University Mathematics Journal, 53(5):1443–1494, 2004

    Viorel Barbu and Roberto Triggiani. Internal stabilization of Navier–Stokes equations with finite- dimensional controllers.Indiana University Mathematics Journal, 53(5):1443–1494, 2004

    work page 2004

  10. [10]

    Feedback stabilization of semilinear heat equations

    Viorel Barbu and Gengsheng Wang. Feedback stabilization of semilinear heat equations. InAbstract and Applied Analysis, volume 2003, pages 697–714. Wiley Online Library, 2003

    work page 2003

  11. [11]

    Boundary stabilization of quasilinear parabolic PDEs that blow up in open loop for arbitrarily small initial conditions

    Mohamed Camil Belhadjoudja, Mohamed Maghenem, Emmanuel Witrant, and Miroslav Krstic. Boundary stabilization of quasilinear parabolic PDEs that blow up in open loop for arbitrarily small initial conditions. 2025. arXiv:2505.10935 [math.AP], v2

    work page arXiv 2025

  12. [12]

    On the theory of dynamic programming.Proceedings of the national Academy of Sciences, 38(8):716–719, 1952

    Richard Bellman. On the theory of dynamic programming.Proceedings of the national Academy of Sciences, 38(8):716–719, 1952

    work page 1952

  13. [13]

    Riccati-based feedback control of the monodomain equations with the fitzhugh–nagumo model.SIAM Journal on Control and Optimization, 52(6):4057–4081, 2014

    Tobias Breiten and Karl Kunisch. Riccati-based feedback control of the monodomain equations with the fitzhugh–nagumo model.SIAM Journal on Control and Optimization, 52(6):4057–4081, 2014

    work page 2014

  14. [14]

    Feedback stabilization of the two-dimensional navier–stokes equations by value function approximation.Applied Mathematics & Optimization, 80(3):599–641, 2019

    Tobias Breiten, Karl Kunisch, and Laurent Pfeiffer. Feedback stabilization of the two-dimensional navier–stokes equations by value function approximation.Applied Mathematics & Optimization, 80(3):599–641, 2019

    work page 2019

  15. [15]

    Universitext

    Haim Brezis.Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, 2010

    work page 2010

  16. [16]

    A classification of linear controllable systems.Kybernetika, 6(3):173–188, 1970

    Pavol Brunovsk` y. A classification of linear controllable systems.Kybernetika, 6(3):173–188, 1970

    work page 1970

  17. [17]

    Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation.Comptes Rendus

    Rémi Buffe and Takéo Takahashi. Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation.Comptes Rendus. Mathématique, 361:1541–1576, 2023

    work page 2023

  18. [18]

    Stabilization by sparse controls for a class of semilinear parabolic equations.SIAM Journal on Control and Optimization, 55(1):512–532, 2017

    Eduardo Casas and Karl Kunisch. Stabilization by sparse controls for a class of semilinear parabolic equations.SIAM Journal on Control and Optimization, 55(1):512–532, 2017

    work page 2017

  19. [19]

    Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: linearized system.Journal of Differential Equations, 350:251–307, 2023

    Jean Cauvin-Vila, Virginie Ehrlacher, and Amaury Hayat. Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: linearized system.Journal of Differential Equations, 350:251–307, 2023

    work page 2023

  20. [20]

    Number 136

    Jean-Michel Coron.Control and nonlinearity. Number 136. American Mathematical Soc., 2007

    work page 2007

  21. [21]

    Stabilization of control systems and nonlinearities

    Jean-Michel Coron. Stabilization of control systems and nonlinearities. InProceedings of the 8th In- ternational Congress on Industrial and Applied Mathematics, pages 17–40. Higher Ed. Press Beijing, 2015

    work page 2015

  22. [22]

    Rapid stabilization of a linearized bi- linear 1-D Schrödinger equation.J

    Jean-Michel Coron, Ludovick Gagnon, and Morgan Morancey. Rapid stabilization of a linearized bi- linear 1-D Schrödinger equation.J. Math. Pures Appl. (9), 115:24–73, 2018

    work page 2018

  23. [23]

    Stabilization of the linearized water tank system.Archive for Rational Mechanics and Analysis, 244(3):1019–1097, 2022

    Jean-Michel Coron, Amaury Hayat, Shengquan Xiang, and Christophe Zhang. Stabilization of the linearized water tank system.Archive for Rational Mechanics and Analysis, 244(3):1019–1097, 2022. F-EQUIV ALENCE FOR PARABOLIC SYSTEMS AND APPLICATIONS TO NONLINEAR PDE 45

    work page 2022

  24. [24]

    Finite-time boundary stabilization of general linear hyperbolic balance laws via fredholm backstepping transformation.Automatica, 84:95–100, 2017

    Jean-Michel Coron, Long Hu, and Guillaume Olive. Finite-time boundary stabilization of general linear hyperbolic balance laws via fredholm backstepping transformation.Automatica, 84:95–100, 2017

    work page 2017

  25. [25]

    Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right.Journal de Mathématiques Pures et Appliquées, 102(6):1080– 1120, 2014

    Jean-Michel Coron and Qi Lü. Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right.Journal de Mathématiques Pures et Appliquées, 102(6):1080– 1120, 2014

    work page 2014

  26. [26]

    Fredholm transform and local rapid stabilization for a Kuramoto- Sivashinsky equation.Journal of Differential Equations, 259(8):3683–3729, 2015

    Jean-Michel Coron and Qi Lü. Fredholm transform and local rapid stabilization for a Kuramoto- Sivashinsky equation.Journal of Differential Equations, 259(8):3683–3729, 2015

    work page 2015

  27. [27]

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

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

    work page 2019

  28. [28]

    Springer Science & Business Media, 1999

    Klaus-Jochen Engel and Rainer Nagel.One-Parameter Semigroups for Linear Evolution Equations, volume 194. Springer Science & Business Media, 1999

    work page 1999

  29. [29]

    A. V. Fursikov. Stabilizability of two-dimensional navier–stokes equations with help of a boundary feedback control.Journal of Mathematical Fluid Mechanics, 3:259–301, September 2001

    work page 2001

  30. [30]

    Fredholm transformation on laplacian and rapid stabilization for the heat equations.Journal of Functional Analysis, 2022

    Ludovick Gagnon, Amaury Hayat, Shengquan Xiang, and Christophe Zhang. Fredholm transformation on laplacian and rapid stabilization for the heat equations.Journal of Functional Analysis, 2022

    work page 2022

  31. [31]

    Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves.Annales de l’Institut Fourier, 2024

    Ludovick Gagnon, Amaury Hayat, Shengquan Xiang, and Christophe Zhang. Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves.Annales de l’Institut Fourier, 2024

    work page 2024

  32. [32]

    Fredholm backstepping for self-adjoint operators and applications to the rapid stabilization of parabolic and fractional parabolic equations in any dimension.Preprint, 2025

    Ludovick Gagnon, Nazim Kacher, and Hoai-Minh Nguyen. Fredholm backstepping for self-adjoint operators and applications to the rapid stabilization of parabolic and fractional parabolic equations in any dimension.Preprint, 2025

    work page 2025

  33. [33]

    A fredholm transformation for the rapid stabilization ofadegenerateparabolicequation.SIAM Journal on Control and Optimization, 59(5):3828–3859, 2021

    Ludovick Gagnon, Pierre Lissy, and Swann Marx. A fredholm transformation for the rapid stabilization ofadegenerateparabolicequation.SIAM Journal on Control and Optimization, 59(5):3828–3859, 2021

    work page 2021

  34. [34]

    Fredholm backstepping and rapid stabilization of general linear systems.preprint, 2024

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

    work page 2024

  35. [35]

    Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes.IEEE Transactions on Automatic Control, 61(11):3301–3314, 2015

    Long Hu, Florent Di Meglio, Rafael Vazquez, and Miroslav Krstic. Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes.IEEE Transactions on Automatic Control, 61(11):3301–3314, 2015

    work page 2015

  36. [36]

    Rapid boundary stabilization of linear distributed systems.SIAM journal on control and optimization, 35(5):1591–1613, 1997

    Vilmos Komornik. Rapid boundary stabilization of linear distributed systems.SIAM journal on control and optimization, 35(5):1591–1613, 1997

    work page 1997

  37. [37]

    Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008

    Miroslav Krstic and Andrey Smyshlyaev.Boundary Control of PDEs: A Course on Backstepping Designs, volume 16 ofAdvances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008

    work page 2008

  38. [38]

    Learning optimal feedback operators and their sparse polynomial approximations.Journal of Machine Learning Research, 24(301):1–38, 2023

    Karl Kunisch, Donato Vásquez-Varas, and Daniel Walter. Learning optimal feedback operators and their sparse polynomial approximations.Journal of Machine Learning Research, 24(301):1–38, 2023

    work page 2023

  39. [39]

    Frequency-domain criterion on the stabilizabil- ity for infinite-dimensional linear control systems.Journal de Mathématiques Pures et Appliquées, 196:103690, 2025

    Karl Kunisch, Gengsheng Wang, and Huaiqiang Yu. Frequency-domain criterion on the stabilizabil- ity for infinite-dimensional linear control systems.Journal de Mathématiques Pures et Appliquées, 196:103690, 2025

    work page 2025

  40. [40]

    Diffusion-induced chaos in reaction systems.Progress of Theoretical Physics Sup- plement, 1978

    Yoshiki Kuramoto. Diffusion-induced chaos in reaction systems.Progress of Theoretical Physics Sup- plement, 1978

    work page 1978

  41. [41]

    1: Abstract parabolic systems, volume 74 ofEncycl

    Irena Lasiecka and Roberto Triggiani.Control theory for partial differential equations: continuous and approximation theories. 1: Abstract parabolic systems, volume 74 ofEncycl. Math. Appl.Cambridge: Cambridge University Press, 2000

    work page 2000

  42. [42]

    2: Abstract hyperbolic-like systems over a finite time horizon, volume 75 of Encycl

    Irena Lasiecka and Roberto Triggiani.Control theory for partial differential equations: continuous and approximation theories. 2: Abstract hyperbolic-like systems over a finite time horizon, volume 75 of Encycl. Math. Appl.Cambridge: Cambridge University Press, 2000

    work page 2000

  43. [43]

    Springer, 1971

    Jacques Louis Lions.Optimal control of systems governed by partial differential equations, volume 170. Springer, 1971

    work page 1971

  44. [44]

    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.Mathematical Control and Related Fields, pages 0–0, 2023

    work page 2023

  45. [45]

    Characterizations of complete stabi- lizability.SIAM Journal on Control and Optimization, 60(4):2040–2069, 2022

    Hanbing Liu, Gengsheng Wang, Yashan Xu, and Huaiqiang Yu. Characterizations of complete stabi- lizability.SIAM Journal on Control and Optimization, 60(4):2040–2069, 2022

    work page 2040

  46. [46]

    Stability enhancement by boundary control in the kuramoto– sivashinsky equation.Nonlinear Analysis: Theory, Methods & Applications, 43(4):485–507, 2001

    Wei-Jiu Liu and Miroslav Krstić. Stability enhancement by boundary control in the kuramoto– sivashinsky equation.Nonlinear Analysis: Theory, Methods & Applications, 43(4):485–507, 2001. 46 VINCENT BOULARD AND AMAURY HAYAT

    work page 2001

  47. [47]

    Feedback law to stabilize linear infinite-dimensional systems.Mathematical Control and Related Fields, 13(3):1160–1183, 2023

    Yaxing Ma, Gengsheng Wang, and Huaiqiang Yu. Feedback law to stabilize linear infinite-dimensional systems.Mathematical Control and Related Fields, 13(3):1160–1183, 2023

    work page 2023

  48. [48]

    Anna Maria Micheletti. Perturbazione dello spettro dell’operatore di laplace, in relazione ad una vari- azione del campo.Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 26(1):151–169, 1972

    work page 1972

  49. [49]

    Rassias Mircea Craioveanu, Mircea Puta.Old and New Aspects in Spectral Geometry

    Themistocles M. Rassias Mircea Craioveanu, Mircea Puta.Old and New Aspects in Spectral Geometry. Springer, 2001

    work page 2001

  50. [50]

    Sourav Mitra. Stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel around a Poiseuille flow by a finite-dimensional boundary feedback.ESAIM: Control, Optimisation and Calculus of Variations, 25, 2019. article en ligne

    work page 2019

  51. [51]

    Rapid stabilization and finite time stabilization of the bilinear Schrödinger equa- tion.arXiv preprint arXiv:2405.10002, 2024

    Hoai-Minh Nguyen. Rapid stabilization and finite time stabilization of the bilinear Schrödinger equa- tion.arXiv preprint arXiv:2405.10002, 2024

    work page arXiv 2024

  52. [52]

    Stabilization of control systems associated with a strongly continuous group.arXiv preprint arXiv:2402.07560, 2024

    Hoai-Minh Nguyen. Stabilization of control systems associated with a strongly continuous group.arXiv preprint arXiv:2402.07560, 2024

    work page arXiv 2024

  53. [53]

    Ortega and Enrique Zuazua

    Jaime H. Ortega and Enrique Zuazua. Generic simplicity of the eigenvalues of the stokes system in two space dimensions.Advances in Differential Equations, 6(8):987–1023, 2001

    work page 2001

  54. [54]

    Springer-Verlag, New York, 1983

    Amnon Pazy.Semigroups of Linear Operators and Applications to Partial Differential Equations, vol- ume 44 ofApplied Mathematical Sciences. Springer-Verlag, New York, 1983

    work page 1983

  55. [55]

    J.-P. Raymond. Stokes and navier–stokes equations with nonhomogeneous boundary conditions.An- nales de l’Institut Henri Poincaré C, Analyse non linéaire, 24(6):921–951, 2007

    work page 2007

  56. [56]

    Feedback boundary stabilization of the two-dimensional Navier–Stokes equa- tions.SIAM Journal on Control and Optimization, 45(3):790–828, 2006

    Jean-Pierre Raymond. Feedback boundary stabilization of the two-dimensional Navier–Stokes equa- tions.SIAM Journal on Control and Optimization, 45(3):790–828, 2006

    work page 2006

  57. [57]

    Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations.Journal de Mathématiques Pures et Appliquées, 87(6):627–669, 2007

    Jean-Pierre Raymond. Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations.Journal de Mathématiques Pures et Appliquées, 87(6):627–669, 2007

    work page 2007

  58. [58]

    Stabilizability of infinite dimensional systems by finite dimensional control

    Jean-Pierre Raymond. Stabilizability of infinite dimensional systems by finite dimensional control. Computational Methods in Applied Mathematics, 19(2):267–282, 2019

    work page 2019

  59. [59]

    Jean-Pierre Raymond and Laetitia Thévenet. Boundary feedback stabilization of the two-dimensional Navier–Stokes equations with finite dimensional controllers.Discrete and Continuous Dynamical Sys- tems - Series A, 27(3):1159–1187, 2010

    work page 2010

  60. [60]

    G. I. Sivashinsky. On flame propagation under conditions of stoichiometry.SIAM Journal on Applied Mathematics, 1980

    work page 1980

  61. [61]

    Sivashinsky

    G.I. Sivashinsky. Nonlinear analysis of hydrodynamic instability in laminar flames—i. derivation of basic equations.Acta Astronautica, 1977

    work page 1977

  62. [62]

    The linear stabilization problem in Hilbert space.Journal of Functional Analysis, 11(3):334–345, 1972

    Marshall Slemrod. The linear stabilization problem in Hilbert space.Journal of Functional Analysis, 11(3):334–345, 1972

    work page 1972

  63. [63]

    Stabilization of semilinear PDEs, and uniform decay under discretization

    Emmanuel Trélat. Stabilization of semilinear PDEs, and uniform decay under discretization. InEvolu- tion equations: long time behavior and control, volume 439, pages 31–76. Cambridge University Press, 2017

    work page 2017

  64. [64]

    Characterization by observability inequalities of controllability and stabilization properties.Pure and Applied Analysis, 2(1):93–122, 2019

    Emmanuel Trélat, Gengsheng Wang, and Yashan Xu. Characterization by observability inequalities of controllability and stabilization properties.Pure and Applied Analysis, 2(1):93–122, 2019

    work page 2019

  65. [65]

    Springer Science & Business Media, 2009

    Marius Tucsnak and George Weiss.Observation and control for operator semigroups. Springer Science & Business Media, 2009

    work page 2009

  66. [66]

    Generic properties of eigenfunctions.American Journal of Mathematics, 98(4):1059– 1078, 1976

    Karen Uhlenbeck. Generic properties of eigenfunctions.American Journal of Mathematics, 98(4):1059– 1078, 1976

    work page 1976

  67. [67]

    Rapid exponential feedback stabilization with unbounded control operators

    Jose Manuel Urquiza. Rapid exponential feedback stabilization with unbounded control operators. SIAM journal on control and optimization, 43(6):2233–2244, 2005

    work page 2005

  68. [68]

    Rapid stabilization in a semigroup framework.SIAM Journal on Control and Opti- mization, 51(5):4169–4188, 2013

    Ambroise Vest. Rapid stabilization in a semigroup framework.SIAM Journal on Control and Opti- mization, 51(5):4169–4188, 2013

    work page 2013

  69. [69]

    Null controllability of a linearized korteweg–de vries equation by backstepping approach.SIAM Journal on Control and Optimization, 57(2):1493–1515, 2019

    Shengquan Xiang. Null controllability of a linearized korteweg–de vries equation by backstepping approach.SIAM Journal on Control and Optimization, 57(2):1493–1515, 2019

    work page 2019

  70. [70]

    Small-time local stabilization of the two-dimensional incompressible Navier–Stokes equations.Annales de l’Institut Henri Poincaré C, 40(6):1487–1511, 2023

    Shengquan Xiang. Small-time local stabilization of the two-dimensional incompressible Navier–Stokes equations.Annales de l’Institut Henri Poincaré C, 40(6):1487–1511, 2023

    work page 2023

  71. [71]

    Quantitative rapid and finite time stabilization of the heat equation.ESAIM: Con- trol, Optimisation and Calculus of Variations, 30:40, 2024

    Shengquan Xiang. Quantitative rapid and finite time stabilization of the heat equation.ESAIM: Con- trol, Optimisation and Calculus of Variations, 30:40, 2024

    work page 2024

  72. [72]

    Springer, 2020

    Jerzy Zabczyk.Mathematical control theory. Springer, 2020

    work page 2020