Depuis une vingtaine d’années, la contrôlabilité de plusieurs exemples de systèmes paraboliques-hyperboliques couplés a été étudiée. Nous initions dans cet article une recherche d’un cadre qui contient et généralise les résultats déjà existants. Nous considérons des systèmes paraboliques-transport, à coefficients constants, couplés par des termes d’ordre et , posés sur le tore de dimension , et avec contrôle interne localisé sur un ouvert du tore. Nous démontrons la contrôlabilité à zéro de ces systèmes en temps optimal (celui attendu en raison des composantes de transport) lorsqu’on contrôle toutes les équations. Lorsque le contrôle agit uniquement sur les composantes hyperboliques (resp. paraboliques), nous démontrons une condition nécessaire et suffisante pour la contrôlabilité à zéro, cette condition étant de type Kalman et portant sur le terme de couplage. Cette étude repose sur une analyse spectrale, elle-même basée sur la théorie perturbative : en hautes fréquences, le spectre se sépare en une branche hyperbolique et une branche parabolique. Le résultat de non-contrôlabilité en temps petit est démontré en construisant des solutions de transport approchées, localisées en hautes fréquences. Le résultat de contrôle en temps grand est démontré en projetant la dynamique sur trois espaces stables, associés respectivement aux hautes fréquences hyperboliques, hautes fréquences paraboliques et basses fréquences, ce qui définit trois systèmes faiblement couplés.
Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the previous results. We consider constant coefficients parabolic-transport systems with coupling of order zero and one, with a locally distributed control in the source term, posed on the one-dimensional torus. We prove the null-controllability, in optimal time (the one expected because of the transport component) when there is as many controls as equations. When the control acts only on the transport (resp. parabolic) component, we prove an algebraic necessary and sufficient condition, on the coupling term, for the null-controllability. The whole study relies on a careful spectral analysis, based on perturbation theory. For high frequencies, the spectrum splits into a parabolic part and a hyperbolic part. The negative controllability result in small time is proved on solutions localized on high hyperbolic frequencies, that solve a pure transport equation up to a compact term. The positive controllability result in large time is proved by projecting the dynamics onto three eigenspaces associated to hyperbolic, parabolic and low frequencies, that defines three weakly coupled systems.
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@article{JEP_2020__7__743_0,
author = {Karine Beauchard and Armand Koenig and K\'evin Le Balc{\textquoteright}h},
title = {Null-controllability of linear~parabolic-transport systems},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {743--802},
publisher = {\'Ecole polytechnique},
volume = {7},
year = {2020},
doi = {10.5802/jep.127},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.127/}
}
TY - JOUR AU - Karine Beauchard AU - Armand Koenig AU - Kévin Le Balc’h TI - Null-controllability of linear parabolic-transport systems JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 743 EP - 802 VL - 7 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.127/ DO - 10.5802/jep.127 LA - en ID - JEP_2020__7__743_0 ER -
%0 Journal Article %A Karine Beauchard %A Armand Koenig %A Kévin Le Balc’h %T Null-controllability of linear parabolic-transport systems %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 743-802 %V 7 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.127/ %R 10.5802/jep.127 %G en %F JEP_2020__7__743_0
Karine Beauchard; Armand Koenig; Kévin Le Balc’h. Null-controllability of linear parabolic-transport systems. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 743-802. doi : 10.5802/jep.127. https://jep.centre-mersenne.org/articles/10.5802/jep.127/
[1] - “Internal controllability of first order quasi-linear hyperbolic systems with a reduced number of controls”, SIAM J. Control Optim. 55 (2017) no. 1, p. 300-323 | DOI | MR | Zbl
[2] - “Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system”, Electron. J. Differential Equations (2000), article ID 22, 15 pages | MR | Zbl
[3] - “A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems”, Differential Equations Appl. 1 (2009) no. 3, p. 427-457 | DOI | MR | Zbl
[4] - “A Kalman rank condition for the localized distributed controllability of a class of linear parbolic systems”, J. Evol. Equ. 9 (2009) no. 2, p. 267-291 | DOI | MR | Zbl
[5] - “Partial null controllability of parabolic linear systems”, Math. Control Relat. Fields 6 (2016) no. 2, p. 185-216 | DOI | MR | Zbl
[6] - “Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary”, SIAM J. Control Optim. 30 (1992) no. 5, p. 1024-1065 | DOI | MR | Zbl
[7] - “Large time asymptotics for partially dissipative hyperbolic systems”, Arch. Rational Mech. Anal. 199 (2011) no. 1, p. 177-227 | DOI | MR | Zbl
[8] - Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011 | Zbl
[9] - “Null controllability of a system of viscoelasticity with a moving control”, J. Math. Pures Appl. (9) 101 (2014) no. 2, p. 198-222 | DOI | MR | Zbl
[10] - “Null controllability of the linearized compressible Navier-Stokes equations using moment method”, J. Evol. Equ. 15 (2015) no. 2, p. 331-360 | DOI | MR | Zbl
[11] - “Null controllability of the linearized compressible Navier-Stokes system in one dimension”, J. Differential Equations 257 (2014) no. 10, p. 3813-3849 | DOI | MR | Zbl
[12] - Control and nonlinearity, Math. Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007 | MR | Zbl
[13] - “Compact perturbations of controlled systems”, Math. Control Relat. Fields 8 (2018) no. 2, p. 397-410 | DOI | MR | Zbl
[14] - “Local exact controllability for the one-dimensional compressible Navier-Stokes equation”, Arch. Rational Mech. Anal. 206 (2012) no. 1, p. 189-238 | DOI | MR | Zbl
[15] - “Remarks on non controllability of the heat equation with memory”, ESAIM Control Optim. Calc. Var. 19 (2013) no. 1, p. 288-300 | DOI | Numdam | MR | Zbl
[16] - “Null controllability of the structurally damped wave equation on the two dimensional torus”, 2019 | HAL
[17] - “Heat equation with memory: lack of controllability to rest”, J. Math. Anal. Appl. 355 (2009) no. 1, p. 1-11 | DOI | MR | Zbl
[18] - Perturbation theory for linear operators, Classics in Math., vol. 132, Springer, Berlin Heidelberg, 1995 | DOI | Zbl
[19] - “Non-null-controllability of the Grushin operator in 2D”, Comptes Rendus Mathématique 355 (2017) no. 12, p. 1215-1235 | DOI | MR | Zbl
[20] - “On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations”, ESAIM Control Optim. Calc. Var. 18 (2012) no. 3, p. 712-747 | DOI | Numdam | MR | Zbl
[21] - “Contrôle exact de l’équation de la chaleur”, Comm. Partial Differential Equations 20 (1995-01) no. 1-2, p. 335-356 | DOI | Zbl
[22] - “Null-controllability of a system of linear thermoelasticity”, Arch. Rational Mech. Anal. 141 (1998) no. 4, p. 297-329 | DOI | MR | Zbl
[23] - “Null controllability of the structurally damped wave equation with moving control”, SIAM J. Control Optim. 51 (2013) no. 1, p. 660-684 | DOI | MR | Zbl
[24] - “Another approach to elliptic boundary problems”, Comm. Pure Appl. Math. 14 (1961) no. 4, p. 711-731 | DOI | MR | Zbl
[25] - Analyse pour l’agrégation, Dunod, Paris, 2013
[26] - “On the controllability of a wave equation with structural damping”, Int. J. Tomogr. Stat. 5 (2007) no. W07, p. 79-84 | MR
[27] - Observation and control for operator semigroups, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2009 | DOI | Zbl
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