Null-controllability of linear parabolic-transport systems
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 743-802.

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.

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 0 et 1, posés sur le tore de dimension 1, 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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.127
Classification: 93B05,  93B07,  93C20,  35M30
Keywords: Parabolic-transport systems, null-controllability, observability
Karine Beauchard 1; Armand Koenig 2; Kévin Le Balc’h 1

1 Univ Rennes, CNRS, IRMAR - UMR 6625 F-35000 Rennes, France
2 Université Côte d’Azur, CNRS, LJAD, France
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Karine Beauchard; Armand Koenig; Kévin Le Balc’h. Null-controllability of linear parabolic-transport systems. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 743-802. doi : 10.5802/jep.127. https://jep.centre-mersenne.org/articles/10.5802/jep.127/

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