Null-controllability of underactuated linear parabolic-transport systems with constant coefficients
[Contrôlabilité à zéro de systèmes paraboliques-transport linéaires ayant moins de contrôles que d’équations]
Koenig, Armand1  ; Lissy, Pierre2, 3
1Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence (France)
2CEREMADE, Université Paris-Dauphine & CNRS UMR 7534, Université PSL, 75016 Paris, (France)
3CERMICS, ENPC, Institut Polytechnique de Paris, CNRS, Marne-la-Vallée, (France)
Annales de l'Institut Fourier, Online first, 43 p.

The goal of the present article is to study controllability properties of mixed systems of linear parabolic-transport equations, with possibly non-diagonalizable diffusion matrix, on the one-dimensional torus. The equations are coupled by zero or first order coupling terms, with constant coupling matrices, without any structure assumptions on them. The distributed control acts through a constant matrix operator on the system, so that there might be notably less controls than equations, encompassing the case of indirect and simultaneous controllability. More precisely, we prove that in small time, such kind of systems are never controllable in any Sobolev spaces, whereas in large time, null-controllability holds, for sufficiently regular initial data, if and only if a spectral Kalman rank condition is verified. We also prove that initial data that are not regular enough are not controllable. Positive results are obtained by using the so-called fictitious control method together with an algebraic solvability argument, whereas the negative results are obtained by using an appropriate WKB construction of approximate solutions for the adjoint system associated to the control problem. As an application to our general results, we also investigate into details the case of $2\times 2$ systems (i.e., one pure transport equation and one parabolic equation).

L’objectif de cet article est d’étudier les propriétés de contrôlabilité des systèmes linéaires couplant des équations de transport et des équations paraboliques, avec éventuellement une matrice de diffusion non diagonalisable, sur le tore de dimension un. Les équations sont couplées par des termes de couplage d’ordre zéro ou un, avec des matrices de couplage constantes, sans aucune hypothèse de structure sur elles. Le contrôle distribué agit sur le système via un opérateur matriciel constant, de sorte qu’il peut y avoir beaucoup moins de contrôles que d’équations, ce qui englobe le cas de la contrôlabilité indirecte et du contrôle simultané. Plus précisément, nous démontrons qu’en temps petit, cette classe de systèmes n’est jamais contrôlable dans aucun espace de Sobolev, alors qu’en temps grand, les conditions initiales assez régulières sont contrôlables à zéro si et seulement si une condition de Kalman spectrale est vérifiée. Nous démontrons également que les données initiales qui ne sont pas suffisamment régulières ne sont pas contrôlables. Les résultats positifs sont obtenus en utilisant la méthode de contrôle fictif ainsi qu’un argument de solvabilité algébrique, tandis que les résultats négatifs sont obtenus en utilisant une construction WKB de solutions approchées pour le système adjoint associé au système de contrôle. Comme application de nos résultats généraux, nous étudions également en détail le cas des systèmes $2\times 2$ (c’est-à-dire une équation de transport pure et une équation parabolique).

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3787
Classification : 93B05, 93B07, 93C20, 35M30
Keywords: Parabolic-transport systems, null-controllability, observability.
Mots-clés : Systèmes paraboliques-transport, contrôlabilité à zéro, observabilité.

Koenig, Armand  1   ; Lissy, Pierre  2 , 3

1 Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence (France)
2 CEREMADE, Université Paris-Dauphine & CNRS UMR 7534, Université PSL, 75016 Paris, (France)
3 CERMICS, ENPC, Institut Polytechnique de Paris, CNRS, Marne-la-Vallée, (France) 
Koenig, Armand; Lissy, Pierre. Null-controllability of underactuated linear parabolic-transport systems with constant coefficients. Annales de l'Institut Fourier, Online first, 43 p.
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