Exact controllability of a cascade system of conservative equations

Lionel Rosier; Luz de Teresa

doi:10.1016/j.crma.2011.01.014

Partial Differential Equations/Optimal Control

Exact controllability of a cascade system of conservative equations
[Contrôlabilité exacte dʼun système en cascade dʼéquations conservatives]

Présenté par : Gilles Lebeau

Lionel Rosier ¹ ; Luz de Teresa ²

¹ Institut Elie-Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 70239, 54506 Vandœuvre-lès-Nancy cedex, France
² Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U. 04510 D.F., Mexico

Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 291-296.

Résumé
Abstract

On considère un système en cascade de deux équations conservatives et lʼon prouve la contrôlabilité du système complet lorsque chaque équation est contrôlable et que le groupe unitaire correspondant à lʼévolution libre est périodique en temps. Ce résultat sʼapplique à des systèmes dʼéquations de Schrödinger ou des ondes. Utilisant la transformée de Kannai, on en déduit quʼun système en cascade dʼéquations de la chaleur est contrôlable à zéro en dimension un, même si les supports du contrôle et du couplage ne sʼintersectent pas.

We consider a cascade system of two conservative equations and prove the controllability of the full system when each equation is controllable, provided that the unitary group corresponding to the free evolution is time-periodic. Applications to systems of Schrödinger (resp. wave) equations are given. With the aid of Kannai transform we infer that a one-dimensional system of heat equations is null controllable even if the supports of the control function and of the coupling term do not intersect.

Reçu le : 2010-12-07
Accepté le : 2011-01-12
Publié le : 2011-02-05

DOI : 10.1016/j.crma.2011.01.014

Affiliations des auteurs :

Lionel Rosier ¹ ; Luz de Teresa ²

¹ Institut Elie-Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 70239, 54506 Vandœuvre-lès-Nancy cedex, France
² Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U. 04510 D.F., Mexico

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     author = {Lionel Rosier and Luz de Teresa},
     title = {Exact controllability of a cascade system of conservative equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {291--296},
     publisher = {Elsevier},
     volume = {349},
     number = {5-6},
     year = {2011},
     doi = {10.1016/j.crma.2011.01.014},
     language = {en},
}

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Lionel Rosier; Luz de Teresa. Exact controllability of a cascade system of conservative equations. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 291-296. doi : 10.1016/j.crma.2011.01.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.014/

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