Comptes Rendus
no. 5-6
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 Rosier1 ; Luz de Teresa2
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.

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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.01.014
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 
@article{CRMATH_2011__349_5-6_291_0,
     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},
}
TY  - JOUR
AU  - Lionel Rosier
AU  - Luz de Teresa
TI  - Exact controllability of a cascade system of conservative equations
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 291
EP  - 296
VL  - 349
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2011.01.014
LA  - en
ID  - CRMATH_2011__349_5-6_291_0
ER  - 
%0 Journal Article
%A Lionel Rosier
%A Luz de Teresa
%T Exact controllability of a cascade system of conservative equations
%J Comptes Rendus. Mathématique
%D 2011
%P 291-296
%V 349
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2011.01.014
%G en
%F CRMATH_2011__349_5-6_291_0
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/

[1] F. Alabau-Boussouira, M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions, C. R. Acad. Sci. Paris, Ser. I (2011), in press.

[2] F. Ammar Khodja; A. Benabdallah; C. Dupaix Null-controllability of some reaction–diffusion systems with one control force, J. Math. Anal. Appl., Volume 320 (2006) no. 2, pp. 928-943

[3] J.-M. Coron; S. Guerrero; L. Rosier Null controllability of a parabolic system with a cubic coupling term, SIAM J. Control Optim., Volume 48 (2010) no. 8, pp. 5629-5653

[4] R. Dáger Insensitizing controls for the 1-D wave equation, SIAM J. Control Optim., Volume 45 (2006) no. 5, pp. 1758-1768

[5] S. Ervedoza; E. Zuazua A systematic method for building smooth controls for smooth data, Discrete Contin. Dynam. Systems Ser. B, Volume 14 (2010), pp. 1375-1401

[6] M. González-Burgos; L. de Teresa Controllability results for cascade systems of m coupled parabolic PDEs by one control force, Port. Math., Volume 67 (2010) no. 1, pp. 91-113

[7] O. Kavian; L. de Teresa Unique continuation principle for systems of parabolic equations, ESAIM: Control Optim. Calc. Var., Volume 16 (2010) no. 2, pp. 247-274

[8] L. Miller The control transmutation and the cost of fast controls, SIAM J. Control Optim., Volume 45 (2006) no. 2, pp. 762-772

[9] L. Rosier; B.-Y. Zhang Control and stabilization of the nonlinear Schrödinger equation on rectangles, M3AS: Math. Models Methods Appl. Sci., Volume 20 (2010) no. 12, pp. 2293-2347

[10] L. Tebou Locally distributed desensitizing controls for the wave equation, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008) no. 7–8, pp. 407-412

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Controllability of cascade coupled systems of multi-dimensional evolution PDEs by a reduced number of controls

Fatiha Alabau-Boussouira

C. R. Math (2012)

Controllability of the Ginzburg–Landau equation

Lionel Rosier; Bing-Yu Zhang

C. R. Math (2008)

Indirect controllability of locally coupled systems under geometric conditions

Fatiha Alabau-Boussouira; Matthieu Léautaud

C. R. Math (2011)