Comptes Rendus
Partial Differential Equations
Application of the exact null controllability of the heat equation to moving sets
[Application de la contrôlabilité exacte à zéro de l'équation de la chaleur au déplacement d'ensembles]
Comptes Rendus. Mathématique, Volume 342 (2006) no. 11, pp. 849-852.

On étudie la contrôlabilité lagrangienne de l'équation de la chaleur en toutes dimensions. En dimension 1, on montre que deux intervalles quelconques sont difféomorphes via le flot de la solution de l'équation de la chaleur avec un contrôle adéquat. En dimension supérieure on prouve un résultat de contrôlabilité similaire pour le flot du gradient, en temps fini fixé pour le cas radial, et en temps assez grand pour le cas convexe.

We study the lagrangian controllability of the heat equation in several dimensions. In dimension one, we prove that any pairs of intervals are diffeomorphic through the flow of the solution of the heat equation via an adequate control. In higher dimensions we prove a similar controllability result for the flow of the gradient of the solution in a radial case in arbitrary finite time, and for convex domains in a sufficiently large time.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.04.001

Thierry Horsin Molinaro 1

1 Université de Versailles-Saint Quentin, laboratoire de mathématiques appliquées, 45, avenue de États-Unis, 78030 Versailles cedex, France
@article{CRMATH_2006__342_11_849_0,
     author = {Thierry Horsin Molinaro},
     title = {Application of the exact null controllability of the heat equation to moving sets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {849--852},
     publisher = {Elsevier},
     volume = {342},
     number = {11},
     year = {2006},
     doi = {10.1016/j.crma.2006.04.001},
     language = {en},
}
TY  - JOUR
AU  - Thierry Horsin Molinaro
TI  - Application of the exact null controllability of the heat equation to moving sets
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 849
EP  - 852
VL  - 342
IS  - 11
PB  - Elsevier
DO  - 10.1016/j.crma.2006.04.001
LA  - en
ID  - CRMATH_2006__342_11_849_0
ER  - 
%0 Journal Article
%A Thierry Horsin Molinaro
%T Application of the exact null controllability of the heat equation to moving sets
%J Comptes Rendus. Mathématique
%D 2006
%P 849-852
%V 342
%N 11
%I Elsevier
%R 10.1016/j.crma.2006.04.001
%G en
%F CRMATH_2006__342_11_849_0
Thierry Horsin Molinaro. Application of the exact null controllability of the heat equation to moving sets. Comptes Rendus. Mathématique, Volume 342 (2006) no. 11, pp. 849-852. doi : 10.1016/j.crma.2006.04.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.001/

[1] C. Conca; J. San Martin; M. Tucsnak Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, Volume 25 (2000) no. 5–6, pp. 1019-1042

[2] J.-M. Coron Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, Volume 5 (1992) no. 3, pp. 295-312

[3] J.-M. Coron Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels, C. R. Acad. Sci. Paris, Volume 317 (1993), pp. 271-276

[4] J.-M. Coron On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. (9), Volume 75 (1996) no. 2, pp. 155-188

[5] J.-M. Coron; E. Trélat Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim., Volume 43 (2004) no. 2, pp. 549-569 (electronic)

[6] R.F. Curtain; H. Zwart An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, vol. 21, Springer-Verlag, New York, 1995

[7] R. Di Perna; P.-L. Lions Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989), pp. 511-547

[8] H.O. Fattorini; D.L. Russell Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., Volume 43 (1971), pp. 272-292

[9] A.V. Fursikov; O.Yu. Imanuvilov Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996

[10] V. Komornik Exact Controllability and Stabilization, The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, 1994

[11] G. Lebeau; L. Robbiano Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, Volume 20 (1995) no. 1–2, pp. 335-356

[12] J. Lewis Capacitary function in convex rings, Arch. Rational Mech. Anal., Volume 66 (1977), pp. 201-224

[13] J.L. Lions Contrôle des systèmes distribués singuliers, Méthodes Mathématiques de l'Informatique, Mathematical Methods of Information Science, vol. 13, Gauthier-Villars, Montrouge, 1983

[14] J. Ortega, L. Rosier, Control of the motion of a ball surrounded by an incompressible perfect fluid, 2006, in preparation

[15] D.L. Russell Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., Volume 20 (1978) no. 4, pp. 639-739

Cité par Sources :

Commentaires - Politique