We consider the heat equation with fast oscillating periodic density, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform controllability result is the first in the multidimensional setting for the heat equation with oscillating density. Finally, we prove that the sequence of null controls converges to the optimal null control of the limit equation when the period tends to zero.
Nous considérons l'équation de la chaleur avec densité périodique rapidement oscillante, et un contrôle interne dans un domaine borné. Nous établissons d'abord des estimations fines de convergence dependant explicitement de la donnée initiale pour l'équation non contrôlée ; ces estimations sont nouvelles, et leur démonstration repose sur une régularisation judicieuse de la donnée initiale. Puis nous utilisons ces estimations pour démontrer que l'équation initiale est uniformément contrôlable à zéro, pourvu q'un contrôle interne, supplémentaire, evanescent, et convenablement choisi, soit ajouté à cette équation. Ce résultat de contrôlabilité uniforme est le premier dans le cadre multidimensionnel pour l'équation de la chaleur avec densité rapidement oscillante. Enfin nous montrons que la suite des contrôles converge vers le contrôle à zero optimal de l'équation limite lorsque la période tend vers zéro.
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Louis Tebou 1
@article{CRMATH_2009__347_13-14_779_0, author = {Louis Tebou}, title = {Uniform null controllability of the heat equation with rapidly oscillating periodic density}, journal = {Comptes Rendus. Math\'ematique}, pages = {779--784}, publisher = {Elsevier}, volume = {347}, number = {13-14}, year = {2009}, doi = {10.1016/j.crma.2009.04.030}, language = {en}, }
Louis Tebou. Uniform null controllability of the heat equation with rapidly oscillating periodic density. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 779-784. doi : 10.1016/j.crma.2009.04.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.030/
[1] Null-controllability of one-dimensional parabolic equations, ESAIM COCV, Volume 14 (2008), pp. 284-293
[2] Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam–New York, 1978
[3] Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl., Volume 71 (1992) no. 3, pp. 197-231
[4] Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., Volume 32 (1974), pp. 45-69
[5] The cost of approximate controllability for heat equations: the linear case, Adv. Differential Equations, Volume 5 (2000) no. 4–6, pp. 465-514
[6] Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996
[7] The wave equation with oscillating density: observability at low frequency, ESAIM Control Optim. Calc. Var., Volume 5 (2000), pp. 219-258
[8] Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics, vol. 323, Springer-Verlag, Berlin–New York, 1973 (in French)
[9] Uniform null-controllability for the one-dimensional heat equation with rapidly oscillating periodic density, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 19 (2002) no. 5, pp. 543-580
[10] Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992
[11] L. Tebou, Uniform null controllability of some parabolic equations with rapidly oscillating periodic coefficients, in preparation
[12] Null controllability of linear and semilinear heat equations in thin domains, Asymptot. Anal., Volume 24 (2000) no. 3–4, pp. 295-317
[13] Approximate controllability for linear parabolic equations with rapidly oscillating coefficients, Control Cybernet., Volume 23 (1994) no. 4, pp. 1-8
[14] Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., Volume 47 (2005) no. 2, pp. 197-243
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