On the regularity of null-controls of the linear 1-d heat equation

Micu, Sorin; Zuazua, Enrique

doi:10.1016/j.crma.2011.05.005

Partial Differential Equations

On the regularity of null-controls of the linear 1-d heat equation
[Sur la regularité des contrôles pour lʼéquation de la chaleur linéaire]

Présenté par : Roland Glowinski

Micu, Sorin ¹ ; Zuazua, Enrique ^2,³

¹ Facultatea de Matematica si Informatica, Universitatea din Craiova, Al. I. Cuza 13, Craiova 200585, Romania
² Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Basque Country, Spain
³ BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160, Derio, Basque Country, Spain

Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 673-677.

Résumé
Abstract

Lʼéquation de la chaleur 1-d dans un intervalle borné est contrôlable depuis le bord. Plus précisément, pour tout $y^{0} \in L^{2} (0, 1)$ il existe un unique contrôle frontière de norme $L^{2} (0, T)$ minimale qui conduit la solution de lʼéquation linéaire de la chaleur à zéro à lʼinstant $T > 0$ . Ce contrôle est donné par la dérivée normale au bord dʼune solution de lʼéquation adjointe avec une donnée initiale ${\hat{φ}}^{0}$ qui minimise une fonction coût quadratique. Dans cet article nous étudions la relation entre la régularité de $y^{0}$ et celle de ${\hat{φ}}^{0}$ . Nous montrons que, si $y^{0}$ a une seule fréquence de Fourier, la fonction ${\hat{φ}}^{0}$ correspondante nʼappartient à aucun espace de Sobolev dʼexposant négatif. Ce fait explique le caractère mal posé du problème et lʼinefficacité de la plupart des algorithmes numériques existants et en particulier la convergence lente des méthodes de régularisation de type Tychonoff.

The 1-d heat equation in a bounded interval is null-controllable from the boundary. More precisely, for each initial data $y^{0} \in L^{2} (0, 1)$ there corresponds a unique boundary control of minimal $L^{2} (0, T)$ -norm which drives the state of the 1-d linear heat equation to zero in time $T > 0$ . This control is given as the normal derivative of a solution of the homogeneous adjoint equation whose initial data ${\hat{φ}}^{0}$ minimizes a suitable quadratic cost functional. In this Note we analyze the relation between the regularity of the initial datum to be controlled $y^{0}$ and that of ${\hat{φ}}^{0}$ . We show that, if $y^{0}$ has only one Fourier mode, the corresponding ${\hat{φ}}^{0}$ does not belong to any Sobolev space of negative exponent. This explains the severe ill-posedness of the problem and the lack of efficiency of most of the existing numerical algorithms for the numerical approximation of the controls, and in particular the slow convergence rate of Tychonoff regularization procedures.

Reçu le : 2011-04-01
Accepté le : 2011-05-10
Publié le : 2011-06-01

DOI : 10.1016/j.crma.2011.05.005

Affiliations des auteurs :

Micu, Sorin ¹ ; Zuazua, Enrique ^{2, 3}

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Micu, Sorin; Zuazua, Enrique. On the regularity of null-controls of the linear 1-d heat equation. Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 673-677. doi : 10.1016/j.crma.2011.05.005. http://www.numdam.org/articles/10.1016/j.crma.2011.05.005/

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