Non-null-controllability of the Grushin operator in 2D

Koenig, Armand

doi:10.1016/j.crma.2017.10.021

Partial differential equations

Non-null-controllability of the Grushin operator in 2D
[Non-contrôlabilité à zéro de l'opérateur de Grushin en dimension 2]

Présenté par : Jean-Michel Coron

Koenig, Armand ¹

¹ Laboratoire de mathématiques Jean-Alexandre-Dieudonné, UMR 7351 CNRS UNS, Université de Nice – Sophia Antipolis, 06108 Nice cedex 02, France

Comptes Rendus. Mathématique, Tome 355 (2017) no. 12, pp. 1215-1235.

Résumé
Abstract

Nous nous intéressons à la contrôlabilité exacte à zéro de l'équation $\partial_{t} f - \partial_{x}^{2} f - x^{2} \partial_{y}^{2} f = 1_{ω} u$ sur $(- 1, 1) \times T$ , avec contrôle u sur ω. Nous démontrons que si ω est le complémentaire d'une bande horizontale, l'équation considérée n'est contrôlable pour aucun temps. L'idée principale est d'interpréter l'inégalité d'observabilité comme une estimation sur les fonctions entières, que nous nions grâce au théorème de Runge. Pour réaliser cette interprétation, nous étudions en particulier la première valeur propre de $- \partial_{x}^{2} + {(n x)}^{2}$ avec conditions de Dirichlet sur $] - 1, 1 [$ , et en obtenons une estimation assez précise, y compris pour certains n complexes.

We are interested in the exact null controllability of the equation $\partial_{t} f - \partial_{x}^{2} f - x^{2} \partial_{y}^{2} f = 1_{ω} u$ , with control u supported on ω. We show that, when ω does not intersect a horizontal band, the considered equation is never null-controllable. The main idea is to interpret the associated observability inequality as an $L^{2}$ estimate on polynomials, which Runge's theorem disproves. To that end, we study in particular the first eigenvalue of the operator $- \partial_{x}^{2} + {(n x)}^{2}$ with Dirichlet conditions on $(- 1, 1)$ , and we show a quite precise estimation it satisfies, even when n is in some complex domain.

Reçu le : 2016-11-10
Accepté le : 2017-10-24
Publié le : 2017-11-21

DOI : 10.1016/j.crma.2017.10.021

Affiliations des auteurs :

Koenig, Armand ¹

¹ Laboratoire de mathématiques Jean-Alexandre-Dieudonné, UMR 7351 CNRS UNS, Université de Nice – Sophia Antipolis, 06108 Nice cedex 02, France

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Koenig, Armand. Non-null-controllability of the Grushin operator in 2D. Comptes Rendus. Mathématique, Tome 355 (2017) no. 12, pp. 1215-1235. doi : 10.1016/j.crma.2017.10.021. http://www.numdam.org/articles/10.1016/j.crma.2017.10.021/

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^☆ This work was partially supported by the ERC advanced grant SCAPDE, seventh framework program, agreement No. 320845.