Non-null-controllability of the Grushin operator in 2D - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Comptes Rendus. Mathématique Année : 2017

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

Armand Koenig

Résumé

We are interested in the exact null controllability of the equation $\partial_t f - \partial_x^2 f - x^2 \partial_y^2f = \mathbf 1_\omega u$, with control $u$ supported on $\omega$. We show that, when $\omega$ 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 entire functions, which Runge's theorem disproves. To that end, we study in particular the first eigenvalue of the operator $-\partial_x^2 + (nx)^2$ with Dirichlet conditions on $(-1,1)$ and we show a quite precise estimation it satisfies, even when $n$ is in some complex domain.
Fichier principal
Vignette du fichier
main.pdf (584.97 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01654043 , version 1 (02-12-2017)

Identifiants

Citer

Armand Koenig. Non-null-controllability of the Grushin operator in 2D. Comptes Rendus. Mathématique, 2017, 355 (12), pp.1215 - 1235. ⟨10.1016/j.crma.2017.10.021⟩. ⟨hal-01654043⟩
147 Consultations
123 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More