Null-controllability properties of the generalized two-dimensional Baouendi–Grushin equation with non-rectangular control sets

Dardé, Jérémi; Koenig, Armand; Royer, Julien

Annales Henri Lebesgue, Tome 6 (2023), pp. 1479-1522

We consider the null-controllability problem for the generalized Baouendi–Grushin equation $({\partial }_t-{\partial }_x^2-q{\left(x\right)}^2{\partial }_y^2)f={\mathbb{1}}_{\omega }u$ on a rectangular domain. Sharp controllability results already exist when the control domain $\omega$ is a vertical strip, or when $q\left(x\right)=x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $\omega$. In some geometries for $\omega$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability.

Our proof relies on several tools: known results when $\omega$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schrödinger operator $-{\partial }_x^2+{\nu }^{2}q{\left(x\right)}^2$ when $Re\left(\nu \right)>0$, pseudo-differential-type operators on polynomials and Runge’s theorem for the lower bound.

Nous considérons le problème de la contrôlabilité à zéro de l’équation de Baouendi–Grushin généralisée $({\partial }_t-{\partial }_x^2-q{\left(x\right)}^2{\partial }_y^2)f={\mathbb{1}}_{\omega }u$ sur un domaine rectangulaire. On connaît déjà des résultats précis de contrôlabilité lorsque le domaine de contrôle $\omega$ est une bande verticale, ou lorsque $q\left(x\right)=x$. Dans cet article, nous démontrons des bornes supérieures et inférieures du temps minimal de contrôlabilité à zéro, pour une classe générale de potentiels $q$ et de domaines de contrôle $\omega$ possiblement non rectangulaires. Pour certaines zones de contrôle, la borne supérieure et la borne inférieure coïncident, auquel cas nous connaissons la valeur exacte du temps minimal de contrôlabilité à zéro.

Notre démonstration s’appuie sur plusieurs outils. Pour la borne supérieure du temps minimal de contrôlabilité, nous utilisons des résultats connus lorsque $\omega$ est une bande verticale et des arguments de troncature. Pour la borne inférieure, nous utilisons une analyse spectrale de l’opérateur de Schrödinger $-{\partial }_x^2+{\nu }^{2}q{\left(x\right)}^2$ lorsque $Re\left(\nu \right)>0$, des opérateurs de type pseudo-différentiel sur les polynômes et le théorème de Runge.

Classification : 35K65, 93B05, 47B28, 47A10
Keywords: null-controllability, observability, degenerate parabolic equations, resolvant estimates

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