On the critical time of observability of the multi-dimensional Baouendi-Grushin equation

Abstract

We investigate the observability properties of the Baouendi-Grushin equation on a tensorized domain $\Omega := \mathcal{B}_R \times \tilde \Omega$, where $\mathcal{B}_R$ is the open ball of radius $R$ in dimension $d \ge 2$, and $\tilde \Omega$ is a smooth, bounded, open set of arbitrary dimension. Our main result is a precise calculation of the minimal observability time $T^*$, for tensorized observation sets of the form $\omega \times \tilde \Omega$, with $\omega \subset \mathcal{B}_R$ (internal observation), and $\Gamma \times \tilde \Omega$, with $\Gamma \subset \partial \mathcal{B}_R$ (boundary observation). The main novelty regards the sufficient condition, that is observability of the system when $T>T^*$. This is established by combining refined observability inequalities on the annulus--or the entire boundary--using Carleman estimates, together with a Lebeau-Robbiano strategy to localize the observation sets.