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 $Ω:= \mathcal{B}_R \times \tilde Ω$, where $\mathcal{B}_R$ is the open ball of radius $R$ in dimension $d \ge 2$, and $\tilde Ω$ 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 $ω\times \tilde Ω$, with $ω\subset \mathcal{B}_R$ (internal observation), and $Γ\times \tilde Ω$, with $Γ\subset \partial \mathcal{B}_R$ (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.

Figures (7)

• Figure 1: Domain $\Omega$, observation set $\omega \times \tilde{\Omega}$
• Figure 2: Domain $\Omega$, observation set $\Gamma_{\!R} \times \tilde{\Omega}$
• Figure 3: Geometrical configuration studied in Beauchard2013.
• Figure 4: Geometrical configuration studied in Koenig2017 -- In this situation, the Baouendi-Grushin equation is not observable for any $T>0$.
• Figure 5: Geometrical configuration studied in dardekoenigroyer23, in which the critical time of observability is precisely determined.

Theorems & Definitions (76)

• Definition 1.1: Internal observability
• Definition 1.2: Boundary observability
• theorem 1.1
• theorem 1.2
• theorem 1.3
• theorem 1.4
• proposition 1.5
• Remark 1.1
• Corollary 1.6
• Corollary 1.7