Observability properties of the singular Grushin equation

TL;DR

This paper investigates the observability of the singular Grushin equation with an inverse square potential, identifying minimal observation times and exploring the influence of singularity dimension on heat equation observability.

Contribution

It provides new insights into the minimal time of observability for the Grushin equation with singularities, utilizing recent Carleman estimates and extending results to almost-Riemannian manifolds.

Findings

Exact minimal observability time in specific configurations

Dependence of minimal observability time on singularity dimension

Application of recent Carleman estimates to singular PDEs

Abstract

We study the observability properties of the Grushin equation with an inverse square potential, whose singularity occurs at the boundary of two-dimensional rectangular domains or in the interior of the domain in higher dimensions. In some specific configurations of the observation set, we obtain the exact minimal time of observability. The analysis we present relies on recent Carleman estimates obtained by K. Beauchard, J. Dard\'e, and S. Ervedoza. As a byproduct of these results, we observe, for the heat equation associated to the Laplace-Beltrami operator on almost-Riemannian manifolds, a dependence of the minimal time of observability on the dimension of the singularity.