Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function

TL;DR

This paper investigates the null controllability of a heat equation with a singular inverse-square potential related to the boundary distance, establishing controllability results for certain parameter ranges using a new Carleman estimate.

Contribution

It proves null controllability for the heat equation with a boundary-distance inverse-square potential when the parameter is at most 1/4, introducing a novel Carleman estimate for this purpose.

Findings

Null controllability holds for μ ≤ 1/4.

For μ > 1/4, solutions blow up and are not controllable.

A new Carleman estimate is developed for the analysis.

Abstract

This article is devoted to the analysis of control properties for a heat equation with singular potential $\mu/\delta^2$, defined on a bounded $C^2$ domain $\Omega\subset\mathbb{R}^N$, where $\delta$ is the distance to the boundary function. More precisely, we show that for any $\mu\leq 1/4$ the system is exactly null controllable using a distributed control located in any open subset of $\Omega$, while for $\mu>1/4$ there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate.