Controllability of the heat equation with an inverse-square potential localized on the boundary

TL;DR

This paper investigates the controllability of the heat equation with a boundary inverse-square potential, extending previous interior singularity results to boundary cases and establishing control to zero for parameters up to a critical Hardy constant.

Contribution

It extends controllability results for the heat equation with boundary inverse-square potential, identifying the maximal parameter range for control, which was not previously established.

Findings

Controllability holds for all μ ≤ N^2/4 with boundary singularity.

The results extend previous interior singularity control results to boundary cases.

Control to zero is achievable in any open subset of the domain.

Abstract

This article is devoted to analyze control properties for the heat equation with singular potential $-\mu/|x|^2$ arising at the boundary of a smooth domain $\Omega\subset \rr^N$, $N\geq 1$. This problem was firstly studied by Vancostenoble and Zuazua [20] and then generalized by Ervedoza [10]in the context of interior singularity. Roughly speaking, these results showed that for any value of parameters $\mu\leq \mu(N):=(N-2)^2/4$, the corresponding parabolic system can be controlled to zero with the control distributed in any open subset of the domain. The critical value $\mu(N)$ stands for the best constant in the Hardy inequality with interior singularity. When considering the case of boundary singularity a better critical Hardy constant is obtained, namely $\mu_{N}:=N^2/4$. In this article we extend the previous results in [18],[8], to the case of boundary singularity. More precisely, we show that for any $\mu \leq \mu_N$, we can lead the system to zero state using a distributed control in any open subset. We emphasize that our results cannot be obtained straightforwardly from the previous works [20], [10].