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

Abstract

This article is devoted to analyze control properties for the heat equation with singular potential $-μ/|x|^2$ arising at the boundary of a smooth domain $Ω\subset \rr^N$, $N\geq 1$. 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 $μ\leq μ(N):=(N-2)^2/4$, the corresponding parabolic system can be controlled to zero with the control distributed in any open subset of the domain. The critical value $μ(N)$ 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 $μ_{N}:=N^2/4$. In this article we extend the previous results in [18],[8], to the case of boundary singularity. More precisely, we show that for any $μ\leq μ_N$, 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].