  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].
