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

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

2012-01-16

arxiv.org/abs/1201.3390v1

Mathematics

Optimization and Control

21 pages

Scientific paper

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 [18] and then generalized by Ervedoza [8]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, excepting some cases where $\Omega$ changes the convexity at origin, in which cases the control is distributed in a semi-annulus surrounding the singularity. We emphasize that our results cannot be obtained straightforwardly from the previous works [18], [8].

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