CONTROLLABILITY OF THE HEAT EQUATION WITH AN INVERSE-SQUARE POTENTIAL LOCALIZED ON THE BOUNDARY?

摘要

This article is devoted to analyzing control properties for the heat equation with singular potential -μ/|x|~2 arising at the boundary of a smooth domain Ω ? R~N, N ≥ 1. This problem was first studied by Vancostenoble and Zuazua [J. Funct. Anal., 254 (2008), pp. 1864-1902] and then generalized by Ervedoza [Comm. Partial Differential Equations, 33 (2008), pp. 1996-2019] in the context of interior singularity. Roughly speaking, these results showed that for any value of parameters μ ≤ 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, μN:= N~2/4. In this article we extend the previous results of Vancostenoble and Zuazua and of Ervedoza to the case of boundary singularity. More precisely, we show that for any μ ≤ μN, 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.