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 $-\mu /|x{|}^2$ arising at the boundary of a smooth domain $Ω\subset \rr^N$, $N\ge 1$. 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 $\mu \le \mu (N):=(N-2{)}^2/4$, 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 $\mu \le {\mu }_N$, 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].

32 pages

References in corpus (4)

• Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results
• Hardy inequality and Pohozaev identity for operators with boundary singularities: some applications
• Hardy's inequality and curvature
• On the Hardy-Poincaré inequality with boundary singularities

Cited by in corpus (4)

• Existence and cost of boundary controls for a degenerate/singular parabolic equation
• New estimates for the Hardy constants of multipolar Schrödinger operators
• Null controllability of one-dimensional parabolic equations by the flatness approach
• Null controllability of the parabolic spherical Grushin equation