# American Institute of Mathematical Sciences

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March  2019, 9(1): 191-219. doi: 10.3934/mcrf.2019011

## Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential

Received  May 2016 Revised  May 2017 Published  November 2018

Fund Project: This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 694126DyCon). Moreover, this work was partially supported by the Grants MTM2014-52347, MTM201792996 of MINECO (Spain), and AFOSR Grant FA9550-18-1-0242.

We analyze controllability properties for the one-dimensional heat equation with singular inverse-square potential
 \begin{align*} u_t-u_{xx}-\frac{\mu}{x^2}u = 0, \;\;\; (x, t)\in(0, 1)\times(0, T).\end{align*}
For any
 $\mu<1/4$
, we prove that the equation is null controllable through a boundary control
 $f\in H^1(0, T)$
acting at the singularity point x = 0. This result is obtained employing the moment method by Fattorini and Russell.
Citation: Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control & Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011
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##### References:
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