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

Abstract

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.