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 $$u_t-u_{xx}-\frac{\mu }{x^2}u=0,(x,t)\in (0,1)\times (0,T).$$ 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.

A second version of the manuscript. The problem treated is slightly different (only one singular potential is considered) and the techniques employed are different too