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

TL;DR

This paper proves boundary controllability for a 1D heat equation with a singular inverse-square potential at the boundary, demonstrating null controllability via boundary control at the singularity for parameters less than 1/4.

Contribution

It establishes null controllability for the heat equation with inverse-square potential at the boundary using the moment method, for all potential strengths below 1/4.

Findings

Null controllability achieved for μ<1/4

Boundary control at the singularity point is effective

Employs the moment method by Fattorini and Russell

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.