Null controllability of the 1D heat equation with interior inverse square potential

TL;DR

This paper proves the null controllability of the 1D heat equation with an interior inverse square potential at x=0, using spectral analysis and moment methods, addressing an open problem from 2015.

Contribution

It introduces a suitable self-adjoint extension for the singular operator and demonstrates null controllability with internal and boundary controls in small time.

Findings

Null controllability achieved with internal control in arbitrary small time.

Null controllability achieved with boundary control on one side.

Spectral analysis of the singular operator using properties of Bessel functions.

Abstract

This paper aims to answer an open problem posed by Morancey in 2015 concerning the null controllability of the heat equation on (-1, 1) with an internal inverse square potential located at x = 0. For the range of singularity under study, after having introduced a suitable self-adjoint extension that enables to transmit information from one side of the singularity to another, we prove null-controllability in arbitrary small time, firstly with an internal control supported in an arbitrary measurable set of positive measure, secondly with a boundary control acting on one side of the boundary. Our proof is mainly based on a precise spectral study of the singular operator together with some recent refinements of the moment method of Fattorini and Russell. This notably requires to use some fine (and sometimes new) properties for Bessel functions and their zeros.