Boundary controllability for a 1D degenerate parabolic equation with drift and a singular potential

TL;DR

This paper establishes the null controllability of a 1D degenerate parabolic equation with drift and a singular potential, using spectral analysis and the moment method to estimate control costs.

Contribution

It introduces a novel spectral decomposition involving Bessel functions for a degenerate PDE with singular potential and derives bounds on controllability costs.

Findings

Spectral decomposition with Bessel functions and zeros.

Upper estimate of controllability cost.

Lower estimate of controllability cost.

Abstract

We prove the null controllability of a one dimensional degenerate parabolic equation with drift and a singular potential. We study the case the potential arises at the left end point and the weighted Dirichlet boundary control is located at this point. We get a spectral decomposition of a suitable operator, defined in a weighted Sobolev space, involving Bessel functions and their zeros, then we use the moment method by Fattorini and Russell to obtain an upper estimate of the cost of controllability. We also obtain a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.