Boundary controllability for a 1D degenerate parabolic equation with a Robin boundary condition

TL;DR

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

Contribution

It introduces a novel approach combining singular Sturm-Liouville theory and spectral decomposition to prove controllability and estimate control costs for degenerate parabolic equations with Robin conditions.

Findings

Proved null controllability of the system.

Derived upper bounds for control cost.

Established lower bounds for control cost.

Abstract

In this paper we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm-Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier-Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.