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

TL;DR

This paper establishes the null controllability of a 1D degenerate parabolic PDE with drift and singular potential, using spectral and moment methods to estimate control costs under Neumann boundary conditions.

Contribution

It introduces a novel approach combining spectral decomposition and moment method to analyze controllability and cost estimates for degenerate PDEs with singular potentials.

Findings

Proves null controllability of the PDE.

Provides upper and lower bounds for control cost.

Employs spectral and moment methods for analysis.

Abstract

We prove the null controllability of a one-dimensional degenerate parabolic equation with drift and a singular potential. Here, we consider a weighted Neumann boundary control at the left endpoint, where the potential arises. We use a spectral decomposition of a suitable operator, defined in a weighted Sobolev space, and 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.