Null controllability of parabolic equations with interior degeneracy and one-sided control

TL;DR

This paper investigates the null controllability of a parabolic PDE with interior degeneracy, showing controllability depends on the degeneracy level and whether controls act on one or both sides of the degeneracy point.

Contribution

It characterizes null controllability for interior degenerate parabolic equations with one-sided control, based on spectral analysis and Bessel functions, distinguishing between weak and strong degeneracy cases.

Findings

Null controllability holds for weak degeneracy ($eta ext{ in } (0,1)$).

Controllability fails for strong degeneracy ($eta ext{ in } [1,2)$) with one-sided control.

Eigenvalues and eigenfunctions are explicitly described using Bessel functions.

Abstract

For $\alpha\in (0,2)$ we study the null controllability of the parabolic operator $$Pu= u_t - (\vert x \vert ^\alpha u_x)_x\qquad (1<x<1),$$ which degenerates at the interior point $x=0$, for locally distributed controls acting only one side of the origin (that is, on some interval $(a,b)$ with $0<a<b<1$). Our main results guarantees that $P$ is null controllable if and only if it is weakly degenerate, that is, $\alpha \in (0,1)$. So, in order to steer the system to zero, one needs controls to act on both sides of the point of degeneracy in the strongly degenerate case $\alpha\in [1,2)$. Our approach is based on spectral analysis and the moment method. Indeed, we completely describe the eigenvalues and eigenfunctions of the associated stationary operator in terms of Bessel functions and their zeroes for both weakly and strongly degenerate problems. Hence, we obtain lower $L^2$ bounds for the eigenfunctions on the control region in the case $\alpha \in [0,1)$ and deduce the lack of observability in the case of $\alpha \in [1,2)$. We also provide numerical evidence to illustrate our theoretical results.