The cost of controlling strongly degenerate parabolic equations

TL;DR

This paper investigates how the cost of controlling a strongly degenerate parabolic equation depends on the degeneracy parameter, showing it grows exponentially as the parameter approaches a critical value or the control time approaches zero.

Contribution

It provides explicit exponential estimates for the controllability cost of a degenerate parabolic equation as the degeneracy parameter approaches a critical value.

Findings

Control cost blows up exponentially as degeneracy parameter approaches 2.

Control cost increases rapidly as control time approaches zero.

Uses advanced mathematical techniques including Bessel functions and moment method.

Abstract

We consider the typical one-dimensional strongly degenerate parabolic operator $Pu= u_t - (x^\alpha u_x)_x$ with $0<x<\ell$ and $\alpha\in(0,2)$, controlled either by a boundary control acting at $x=\ell$, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter $\alpha$. We prove that the control cost blows up with an explicit exponential rate, as $e^{C/((2-\alpha)^2 T)}$, when $\alpha \to 2^-$ and/or $T\to 0^+$. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, G\"uichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions $J_\nu$ of large order $\nu$ (obtained by ordinary differential equations techniques).