The cost of controlling strongly degenerate parabolic equations

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).