The cost of controlling strongly degenerate parabolic equations^*

¹ Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy.
² Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse; CNRS UPS IMT 31062 Toulouse Cedex 9, France.

^** Corresponding author: cannarsa@mat.uniroma2.it

Received: 7 July 2017
Accepted: 18 January 2018

Abstract

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = u_t − (x^αu_x)_x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, 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 α. We prove that the control cost blows up with an explicit exponential rate, as e^{C/((2−α)2T)}, when α → 2⁻ and/or T → 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, Güichal, 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_ν of large order ν (obtained by ordinary differential equations techniques).