The cost of controlling strongly degenerate parabolic equations

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut - (xalphaux)x with 0 < x < l and alpha is an element of (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 alpha. We prove that the control cost blows up with an explicit exponential rate, as eC/((2-alpha)2T), when alpha -> 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, Guichal, 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 Jnu of large order nu (obtained by ordinary differential equations techniques).