Sharp Estimate of the Cost of Controllability for a Degenerate Parabolic Equation with Interior Degeneracy

This work is motivated by the study of null controllability for the typical degenerate parabolic equation with interior degeneracy and one-sided control: u(t) - (vertical bar x vertical bar(alpha)u(x))(x) = h(x, t)chi((a,b)), x is an element of(-1, 1), with 0 < a < b < 1. It was proved in [7] that this equation is null controllable (in any positive time T) if and only if alpha < 1, and that the cost of null controllability blows up as alpha -> 1(-). This is related to the following property of the eigenvalues: the gap between an eigenvalue of odd order and the consecutive one goes to 0 as alpha -> 1(-) (see [7]). The goal of the present work is to provide optimal upper and lower estimates of the null controllability cost, with respect to the degeneracy parameter (when alpha -> 1(-)) and in short time (when T -> 0(+)). We prove that the null controllability cost behaves as 1/1-alpha as a alpha -> 1(-) and as e(1/T) as T -> 0(+). Our analysis is based on the construction of a suitable family biorthogonal to the sequence (e(lambda nt)) n in L-2(0, T), under some general gap conditions on the sequence (lambda(n))(n), conditions that are suggested by a motivating example.