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

Abstract

This work is motivated by the study of null controllability for the typical degenerate parabolic equation with interior degeneracy and one-sided control:
ut − (|x|
αux)x = h(x, t)χ(a,b)
, x ∈ (−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 α < 1, and that the cost of null controllability blows up as α → 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 α → 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 α → 1 −) and in short time (whenT → 0
+). We prove that the null controllability cost behaves as 1 1−α as α → 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 λnt )n in L2 (0, T), under some general gap conditions on the sequence (λn)n, conditions that are suggested by a motivating example.