Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences

Let (A, D (A)) be a diagonalizable generator of a C-0-semigroup of contractions on a complex Hilbert space X, B is an element of L(C, Y), Y being some suitable extrapolation space of X, and u is an element of L-2(0, T; C). Under some assumptions on the sequence of eigenvalues Lambda = {lambda(k)}(k >= 1) subset of C of (A, D(A)), we prove the existence of a minimal time T-0 is an element of[0, infinity] depending on Bernstein's condensation index of Lambda and on B such that y' = Ay + Bu is null-controllable at any time T > T-0 and not null-controllable for T < T-0. As a consequence, we solve controllability problems of various systems of coupled parabolic equations. In particular, new results on the boundary controllability of one-dimensional parabolic systems are derived. These seem to be difficult to achieve using other classical tools. (C) 2014 Elsevier Inc. All rights reserved.