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

Abstract

Let (A, D(A)) a diagonalizable generator of a C0−semigroup of contractions on a complex Hilbert space X, B2L(C, Y ), Y being some suitable extrapolation space of X, and u 2 L2 (0, T; C). Under some assumptions on the sequence of eigenvalues Λ = {λk}k≥1 ⇢ C of (A, D(A)), we prove the existence of a minimal time T0 2 [0, 1] depending on Bernstein’s condensation index of Λ and on B such that y 0 = Ay+Bu is null-controllable at any time T >T0 and not null-controllable for T <T0. 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.