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

Let (A,D(A))(A,D(A)) be a diagonalizable generator of a C0C0-semigroup of contractions on a complex Hilbert space XX, B∈L(C,Y)B∈L(C,Y), Y   being some suitable extrapolation space of XX, and u∈L2(0,T;C)u∈L2(0,T;C). Under some assumptions on the sequence of eigenvalues Λ={λk}k≥1⊂CΛ={λk}k≥1⊂C of (A,D(A))(A,D(A)), we prove the existence of a minimal   time T0∈[0,∞]T0∈[0,∞] depending on Bernstein's condensation index of Λ   and on BB such that y′=Ay+Buy′=Ay+Bu is null-controllable at any time T>T0T>T0 and not null-controllable for T