A spectral approach of the null controllability of coupled non ...

Spectral properties of A Parabolic controllability properties An applicationSpectral properties of A: resolvent estimateTheorem (Markus, Katsnel’son, Matsaev, Agranovich,Dzhanlatyan... 1970-2000...)Let 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ k ≤ · · · be the spectrum of A 0(selfadjoint). Suppose that ∃p > 0 s.t. lim sup j→∞ λ j j −p > 0.Then, setting β = max{0, p −1 − (1 − q)} there exists a sequence0 < α 0 ≤ α 1 ≤ · · · ≤ α k ≤ · · · → ∞ such that‖R A (z)‖ L(H) ≤ e Cα k β , k ∈ N, Re(z) = α k . (2)Example:If A = −∆ + b · ∇ + c, then q = 1/2, p = 2/n (Weyl). Thusβ = max{0, (n − 1)/2}