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

Spectral properties of A Parabolic controllability properties An applicationSpectral properties of A: localization of the spectrumPropositionLet A = A 0 + A 1 be an operator on H. Assume that(a) Re(Au, u) H ≥ λ 0 ‖u‖ 2 Hfor all u ∈ D(A),(b) A 0 : D(A 0 ) ⊂ H → H is selfadjoint, positive, with a densedomain and compact resolvent,(c) A 1 : D(A 1 ) ⊂ H → H satisfies|(A 1 u, u) H | ≤ C‖A 1/20u‖ 2qH ‖u‖2−2q Hfor q ∈ [0, 1)We then have,(i) D(A) = D(A 0 ) ⊂ D(A 1 ) ⊂ H and A has a dense domain anda compact resolvent,(ii) Sp(A) ⊂ P q K 0= {z ∈ C, Re(z) ≥ 0, | Im(z)| < K 0 Re(z) q }(iii) ‖R A (z)‖ L(H) ≤2d(z,Sp(A 0 )) ≤ 2d(z,R +) for z ∈ C \ Pq K 0,