A spectral approach of the null controllability of coupled non ...
A spectral approach of the null controllability of coupled non ...
A spectral approach of the null controllability of coupled non ...
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Spectral properties <strong>of</strong> A Parabolic <strong>controllability</strong> properties An applicationNotation:Spectral inequality for A• H = state space; Y = observation (control) space• B ∗ ∈ L(H, Y ) a bounded observation operator• θ = max{1/2, p −1 − (1 − q)}TheoremSuppose that ∀v ∈ H 2 (0, T 0 ) satisfying v(0) = 0, we have forsome ν ∈ (0, 1]‖v‖ H 1 (ζ,T 0 −ζ) ≤ C‖v‖ 1−νH 1 (0,T 0 )(‖B ∗ ∂ t v(0)‖ Y + ‖(−∂ 2 t + A ∗ )v‖ H 0 (0,T 0 )) ν.Then, ∃C, D > 0 such that ∀α > 0, ∀w ∈ Π ∗ αH,‖w‖ H ≤ Ce Dαθ ‖B ∗ w‖ Y.If A = −∆ + b · ∇ + c, <strong>the</strong>n θ = max{1/2, (n − 1)/2}