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 applicationFinal controlTheoremSuppose that θ < 1. Then, for every T > 0, u 0 ∈ H, <strong>the</strong>re exists acontrol function g ∈ L 2 (0, T ; Y ) such that <strong>the</strong> solution u <strong>of</strong> <strong>the</strong>problem{∂t u + Au = Bg,satisfies u(T ) = 0.u |t=0 = u 0 ∈ H,Idea <strong>of</strong> <strong>the</strong> pro<strong>of</strong> (Lebeau-Robbiano ’95): [0, T ] = ⋃ j∈N [a j, a j+1 ],a j+1 = a j + 2T j• if t ∈ (a j , a j + T j ]: g =partial control → cost e Dαθ j• if t ∈ (a j + T j , a j+1 ]: g = 0, parabolic dissipation e −Cα j