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

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Spectral properties of A Parabolic controllability properties An applicationFinal controlTheoremSuppose that θ < 1. Then, for every T > 0, u 0 ∈ H, there exists acontrol function g ∈ L 2 (0, T ; Y ) such that the solution u of theproblem{∂t u + Au = Bg,satisfies u(T ) = 0.u |t=0 = u 0 ∈ H,Idea of the proof (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
Spectral properties of A Parabolic controllability properties An applicationFinal controlIm(z) ∂P q K 0Re(z)αα 0α 1λ 0Γ αα k−1 α kFigure: Elimination and dissipation
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A spectral approach of the null controllability of coupled non ...

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