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Article Dans Une Revue Dynamics of Partial Differential Equations Année : 2020

Simultaneous global exact controllability in projection of infinite 1D bilinear Schrödinger equations

Résumé

The aim of this work is to study the controllability of infinite bilinear Schr\"odinger equations on a segment. We consider the equations (BSE) $i\partial_t\psi^{j}=-\Delta\psi^j+u(t)B\psi^j$ in the Hilbert space $L^2((0,1),\mathbb{C})$ for every $j\in\mathbb{N}^*$. The Laplacian $-\Delta$ is equipped with Dirichlet homogeneous boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We prove the simultaneous local and global exact controllability of infinite (BSE) in projection. The local controllability is guaranteed for any positive time and we provide explicit examples of $B$ for which our theory is valid. In addition, we show that the controllability of infinite (BSE) in projection onto suitable finite dimensional spaces is equivalent to the controllability of a finite number of (BSE) (without projecting). In conclusion, we rephrase our controllability results in terms of density matrices.
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Dates et versions

hal-01481873 , version 1 (03-03-2017)
hal-01481873 , version 2 (23-11-2017)
hal-01481873 , version 3 (02-06-2018)
hal-01481873 , version 4 (03-06-2019)
hal-01481873 , version 5 (16-07-2020)

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Paternité - Pas d'utilisation commerciale

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Citer

Alessandro Duca. Simultaneous global exact controllability in projection of infinite 1D bilinear Schrödinger equations. Dynamics of Partial Differential Equations, 2020, 17 (3), pp.275-306. ⟨10.4310/DPDE.2020.v17.n3.a4⟩. ⟨hal-01481873v5⟩
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