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

The aim of this work is to study the controllability of infinite bilinear Schrödinger 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 $j \in \mathbb{N}^{\ast}$. 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 $T \gt 0$. 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.

Keywords

Schrödinger equation, simultaneous control, global exact controllability, moment problem, perturbation theory, density matrices

2010 Mathematics Subject Classification

Primary 93B05, 93C20. Secondary 35Q41, 81Q15.

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Received 21 November 2019

Published 7 July 2020