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arXiv · 2504.16641

$H^1$ local exact controllability of some one-dimensional bilinear Schr{ö}dinger equations

Nabile Boussaïd·Alessandro Duca
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Abstract

The local exact controllability of the one-dimensional bilinear Schr{ö}dinger equation with Dirichlet boundary conditions has been extensively studied in subspaces of H 3 since the seminal work of K. Beauchard. Our first objective is to revisit this result and establish the controllability in H 1 0 for suitable discontinuous control potentials. In the second part, we consider the equation in the presence of periodic boundary conditions and a constant magnetic field. We prove the local exact controllability of periodic H 1 -states, thanks to a Zeeman-type effect induced by the magnetic field which decouples the resonant spectrum. Finally, we discuss open problems and partial results for the Neumann case and the harmonic oscillator.

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Nabile Boussaïd, Alessandro Duca. 2025-04-23. $H^1$ local exact controllability of some one-dimensional bilinear Schr{ö}dinger equations. https://arxiv.org/abs/2504.16641

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