Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schr\"odinger equations

TL;DR

This paper proves that an arbitrary number of 1D bilinear Schrödinger equations can be simultaneously driven to any desired state in large time using a single control, extending previous results to multiple equations and general potentials.

Contribution

It establishes the first global exact controllability result for multiple 1D Schrödinger equations with a single control and generic potentials, combining several control strategies.

Findings

Proves global exact controllability for multiple Schrödinger equations.

Extends controllability results to arbitrary potentials and multiple particles.

Uses a novel combination of Coron's return method, Lyapunov strategy, and compactness.

Abstract

We consider a system of an arbitrary number of \textsc{1d} linear Schr\"odinger equations on a bounded interval with bilinear control. We prove global exact controllability in large time of these $N$ equations with a single control. This result is valid for an arbitrary potential with generic assumptions on the dipole moment of the considered particle. Thus, even in the case of a single particle, this result extends the available literature. The proof combines local exact controllability around finite sums of eigenstates, proved with Coron's return method, a global approximate controllability property, proved with Lyapunov strategy, and a compactness argument.