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

Nabile Boussaïd, Alessandro Duca

TL;DR

This work investigates local exact controllability of a one-dimensional bilinear Schrödinger equation under Dirichlet and periodic boundary conditions, incorporating discontinuous control potentials and a constant magnetic field. It demonstrates local exact controllability in $H^1_0$ for Dirichlet via a diffusion of spectral couplings and a moment-problem—solvable through an inverse mapping theorem—alongside local controllability in $H^1_p$ under Zeeman-type spectral decoupling in the periodic setting. The results rely on careful well-posedness in low-regularity spaces, endpoint-map $C^1$ regularity, and spectral/gauge-based arguments, with explicit coupling conditions $|ig<\mu\phi_l,\phi_k\big>|\ge C/(|k|+1)$ and non-resonant spectra. In the harmonic oscillator, the authors obtain global exact controllability for the linearized system within a weighted Hermite-space framework, clarifying the reachable subspaces and outlining open problems for Neumann boundaries and full nonlinear controllability.

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.

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

TL;DR

This work investigates local exact controllability of a one-dimensional bilinear Schrödinger equation under Dirichlet and periodic boundary conditions, incorporating discontinuous control potentials and a constant magnetic field. It demonstrates local exact controllability in for Dirichlet via a diffusion of spectral couplings and a moment-problem—solvable through an inverse mapping theorem—alongside local controllability in under Zeeman-type spectral decoupling in the periodic setting. The results rely on careful well-posedness in low-regularity spaces, endpoint-map regularity, and spectral/gauge-based arguments, with explicit coupling conditions and non-resonant spectra. In the harmonic oscillator, the authors obtain global exact controllability for the linearized system within a weighted Hermite-space framework, clarifying the reachable subspaces and outlining open problems for Neumann boundaries and full nonlinear controllability.

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.

Paper Structure

This paper contains 17 sections, 19 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. The Dirichlet case
  3. Well-posedness
  4. Local exact controllability
  5. The periodic case
  6. Spectral properties
  7. Well-posedness
  8. Local exact controllability
  9. The Neumann case
  10. Well-posedness
  11. An open problem: the local exact controllability in H1
  12. Quantum Harmonic oscillator
  13. Spectral properties
  14. Well-posedness
  15. Controllability of the linearised system
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $A:=\{a_j\}_{j\leq N}\subset (0,1)$ with $N\in{\mathbb N}^*$ and $\mu\in H^1(A)$. 1. Let $T>0$, $\psi_0\in H^1_0$ and $u\in L^2( (0,T) ,{\mathbb R})$. Then, there exists a unique mild solution $\psi \in C( [0,T] ,H^1_0)$ to the problem 0.1_d that is a solution to the Duhamel formula: 2. Let $l\in{\mathbb N}^*$ such that Assume there exists a constant $C>0$ such that For any $T>0$, there e

Theorems & Definitions (40)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['well_d']}
  • Proposition 2.3
  • Theorem 2.4
  • ...and 30 more