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Small-time approximate controllability of the logarithmic Schr\''dinger equation

Karine Beauchard, Rémi Carles, Eugenio Pozzoli

TL;DR

This work proves that logarithmic Schrödinger equations with bilinear controls on $\mathbb{T}^d$ or $\mathbb{R}^d$ are small-time $L^2$-approximately controllable, extending previous linear results to a nonlinear, non-Lipschitz setting. The authors adapt a linear-control strategy based on small-time controllability of phases and gradient flows, employing a WKB-inspired analysis and a nonlinear Trotter–Kato-type splitting to handle the logarithmic nonlinearity. The main contributions include establishing $L^2$-STAR controllability for phase multipliers and for flows of gradient vector fields, and combining these to obtain STAC for the nonlinear log-NLS in both compact and non-compact settings, under suitable hypotheses on the potential. A negative result is also shown: Gaussian-state invariance under quadratic potentials prevents small-time controllability in certain cases, delineating the boundary of STAC in this nonlinear bilinear framework. These results advance the understanding of rapid quantum-state steering under non-polynomial nonlinearities and broaden the toolbox for nonlinear bilinear control of dispersive PDEs.

Abstract

We consider Schr{ö}dinger equations with logarithmic nonlinearity and bilinear controls, posed on $\mathbb{T}^d$ or $\mathbb{R}^d$. We prove their small-time global $L^2$-approximate controllability. The proof consists in extending to this nonlinear framework the approach introduced by the first and third authors in \cite{beauchard-pozzoli2} to control the linear equation: it combines the small-time controllability of phases and gradient flows. Due to the nonlinearity, the required estimates are more difficult to establish than in the linear case. The proof here is inspired by WKB analysis. This is the first result of (small-time) global approximate controllability, for nonlinear Schr{ö}dinger equations, with bilinear controls.

Small-time approximate controllability of the logarithmic Schr\''dinger equation

TL;DR

This work proves that logarithmic Schrödinger equations with bilinear controls on or are small-time -approximately controllable, extending previous linear results to a nonlinear, non-Lipschitz setting. The authors adapt a linear-control strategy based on small-time controllability of phases and gradient flows, employing a WKB-inspired analysis and a nonlinear Trotter–Kato-type splitting to handle the logarithmic nonlinearity. The main contributions include establishing -STAR controllability for phase multipliers and for flows of gradient vector fields, and combining these to obtain STAC for the nonlinear log-NLS in both compact and non-compact settings, under suitable hypotheses on the potential. A negative result is also shown: Gaussian-state invariance under quadratic potentials prevents small-time controllability in certain cases, delineating the boundary of STAC in this nonlinear bilinear framework. These results advance the understanding of rapid quantum-state steering under non-polynomial nonlinearities and broaden the toolbox for nonlinear bilinear control of dispersive PDEs.

Abstract

We consider Schr{ö}dinger equations with logarithmic nonlinearity and bilinear controls, posed on or . We prove their small-time global -approximate controllability. The proof consists in extending to this nonlinear framework the approach introduced by the first and third authors in \cite{beauchard-pozzoli2} to control the linear equation: it combines the small-time controllability of phases and gradient flows. Due to the nonlinearity, the required estimates are more difficult to establish than in the linear case. The proof here is inspired by WKB analysis. This is the first result of (small-time) global approximate controllability, for nonlinear Schr{ö}dinger equations, with bilinear controls.
Paper Structure (30 sections, 22 theorems, 102 equations)

This paper contains 30 sections, 22 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Models
  3. An equation posed on $M={\mathbb T}^d={\mathbb R}^d/(2\pi{\mathbb Z})^d$.
  4. An equation posed on $M={\mathbb R}^d$.
  5. Bibliographical comments about bilinear control
  6. Exact controllability
  7. Approximate controllability (in potentially large time)
  8. Obstructions to the small-time approximate controllability
  9. Small-time approximate controllability
  10. Background on the logarithmic Schrödinger equation
  11. Main results
  12. Proof strategy
  13. Structure of the article
  14. Well-posedness of the Cauchy problem
  15. A special property of the logarithmic nonlinearity
  16. ...and 15 more sections

Key Result

Theorem 1.4

Let $d \in {\mathbb N}^*$ and $\lambda \in {\mathbb R}$. If $V \in L^\infty({\mathbb T}^d,{\mathbb R})$ then system logNLS_tore is small-time $L^2$-approximately controllable.

Theorems & Definitions (49)

  • Definition 1.1: Exact controllability
  • Definition 1.2: Approximate controllability
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7: $L^2$-STAR maps
  • Lemma 1.8
  • Definition 1.9: Vector fields and flows $\Phi_f^s$
  • Definition 1.10
  • ...and 39 more