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Observability and controllability for Schrödinger equations in the semi-periodic setting

Jingrui Niu, Zehua Zhao

TL;DR

The paper extends control theory for Schrödinger equations to semi-periodic waveguides $\\mathbb{R}^m\\times\\mathbb{T}^n$ by proving local exact controllability for the cubic NLS on $\\mathbb{R}^2\\times\\mathbb{T}$ under a geometric control condition. The approach combines linear observability, established via a Floquet-Bloch reduction to torus problems and $H^s$ resolvent-type estimates, with a nonlinear fixed-point argument in Bourgain spaces to handle the cubic term. Key steps include deriving null controllability for the linear problem through the Hilbert Uniqueness Method and then leveraging a decomposition $u=v+\\Psi$ to obtain a contraction mapping for small data. The results bridge noncompact and periodic geometries, expanding the applicability of controllability results to waveguide models relevant in nonlinear optics and signal processing.

Abstract

Strichartz estimates, well-posedness theory and long time behavior for (nonlinear) Schrödinger equations on waveguide manifolds $\mathbb{R}^m \times \mathbb{T}^n$ are intensively studied in recent decades while the corresponding control theory and observability estimates remain incomplete. The purpose of this short paper is to investigate the observability and controllability for Schrödinger equations in the waveguide (semi-periodic) setting. Our main result establishes local exact controllability for the cubic nonlinear Schrödinger equations (NLS) on $\mathbb{R}^2 \times \mathbb{T}$, under certain geometric conditions on the control region. To address the nonlinear control problem, we begin by analyzing the observability properties of the linear Schrödinger operator on a general waveguide manifold $\mathbb{R}^m \times \mathbb{T}^n$. Utilizing $H^s$ estimates of the Hilbert Uniqueness Method (HUM) operator and Bourgain spaces, we then prove local exact controllability through a fixed-point method.

Observability and controllability for Schrödinger equations in the semi-periodic setting

TL;DR

The paper extends control theory for Schrödinger equations to semi-periodic waveguides by proving local exact controllability for the cubic NLS on under a geometric control condition. The approach combines linear observability, established via a Floquet-Bloch reduction to torus problems and resolvent-type estimates, with a nonlinear fixed-point argument in Bourgain spaces to handle the cubic term. Key steps include deriving null controllability for the linear problem through the Hilbert Uniqueness Method and then leveraging a decomposition to obtain a contraction mapping for small data. The results bridge noncompact and periodic geometries, expanding the applicability of controllability results to waveguide models relevant in nonlinear optics and signal processing.

Abstract

Strichartz estimates, well-posedness theory and long time behavior for (nonlinear) Schrödinger equations on waveguide manifolds are intensively studied in recent decades while the corresponding control theory and observability estimates remain incomplete. The purpose of this short paper is to investigate the observability and controllability for Schrödinger equations in the waveguide (semi-periodic) setting. Our main result establishes local exact controllability for the cubic nonlinear Schrödinger equations (NLS) on , under certain geometric conditions on the control region. To address the nonlinear control problem, we begin by analyzing the observability properties of the linear Schrödinger operator on a general waveguide manifold . Utilizing estimates of the Hilbert Uniqueness Method (HUM) operator and Bourgain spaces, we then prove local exact controllability through a fixed-point method.

Paper Structure

This paper contains 17 sections, 23 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. The statement of the main results
  4. Strategy of the proofs
  5. Organization of the rest of this paper
  6. Notations
  7. Linear observability and Stationary estimates
  8. The Floquet-Bloch transform
  9. Observability for partial-twisted Laplacian
  10. Stationary estimate: proof of Theorem \ref{['thm2']}
  11. Observability in $\mathbb{R}^m\times\mathbb{T}^n$: Proof of Theorem \ref{['thm3']}
  12. Well-posedness of NLS
  13. Local null controllability of NLS: Proof of Theorem \ref{['thm4']}
  14. Control for linear problem: Hilbert uniqueness method
  15. Proof of Theorem \ref{['thm4']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $T>0$, and $\epsilon=\pm1$. Let $\Omega=(\Omega_1,\Omega_2)$ satisfy the condition (G). For any $s\geq 1$, there exists $\delta>0$, such that for all $u_0,u_{f}\in H^s(\mathbb{R}^2\times\mathbb{T})$ satisfying that $\|u_0\|_{H^s(\mathbb{R}^2\times\mathbb{T})}+\|u_f\|_{H^s(\mathbb{R}^2\times\math fulfils $u|_{t=T}=u_f$.

Theorems & Definitions (41)

  • Theorem 1.1: Exact controllability
  • Theorem 1.2: Null controllability
  • Theorem 1.3: Stationary estimate
  • Remark 1.4
  • Theorem 1.5: Observability
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • ...and 31 more