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

2025/02/13 by Jingrui Niu, Niu, Jingrui, Zehua Zhao +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2502.09312

openalex publication_date 2025/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Strichartz estimates, well-posedness theory and long time behavior for (nonlinear) Schrödinger equations on waveguide manifolds ℝ^m × \mathbbTⁿ 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 ℝ² × \mathbbT, 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 ℝ^m × \mathbbTⁿ. 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.

Related

• Controllability of the Schrödinger equation on unbounded domains without geometric control condition
• Internal control of the Schrödinger equation
• Control and Stabilization of the Nonlinear Schroedinger Equation on Rectangles
• Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3
• On bilinear Strichartz estimates on waveguides with applications