Observability and controllability for Schr\"odinger equations in the semi-periodic setting

TL;DR

This paper establishes local exact controllability for nonlinear Schr"odinger equations on semi-periodic waveguides, advancing control theory in settings with mixed Euclidean and periodic geometries.

Contribution

It provides the first controllability results for NLS on waveguides, analyzing linear observability and applying fixed-point methods for nonlinear control.

Findings

Proves local exact controllability for cubic NLS on b2 d7 a0T.

Analyzes observability of linear Schrf6dinger operators on waveguides.

Uses Bourgain spaces and HUM estimates for control results.

Abstract

Strichartz estimates, well-posedness theory and long time behavior for (nonlinear) Schr\"odinger 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 Schr\"odinger equations in the waveguide (semi-periodic) setting. Our main result establishes local exact controllability for the cubic nonlinear Schr\"odinger 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 Schr\"odinger 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.