Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback

Haoyue Cui; Dongyi Liu; Genqi Xu

doi:10.3934/mcrf.2018015

Article Contents

2018, Volume 8, Issue 2: 383-395. Doi: 10.3934/mcrf.2018015

This issue Previous Article Next Article Compact perturbations of controlled systems

Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback

1.
Department of Mathematics, Tianjin University of Commerce, Tianjin 300134, China
2.
School of Mathematics, Tianjin University, Tianjin 300354, China

* Corresponding author: Dongyi Liu
* Corresponding author: Dongyi Liu

Received: January 31, 2016

Revised: February 28, 2017

Published: May 2018

This research is supported by the Natural Science Foundation of China grant NSFC-61573252.

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Abstract

Design of controller subject to a constraint for a Schrödinger equation is considered based on the energy functional of the system. Thus, the resulting closed-loop system is nonlinear and its well-posedness is proven by the nonlinear monotone operator theory and a complex form of the nonlinear Lax-Milgram theorem. The asymptotic stability and exponential stability of the system are discussed with the LaSalle invariance principle and Riesz basis method, respectively. In the end, a numerical simulation illustrates the feasibility of the suggested feedback control law.

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Mathematics Subject Classification: Primary: 93D15, 93D20; Secondary: 35B40.

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Figure 1. Real part of $w(x, t)$

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Figure 2. Imaginary part of $w(x, t)$

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Figure 3. Real and imaginary parts of $w(1, t)$

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