Uniform stabilization of 1-D Schrödinger equation with internal difference-type control

Xiaorui Wang; Genqi Xu; Hao Chen

doi:10.3934/dcdsb.2021022

Article Contents

2021, Volume 26, Issue 12: 6359-6376. Doi: 10.3934/dcdsb.2021022 open access

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Uniform stabilization of 1-D Schrödinger equation with internal difference-type control

1.
Department of Mathematics and Statistics, Qinghai Nationalities University, Xining, Qinghai 810007, China
2.
School of Mathematics, Tianjin University, Tianjin, 300354, China
3.
School of Mechatronical Engineering, Beijing Institute of Technology, Beijing 100081, China

^* Corresponding author: Xiaorui Wang
^* Corresponding author: Xiaorui Wang

Received: March 31, 2020

Revised: October 31, 2020

Early access: January 2021

Published: December 2021

This project is partially supported by the National Natural Science Foundation in China (NSFC 61773277), and partially supported by NSF of Qinghai Province (2017-ZJ-908)

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Abstract

In this paper, we consider the stabilization problem of 1-D Schrödinger equation with internal difference-type control. Different from the other existing approaches of controller design, we introduce a new approach of controller design so called the parameterization controller. At first, we rewrite the system with internal difference-type control as a cascaded system of a transport equation and Schödinger equation; Further, to stabilize the system under consideration, we construct a target system that has exponential stability. By selecting the solution of nonlocal and singular initial value problem as parameter function and defining a bounded linear transformation, we show that the transformation maps the closed-loop system to the target system; Finally, we prove that the transformation is bounded inverse. Hence the closed-loop system is equivalent to the target system.

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Mathematics Subject Classification: Primary: 35J10, 93C20; Secondary: 93D15.

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Figure 1. The dynamic behaviour of system (1) for $ \alpha = \beta = 0 $

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Figure 2. The dynamic behaviour of system (1) for $ \alpha = 1, \beta = 0 $ under $ U(t) = -kw(x,t) $

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Figure 3. The dynamic behaviour of system (1) for $ \alpha = 2, \beta = 1 $ under $ U(t) = -kw(x,t) $

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Figure 4. The dynamic behaviour of system (1) for $ \alpha = 1, \beta = 0 $ under the control (3)

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