# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021022
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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

Received  April 2020 Revised  November 2020 Early access January 2021

Fund Project: 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)

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.

Citation: Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021022
##### References:

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##### References:
The dynamic behaviour of system (1) for $\alpha = \beta = 0$
The dynamic behaviour of system (1) for $\alpha = 1, \beta = 0$ under $U(t) = -kw(x,t)$
The dynamic behaviour of system (1) for $\alpha = 2, \beta = 1$ under $U(t) = -kw(x,t)$
The dynamic behaviour of system (1) for $\alpha = 1, \beta = 0$ under the control (3)
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