Open AccessUniform stabilization of 1-D Schrödinger equation with internal difference-type control
Author(s) -
Xiaorui Wang,
Genqi Xu,
Hao Chen
Publication year - 2021
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2021022
Subject(s) - bounded function , transformation (genetics) , mathematics , controller (irrigation) , control theory (sociology) , stability (learning theory) , function (biology) , type (biology) , mathematical analysis , control (management) , computer science , ecology , biochemistry , chemistry , artificial intelligence , machine learning , evolutionary biology , biology , agronomy , gene
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.