# American Institute of Mathematical Sciences

December  2021, 26(12): 6359-6376. doi: 10.3934/dcdsb.2021022

## 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 Published  December 2021 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, 2021, 26 (12) : 6359-6376. doi: 10.3934/dcdsb.2021022
##### References:

show all references

##### 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)
 [1] Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161 [2] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [3] Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016 [4] César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka, Thales Maier Souza. Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1367-1386. doi: 10.3934/cpaa.2020067 [5] Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188 [6] Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066 [7] Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273 [8] Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033 [9] Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275 [10] Xuecheng Wang. Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 5037-5048. doi: 10.3934/dcds.2017217 [11] Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206 [12] Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085 [13] Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843 [14] Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174 [15] Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093 [16] Lei Qiao. Matsaev's type theorems for solutions of the stationary Schrödinger equation and its applications. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5709-5720. doi: 10.3934/dcds.2016050 [17] Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $L^2$-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122 [18] Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014 [19] Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919 [20] Türker Özsarı, Kemal Cem Yılmaz. Stabilization of higher order Schrödinger equations on a finite interval: Part II. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021037

2020 Impact Factor: 1.327