    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:
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##### References:
  H. Chen, Y. Xie and G. Xu, Rapid stabilization of multi-dimensional Schrödinger equation with the internal delay control, International Journal of Control, 92 (2019), 2521-2531.  doi: 10.1080/00207179.2018.1444283.  Google Scholar  H.-Y. Cui, Z.-J. Han and G.-Q. Xu, Stabilization for Schrödinger equation with a time delay in the boundary input, Applicable Analysis, 95 (2016), 963-977.  doi: 10.1080/00036811.2015.1047830.  Google Scholar  H. Cui, D. Liu and G. Xu, Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback, Mathematical Control and Related Fields, 8 (2018), 383-395.  doi: 10.3934/mcrf.2018015.  Google Scholar  R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equation, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar  R. Datko, Two examples of ill-posedness with respect to small time delays in stabilized elastic systems, IEEE Trans. Autom. Control, 38 (1993), 163-166.  doi: 10.1109/9.186332.  Google Scholar  R. Datko, Two examples of ill-posedness with respect to time delays revisited, IEEE Trans. Autom. Control, 42 (1997), 511-515.  doi: 10.1109/9.566660.  Google Scholar  X. Feng, G. Xu and Y. Chen, Rapid stabilization of an Euler-Bernoulli beam with the internal delay control, International Journal of Control, 92 (2019), 42-55.  doi: 10.1080/00207179.2017.1286693.  Google Scholar  I. Gumowski and S. Mira, Optimization in Control Theory and Practice, Cambridge University Press, Cambridge, 1968.   Google Scholar  B.-Z. Guo and K.-Y. Yang, Output feedback stabilization of a one-dimensional Schrödinger equation by boundary observation with time delay, IEEE Trans. Autom. Control, 55 (2010), 1226-1232.  doi: 10.1109/TAC.2010.2042363.  Google Scholar  Y. Li, H. Chen and Y. Xie, Stabilization with arbitrary convergence rate for the Schrödinger equation subjected to an input time delay, J. Syst. Sci. Complex, (2020). doi: 10.1007/s11424-020-9294-6. Google Scholar  J.-J. Liu and J.-M. Wang, Output-feedback stabilization of an anti-stable Schrödinger equation by boundary feedback with only displacement obeservation, J. Dyn. Control Syst., 19 (2013), 471-482.  doi: 10.1007/s10883-013-9189-0.  Google Scholar  E. Machtygier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256. Google Scholar  S. Nicaise and S.-E. Rebiai, Stability of the Schödinger equation with a delay term in the boundary or internal feedbacks, Portugaliae Mathematica, 68 (2011), 19-39.  doi: 10.4171/PM/1879.  Google Scholar  S. Nicaise and J. Valein, Stabilization of second-order evolution equations with time unbounded feedback with delay, ESAIM: Control Optim. Calc. Var., 16 (2010), 420-456.  doi: 10.1051/cocv/2009007.  Google Scholar  S. Nicaise and C. Pignotti, Stabilization of second-order evolution equations with time delay, Math Control Signals Syst., 26 (2014), 563-588.  doi: 10.1007/s00498-014-0130-1.  Google Scholar  Y. F. Shang and G. Q. Xu, The stability of a wave equation with delay-dependent position, IMA Journal of Mathematical Control and Information, 28 (2011), 75-95.  doi: 10.1093/imamci/dnq026.  Google Scholar  Y. F. Shang and G. Q. Xu, Stabilization of an Euler-Bernoulli beam with input delay in the boundary control, Syst. Control Lett., 61 (2012), 1069-1078.  doi: 10.1016/j.sysconle.2012.07.012.  Google Scholar  Y. F. Shang and G. Q. Xu, Dynamic feedback control and exponential stabilization of a compound system, J. Math Anal. Appl., 442 (2015), 858-879.  doi: 10.1016/j.jmaa.2014.09.013.  Google Scholar  Y. F. Shang, G. Q. Xu and X. Li, Output-based stabilization for a one-dimensional wave equation with distributed input delay in the boundary control, IMA Journal of Mathematical Control and Information, 33 (2016), 95-119.  doi: 10.1093/imamci/dnu030.  Google Scholar  K. Sriram and M. S. Gopinathan, A two variable delay modle for the circadian rhythm of Neurospora crassa, J. Theor. Biol., 231 (2004), 23-38.  doi: 10.1016/j.jtbi.2004.04.006.  Google Scholar  J. Srividhya and M. S. Gopinathan, A simple time delay model for eukaryotic cell cycle, J. Theor. Biol., 241 (2006), 617-627.  doi: 10.1016/j.jtbi.2005.12.020.  Google Scholar  G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave system with input delay in the boundary control, ESAIM: Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.  Google Scholar  K.-Y. Yang and C.-Z. Yao, Stabilization of one-dimensional Schrödinger equation with variable coefficient under delayed boundary output, Asian. J. Control., 15 (2013), 1531-1537. Google Scholar 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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