American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2021022

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

Xiaorui Wang 1,2,, , Genqi Xu 2, and  Hao Chen 3,

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  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.

Keywords: Schrödinger equation, deference-type control, parameterization controller, exponential stabilization.
Mathematics Subject Classification: Primary: 35J10, 93C20; Secondary: 93D15.
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:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8] I. Gumowski and S. Mira, Optimization in Control Theory and Practice, Cambridge University Press, Cambridge, 1968.   Google Scholar
[9]

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

[10]

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

[11]

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

[12]

E. Machtygier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256.   Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8] I. Gumowski and S. Mira, Optimization in Control Theory and Practice, Cambridge University Press, Cambridge, 1968.   Google Scholar
[9]

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

[10]

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

[11]

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

[12]

E. Machtygier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256.   Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392
[2]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125
[3]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018
[4]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[5]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376
[6]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[7]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[8]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284
[9]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[10]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[11]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298
[12]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[13]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[14]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294
[15]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298
[16]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[17]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002
[18]

Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021008
[19]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[20]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

2019 Impact Factor: 1.27

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences