Figure 1.
The dynamic behaviour of system (1) for $ \alpha = \beta = 0 $
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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 |
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 and Continuous Dynamical Systems - B,
2021, 26
(12)
: 6359-6376.
doi: 10.3934/dcdsb.2021022
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References:
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Rapid stabilization of multi-dimensional Schrödinger equation with the internal delay control, International Journal of Control, 92 (2019), 2521-2531.
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Stabilization for Schrödinger equation with a time delay in the boundary input, Applicable Analysis, 95 (2016), 963-977.
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Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback, Mathematical Control and Related Fields, 8 (2018), 383-395.
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An example on the effect of time delays in boundary feedback stabilization of wave equation, SIAM J. Control Optim., 24 (1986), 152-156.
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Two examples of ill-posedness with respect to small time delays in stabilized elastic systems, IEEE Trans. Autom. Control, 38 (1993), 163-166.
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R. Datko,
Two examples of ill-posedness with respect to time delays revisited, IEEE Trans. Autom. Control, 42 (1997), 511-515.
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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.
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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.
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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. |
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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.
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Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256.
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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. |
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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.
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S. Nicaise and C. Pignotti,
Stabilization of second-order evolution equations with time delay, Math Control Signals Syst., 26 (2014), 563-588.
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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.
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| [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.
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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.
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| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [8] |
I. Gumowski and S. Mira, Optimization in Control Theory and Practice, Cambridge University Press, Cambridge, 1968.
|
| [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. |
| [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. |
| [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. |
| [12] |
E. Machtygier and E. Zuazua,
Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
Figure 2.
The dynamic behaviour of system (1) for $ \alpha = 1, \beta = 0 $ under $ U(t) = -kw(x,t) $
Figure 3.
The dynamic behaviour of system (1) for $ \alpha = 2, \beta = 1 $ under $ U(t) = -kw(x,t) $
Figure 4.
The dynamic behaviour of system (1) for $ \alpha = 1, \beta = 0 $ under the control (3)
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