带有时滞边界反馈的退化波动方程的镇定

数学物理学报 ›› 2024, Vol. 44 ›› Issue (1): 133-139.

带有时滞边界反馈的退化波动方程的镇定

白晋彦¹(),柴树根^2,^*()

1.晋中学院数学系山西晋中 030619
2.山西大学数学科学学院太原 030006

收稿日期:2023-01-25 修回日期:2023-09-11 出版日期:2024-02-26 发布日期:2024-01-10
通讯作者: 柴树根, E-mail：sgchai@sxu.edu.cn
作者简介:白晋彦, E-mail：bjy207@163.com;
基金资助:
国家自然科学基金(12271316);国家自然科学青年基金(12001343);山西省自然科学基金(201901D111042);晋中学院技术创新团队建设计划(jzxyjscxtd202111)

Stabilization of Degenerate Wave Equations with Delayed Boundary Feedback

Bai Jinyan¹(),Chai Shugen^2,^*()

1. Department of Mathematics, Jinzhong University, Shanxi Jinzhong 030619
2. School of Mathematical Sciences, Shanxi University, Taiyuan 030006

Received:2023-01-25 Revised:2023-09-11 Online:2024-02-26 Published:2024-01-10
Supported by:
NSFC(12271316);NSFY(12001343);Natural Science Foundation of Shanxi Province(201901D111042);Technical Innovation Team of Jinzhong University(jzxyjscxtd202111)

摘要/Abstract

摘要：

该文在一维空间中研究了带有时滞边界反馈的退化波动方程的镇定问题. 首先运用半群理论证明了其解的适定性, 然后通过选用合适的乘子以及构造合适的 Lyapunov 泛函，证明了该退化波动方程解的指数稳定性结果.

关键词: 退化波方程, 时滞, 反馈镇定

Abstract:

In this paper, we study the stabilization of degenerate wave equations with time-delay boundary feedback. Firstly, the well-posedness of the solution is proved by using semigroup theory. And then the exponential stability of the solution is proved by selecting suitable multipliers and constructing suitable Lyapunov functional.

Key words: Degenerate wave equation, Time delay, Feedback stabilization

中图分类号:

O231.4

白晋彦, 柴树根. 带有时滞边界反馈的退化波动方程的镇定[J]. 数学物理学报, 2024, 44(1): 133-139.

Bai Jinyan, Chai Shugen. Stabilization of Degenerate Wave Equations with Delayed Boundary Feedback[J]. Acta mathematica scientia,Series A, 2024, 44(1): 133-139.

参考文献 19

[1]	Citti G, Manfredini M. A degenerate parabolic equation arising in image processing. Commun Appl Anal, 2004, 8(1): 125-141
[2]	Ghil M. Climate stability for a Sellers type mode. J Atmo Sci, 1976, 29(5): 483-493
[3]	Ethier S N, Kurtz T G. Fleming-Viot processes in population genetics. SIAM J Control Optim, 1993, 31(2): 345-386 doi: 10.1137/0331019
[4]	Alabau-Boussouira F, Cannarsa P, Leugering G. Control and stabilization of degenerate wave equation. SIAM J Control Optim, 2017, 55(3): 2052-2087 doi: 10.1137/15M1020538
[5]	Zhang M M, Gao H. Null controllability of some degenerate wave equations. J Syst Sci Complex, 2017, 30(5): 1027-1041 doi: 10.1007/s11424-016-5281-3
[6]	Zhang M M, Gao H. Persistent regional null controllability of some degenerate wave equations. Math Meth Appl Sci, 2017, 40(16): 5821-5830 doi: 10.1002/mma.v40.16
[7]	Zhang M M, Gao H. Interior controllability of semi-linear degenerate wave equations. J Math Anal Appl, 2018, 457(1): 10-22 doi: 10.1016/j.jmaa.2017.07.057
[8]	Bai J Y, Chai S G. Exact controllability for some degenerate wave equations. Math Meth Appl Sci, 2020, 43(12): 7292-7302 doi: 10.1002/mma.v43.12
[9]	Bai J Y, Chai S G. Exact controllability for a one-dimensional degenerate wave equation in domains with moving boundary. Appl Math Lett, 2021, 119: Article ID 107235
[10]	Gumowski I, Mira C. Optimization in Control Theory and Practice. Cambridge: Cambridge University Press, 1968
[11]	Datko R, Lagness J, Poilis M P. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J Control Optim, 1986, 24(1): 152-156 doi: 10.1137/0324007
[12]	Datko R. Not all feedback stabilized hyperbolic systems are robust with respect to small time delay in their feedbacks. SIAM J Control Optim, 1988, 26(3): 697-713 doi: 10.1137/0326040
[13]	Wang J N, Yang D Z. Stability and bifurcation of a pathogen-immune model with delay and diffusion effects. Acta Math Sci, 2021, 41A(4): 1204-1217
[14]	Xu G Q, Yung S P, Li L K. Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim Calc Var, 2006, 12(4): 770-785 doi: 10.1051/cocv:2006021
[15]	Nicaise S, Pignotti C. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J Control Optim, 2006, 45(5): 1561-1585 doi: 10.1137/060648891
[16]	Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Differential Integral Equ, 2008, 21(9/10): 935-958
[17]	Ait Benhassi E M, Ammari K, Boulite S, Maniar L. Feedback stabilization of a class of evolution equations with delay. J Evol Equ, 2009, 9(1): 103-121 doi: 10.1007/s00028-009-0004-z
[18]	Nicaise S, Valein J. Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM Control Optim Calc Var, 2010, 16(2): 420-456 doi: 10.1051/cocv/2009007
[19]	Ammari K, Nicaise S, Pignotti C. Feedback boundary stabilization of wave equations with interior delay. Syst Control Lett, 2010, 59(10): 623-628 doi: 10.1016/j.sysconle.2010.07.007