强耦合变系数波动方程的间接边界镇定

强耦合变系数波动方程的间接边界镇定

Indirect Boundary Stabilization of Strongly Coupled Variable Coefficient Wave Equations

摘要/Abstract

图/表 1

参考文献 26

Metrics

数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 389-407.

崔佳楠^1,²(),柴树根^1,^*()

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

收稿日期:2024-05-28 修回日期:2024-10-30 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 柴树根 E-mail:1391958885@qq.com;sgchai@sxu.edu.cn
作者简介:崔佳楠，E-mail:1391958885@qq.com
基金资助:
国家自然科学基金(12271316);晋中学院博士专项基金(23E00611)

Jianan Cui^1,²(),Shugen Chai^1,^*()

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

Received:2024-05-28 Revised:2024-10-30 Online:2025-04-26 Published:2025-04-09
Contact: Shugen Chai E-mail:1391958885@qq.com;sgchai@sxu.edu.cn
Supported by:
NSFC(12271316);Jinzhong University Research Funds for Doctor(23E00611)

摘要：

该文旨在研究带变系数和边界阻尼的强耦合波动方程的间接镇定. 值得注意的是, 系统中只有一个方程直接受到边界阻尼的影响. 利用黎曼几何方法和高阶能量方法, 证明了全局耦合系统的衰减速率受边界条件类型的影响. 研究结果表明, 当无阻尼方程具有 Dirichlet 边界条件时, 系统表现出指数稳定性, 而当无阻尼方程具有 Neumann 边界条件时, 系统仅有多项式稳定性. 最后, 在 Dirichlet 和 Neumann 边界条件下建立了局部耦合系统的指数稳定性.

关键词: 间接镇定, 强耦合, 非均匀介质, 边界反馈

Abstract:

In this paper, the indirect stabilization of strongly coupled wave equations with variable coefficients and boundary damping is studied. It is important to note that only one equation in the system is directly affected by boundary damping. By using Riemannian geometry method and higher order energy method, it is proved that the decay rate of the globally coupled system is affected by the type of boundary conditions. The results show that when the undamped equations have Dirichlet boundary conditions, the system exhibits exponential stability, while when the undamped equations have Neumann boundary conditions, the system has only polynomial stability. Finally, the exponential stability of the locally coupled system is established under Dirichlet and Neumann boundary conditions.

Key words: indirect stabilization, strong coupling, inhomogeneous media, boundary feedback

中图分类号:

O231.4

崔佳楠,柴树根. 强耦合变系数波动方程的间接边界镇定[J]. 数学物理学报, 2025, 45(2): 389-407.

Jianan Cui,Shugen Chai. Indirect Boundary Stabilization of Strongly Coupled Variable Coefficient Wave Equations[J]. Acta mathematica scientia,Series A, 2025, 45(2): 389-407.

图1

在 $\Gamma_0$ 上, $(x-x_0)\cdot \nu(x)\leq 0. \ \Gamma_0$ 被称为边界 $\Gamma$ 上的不可控部分, 而在 $\Gamma_1$ 上满足 $(x-x_0)\cdot \nu(x)>0$ . 这种区域称为星形域."

图1

[1]	Rao B P. On the sensitivity of the transmission of boundary dissipation for strongly coupled and indirectly damped systems of wave equations. Z Angew Math Phys, 2019, 70(3): 1-25
[2]	Chai S G, Liu K S. Boundary stabilization of the transmission of wave equations with variable coefficients. Chinese Ann Math A, 2005, 26(5): 605-612
[3]	Chai S G, Guo Y X. Boundary stabilization of wave equations with variable coefficients and memory. Differ Integral Equ, 2004, 17(5/6): 669-680
[4]	Liu Y X. Polynomial decay of a variable coefficient wave equation with an acoustic undamped boundary condition. J Math Anal Appl, 2019, 479(2): 1641-1652
[5]	Ning Z H. Asymptotic behavior of the nonlinear Schrödinger equation on exterior domain. Math Re Lett, 2020, 27(1): 1825-1866
[6]	Zhao X P, Ning Z H, Shen S X. Stabilization of the wave equation with variable coefficients and a delay in dissipative internal feedback. J Math Anal Appl, 2013, 405(1): 148-155
[7]	Guo B Z, Shao Z C. On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback. Nonlinear Anal-Theor, 2009, 71(12): 5961-5978
[8]	Ning Z H, Yan Q X. Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback. J Math Anal Appl, 2010, 367(1): 167-173
[9]	Guo Z L, Chai S G. Stabilization of the transmission wave equation with variable coefficients and interior delay. J Geom Anal, 2022, 2: 33-53
[10]	Feng S J, Feng D X. Nonlinear boundary stabilization of wave equations with variable coefficients. Chinese Ann Math, 2003, 24(2): 239-248
[11]	Yao P F. On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J Control Optim, 1999, 37(5): 1568-1599
[12]	Lasiecka I, Triggiani R, Yao P F. Exact controllability for second-order hyperbolic equations with variable coefficients-principal part and first-order term. Nonlinear Anal-Theor, 1997, 30(1): 111-122
[13]	Lasiecka I, Triggiani R, Yao P F. Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J Math Anal Appl, 1999, 235(1): 13-57
[14]	Russell D L. A general framework for the study of indirect damping mechanisms in elastic systems. J Math Anal Appl, 1993, 173(2): 339-358
[15]	Alabau-Boussouira F. Stabilisation frontiere indirecte de systemes faiblement couplés. Compt Rendus Acad Sci Math, 1999, 328(11): 1015-1020
[16]	Alabau-Boussouira F, Cannarsa P, Komornik V. Indirect internal stabilization of weakly coupled evolution equations. J Evol Equ, 2002, 2(2): 127-150
[17]	Alabau-Boussouira F, Cannarsa P, Guglielmi R. Indirect stabilization of weakly coupled systems with hybrid boundary conditions. Math Control Relat F, 2011, 1: 413-436
[18]	Alabau-Boussouira F, Léautaud M. Indirect stabilization of locally coupled wave-type systems. ESAIM Contr Optim Ca, 2012, 18(2): 548-582
[19]	Alabau-Boussouira F, Wang Z Q, Yu L X. A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities. ESAIM Contr Optim Ca, 2017, 23(2): 721-749
[20]	Cavalcanti M M, Domingos Cavalcanti V N, Mansouri S, et al. Asymptotic stability for a strongly coupled Klein-Gordon system in an inhomogeneous medium with locally distributed damping. J Differ Equ, 2020, 268(2): 447-489
[21]	Cavalcanti M M, Corrêa W J, Domingos Cavalcanti V N, et al. Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping. Z Angew Math Phys, 2021, 72: 1-20
[22]	Wu H, Shen C L, Yu Y L. An Introduction to Riemannian Geometry. Beijing: Peking University Press, 1989
[23]	Yao P F. Modeling and Control in Vibrational and Structural Dynamics:A Differential Geometric Approach. Boca Raton: CRC Press, 2011
[24]	Yao P F. Observability inequalities for the Euler-Bernoulli plate with variable coefficients. Contemp Math-Singap, 2000, 268: 383-406
[25]	Martinez P. Stabilisation de systèmes distribués semilinéaires: domaines presque étoilés et inégalités intégrales généralisées. France: University of Strasbourg, 1998
[26]	Martinez P. A new method to obtain decay rate estimates for dissipative systems. ESAIM Contr Optim Ca, 1999, 4: 419-444

Just accepted

Online first

Just accepted

Online first

Viewed

Full text

	From	local

	Times	77
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	29

From	Others	local

Times	28	1
Rate	97%	3%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed