具有不同类型阻尼弱耦合板方程的间接镇定和最优衰减率

数学物理学报 ›› 2023, Vol. 43 ›› Issue (6): 1681-1698.

具有不同类型阻尼弱耦合板方程的间接镇定和最优衰减率

韩忠杰¹,贺以恒¹,赵志学^2,^*()

¹天津大学数学学院天津 300354
²天津师范大学数学科学学院天津 300387

收稿日期:2021-12-23 修回日期:2023-04-10 出版日期:2023-12-26 发布日期:2023-11-16
通讯作者: *赵志学，E-mail: zxzhao@amss.ac.cn
基金资助:
国家自然科学基金(62073236);国家自然科学基金(11901433)

Indirect Stabilization and Optimal Decay Rates of Weakly Coupled Plates with Various Types of Damping

Han Zhongjie¹,He Yiheng¹,Zhao Zhixue^2,^*()

¹School of Mathematics, Tianjin University, Tianjin 300354
²School of Mathematical Science, Tianjin Normal University, Tianjin 300387

Received:2021-12-23 Revised:2023-04-10 Online:2023-12-26 Published:2023-11-16
Supported by:
NSFC(62073236);NSFC(11901433)

摘要/Abstract

摘要：

该文主要研究弱耦合弹性板系统的稳定性和最优衰减率, 其中系统中仅有一块板带有阻尼 (粘性阻尼、结构阻尼或 Kelvin-Voigt 阻尼). 基于频域方法和对系统算子细致的谱分析, 导出了系统的最优多项式衰减率. 此外, 还确定了系统的最优衰减率与阻尼阶数之间的关系, 并且发现了一个有趣的现象, 即间接阻尼的阶数越高, 弱耦合板系统衰减越慢. 最后, 通过数值模拟对理论结果进行了验证.

关键词: 耦合弹性板, 间接镇定, 最优多项式衰减, 谱分析, 频域方法

Abstract:

This work is concerned with the stabilization and optimal decay rates of a weakly coupled (coupling through displacements) elastic plate system, where the damping (viscous damping, structural damping or Kelvin-Voigt damping) is actuated at only one of the two plates. The optimal polynomial decay rate is derived based on the frequency domain approach and detailed spectral analysis of the system operator. Moreover, the relationship between the optimal decay rates of the system and the order of damping is identified. Besides, an interesting phenomenon is found that the higher the order of indirect damping is actuated, the slower polynomial decay rate of the weakly coupled plate system achieves. Finally, some numerical simulations are presented to verify the theoretical results.

Key words: Coupled elastic plate, Indirect stabilization, Optimal polynomial decay, Spectral analysis, Frequency domain analysis

中图分类号:

O231.4

韩忠杰, 贺以恒, 赵志学. 具有不同类型阻尼弱耦合板方程的间接镇定和最优衰减率[J]. 数学物理学报, 2023, 43(6): 1681-1698.

Han Zhongjie, He Yiheng, Zhao Zhixue. Indirect Stabilization and Optimal Decay Rates of Weakly Coupled Plates with Various Types of Damping[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1681-1698.

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