具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定

数学物理学报 ›› 2024, Vol. 44 ›› Issue (4): 946-959.

具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定

章春国^1,^*(),孙宝楠¹,付煜之¹,于欣²

¹杭州电子科技大学数学系杭州 310018
²浙大宁波理工学院计算机与数据工程学院浙江宁波 315100

收稿日期:2023-04-29 修回日期:2023-10-07 出版日期:2024-08-26 发布日期:2024-07-26
通讯作者: *章春国, E-mail: cgzhang@hdu.edu.cn
基金资助:
国家自然科学基金(61503103);浙江省自然科学基金重点项目(LZ21A010001)

Stabilization of 2-D Mindlin Timoshenko Plate Systems with Local Damping

Zhang Chunguo^1,^*(),Sun Baonan¹,Fu Yuzhi¹,Yu Xin²

¹Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018
²School of Computer and Data Engineering, Ningbo Tech University, Zhejiang Ningbo 315100

Received:2023-04-29 Revised:2023-10-07 Online:2024-08-26 Published:2024-07-26
Supported by:
NSFC(61503103);key project of Zhejiang Provencial Natural Science Foundation(LZ21A010001)

摘要/Abstract

摘要：

该文研究了具有局部阻尼的二维 Mindlin-Timoshenko 板系统, 首先将原系统转化成抽象柯西问题, 运用算子半群理论得到了系统的适定性. 借助系统频域稳定性结果, 通过引入几何光学条件结合乘子技巧得到了系统的一致指数稳定性.

关键词: Mindlin-Timoshenko 板系统, 局部阻尼, 分片乘子法, 一致指数稳定性

Abstract:

In this paper, the two-dimensional Mindlin-Timoshenko plate system with local damping is studied. First, the original system is transformed into an abstract Cauchy problem, and the well posedness of the system is obtained by using operator semigroup theory. With the help of the frequency domain stability results of the linear system, the uniform exponential stability of the system is obtained by introducing geometric optical conditions and multiplier techniques.

Key words: Mindlin-Timoshenko plate system, Local damping, Piecewise multiplier method, Uniform exponential stability.

中图分类号:

O231.4

章春国, 孙宝楠, 付煜之, 于欣. 具有局部阻尼的二维 Mindlin-Timoshenko 板系统的镇定[J]. 数学物理学报, 2024, 44(4): 946-959.

Zhang Chunguo, Sun Baonan, Fu Yuzhi, Yu Xin. Stabilization of 2-D Mindlin Timoshenko Plate Systems with Local Damping[J]. Acta mathematica scientia,Series A, 2024, 44(4): 946-959.

图/表 1

参考文献 16

[1]	Lagnese J E. Boundary Stabilization of Thin Plates. Philadelhia, PA: Society for Industrial and Applied Mathematics, 1989
[2]	Nicaise S. Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks. Mathematical Control and Related Fields, 2011, 1(3): 331-352
[3]	Rivera J E M, Oquendo H P O. Asymptotic behavior of a Mindlin-Timoshenko plate with viscoelastic dissipation on the boundary. Funkcialaj Ekvacioj, 2003, 46(3): 363-382
[4]	Dalsen G V. Stabilization of a thermoelastic Mindlin-Timoshenko plates model revisited. Zeitschrift fur angewandte Mathematik und Physik, 2013, 64(4): 1305-1325
[5]	Dalsen G V. Polynomial decay rate of a thermoelastic Mindlin-Timoshenko plates model with Dirichlet boundary conditions. Zeitschrift fur angewandte Mathematik und Physik, 2015, 66(1): 113-128
[6]	Louis T. Indirect stabilization of a Mindlin-Timoshenko plates. Journal of Mathematical Analysis and Applications, 2017, 449(2): 1880-1891
[7]	章春国, 付煜之, 刘宇标. 二维 Mindlin-Timoshenko 板系统的稳定性与最优性. 数学物理学报, 2021, 41A(5): 1465-1491
	Zhang C G, Fu Y Z, Liu Y B. Stability and optimality of 2-D Mindlin-Timoshenko plate system. Acta Math Sci, 2021, 41A(5): 1465-1491
[8]	Liu Y, Zhang C, Chen T. Stabilization of 2-d Mindlin-Timoshenko plates with localized acoustic boundary feedback. Journal of Industrial and Management Optimization, 2022, 18(2): 1009-1034
[9]	Liu K. Locally distributed control and damping for the conservative systems. SIAM Journal on Control and Optimization, 1997, 35(5): 1574-1590
[10]	Martinez P. A new method to obtain decay rate estimates for dissipative systems. ESAIM: Control, Optimisation and Calculus of Variations, 1999, 4: 419-444
[11]	Liu K, Liu Z. Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping. SIAM Journal on Control and Optimization, 1998, 36(3): 1086-1098
[12]	Fatiha A B. Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation. Journal of Evolution Equations, 2006, 6(1): 95-112
[13]	Hong G, Hong H. Stabilization of transmission system of Kirchhoff plates and wave equations with a localized Kelvin-Voigt damping. Journal of Evolution Equations, 2021, 20: 2239-2264
[14]	Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer Science and Business Media, 2012
[15]	Adams R A. Sobolev Spaces. New York: Acadamic Press, 1975
[16]	Huang F. Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann of Differ Eqs, 1985, 1: 43-56