带分数阶磁效应的压电梁在有/无热效应时的稳定性

数学物理学报 ›› 2023, Vol. 43 ›› Issue (2): 355-376.

带分数阶磁效应的压电梁在有/无热效应时的稳定性

安雁宁¹,刘文军^1,^2,^3,^*(),孔奥文¹

¹南京信息工程大学数学与统计学院南京 210044
²南京信息工程大学江苏省应用数学中心南京210044
³南京信息工程大学江苏省系统建模与数据分析国际合作联合实验室南京210044

收稿日期:2021-09-29 修回日期:2022-10-17 出版日期:2023-04-26 发布日期:2023-04-17
通讯作者: 刘文军，E-mail：wjliu@nuist.edu.cn
基金资助:
国家自然科学基金(12271261);江苏省重点研发计划(社会发展)(BE2019725);江苏省青蓝工程项目和江苏省研究生科研与实践创新计划(KYCX22_1125)

Stability of Piezoelectric Beams with Magnetic Effects of Fractional Derivative Type and with/without Thermal Effects

An Yanning¹,Liu Wenjun^1,^2,^3,^*(),Kong Aowen¹

¹School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044
²Center for Applied Mathematics of Jiangsu Province, Nanjing University of Information Science and Technology, Nanjing 210044
³Jiangsu International Joint Laboratory on System Modeling and Data Analysis, Nanjing University of Information Science and Technology, Nanjing 210044

Received:2021-09-29 Revised:2022-10-17 Online:2023-04-26 Published:2023-04-17
Supported by:
National Natural Science Foundation of China(12271261);Key Research and Development Program of Jiangsu Province (Social Development)(BE2019725);Qing Lan Project of Jiangsu Province and the Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX22_1125)

摘要/Abstract

摘要：

该文考虑了具有分数阶磁效应的一维压电梁系统的适定性及稳定性. 首先, 通过引入新函数将原系统转换为不含分数阶边界项的等价系统, 并利用Lumer-Philips定理证明了该系统的适定性. 然后, 基于谱分析证得无热效应的压电梁系统的非指数稳定性, 并借助Borichev-Tomilov定理[33]进一步推得系统是多项式稳定的. 此外, 该文又讨论了有热效应的压电梁系统的适定性, 并借助扰动泛函方法证明了压电梁系统在带有热效应时的指数稳定性.

关键词: 压电梁, 渐近行为, 分数阶导数, 半群理论

Abstract:

In this paper, we consider the well-posedness and asymptotic behavior of a one-dimensional piezoelectric beam system with control boundary conditions of fractional derivative type, which represent magnetic effects on the system. By introducing two new matrixs to deal with control boundary conditions of fractional derivative type, we obtain a new equivalent system, so as to show the well-posedness of the system by using Lumer-Philips theorem. We then prove the lack of exponential stability by a spectral analysis, and obtain the polynomial stability of the system without thermal effects by using a result of Borichev and Tomilov (Math. Ann. 347 (2010), 455-478). To find a more stable system, we then consider the stability of the above system with thermal effects described by Fourier's law, and achieve the exponential stability for the system by using the perturbed functional method.

Key words: Piezoelectric beams, Asymptotic behavior, Fractional derivative, Semigroup method

中图分类号:

O175.21

安雁宁, 刘文军, 孔奥文. 带分数阶磁效应的压电梁在有/无热效应时的稳定性[J]. 数学物理学报, 2023, 43(2): 355-376.

An Yanning, Liu Wenjun, Kong Aowen. Stability of Piezoelectric Beams with Magnetic Effects of Fractional Derivative Type and with/without Thermal Effects[J]. Acta mathematica scientia,Series A, 2023, 43(2): 355-376.

参考文献 33

[1]	Dagdeviren C, Duk Yang B, Su Y, et al. Conformal piezoelectric energy harvesting and storage from motions of the heart, lung and diaphragm. P Natl Acad Sci USA, 2014, 111(5): 1927-1932 doi: 10.1073/pnas.1317233111
[2]	Gu G Y, Zhu L M, Su C Y, et al. Modeling and control of piezo-actuated nanopositioning stages: A survey. IEEE T Autom Sci Eng, 2014, 13(1): 313-332
[3]	Cuc A, Giurgiutiu V, Joshi S, Tidwell Z. Structural health monitoring with piezoelectric wafer active sensors for space applications. AIAA J, 2007, 45(12): 2838-2850 doi: 10.2514/1.26141
[4]	Erturk A, Inman D J. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. J Vib Acoust, 2008, 130( 4): 041002
[5]	Yang J S. A review of a few topics in piezoelectricity. Appl Mech Rev Nov, 2006, 59(6): 335-345
[6]	Banks H T, Smith R C, Wang Y. Smart Material Structures-Modeling, Estimation and Control. Chichester, Wiley, 1996
[7]	Dietl J M, Wickenheiser A M, Garcia E. A Timoshenko beam model for cantilevered piezoelectric energy harvesters. Smart Mater Struct, 2010, 19( 5): 055018
[8]	Morris K A, Özer A Ö. Strong stabilization of piezoelectric beams with magnetic effects. 52nd IEEE Conference on Decision and Control. IEEE, 2013: 3014-3019
[9]	Morris K A, Özer A Ö. Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects. SIAM J Control Optim, 2014, 52(4): 2371-2398 doi: 10.1137/130918319
[10]	Ramos A J A, Freitas M M, Almeida D S, et al. Equivalence between exponential stabilization and boundary observability for piezoelectric beams with magnetic effect. Z Angew Math Phys, 2019, 70(2): 1-14 doi: 10.1007/s00033-018-1046-2
[11]	Benaissa A, Benazzouz S. Well-posedness and asymptotic behavior of Timoshenko beam system with dynamic boundary dissipative feedback of fractional derivative type. Z Angew Math Phys, 2017, 68(4): 1-38 doi: 10.1007/s00033-016-0745-9
[12]	Benaissa A, Gaouar S. Asymptotic stability for the Lamé system with fractional boundary damping. Comput Math Appl, 2019, 77(5): 1331-1346
[13]	Maryati T, Muñoz Rivera J, Poblete V, Vera O. Asymptotic behavior in a laminated beams due interfacial slip with a boundary dissipation of fractional derivative type. Appl Math Optim, 2021, 84(1): 85-102 doi: 10.1007/s00245-019-09639-1
[14]	Akil M, Chitour Y, Ghader M, Wehbe A. Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary. Asymptot Anal, 2020, 119(3-4): 221-280
[15]	Maryati T K, Muñoz Rivera J E, Rambaud A, Vera O. Stability of an $N$-component Timoshenko beam with localized Kelvin-Voigt and frictional dissipation. Electron J Differential Equations, 2018, 136: 1-18
[16]	Rivera J M, Poblete V, Vera O. Stability for an Klein-Gordon matrix type with a boundary dissipation of fractional derivative type. Asymptot Anal, 2022, 127(3): 249-273
[17]	Bouzettouta L, Abdelhak D. Exponential stabilization of the full von Kármán beam by a thermal effect and a frictional damping and distributed delay. J Math Phys, 2019, 60( 4): 041506
[18]	Park J H, Kang J R. Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation. IMA J Appl Math, 2011, 76(2): 340-350 doi: 10.1093/imamat/hxq040
[19]	Belhannache F, Messaoudi S A. On the general stability of a viscoelastic wave matrix with an integral condition. Acta Math Appl Sin Engl Ser, 2020, 36(4): 857-869 doi: 10.1007/s10255-020-0979-3
[20]	Hao J, Rao B. Influence of the hidden regularity on the stability of partially damped systems of wave matrixs. J Math Pures Appl, 2020, 143(9): 257-286 doi: 10.1016/j.matpur.2020.09.004
[21]	Keddi A A, Apalara T A, Messaoudi S A. Exponential and polynomial decay in a thermoelastic-Bresse system with second sound. Appl Math Optim, 2018, 77(2): 315-341 doi: 10.1007/s00245-016-9376-y
[22]	Liu W, Kong X, Li G. Lack of exponential decay for a laminated beam with structural damping and second sound. Ann Polon Math, 2020, 124(3): 281-289 doi: 10.4064/ap181224-17-9
[23]	Alves M S, Monteiro R N. Stabilization for partially dissipative laminated beams with non-constant coefficients. Z Angew Math Phys, 2020, 71(5): 1-15 doi: 10.1007/s00033-019-1224-x
[24]	Mustafa M I. Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems. Z Angew Math Phys, 2021, 72(2): Paper No. 67
[25]	Cardozo C L, Jorge Silva M A, Ma T F, Muñoz Rivera J E. Stability of Timoshenko systems with thermal coupling on the bending moment. Math Nachr, 2019, 292(12): 2537-2555 doi: 10.1002/mana.201800546
[26]	Feng B, Yang X G. Long-time dynamics for a nonlinear Timoshenko system with delay. Appl Anal, 2017, 96(4): 606-625 doi: 10.1080/00036811.2016.1148139
[27]	Liu W, Zhao W. Stabilization of a thermoelastic laminated beam with past history. Appl Math Optim, 2019, 80(1): 103-133 doi: 10.1007/s00245-017-9460-y
[28]	Liu W, Chen D, Chen Z. Long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law. Acta Math Sci, 2021, 41B(2): 609-632
[29]	Jiang J. Boundedness and exponential stabilization in a parabolic-elliptic Keller-Segel model with signal-dependent motilities for local sensing chemotaxis. Acta Math Sci, 2022, 42B(3): 825-846
[30]	欧阳成, 汪维刚, 莫嘉琪. 分数阶广义扰动热波方程. 数学物理学报, 2020, 40A(2): 452-459
	Ouyang C, Wang W G, Mo J Q. The fractional generalized disturbed thermal wave matrix. Acta Math Sci, 2020, 40A(2): 452-459
[31]	Mbodje B. Wave energy decay under fractional derivative controls. IMA J Math Control Inform, 2006, 23(2): 237-257 doi: 10.1093/imamci/dni056
[32]	Engel K J, Nagel R. One-Parameter Semigroups for Linear Evolution Equations. New York: Springer, 2000
[33]	Borichev A, Tomilov Y. Optimal polynomial decay of functions and operator semigroups. Math Ann, 2010, 347(2): 455-478 doi: 10.1007/s00208-009-0439-0