Timoshenko梁的边界最优控制

数学物理学报 ›› 2018, Vol. 38 ›› Issue (3): 454-466.

Timoshenko梁的边界最优控制

章春国, 刘宇标, 刘维维

杭州电子科技大学数学系杭州 310018

收稿日期:2017-02-25 修回日期:2017-10-01 出版日期:2018-06-26 发布日期:2018-06-26
作者简介:章春国,E-mail:cgzhang@hdu.edu.cn
基金资助:
国家自然科学基金（61374096）

Boundary Optimal Control for the Timoshenko Beam

Zhang Chunguo, Liu Yubiao, Liu Weiwei

Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018

Received:2017-02-25 Revised:2017-10-01 Online:2018-06-26 Published:2018-06-26
Supported by:
Supported by the NSFC (61374096)

摘要/Abstract

摘要： 该文研究Timoshenko梁的边界最优控制问题.首先运用线性算子半群理论得到了系统的适定性；进而针对目标函数，借助伴随系统得到系统的最优性（最大值）原理和相应的最优控制；最后通过齐次系统特征函数渐近展开，给出最优控制与最优轨线详尽的计算步骤.

关键词: Timoshenko梁, 边界最优控制, 伴随系统, 最大值原理

Abstract: This paper studies the boundary optimal control problem governed by a Timoshenko beam equation. Firstly, the well posedness of the system is token by using the theory of the linear operator semigroup. Based on the objective function, the optimality principle (the maximum principle) and the corresponding optimal control are obtained by using the adjoint system. Finally, by the eigenfunction asymptotic expansion of the homogeneous system, the exhaustive calculation steps of optimal control and optimal trajectory are given.

Key words: Timoshenko beam, Boundary optimal control, Adjoint system, Maximum principle

中图分类号:

O231.4

章春国, 刘宇标, 刘维维. Timoshenko梁的边界最优控制[J]. 数学物理学报, 2018, 38(3): 454-466.

Zhang Chunguo, Liu Yubiao, Liu Weiwei. Boundary Optimal Control for the Timoshenko Beam[J]. Acta mathematica scientia,Series A, 2018, 38(3): 454-466.

参考文献

[1] Ammari K, Tucsnak M. Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J Cont Optim, 2000, 39(4):1160-1181
[2] Liu K S, Liu Z Y. Boundary stabilization of a nonhomogeneous beam with rotatory inertia at the tip. Journal of Computational and Applied Mathematics, 2000, 114:1-10
[3] Liu K S, Liu Z Y, Rao B P. Exponential stability of an abstract non-dissipative linear system. SIAM J Cont Optim, 2001, 40(1):149-165
[4] 章春国, 赵宏亮, 刘康生. 具有局部分布反馈与边界反馈耦合控制的非均质Timoshenko梁的指数镇定. 数学年刊(A辑),2003, 24(6):757-764 Zhang C G, Zhao H L, Liu K S. The exponential stabilization of nonhomogeneous Timoshenko beam with one locally distributed control and one boundary control. Chinese Ann Math Ser A, 2003, 24(6):757-764
[5] Zhang C G. Boundary feedback stabilization of the undamped Timoshenko beam with both ends free. J Math Anal Appl, 2007, 326:488-499
[6] Zhang C G, Hu H X. Exact controllability of a Timoshenko beam with dynamical boundary. J Math Kyoto Univ, 2007, 47(3):643-655
[7] Sadek I S, Kucuk I I, Sloss J M, Zeini E E, Adali S. Optimal boundary control of dynamics responses of piezo actuating micro-beams. Applied Mathematical Modelling, 2009, 33(8):3343-3353
[8] Ozkan Ozer A, Scott W H. Exact controllability of a Rayleigh beam with a single boundary control. Math Control Signals Syst, 2011, 23:199-222
[9] 章春国. 具有局部记忆阻尼的非均质Timoshenko梁的稳定性. 数学物理学报,2012, 32A(1):186-200 Zhang C G. Stability for the nonhomogeneous Timoshenko beam with local memory damping. Acta Math Sci, 2012, 32A(1):186-200
[10] Sadek I S, Sloss J M, Bruch J C J, Adali S. Optimal control of a Timoshenko beam by distributed force. J Optim Theory Appl, 1986, 50(3):451-461
[11] Zelikin M I, Manita L A. Optimal control for a Timoshenko beam. C R Mecanique, 2006, 334:292-297
[12] Zhang C G, He Z R. Optimality conditions for a Timoshenko beam equation with periodic constraint. Z Angew Math Phys, 2014, 65(2):315-324
[13] Adams R A. Sobolev Spaces. New York:Acadamic Press, 1975
[14] Pazy A. Semigroups of Linear Operators and Applications to Partical Differential Euqations. New York:Springer-Verlag, 1983
[15] Liu K S. Locally distributed control and damping for the conservative systems. SIAM J Cont Optim, 1997, 35(5):1574-1590