一类内部控制的非均质Timoshenko梁系统最优性分析

钱 嘉祺
杭州电子科技大学

摘要

本文考察了一类具有有界变差函数内部控制的Timoshenko梁系统的稳定性.首先，运用算子半群理论证明了该系统能量是衰减的，然后利用滚动时域法，将无限时域的最优性问题转化为有限时域的最优性问题，运用乘子法，得到系统最优轨线是指数衰减的.最后拓展到无限时域，得到了系统的次优性条件，证明了最优轨线的指数衰减性。

关键词

有界变差函数;Timoshenko梁;滚动时域法;最优性分析;指数衰减.

参考

[1] Kunisch, Karl, Casas, et al. OPTIMAL CONTROL OF SEMILINEAR PARABOLIC EQUATIONS BY BV-FUNCTIONS[J]. SIAM Journal on Control and Optimization, 2017, 55(3):1752-1788. [2] Chen S , Liu L Z . Spectrum and Stability for Elastic Systems with Global or Local Kelvin--Voigt Damping[J]. Siam Journal on Applied Mathematics, 1998, 59(2):651-668. [3] 严佳梦, 鲁轶戈, 章春国. 具有动态边界反馈的Timoshenko梁的能量衰减估计[J]. 杭州电子科技大学学报:自然科学版, 2021, 41(6):7. [4] Azmi B, Kunisch K. Receding horizon control for the stabilization of the wave equation[J]. Discrete and Continuous Dynamical Systems, 2017, 38(2):449-484. [5] 周凯丽. Timoshenko梁点控制的稳定性和分布控制的最优性[D].杭州电子科技大学,2019.DOI:10.27075/d.cnki.ghzdc.2019.000108. [6] Russell D L, Lions J L, Mitter S K . Optimal Control of Systems Governed by Partial Differential Equations[J]. The American Mathematical Monthly, 1972, 79(9):1049. [7] Zhang C G, Hu H X.Exact controllability of a Timoshenko beam with dynamical boundary[J].Kyoto Journal of mathematics,2007,47(2007):643-655. [8] 章春国, 付煜之, 刘宇标. 二维Mindlin-Timoshenko板系统的稳定性与最优性[J]. 数学物理学报:A辑, 2021, 41(5):27. [9] Bardos C, Leb Ea U G, Rauch J.Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary[J]. SIAM Journal on Control and Optimization, 1992. [10] Azmi B, Kunisch K. On the Stabilizability of the Burgers Equation by Receding Horizon Control[J]. SIAM Journal on Control & Optimization,2016,54(3):1378-1405.