A study on the dynamic control of large flexible structures with BIEM

1. 利用BIEMA计算机程序对二维波动方程进行了时域控制研究。然后将该代码修改为一个程序，以借助HUM（希尔伯特唯一性方法）获得波动方程的精确控制。这种方法与有限元法不同，它不需要分析后向问题。所得到的代码具有简单、速度快、精度高等特点。我们发现用这种方法可以得到轴对称问题的解。然而，在具有许多自由度的未知函数的非轴对称问题中，使用Tikhonov正则化是必要的。虽然最初的计划包括约束的控制问题，但我们认为考虑最小化场变量（压力）和边界控制的L^2规范之和会更有价值。这是因为人们对降低噪声控制问题中的压力级感兴趣。一些理论上的考虑将这个问题简化为一个类似于所谓最优系统的偏微分方程系统。然而，我们发现，用BIEM对该问题进行数值计算是相当不稳定的，并且只能在简单的问题中得到数值解。用BIEMAn分析法对平板方程进行控制的研究。是在平板动力学方程的背景下进行的。在这个问题中，狄利克雷控制通过规定适当的边界挠度和斜率来驱动板的静止。对轴对称问题进行了数值研究。结果表明，时间形状函数的选择控制着数值分析的稳定性。在Tikhonov正则化的帮助下，我们可以用Dirichlet （Neumann）数据的分段线性（常数）形状函数得到一个很好的结果。

1. Study of the control for the wave equation with BIEMA computer code for 2 dimensional wave equation in time domain is developed. This code is then modified into a program to obtain the exact control for the wave equation with the help of HUM (the Hilbert uniqueness method). This approach is in contrast to FEM in that it does not require the analysis of the backward problem. Also, the obtained code is remarkable in terms of its simplicity high speed and high accuracy. We found that the solution of axisymmetric problems can be obtained with this approach. In non-axisymmetric problems with many degrees of freedom for the unknown functions, however, the use of Tikhonov's regularisation is necessary.Although the original plan included control problems with constrains, we felt it would be more worth-while to consider minimization of the sum of L^2 norms of the field variable (pressure) and the boundary control. This is because one is interested in reducing the pressure level in noise control problems. Some theoretical consideration reduces this problem to a system of PDEs similar to the so called optimal system. As we found, however, the numerical calculation for this problem with BIEM is rather unstable, and could obtain numerical solutions only in simple problems.2. Study of the control for the plate equation with BIEMAn analysis similar to those in 1.was carried out in the context of the dynamical equation of plate. The Dirichlet control in this problem drives the plate to rest by prescribing appropriate boundary deflection and slope. A numerical study is carried out in an axisymmetric problem. It is found that the choice of the time shape functions controls the stability of the numerical analysis. We could obtain a good result with piecewise linear (constant) shape functions for the Dirichlet (Neumann) data, with the help of the Tikhonov regularisation.