标题： 波动方程边界控制的谱配置方法数值逼近 显示英文标题

标题： Numerical approximation of the boundary control for the wave equation with a spectral collocation method

摘要： 我们提出一种谱配置方法来近似方形区域中波动方程的精确边界控制。 其思想是引入一个合适的近似控制问题，并在空间多项式次数为N的有限维空间中求解该问题。 我们证明可以选取与近似控制问题相关的参数N的离散控制序列，使得当N趋于无穷时，这些控制收敛于连续波动方程的控制。 与其他文献中尝试的数值近似不同，这种方法不需要正则化技术，并且可以轻松适应其他控制性已知的连续模型方程和系统。 该方法通过一维和二维方形区域中的多个示例进行说明。 我们还提供了谱方法固有高度精确近似的数值证据。 显示英文摘要

摘要： We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of polynomials of degree N in space. We prove that we can choose a sequence of discrete controls depending on the parameter N associated with the approximate control problem in such a way that they converge, as N goes to infinity, to a control of the continuous wave equation. Unlike other numerical approximations tried in the literature, this one does not require regularization techniques and can be easily adapted to other equations and systems where the controllability of the continuous model is known. The method is illustrated with several examples in 1-d and 2-d in a square domain. We also give numerical evidence of the highly accurate approximation inherent to spectral methods.