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.

翻译：我们提出了一种谱配点方法来逼近正方形区域中波动方程的精确边界控制。其思想是引入一个适当的近似控制问题，在空间中以N次多项式的有限维空间中解决。我们证明，我们可以选择一系列离散控制，这些离散控制取决于与近似控制问题相关的参数N，以便它们在N趋于无穷大时收敛到连续波动方程的控制。与文献中尝试过的其他数值逼近方法不同，这种方法不需要正则化技术，并且可以轻松适应已知连续模型的其他方程和系统的控制性。该方法在正方形区域的一维和二维中用多个示例进行说明。我们还提供了谱方法内在高精度逼近的数值证据。