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：我们提出一种谱配点法，用于逼近正方形域上波动方程的精确边界控制。该方法的思路是引入一个适当的近似控制问题，并在空间上N次多项式构成的有限维空间中求解该问题。我们证明：能够选取一组与近似控制问题相关的、依赖于参数N的离散控制序列，使得当N趋于无穷时，该序列收敛于连续波动方程的控制解。与文献中尝试的其他数值逼近方法不同，本方法无需正则化技术，且可轻松推广至其他已知连续模型可控性的方程与系统。通过正方形域上的一维和二维数值算例验证了该方法的有效性，同时给出了谱方法固有高精度逼近特性的数值证据。