This article deals with the approximation of the boundary controls of a 1-D linear wave equation with a variable potential by using a finite difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh-size and the convergence of the approximate controls cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the approximation of the controls. Under certain conditions on the potential, we prove that the convergence of the scheme is ensured if the highest frequencies of the discrete initial data have been previously filtered out. Several filtration procedures are proposed and analyzed. Moreover, we identify a class of (regular) continuous initial data which can be controlled uniformly without any special treatment.

中文翻译：

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具有势能的波动方程控制的近似

本文使用有限差分空间半离散格式处理具有可变电位的一维线性波动方程的边界控制的近似。由于高频数值寄生振荡，半离散模型在网格尺寸方面不是均匀可控的，并且不能保证近似控制的收敛性。在本文中，我们分析了要控制的初始数据及其离散化如何影响控制的逼近。在一定的势条件下，我们证明如果离散初始数据的最高频率已经被预先过滤掉，则保证了该方案的收敛性。提出并分析了几种过滤程序。而且，