In this paper, we treat the problem of uniform exact boundary controllability for the finite-difference space semi-discretization of the $ 1 $-$ d $ coupled wave equations with a control acting only in one equation. First, we show how, after filtering the high frequencies of the discrete initial data in an appropriate way, we can construct a sequence of uniformly (with respect to the mesh size) bounded controls. Thus, we prove that the weak limit of the aforementioned sequence is a control for the continuous system. The proof of our results is based on the moment method and on the construction of an explicit biorthogonal sequence.

中文翻译：

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半离散的均匀间接边界可控性 \ begin {document} $ 1 $ \ end {document}--\ begin {document} $ d $ \ end {document} 耦合波方程

在本文中，我们处理了仅在一个方程中起作用的$ 1 $-$ d $耦合波动方程的有限差分空间半离散化的一致精确边界可控制性的问题。首先，我们展示了如何以适当的方式对离散的初始数据进行高频滤波之后，我们可以构造一个均匀（相对于网格大小）有界控件的序列。因此，我们证明了上述序列的弱极限是对连续系统的控制。我们的结果证明基于矩量法和显式双正交序列的构建。