Uniform indirect boundary controllability of semi-discrete <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ d $\end{document}</tex-math></inline-formula> coupled wave equations

Abstract

<p style='text-indent:20px;'>In this paper, we treat the problem of uniform exact boundary controllability for the finite-difference space semi-discretization of the <inline-formula><tex-math id="M3">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula> 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.