Uniform boundary stabilization for the finite difference semi-discretization of 2-D wave equation

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

“…The indirect controllability problem for system (4) is formulated as follows: Given an initial data y 0 j , y 1 j , q 0 j , q 1 j 1≤j≤N ∈ C 4N , does there exist a control function v h ∈ L 2 (0, T ) such that the solution of equation ( 4) satisfies y j (T ) = y j (T ) = q j (T ) = q j (T ) = 0 (5) for all j = 1, . .…”

“…This phenomenon is due to the fact that the semi-discrete dynamics lead to high-frequency spurious solutions which propagate with arbitrary small velocity and make the discrete controls diverge when the mesh-size h goes to zero. We notice that the uniform controllability question is closely related to the uniform observability [3,8,16] and the uniform stabilization problems for the discrete systems [5,9,25].…”