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

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.

Uniform boundary controllability of a semi-discrete 1-D coupled wave equations with vanishing viscosity

“…The multiplier. We use the same multiplier function as in [5] and we will be proving that it also fits a few criteria that will benefit us. It is constructed in [5] by first considering the increasing sequence (ρ h,n ) n≥1 defined by ρ h,n = eϕ h (n)…”

“…Since the main issue is the existence of the spurious high frequencies generated by the discretization process, the idea of eliminating or reducing them one way or another arises naturally. In [5], an appropriate filtering technique was considered in order to restore the uniform controllability of system (2). Based on this technique, the existence of a uniformly bounded sequence of controls (ν h ) h>0 is proved by filtering high frequency discrete initial data.…”

“…F 2N is the 2N -th vector of the canonical basis of R 4N and A h is the matrix defined previously in (5). System ( 8) is controllable to zero if for any initial state X ∈ C 4N there is f h ∈ L 2 (0, T ) such that U h (T ) = 0, where U h is the solution of ( 8) with U h (0) = X.…”

“…is indeed a control for system (8) that steers a given state U h to zero. Following the ideas in [5], we shall prove the existence of a biorthogonal sequence Θ h s,n (t) s=±1 1≤|n|≤N such that the sequence of the discretized controls {ν h } h>0 converges, in a sense that will be precised later on, to a control v of the continuous system, where f h = ν h +εν h is defined in (15). For this convergence to hold, some estimations of the L ∞ -norms of the functions in the biorthogonal sequence should be made.…”

Ingham type approach for uniform observability inequality of the semi-discrete coupled wave equations

“…Almeida Júnior et al [1] first contributed in that direction for uniform boundary observability. See also Akri and Maniar [2,3] and Xu [17]. To fix our ideas we address the problem of the well known observability inequality in one-dimensional setting.…”