Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations

Abdeladim El Akri; Lahcen Maniar

doi:10.3934/mcrf.2020015

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2020, Volume 10, Issue 4: 669-698. Doi: 10.3934/mcrf.2020015

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Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations

Abdeladim El Akri^, and
Lahcen Maniar

Université Cadi Ayyad, Faculté des Sciences Semlalia, LMDP, UMMISCO (IRD- UPMC), Marrakech 40000, B.P. 2390, Maroc

^* Corresponding author: Abdeladim El Akri
^* Corresponding author: Abdeladim El Akri

The first author would like to thank S. Micu for fruitful discussions on several parts of this paper during his visit to Craiova University

Received: December 31, 2018

Revised: September 30, 2019

Early access: December 2019

Published: October 2020

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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.

Keywords:

Coupled wave equations,
uniform indirect exact boundary controllability,
space semi-discretization,
finite differences,
moment problem,
biorthogonal sequence,
filtered spaces.

Mathematics Subject Classification: Primary: 93B05, 35L05, 30E05; Secondary: 65M06.

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References

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