Boundary observability and exact controllability of strongly coupled wave equations

Ali Wehbe; Marwa Koumaiha; Layla Toufaily

doi:10.3934/dcdss.2021091

Article Contents

2022, Volume 15, Issue 5: 1269-1305. Doi: 10.3934/dcdss.2021091

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Boundary observability and exact controllability of strongly coupled wave equations

1.
Lebanese University, Faculty of sciences 1 and EDST, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon
2.
Lebanese International University, Department of mathematics and physics, Beirut, Lebanon, Lebanese University, Faculty of Business, Section 5, Nabateieh, Lebanon

^* Corresponding author: Ali Wehbe
^* Corresponding author: Ali Wehbe

Received: March 2021

Revised: May 2021

Early access: August 2021

Published: May 2022

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Abstract

In this paper, we study the exact controllability of a system of two wave equations coupled by velocities with boundary control acted on only one equation. In the first part of this paper, we consider the $ N $-d case. Then, using a multiplier technique, we prove that, by observing only one component of the associated homogeneous system, one can get back a full energy of both components in the case where the waves propagate with equal speeds (i.e. $ a = 1 $ in (1)) and where the coupling parameter $ b $ is small enough. This leads, by the Hilbert Uniqueness Method, to the exact controllability of our system in any dimension space. It seems that the conditions $ a = 1 $ and $ b $ small enough are technical for the multiplier method. The natural question is then : what happens if one of the two conditions is not satisfied? This consists the aim of the second part of this paper. Indeed, we consider the exact controllability of a system of two one-dimensional wave equations coupled by velocities with a boundary control acted on only one equation. Using a spectral approach, we establish different types of observability inequalities which depend on the algebraic nature of the coupling parameter $ b $ and on the arithmetic property of the wave propagation speeds $ a $.

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Mathematics Subject Classification: Primary: 93B05, 35P15; Secondary: 93B07.

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References

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