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Article Contents

# Boundary observability and exact controllability of strongly coupled wave equations

• * Corresponding author: Ali Wehbe
• 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 $a$.

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

 Citation:

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