Boundary observability and exact controllability of strongly coupled wave equations

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-d case. Then, using a multiplier technique, we prove that, by observing only one component of the associated homogeneous system, one can get back a full energy of both components in the case where the waves propagate with equal speeds (<i>i.e.</i> <inline-formula><tex-math id="M2">\begin{document}$ a = 1 $\end{document}</tex-math></inline-formula> in (1)) and where the coupling parameter <inline-formula><tex-math id="M3">\begin{document}$ b $\end{document}</tex-math></inline-formula> is small enough. This leads, by the Hilbert Uniqueness Method, to the exact controllability of our system in any dimension space. It seems that the conditions <inline-formula><tex-math id="M4">\begin{document}$ a = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ b $\end{document}</tex-math></inline-formula> small enough are technical for the multiplier method. The natural question is then : what happens if one of the two conditions is not satisfied? This consists the aim of the second part of this paper. Indeed, we consider the exact controllability of a system of two one-dimensional wave equations coupled by velocities with a boundary control acted on only one equation. Using a spectral approach, we establish different types of observability inequalities which depend on the algebraic nature of the coupling parameter <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> and on the arithmetic property of the wave propagation speeds <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula>.</p>