About the stability to Timoshenko system with pointwise dissipation

Abstract

<p style='text-indent:20px;'>In this paper we study the Timoshenko model over the interval <inline-formula><tex-math id="M1">\begin{document}$ (0, \ell) $\end{document}</tex-math></inline-formula> with pointwise dissipation at <inline-formula><tex-math id="M2">\begin{document}$ \xi\in (0, \ell) $\end{document}</tex-math></inline-formula>. We prove that this dissipation produces exponential stability when <inline-formula><tex-math id="M3">\begin{document}$ \xi\in \mathbb{Q}\ell $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \xi\ne \frac{n}{2m+1}\ell $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ n, m\in \mathbb{N} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M7">\begin{document}$ 2m+1 $\end{document}</tex-math></inline-formula> are co-prime.</p>