About the stability to Timoshenko system with pointwise dissipation

In this paper we study the Timoshenko model over the interval \begin{document}$ (0, \ell) $\end{document} with pointwise dissipation at \begin{document}$ \xi\in (0, \ell) $\end{document} . We prove that this dissipation produces exponential stability when \begin{document}$ \xi\in \mathbb{Q}\ell $\end{document} and \begin{document}$ \xi\ne \frac{n}{2m+1} $\end{document} , where \begin{document}$ n, m\in \mathbb{N} $\end{document} and \begin{document}$ n $\end{document} , and \begin{document}$ 2m+1 $\end{document} are co-prime.