About the stability to Timoshenko system with pointwise dissipation

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