Non-structural controllability of linear elastic systems with structural damping

13 pages, a4paper, no figures. Note added in proof: After our article was accepted for publication, we became aware of a paper to appear in Asymptotic Analysis: "Internal null-controllability for a structurally damped beam equation" by Julian Edward and Louis Tebou. This paper concerns (18) when $M$ is a segment and focuses on the limit $\rho\to 0$. We claim that our Theorem~2 generalizes the main result of this paper (n.b. $L_{\Omega}1 and some C_p>0. Indeed, this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non-structural controls satisfying a spectral observability condition. Moreover, under some geometric optics condition on the subdomain allowing to apply the control transmutation method, this estimate is improved into p=1 and the dependence of C_p on the subdomain is made explicit. These results are analogous to the optimal ones known for the heat flow