The aim of this very important paper is to study the internal stabilization of a grid-like body and to obtain the exact controllability of its vibrations by using the principle of D. L. Russel. The paper is the first in this consideration. The author combines stabilization techniques and asymptotic analysis techniques to solve the problems which depend on three small parameters characterizing the perforated body. Main result: The existence and uniqueness results are presented. The author precisely proves the uniform stabilization of the vibrations of the grid-like body. The precise hypotheses on initial data which permit to let the three parameters go to zero successively are given. The author also proves that, in order to have less calculations, it is more convenient during the study of the convergence process, first to let the thickness of the grid go to zero, then the size of the holes and finally the thickness of the bars. The principle of D. L. Russell to establish exact controllability result is used. Finally, the convergence results related to controllability prolem are proven. Moreover, some remarks are given.