The Linear Quadratic Regulator (LQR) is a well-known design method for control systems. The theory behind it can be adapted in an simple way to deal with other optimal control problems of great interest for the community, as witnessed in the literature. Our aim is the optimal control of the vibrations for flexible structure models. The optimal control problem is posed over infinite-dimensional spaces, formulated as (view document) where Y, U are separable Hilbert spaces, of observation and control respectively, and the operators Q, R, T, F and C, reflect the control, observation, design and dynamics associated with the model. In order to solve this problem we must establish a formulation over finitedimensional spaces, with a solution that converges to the solution of the continuous problem. For this purpose we use a finite element scheme, arriving to the classical statement of the LQR problem in finite-dimensional spaces. In the literature there are many books and papers dealing with existence, stability and convergence issues, but the only approach that states optimal convergence rates for the approximated control problem is due to Lasiecka and Triggiani. Our goal then is to perform an error analysis from a theoretical and a computational point of view under this framework. We apply the results of these two authors for two flexible strcuture models: the highly-damped wave equation, and the Timoshenko model. For the highly-damped wave equation we obtain theoretical orders of convergence for a distributed and a point control problem, which are computationally validated. For the Timoshenko beam model, our motivation is related to the study of convergence rates for the feedback gain.