Quadratic cost optimal controls can be solved by penalizing the governing linear differential equation , . In this paper, we study the numerical analysis of this approach using finite elements. We formulate the geometric
condition(H) which requires that pairs of certain related finite-dimensional approximation spaces form angles which are bounded away from the hare independent in the error bounds, thereby giving sharp asymptotic error estimates. This condition (H) is shown to be also a necessary condition for such independence. Examples and numerical evidence are also provided.