In this paper, we consider the error estimates of the numerical solutions of a class of fourth order linear-quadratic elliptic optimal control problems by using mixed finite element methods. The state and co-state are approximated by the order $k$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order $k(k{geq}1)$. $L^2$ and $L^{infty}$-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for fourth order elliptic control problems.