  We consider a distributed optimal control problem governed by an elliptic
convection diffusion PDE, and propose a hybridizable discontinuous Galerkin
(HDG) method to approximate the solution. We use polynomials of degree $k+1$
and $k \ge 0$ to approximate the state, dual state, and their fluxes,
respectively. Moreover, we use polynomials of degree $k$ to approximate the
numerical traces of the state and dual state on the faces, which are the only
globally coupled unknowns. We prove optimal a priori error estimates for all
variables when $ k > 0 $. Furthermore, from the point of view of the number of
degrees of freedom of the globally coupled unknowns, this method achieves
superconvergence for the state, dual state, and control when $k\geq 1$. We
illustrate our convergence results with numerical experiments.
