We propose an embedded discontinuous Galerkin (EDG) method to approximate
the solution of a distributed control problem governed by convection diffusion PDEs, and obtain
optimal a priori error estimates for the state, dual state, their fluxes, and the control. Moreover,
we prove the optimize-then-discretize (OD) and discrtize-then-optimize (DO) approaches coincide.
Numerical results confirm our theoretical results.