  We propose a new hybridizable discontinuous Galerkin (HDG) method to
approximate the solution of a Dirichlet boundary control problem governed by an
elliptic convection diffusion PDE. Even without a convection term, Dirichlet
boundary control problems are well-known to be very challenging theoretically
and numerically. Although there are many works in the literature on Dirichlet
boundary control problems for the Poisson equation, the authors are not aware
of any existing theoretical or numerical analysis works for convection
diffusion Dirichlet control problems. We make two contributions. First, we
obtain well-posedness and regularity results for the Dirichlet control problem.
Second, under certain assumptions on the domain and the target state, we obtain
optimal a priori error estimates in 2D for the control for the new HDG method.
As far as the authors are aware, there are no existing comparable results in
the literature. We present numerical experiments to demonstrate the performance
of the HDG method.
