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We propose a new hybridizable discontinuous Galerkin (HDG) method toapproximate the solution of a Dirichlet boundary control problem governed by anelliptic convection diffusion PDE. Even without a convection term, Dirichletboundary control problems are well-known to be very challenging theoreticallyand numerically. Although there are many works in the literature on Dirichletboundary control problems for the Poisson equation, the authors are not awareof any existing theoretical or numerical analysis works for convectiondiffusion Dirichlet control problems. We make two contributions. First, weobtain well-posedness and regularity results for the Dirichlet control problem.Second, under certain assumptions on the domain and the target state, we obtainoptimal 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 inthe literature. We present numerical experiments to demonstrate the performanceof the HDG method.

A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs I: High Regularity