We study goal-oriented a posteriori error estimates for the numerical approximation of Dirichlet boundary control problem governed by a convection diffusion equation with pointwise control constraints on a two dimensional convex polygonal domain. The local discontinuous Galerkin method is used as a discretization technique since the control variable is involved in a variational form in a natural sense. We derive primal-dual weighted error estimates for the objective functional with an error term representing the mismatch in the complementary system due to the discretization. Numerical examples are presented to illustrate the performance of the proposed estimator. (C) 2020 Elsevier B.V. All rights reserved.