  A finite element analysis of a Dirichlet boundary control problem governed by
the linear parabolic equation is presented in this article. The Dirichlet
control is considered in a closed and convex subset of the energy space
$H^1(\Omega \times(0,T)).$ We prove well-posedness and discuss some regularity
results for the control problem. We derive the optimality system for the
optimal control problem. The first order necessary optimality condition results
in a simplified Signorini type problem for control variable. The space
discretization of the state variable is done using conforming finite elements,
whereas the time discretization is based on discontinuous Galerkin methods. To
discretize the control we use the conforming prismatic Lagrange finite
elements. We derive an optimal order of convergence of error in control, state,
and adjoint state. The theoretical results are corroborated by some numerical
tests.
