  In this paper, we consider the finite element approximation to a parabolic
Dirichlet boundary control problem and establish new a priori error estimates.
In the temporal semi-discretization we apply the DG(0) method for the state and
the variational discretization for the control, and obtain the convergence
rates $O(k^{\frac{1}{4}})$ and $O(k^{\frac{3}{4}-\varepsilon})$
$(\varepsilon>0)$ for the control for problems posed on polytopes with $y_0\in
L^2(\Omega)$, $y_d\in L^2(I;L^2(\Omega))$ and smooth domains with $y_0\in
H^{\frac{1}{2}}(\Omega)$, $y_d\in L^2(I;H^1(\Omega))\cap
H^{\frac{1}{2}}(I;L^2(\Omega))$, respectively. In the fully discretization of
the optimal control problem posed on polytopal domains, we apply the
DG(0)-CG(1) method for the state and the variational discretization approach
for the control, and derive the convergence order $O(k^{\frac{1}{4}}
+h^{\frac{1}{2}})$, which improves the known results by removing the mesh size
condition $k=O(h^2)$ between the space mesh size $h$ and the time step $k$. As
a byproduct, we obtain a priori error estimate $O(h+k^{1\over 2})$ for the
fully discretization of parabolic equations with inhomogeneous Dirichlet data
posed on polytopes, which also improves the known error estimate by removing
the above mesh size condition.
