  Finite element approximations of Dirichlet boundary control problems governed
by parabolic PDEs on convex polygonal domains are studied in this paper. The
existence of a unique solution to optimal control problems is guaranteed based
on very weak solution of the state equation and $L^2(0,T;L^2(\Gamma))$ as
control space. For the numerical discretization of the state equation we use
standard piecewise linear and continuous finite elements for the space
discretization of the state, while a dG(0) scheme is used for time
discretization. The Dirichlet boundary control is realized through a space-time
$L^2$-projection. We consider both piecewise linear, continuous finite element
approximation and variational discretization for the controls and derive a
priori $L^2$-error bounds for controls and states. We finally present numerical
examples to support our theoretical findings.
