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

A finite element method for Dirichlet boundary control problems governed by parabolic PDEs