For the modeling and numerical approximation of problems with time-dependent Dirichlet
boundary conditions, one can call on several consistent and inconsistent approaches.
We show that spatially discretized boundary control problems can be brought into a
standard state space form accessible for standard optimization and model reduction
techniques. We discuss several methods that base on standard finite element discretizations,
propose a newly developed problem formulation, and investigate their performance in
numerical examples. We illustrate that penalty schemes require a wise choice of the
penalization parameters in particular for iterative solves of the algebraic equations.
Incidentally, we confirm that standard finite element discretizations of higher order
may not achieve the optimal order of convergence in the treatment of boundary forcing
problems and that convergence estimates by the common method of manufactured solutions can be misleading.