  We study Dirichlet boundary control of Stokes flows in 2D polygonal domains.
We consider cost functionals with two different boundary control regularization
terms: the $L^2$ norm and an energy space seminorm. We prove well-posedness and
regularity results for both problems, develop finite element discretizations
for both problems, and prove finite element error estimates for the latter
problem. The motivation to study the energy space problem follows from our
analysis: we prove that the choice of the control space ${\bm L}^2(\Gamma)$ can
lead to an optimal control with discontinuities at the corners, even when the
domain is convex. We observe this phenomenon in numerical experiments. This
behavior does not occur in Dirichlet boundary control problems for the Poisson
equation on convex polygonal domains, and may not be desirable in real
applications. For the energy space problem, we derive the first order
optimality conditions, and show that the solution of the control problem is
more regular than the solution of the problem with the ${\bm L}^2(\Gamma)$
regularization. We also prove a priori error estimates for the control in the
energy norm, and present several numerical experiments for both control
problems on convex and nonconvex domains.
