Analysis and Approximations of Dirichlet Boundary Control of Stokes Flows in the Energy Space

We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the L² () -norm and an energy space seminorm. We prove well-posedness, provide first order optimality conditions, derive regularity results, and develop finite element discretizations for both problems, and we also 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 L² () can lead to an optimal control with discontinuities at the corners, even when the domain is convex. This phenomenon is also observed in numerical experiments. This behavior does not occur in Dirichlet boundary control problems for the Poisson equation on convex polygonal domains, and it may not be desirable in real applications. For the energy space problem, we show that the solution of the control problem is more regular than the solution of the problem with the L² () -regularization. The improved regularity enables us to prove a priori error estimates for the control in the energy norm. We present several numerical experiments for both control problems on convex and nonconvex domains.