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In this work we study optimal boundary control problems in energy spaces, their construction of robust preconditioners and applications to arterial blood ow. More precisely we consider the unconstrained optimal Dirichlet and Neumann boundary control problems for the Poisson equation. In both cases it turns out that the control can be eliminated and thus a variational formulation in saddle point structure is obtained. Further, the construction of corresponding robust preconditioners for optimal boundary control problems is investigated. We observe that the optimal boundary control problems are related to biharmonic equation of rst kind. For the preconditioning we consider either a preconditioner motivated from boundary element methods or a multilevel preconditioner of BPX type. As an application we study the optimal Dirichlet boundary control problem for arterial blood fow. In particular, we are interested in the optimal in ow prole into an arterial system, motivated for instance by an articial heart pump. Also, we investigate on hemodynamic indicators, for the identication of potential risk factors for aneurysms. Several numerical examples illustrate the obtained theoretical results