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Optimal Boundary Control for the Evolutionary Navier--Stokes System: The Three-Dimensional Case

Optimal boundary control problems for the three-dimensional, evolutionary Navier--Stokes equations in the exterior of a bounded domain are studied. Control is effected through the Dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with almost minimal possible regularity. The control objective is to minimize the drag functional. The existence of an optimal solution is proved. A strong form of an optimality system of equations is derived on the basis of regularity results established in this work for the adjoint Oseen equations with regular initial data which do not satisfy the compatibility conditions.