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We study optimal boundary control problems for the Navier-Stokes equations in an unbounded domain. The control is of Dirichlet type, i.e., the boundary velocity field. The drag functional is used as an example of control objectives. We identify the trace space for the velocity fields possessing finite energy, we prove the existence of a solution for the Navier-Stokes equations with boundary data belonging to the trace space, we establish the existence of an optimal solution over the control set, and we derive an optimality system of equations in the weak sense by using the Lagrange multiplier principles. The strong form of the optimality system of equations is also obtained and is described by a system of partial differential equations with boundary values.

Optimal dirichlet control and inhomogeneous boundary value problems for the unsteady Navier-Stokes equations