Optimization Problems for the Navier-Stokes Equations with Regular Boundary Controls

We study the necessary conditions for an optimal boundary control problem associated with the stationary Navier-Stokes equations with regular controls. The control is the velocity on part or all of the boundary of the given flow domain. We present a rigorous justification for the use of the Lagrange multiplier rule to derive first order necessary conditions for optimality; these are expressed as a system of partial differential equations. We study the regularity of solutions of this system, and, finally, we give some examples of specific functionals useful in applications.