Analysis and Finite Element Approximation of Optimal Control Problems for the Stationary Navier-Stokes Equations with Dirichlet Controls

Optimal control problems for the stationary Navier-Stokes equations are examined from analytical and numerical points of view. The controls considered are of Dirichlet type, that is, control is effected through the velocity field on (or the mass flux through) the boundary; the functionals minimized are either the viscous dissipation or the L4-distance of candidate flows to some desired flow. We show that optimal solutions exist and justify the use of Lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced. We study the regularity of solutions of this system. The n, finite element approximations of solutions of the optimality system are defined and optimal error estimates are derived.