Optimal control of the stationary Navier-Stokes equations with mixed control-state constraints

In this paper we consider the distributed optimal control of the Navier-Stokes equations in the presence of pointwise mixed control-state constraints. After deriving a first order necessary condition, the regularity of the mixed constraint multiplier is investigated. Second order sufficient optimality conditions are studied as well. In the last part of the paper, a semismooth Newton method is applied for the numerical solution of the control problem. The convergence of the method is proved and numerical experiments are carried out.