Regularized state-constrained boundary optimal control of the Navier–Stokes equations

The numerical solution of the Dirichlet boundary optimal control problem of the Navier–Stokes equations in presence of pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau–Yosida regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each regularized problem a semi-smooth Newton method is applied.