State-constrained optimal control of the stationary Navier-Stokes equations

In this paper, the optimal control problem of the stationary Navier-Stokes equations in the presence of state constraints is investigated. We prove the existence of an optimal solution and derive first order necessary optimality conditions. The regularity of the adjoint state and the state constraint multiplier is also studied. Finally, the Lipschitz stability of the optimal control, state and adjoint variables with respect to perturbations is proved