Optimal Dirichlet Boundary Control of Stationary Navier-Stokes Equations with State Constraint

We investigate optimal boundary control of the steady-state Navier-Stokes equations. The control goal is to increase the lift while maintaining a drag constraint. The resulting problem has control as well as state constraints. We use as control space L 2 (Γ), which makes it necessary to work with very weak solutions of the Navier-Stokes equations. Moreover, the low regularity of y ∈ L p (Ω) n states forces to reformulate cost functional and state constraint, which results in a problem with nonlinear and mixed control-state constraint. We derive first-order necessary and second-order sufficient optimality conditions for this optimal control problem. Moreover, we report on numerical experiments on the solution of the first-order optimality system.