Optimal Control of the Navier-Stokes equations via Pressure Boundary Conditions

In this work we study an optimal control problem subject to the instationary Navier-Stokes equations, where the control enters via an inhomogeneous Neumann/Do-Nothing boundary condition. Despite the Navier-Stokes equations with these boundary conditions not being well-posed for large times and/or data, we obtain wellposedness of the optimal control problem by choosing a proper tracking type term. In order to discuss the regularity of the optimal control, state and adjoint state, we present new results on $L^2(I;H^2(\Omega))$ regularity of solutions to a Stokes problem with mixed inhomogeneous boundary conditions.