Optimal control of the Navier-Stokes equations via pressure boundary conditions

In this work, we studied an optimal control problem subject to the unsteady 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 obtained well-posedness of the optimal control problem by choosing a proper tracking type term. Moreover, we derived first- and second-order optimality conditions. In order to discuss the regularity of the optimal control, state, and adjoint state, we presented new results on $ L^2(I;H^2( \Omega)) $ regularity of solutions to a Stokes problem with mixed inhomogeneous boundary conditions.