A remark on second order methods in control of fluid flow

We present the functional analytic framework for a tracking-type control problem of the instationary Navier-Stokes equations. As solution methods for the control problem we comparatively discuss Newton's method as an example of the black box approach, and the SQP-method as one of the all-at-once methods. It is argued that black box approaches in general outperform all-at-once methods since the numerical effort necessary to numerically solve linearizations of the instationary Navier-Stokes equations compares to that of the nonlinear Navier-Stokes system. We report a numerical comparison which illustrates our argumentation. For both approaches local convergence results are cited.