We present a multigrid solution concept for the optimal distributed  control of
the time-dependent Navier--Stokes equation. This problem is described by a fully
coupled KKT system  in space and  time involving primal  and dual variables  for
velocity and pressure.

In this talk we present basic  concepts and ingredients which are necessary for
setting up a hierarchical solver for such systems. The underlying KKT system  is
discretised in  a monolithic  way on  the whole  space-time domain  using finite
elements  in space  and a  one-step-$/theta$-schemes in  time. A  global Newton
solver is applied  to solve for  the nonlinearity, while  a space-time multigrid
solver is  used for  the linear  subproblems. We  obtain a  robust solver  whose
convergence behaviour  is quite  independent of  the number  of unknowns  of the
discrete problem and robust with respect to the considered flow configuration. A
set of numerical examples demonstrates the feasibility of this approach.