  The focus of this work is on the construction and analysis of optimal-order
multigrid preconditioners to be used in the Newton-Krylov method for a
distributed optimal control problem constrained by the stationary Navier-Stokes
equations. As in our earlier work [7] on the optimal control of the stationary
Stokes equations, the strategy is to eliminate the state and adjoint variables
from the optimality system and solve the reduced nonlinear system in the
control variables. While the construction of the preconditioners extends
naturally the work in [7], the analysis shown in this paper presents a set of
significant challenges that are rooted in the nonlinearity of the constraints.
We also include numerical results that showcase the behavior of the proposed
preconditioners and show that for low to moderate Reynolds numbers they can
lead to significant drops in number of iterations and wall-clock savings.
