  In this work, we study a boundary control problem for the evolutionary
Navier-Stokes equations, under mixed boundary conditions, in two dimensions.
The cost functional here considered is of quadratic type, depending on both
state and control variables. We provide a comprehensive theoretical framework
to address the analysis and the derivation of a system of first-order
optimality conditions that characterizes the solution of the control problem.
We take advantage of an adequate treatment of the Dirichlet control through the
study of the reduced functional. Despite the fact that this approach is quite
common, a detailed analysis for the case of mixed boundary conditions with is
still lacking. Finally, solution-finding algorithms of descent type are
proposed and illustrated with several simulations.
