The velocity tracking problem for Navier-Stokes flows with bounded distributed controls

We present some systematic approaches to the mathematical analysis and nume rical approximation of the time dependent optimal control problem of tracki ng the velocity for Navier-Stokes flows in bounded two-dimensional domains with bounded distributed controls. We study the existence of optimal soluti ons and derive an optimality system from which optimal solutions may be det ermined. We also define and analyze semidiscrete-in-time and fully space-ti me discrete approximations of the optimality system and a gradient method f or the solution of the fully discrete system. The results of some computati onal experiments are provided.