The velocity tracking problem for Navier-Stokes flows with boundary control

We present some systematic approaches to the mathematical formulation and n umerical approximation of the time-dependent optimal control problem of tra cking the velocity for Navier Stokes flows in a bounded, two-dimensional do main with boundary control. We study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined . We also de ne and analyze semidiscrete-in-time and full space-time discre te approximations of the optimality system and a gradient method for the so lution of the fully discrete system. The results of some computational expe riments are provided.