We present some systematic approaches to the mathematical formulation and numerical approximation of the time-dependent optimal control problem of tracking the velocity for Navier Stokes flows in a bounded, two-dimensional domain 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 discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system. The results of some computational experiments are provided.