Abstract
We present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for Navier--Stokes flows in bounded two-dimensional domains with bounded distributed controls. We study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined. We also define and analyze semidiscrete-in-time and fully 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.
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