The velocity tracking problem is motivated by the desire to match a desired target flow with a flow which can be controlled through time dependent distributed forces or time dependent boundary conditions.

The flow model is the Navier-Stokes equations for a viscous incompressible fluid and different kinds of controls are studied. Optimal distributed and boundary controls minimizing a quadratic functional and an optimal bounded distributed control are investigated. The distributed optimal and the bounded control are compared with a linear feedback control.

Here, a unified mathematical formulation, covering several specific classes of meaningful control problems in bounded domains, is presented with a complete and detailed analysis of all these time dependent optimal control velocity tracking problems. We concentrate not only on questions such as existence and necessary first order conditions but also on discretization and computational aspects.

The first order necessary conditions are derived in the continuous, in the semidiscrete time approximation and in the fully finite element discrete case. This derivation is needed to obtain an accurate meaningful numerical algorithm with a satisfactory convergence rate. The gradient algorithm is used and several numerical computations are performed to compare and understand the limits imposed by the theory. Some computational aspects are discussed without which problems of any realistic size would remain intractable.