We consider the mathematical formulation, analysis, and the numerical solution of a time-dependent optimal control problem associated with the tracking of the velocity of a Navier--Stokes flow in a bounded two-dimensional domain through the adjustment of a distributed control. The existence of optimal solutions is proved and the first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time and fully discrete space-time approximations are defined and their convergence to the exact optimal solutions is shown. A gradient method for the solution of the fully discrete equations is examined, as are its convergence properties. Finally, the results of some illustrative computational experiments are presented.

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.

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 define 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.