Abstract. 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 NavierStokes 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.Key words. optimal control, Navier-Stokes equations, finite elements, fluid mechanics
AMS subject classifications. 35B40, 35B37, 35Q30, 65M60PII. S00361429973294141. Introduction. The purpose of the velocity tracking problem is to steer, over time, a velocity field to a given target velocity field. In this paper, we consider controls that act as a distributed body force; the state of the system, i.e., the velocity and pressure fields, is the solution of an initial-boundary value problem of the Navier-Stokes system of partial differential equations that models the evolution of viscous, incompressible flows. The cost or objective functional is a quadratic functional involving the state and the control variables; the functional measures, in an appropriate norm, the distance between the flow velocity and the target velocity fields, and through a penalty term, also measures the cost of control. Thus, the minimization of the functional is used to both drive the flow towards the target flow and to limit the cost of control.Several treatments of similar optimal control problems can be found in the literature, most notably in [1], [5], [6], and [7]. Our work differs from these in that we use a different functional that we show does a better job of tracking the target velocity field; also, compared to previous authors, we use general, e.g., nonseparable, controls and weaker hypotheses on the domain. The numerical treatment of the velocity tracking problem is also an outstanding problem and other algorithms have been proposed. For example, a quasi-optimal control has been studied in [12] and [13].In this paper, we will formulate the problem in a convenient and precise mathematical way; then, we will prove the existence of optimal controls and characterize optimal controls by deriving the first-order necessary conditions associated with the problem. We then examine semidiscrete-in-time and fully discrete in space and time
