Dynamics for controlled navier-stokes systems with distributed controls

The long-time behavior of solutions for an optimal distributed control problem associated with the Navier--Stokes equations is studied. First, a linear feedback solution for the Navier--Stokes equations is constructed; this feedback solution possesses decay (in time) properties. Then, some preliminary estimates for the long-time behavior of all solutions of the Navier--Stokes equations are derived. Next, the existence of a solution for the optimal control problem is proved. Finally, the long-time decay properties for the optimal solutions are established.