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 constr ucted; this feedback solution possesses decay (in time) properties. Th en, some preliminary estimates for the long-time behavior of all solut ions of the Navier-Stokes equations are derived. Next, the existence o f a solution for the optimal control problem is proved. Finally, the l ong-time decay properties for the optimal solutions are established.