ИСТИНА

Two control problems for the Navier-Stokes equations will be considered. The first one is the exact controllability problem that is formulated as follows: Given initial condition v_0(x) and restiction v(T,x) of the solution at prescribed time T>0. Find control function u(t,x) such that solution of controlled Navier-Stokes equations that goes from v_0(x) coincides with v(T,x) at time t=T. To solve this problem the distributed control is used, i.e. vector field u(t,x) from right side of the system supported in an arbitrary fixed subdomain of the space domain where Navier-Stokes system is defined. Besides, the problem of stabilization of solution by feedback control will be considered. In this problem instead of v(T,x) the steady-state solution w(x) is given, and one should choose such control that the solution which goes from v_0(x) tends to w(x) exponentialy as time t tends to infinity.The key moment here is that the control u chould be feedback, i.e. it can react on unpredictable fluctuation of the solution dumping them. The main ideas of solution to the problems mentioned above will be discussed.