LOCAL EXACT CONTROLLABILITY OF THE 2-DIMENSIONAL NAVIER-STOKES EQUATIONS

Let Omega subset of R(2) be a bounded domain with boundary partial der ivative Omega consisting of two disjoint closed curves Gamma(0) and Ga mma(1) such that Gamma(0) is connected and Gamma(1) not equal empty se t . The Navier-Stokes system partial derivative(t) upsilon(t, x) - Del ta upsilon + (upsilon, del)upsilon + del p = f(t, x), div upsilon = 0 is considered in Omega with boundary and initial conditions (upsilon, nu)\(Gamma 0) = rot upsilon\(Gamma 0) = 0 and upsilon\(t = 0) = upsilo n(0)(x) (here t is an element of (0, T), x is an element of Omega, and nu is the outward normal to Gamma(0)) Let <(v)over cap (t, x)> be a s olution of this system such that <(upsilon)over cap> satisfies the ind icated boundary conditions on Gamma(0) and \\<(upsilon)over cap (0, .) > - upsilon(0)\\w(22(Omega)) < epsilon, where epsilon = <epsilon((upsi lon))over cap> much less than 1. Then the existence of a control u(t,x ) on (0,T) x Gamma(1) with the following properties is proved: the sol ution upsilon(t,x) of the Navier-Stokes system such that (upsilon, nu) \Gamma(0) = rot upsilon\(Gamma 0) = 0, upsilon\(t = 0) = upsilon(0)(x) , and upsilon\(Gamma 1) = u coincides with <(upsilon)over cap (T, .)> for t = T, that is, upsilon(T, x) = <(upsilon)over cap (T,x)>. In part icular, if f and <(upsilon)over cap> do not depend on t and <(upsilon) over cap (x)> is an unstable steady-state solution, then it follows fr om the above result that one can suppress the occurrence of turbulence by some control alpha On Gamma(1). An analogous result is established in the case when Gamma(0) = partial derivative Omega and alpha(t,x) i s a distributed control concentrated in an arbitrary subdomain omega s ubset of Omega.