Local exact controllability of the two-dimensional Navier-Stokes equations

Let Omega subset of R(2) be a bounded domain with boundary partial derivative Omega consisting of two disjoint closed curves Gamma(0) and Gamma(1) such that Gamma(0) is connected and Gamma(1) not equal empty set . The Navier-Stokes system partial derivative(t) upsilon(t, x) - Delta 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) = upsilon(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 solution of this system such that <(upsilon)over cap> satisfies the indicated boundary conditions on Gamma(0) and \\<(upsilon)over cap (0, .)> - upsilon(0)\\w(22(Omega)) < epsilon, where epsilon = <epsilon((upsilon))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 solution 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 particular, 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 from 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) is a distributed control concentrated in an arbitrary subdomain omega subset of Omega.