Av. Fursikov et Oy. Emanuilov, LOCAL EXACT CONTROLLABILITY OF THE 2-DIMENSIONAL NAVIER-STOKES EQUATIONS, Sbornik. Mathematics, 187(9-10), 1996, pp. 1355-1390
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.