Ji. Diaz et Av. Fursikov, APPROXIMATE CONTROLLABILITY OF THE STOKES SYSTEM ON CYLINDERS BY EXTERNAL UNIDIRECTIONAL FORCES, Journal de mathematiques pures et appliquees, 76(4), 1997, pp. 353-375
Wt,give some negative and positive results on the approximate controll
ability of the Stokes system formulated on a cylinder Omega = G x R of
R-3 when the control is a density of external unidirectional forces.
We distinguish the case where the direction of the controls e is paral
lel to the cylinder generatrix (e = e(3)) from the one where e is orth
ogonal to this generatrix (e = e(1)). A negative result in the case of
e = e(3) is proved for periodic boundary conditions on x(3), and homo
geneous Dirichlet conditions on partial derivative G x R where G is a
general set of R-2. In contrast to that, the approximate controllabili
ty is proved for homogeneous Dirichlet conditions on partial derivativ
e Omega (i.e. zero on partial derivative G x R and solutions in (L-2(G
x R))(3) for any t), when G is a rectangle and e = e(1) is orthogonal
to the cylinder generatrix.