APPROXIMATE CONTROLLABILITY OF THE STOKES SYSTEM ON CYLINDERS BY EXTERNAL UNIDIRECTIONAL FORCES

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.