We study controllability issues for the 2D Euler and Navier-
Stokes (NS) systems under periodic boundary conditions. These systems
describe motion of homogeneous ideal or viscous incompressible fluid on
a two-dimensional torus T^2. We assume the system to be controlled by
a degenerate forcing applied to fixed number of modes.
In our previous work [3, 5, 4] we studied global controllability by
means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing
viscosity (\nu > 0). Methods of dfferential geometric/Lie algebraic
control theory have been used for that study. In [3] criteria for
global controllability of nite-dimensional Galerkin approximations of
2D and 3D NS systems have been established. It is almost immediate
to see that these criteria are also valid for the Galerkin approximations
of the Euler systems. In [5, 4] we established a much more intricate suf-
cient criteria for global controllability in finite-dimensional observed
component and for L2-approximate controllability for 2D NS system.
The justication of these criteria was based on a Lyapunov-Schmidt
reduction to a finite-dimensional system. Possibility of such a reduction
rested upon the dissipativity of NS system, and hence the previous
approach can not be adapted for Euler system.
In the present contribution we improve and extend the controllability
results in several aspects: 1) we obtain a stronger sufficient condition for
controllability of 2D NS system in an observed component and for L2-
approximate controllability; 2) we prove that these criteria are valid for
the case of ideal incompressible uid (\nu = 0); 3) we study solid controllability
in projection on any finite-dimensional subspace and establish a
sufficient criterion for such controllability.
