Controllability of 2D Euler and Navier-Stokes Equations by Degenerate Forcing

We study controllability issues for the 2D Euler and Navier-Stokes (NS) systems under periodic boundary conditions. These systems describe the motion of the homogeneous ideal or viscous incompressible fluid on a two-dimensional torus T2. We assume the system to be controlled by a degenerate forcing applied to a 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 (ν >0). Methods of differential geometric/Lie algebraic control theory have been used for that study. In [3] criteria for global controllability of finite-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 sufficient criteria for global controllability in a finite-dimensional observed component and for L_2-approximate controllability for the 2D NS system. The justification 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 the NS system, and hence the previous approach can not be adapted for the 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 the 2D NS system in an observed component and for L2-approximate controllability; 2) we prove that these criteria are valid for the case of an ideal incompressible fluid (ν = 0); 3) we study solid controllability in projection on any finite-dimensional subspace and establish a sufficient criterion for such controllability.