Navier-Stokes Equation on the Rectangle

We study controllability issues for the Navier-Stokes Equation on a two dimensional rectangle with so-called Lions boundary conditions. Rewriting the Equation using a basis of harmonic functions we arrive to an infinite-dimensional system of ODEs. Methods of Geometric/Lie Algebraic Control Theory are used to prove controllability by means of low modes forcing of finite-dimensional Galerkin approximations of that system. Proving the continuity of the ``control $\mapsto$ solution’’ map in the so-called relaxation metric we use it to prove both solid controllability on observed component and ${\mathbf{L}}^2$-approximate controllability of the Equation (full system) by low modes forcing.