Controlling the Dynamics of the 2-D Navier-Stokes Equations

The dynamics of the two-dimensional (2-d) Navier-Stokes (N-S) equations with spatially periodic and temporally steady forcing $f=(\frac{1}{Re} k^3 \sin ky, 0)$ is analyzed. First, a system of nine-dimensional nonlinear dynamical system is obtained by a truncation of the 2-d N-S equations for various values of $k$. We show that for $k=4$, the dynamics transforms from periodic solutions to chaotic attractors through a sequence of bifurcations including a period doubling scenarios. Then, a state feedback control is designed to drive the state of the system to any desired state.