Dynamics and control of the seven-mode truncation system of the 2-d Navier–Stokes equations

• Performed a Galerkin projection on the 2-d Navier–Stokes equations to obtain a reduced system of ODEs.
• Analyzed the dynamics and designed control laws for the reduced ODE system.
• Presented numerical simulations to validate the theoretical developments.

This paper deals with the dynamics and control problem of the two dimensional Navier–Stokes equations. A seventh order system of nonlinear ordinary differential equations which approximates the behavior of the Navier–Stokes equations is obtained by using the Fourier Galerkin method. Extensive simulations show that the obtained system is able to display the different behaviors of the Navier–Stokes equations. Then the paper proposes two Lyapunov based controllers to either control the system of ordinary differential equations to a desired fixed point or to synchronize two ordinary differential equations systems obtained from the two dimensional Navier–Stokes equations under different initial conditions. The proposed control schemes are simulated using the MATLAB software and the simulation results show their effectiveness.