Dynamics and control of the 2-d Navier–Stokes equations

This paper deals with the dynamics and control of the two-dimensional (2-d) Navier–Stokes (N–S) equations with a spatially periodic and temporally steady forcing term. First, we construct a dynamical system of nine nonlinear differential equations by Fourier expansion and truncation of the 2-d N–S equations. Then, we study the dynamics of the obtained reduced order system by analyzing the system’s attractors for different values of the Reynolds number, ReRe. By applying the symmetry of the equations on one of the system’s attractors, a symmetric limit trajectory that is part of the dynamics is obtained. Moreover, a Lyapunov based control strategy to control the dynamics of the system for a given ReRe is designed. Finally, numerical simulations are undertaken to validate the theoretical developments.