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, R-e. 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 R-e is designed. Finally, numerical simulations are undertaken to validate the theoretical developments. (C) 2014 Elsevier Inc. All rights reserved.