Finite dimensional feedback control of the 1-d Kuramoto-Sivashinsky equation

A finite-dimensional feedback control scheme of the Kuramoto-Sivashinsky (K-S) partial differential equation (PDE) with periodic boundary conditions is presented. First, the dynamical behavior of the K-S equation at the bifurcation parameter α = 17.75 representing a homoclinic connection in phase space is investigated, where a pseudo-spectral Galerkin method is used to solve the one-dimensional (1-d) K-S equation. Then, the Karhunen-Loeve (K-L) decomposition is applied on the numerical simulation data to extract the coherent structures or eigenfunctions of the dynamics of the equation. Projecting the one-dimensional K-S equation along the three most energetic K-L eigenfunctions. a system of ordinary differential equations (ODEs) is obtained. A control scheme is designed for the system of ODEs with the task of stabilizing the dynamics of the K-S equation to a stable solution. Theoretical development of the control scheme is illustrated through numerical simulations.