IJPAM: Volume 7, No. 3 (2003)

CONTROLLING THE DYNAMICS OF
THE KURAMOTO-SIVASHINSKY EQUATION

Nejib Smaoui$^1$, Mohamed Zribi$^2$
$^1$Department of Mathematics and Computer Science
Faculty of Science
Kuwait University
P.O. Box 5969, Safat 13060, KUWAIT
e-mail: smaoui@mcs.sci.kuniv.edu.kw
$^{2}$Department of Electrical Engineering
Kuwait University
P.O. Box 5969, Safat 13060, KUWAIT
e-mail: mzribi@eng.kuniv.edu.kw


Abstract.Finite-dimensional feedback control schemes of the Kuramoto-Sivashinsky (K-S) partial differential equation (PDE) with periodic boundary conditions are presented. First, the dynamical behavior of the K-S equation at the bifurcation parameter $\alpha =17.75$ 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-Loéve (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. Three different control schemes are 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 schemes are illustrated through numerical simulations.

Received: April 12, 2003

AMS Subject Classification: 35B37, 35K55, 37N35

Key Words and Phrases: Kuramoto-Sivashinsky equation, Karhunen-Loéve decomposition, control

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2003
Volume: 7
Issue: 3