Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 621672, 19 pages
Distributed Control of the Generalized Korteweg-de Vries-Burgers Equation
Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
Received 24 January 2008; Accepted 10 May 2008
Academic Editor: Giuseppe Rega
Copyright © 2008 Nejib Smaoui and Rasha H. Al-Jamal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The paper deals with the distributed control of the generalized Kortweg-de Vries-Burgers equation (GKdVB) subject to periodic boundary conditions via the Karhunen-Loève (K-L) Galerkin method. The decomposition procedure of the K-L method is presented to illustrate the use of this method in analyzing the numerical simulations data which represent the solutions to the GKdVB equation. The K-L Galerkin projection is used as a model reduction technique for nonlinear systems to derive a system
of ordinary differential equations (ODEs) that mimics the dynamics of the GKdVB equation. The data coefficients derived from the ODE system are then used to approximate the solutions of the GKdVB equation. Finally, three state feedback linearization control schemes with the objective of enhancing the stability of the GKdVB equation are proposed. Simulations of the controlled system are given to illustrate the developed theory.