Journal of Applied Mathematics
Volume 2012 (2012), Article ID 389450, 12 pages
http://dx.doi.org/10.1155/2012/389450
Research Article

Adaptive Stabilization of the Korteweg-de Vries-Burgers Equation with Unknown Dispersion

Xiaoyan Deng,1,2 Lixin Tian,1,2 and Wenxia Chen1,2

1Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, China
2Nonlinear Scientific Research Center, Jiangsu University, Jiangsu, Zhenjiang 212013, China

Received 4 July 2012; Revised 10 September 2012; Accepted 10 September 2012

Academic Editor: Junjie Wei

Copyright © 2012 Xiaoyan Deng et al. 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.

Abstract

This paper studies the adaptive control problem of the Korteweg-de Vries-Burgers equation. Using the Lyapunov function method, we prove that the closed-loop system including the parameter estimator as a dynamic component is globally stable. Furthermore, we show that the state of the system is regulated to zero by developing an alternative to Barbalat's lemma which cannot be used in the present situation. The closed-loop system is shown to be well posed.