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IMA Journal of Mathematical Control and Information Advance Access published online on March 30, 2006
IMA Journal of Mathematical Control and Information, doi:10.1093/imamci/dnl009
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Article

Non-linear robust boundary control of the Kuramoto-Sivashinsky equation

Rathinasamy Sakthivel 1 * and Hiroshi Ito 1

1 Department of Systems Innovation and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan

* To whom correspondence should be addressed.
Rathinasamy Sakthivel, E-mail: krsakthivel{at}rediffmail.com


  Abstract

This paper considers the problem of robust global stabilization of the Kuramoto-Sivashinsky equation subject to Neumann and Dirichlet boundary conditions. The aim is to derive non-linear robust boundary control laws which make the system robustly globally asymptotically stable in spite of uncertainty in both the instability parameter and the anti-diffusion parameter. A unique approach this paper introduces for achieving the required robustness is spatially dependent scaling of uncertain elements in Lyapunov-based stabilization. An important advantage of this approach is flexibility to obtain robust control laws with small control effort. The control laws can be implemented as Dirichlet-like boundary control as well as Neumann-like boundary control. Furthermore, it is shown that they guarantee the stability and boundedness in terms of both L2 and L{infty}.

Keywords: Kuramoto-Sivashinsky equation; robust global stabilization; spatially dependent scaling; non-linear boundary control; Lyapunov function.
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