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IMA Journal of Mathematical Control and Information Advance Access originally published online on March 30, 2006
IMA Journal of Mathematical Control and Information 2007 24(1):47-55; 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.

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

Rathinasamy Sakthivel** and Hiroshi Ito***

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

** Email: krsakthivel{at}rediffmail.com

*** Email: hiroshi{at}ces.kyutech.ac.jp


  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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