Nonlinear Dynamics 37: 75–86, 2004. C  2004 Kluwer Academic Publishers. Printed in the Netherlands. Nonlinear Boundary Control of the Generalized Burgers Equation NEJIB SMAOUI Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat, 13060, Kuwait (e-mail: smaoui@mcs.sci.kuniv.edu.kw; fax: +965-481-7201) (Received: 29 October 2003; accepted: 24 February 2004) Abstract. In this paper, the adaptive and non-adaptive stabilization of the generalized Burgers equation by nonlinear boundary control are analyzed. For the non-adaptive case, we show that the controlled system is exponentially stable in L 2 . As for the adaptive case, we present a novel and elegant approach to show the L 2 regulation of the solution of the generalized Burgers system. Numerical results supporting and reinforcing the analytical ones of both the controlled and uncontrolled system for the non-adaptive and adaptive cases are presented using the Chebychev collocation method with backward Euler method as a temporal scheme. Key words: adaptive and non-adaptive control, generalized Burgers equation, stabilization 1. Introduction Burgers equation, a simple, one-dimensional, partial differential equation (PDE), which contains many features of fluid dynamics, has recently received much attention and interest as a first step towards developing methods for flow control [1–11]. Many people from the mathematical and control commu- nities used Burgers equation as a model for their analytical and numerical studies of high-dimensional nonlinear PDEs like the Navier–Stokes equations [12, 13], where most of these studies involved non- adaptive control. Adaptive control was also used to investigate different distributed parameter systems [14–18]. The main difference between adaptive and non-adaptive control is that in adaptive control, good control performance can be directly achieved even in the presence of undesirable or unpredictable disturbances. Up to 1998, a tremendous progress had been achieved in local stabilization and global analysis of the attractors of Burgers equation [2, 3, 5, 8, 10]. In 1999, nonlinear boundary control laws that achieve global asymptotic stability were derived by Krsti´ c [4] for both the viscous and inviscid Burgers equation, and in 2001, adaptive control of Burgers equation with unknown viscosity was investigated by Liu and Krsti´ c [11] to regulate the solution of the closed-loop system to zero in L 2 sense using an extension to Barbalat’s lemma. In this paper, we consider the adaptive and non-adaptive control of the generalized Burgers equation w t (x , t ) = νw xx (x , t ) − w(x , t )w x (x , t ) + mw(x , t ) x ∈ (0, 2π ) t > 0 (1) Subject to: aw x (0, t ) + bw(0, t ) = u 1 (t ), cw x (2π, t ) + d w(2π, t ) = u 2 (t ), (2) y (t ) =  w (0, t ) w (2π, t )  , (3)