IJPAM: Volume 20, No. 3 (2005)

BOUNDARY CONTROL OF THE FORCED
VISCOUS BURGERS EQUATION

Mohamed Zribi$^1$, Nejib Smaoui$^2$
$^1$Department of Electrical Engineering
Kuwait University
P.O. Box 5969, Safat, 13060, KUWAIT
e-mail: mzribi@eng.kuniv.edu.kw
$^2$Department of Mathematics and Computer Science
Kuwait University
P.O. Box 5969, Safat, 13060, KUWAIT
e-mail: smaoui@mcs.sci.kuniv.edu.kw


Abstract.In this paper, the control problem of the forced Burgers equation:

\begin{eqnarray*} \frac{\partial u}{\partial t}\, = \nu \frac{\partial^{2}u}{\pa... ... + mu + f(x), \hspace*{0.3in} 0 < x < 2\pi, \hspace*{0.2in} t >0 \end{eqnarray*}

subject to Neumann boundary conditions:

\begin{eqnarray*} \displaystyle \frac{\partial u}{\partial x}(0,t)=\tilde{u}_{1}... ...al u}{\partial x}(2\pi,t)= \tilde{u}_{2}(t) \hspace*{0.3in} t>0. \end{eqnarray*}

and the initial condition:

\begin{eqnarray*}u(x,0)=u_{0}(x), \hspace{0.5in}x \in (0,2\pi) \ \ t>0,\end{eqnarray*}

where $\nu$ is a positive constant, the parameter $m\in$ I R, and $\tilde{u}_1(t)$, $\tilde{u}_2(t)$ are two control inputs is considered. We show that the controlled forced Burgers equation is exponentially stable when $f \in L^2(0,2\pi)$, and the viscosity $\nu > 4\pi^2(2m+1)$.

Received: March 29, 2005

AMS Subject Classification: 34H05, 35B37, 35Q53, 93D15

Key Words and Phrases: boundary control, forced Burgers equation, stabilization

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 20
Issue: 3