ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2010 - Volume 16, Issue 04  

Research Article

Nonlinear feedback stabilization of a two-dimensional Burgers equation

Thevenet, Laetitiaa1, Buchot, Jean-Mariea1 and Raymond, Jean-Pierrea1

a1 Université de Toulouse, UPS, Institut de Mathématiques, UMR CNRS 5219, 31062 Toulouse Cedex 9, France. Laetitia.Thevenet@math.univ-toulouse.fr; jean-marie.buchot@math.univ-toulouse.fr; raymond@math.univ-toulouse.fr

Abstract

In this paper, we study the stabilization of a two-dimensional Burgers equation around a stationary solution by a nonlinear feedback boundary control. We are interested in Dirichlet and Neumann boundary controls. In the literature, it has already been shown that a linear control law, determined by stabilizing the linearized equation, locally stabilizes the two-dimensional Burgers equation. In this paper, we define a nonlinear control law which also provides a local exponential stabilization of the two-dimensional Burgers equation. We end this paper with a few numerical simulations, comparing the performance of the nonlinear law with the linear one.

(Received December 2 2008)

(Revised May 5 2009)

(Online publication July 31 2009)

Key Words:

  • Dirichlet control;
  • Neumann control;
  • feedback control;
  • stabilization;
  • Burgers equation;
  • Algebraic Riccati equation

Mathematics Subject Classification:

  • 93B52;
  • 93C20;
  • 93D15