Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-20T22:01:31.767Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear feedback stabilization of a two-dimensional Burgers equation

Published online by Cambridge University Press:  31 July 2009

Laetitia Thevenet ,
Jean-Marie Buchot  and
Jean-Pierre Raymond
Laetitia Thevenet
Affiliation:
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
Jean-Marie Buchot
Affiliation:
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
Jean-Pierre Raymond
Affiliation:
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
Get access

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.

Keywords

Dirichlet controlNeumann controlfeedback controlstabilizationBurgers equationAlgebraic Riccati equation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 929 - 955
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Badra, Stabilisation par feedback et approximation des équations de Navier-Stokes. Ph.D. Thesis, Université Paul Sabatier, Toulouse, France (2006).
M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations. SIAM J. Control. (to appear).
Beeler, S.C., Tran, H.T. and Banks, H.T., Feedback control methodologies for nonlinear systems. J. Optim. Theory Appl. 107 (2000) 1–33. CrossRef
Ben Belgacem, F., El Fekik, H. and Raymond, J.-P., A penalized Robin approach for solving a parabolic equation with non smooth Dirichlet boundary conditions. Asymptotic Anal. 34 (2003) 121–136.
A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. 1. Birkhäuser (1992).
A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. 2. Birkhäuser (1993).
Fernandez-Cara, E., Guerrero, S., Imanuvilov, O. Yu. and Puel, J.-P., Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. CrossRef
Fernandez-Cara, E., Gonzalez-Burgos, M., Guerrero, S. and Puel, J.-P., Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case. ESAIM: COCV 12 (2006) 466–483 (electronic). CrossRef
Grubb, G. and Solonnikov, V.A., Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods. Math. Scand. 69 (1991) 217–290. CrossRef
L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equations. Springer (1997).
Krstic, M., Magnis, L. and Vazquez, R., Nonlinear stabilization of shock-like unstable equilibria in the viscous Burgers PDE. IEEE Trans. Automat. Contr. 53 (2008) 1678–1683. CrossRef
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Vol. 1. Cambridge University Press (2000).
Laub, A.J., Schur, A method method for solving algebraic Riccati equations. IEEE Trans. Automat. Contr. 24 (1979) 913–921. CrossRef
Lions, J.-L., Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs. J. Math. Soc. Japan 14 (1962) 233–241. CrossRef
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes, Vol. 2. Dunod, Paris (1968).
Raymond, J.-P., Boundary feedback stabilization of the two dimensional Navier-Stokes equations. SIAM J. Control Optim. 45 (2006) 790–828. CrossRef