Nonlinear feedback stabilization of a two-dimensional Burgers equation
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, p. 929-955

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.

DOI : https://doi.org/10.1051/cocv/2009028
Classification:  93B52,  93C20,  93D15
Keywords: Dirichlet control, Neumann control, feedback control, stabilization, Burgers equation, algebraic Riccati equation
@article{COCV_2010__16_4_929_0,
author = {Thevenet, Laetitia and Buchot, Jean-Marie and Raymond, Jean-Pierre},
title = {Nonlinear feedback stabilization of a two-dimensional Burgers equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {16},
number = {4},
year = {2010},
pages = {929-955},
doi = {10.1051/cocv/2009028},
zbl = {1202.93129},
mrnumber = {2744156},
language = {en},
url = {http://www.numdam.org/item/COCV_2010__16_4_929_0}
}

Thevenet, Laetitia; Buchot, Jean-Marie; Raymond, Jean-Pierre. Nonlinear feedback stabilization of a two-dimensional Burgers equation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 929-955. doi : 10.1051/cocv/2009028. http://www.numdam.org/item/COCV_2010__16_4_929_0/

[1] M. Badra, Stabilisation par feedback et approximation des équations de Navier-Stokes. Ph.D. Thesis, Université Paul Sabatier, Toulouse, France (2006).

[2] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations. SIAM J. Control. (to appear). | Zbl pre05719723

[3] S.C. Beeler, H.T. Tran and H.T. Banks, Feedback control methodologies for nonlinear systems. J. Optim. Theory Appl. 107 (2000) 1-33. | Zbl 0971.49023

[4] F. Ben Belgacem, H. El Fekik and J.-P. Raymond, A penalized Robin approach for solving a parabolic equation with non smooth Dirichlet boundary conditions. Asymptotic Anal. 34 (2003) 121-136. | Zbl 1043.35014

[5] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. 1. Birkhäuser (1992). | Zbl 0790.93016

[6] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. 2. Birkhäuser (1993). | Zbl 0790.93016

[7] E. Fernandez-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | Zbl pre02164954

[8] E. Fernandez-Cara, M. Gonzalez-Burgos, S. Guerrero and J.-P. Puel, Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case. ESAIM: COCV 12 (2006) 466-483 (electronic). | Numdam | Zbl 1106.93010

[9] G. Grubb and V.A. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods. Math. Scand. 69 (1991) 217-290. | Zbl 0766.35034

[10] L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equations. Springer (1997). | Zbl 0881.35001

[11] M. Krstic, L. Magnis and R. Vazquez, Nonlinear stabilization of shock-like unstable equilibria in the viscous Burgers PDE. IEEE Trans. Automat. Contr. 53 (2008) 1678-1683.

[12] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Vol. 1. Cambridge University Press (2000). | Zbl 0942.93001

[13] A.J. Laub, A Schur method method for solving algebraic Riccati equations. IEEE Trans. Automat. Contr. 24 (1979) 913-921. | Zbl 0424.65013

[14] J.-L. Lions, Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs. J. Math. Soc. Japan 14 (1962) 233-241. | Zbl 0108.11202

[15] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes, Vol. 2. Dunod, Paris (1968). | Zbl 0165.10801

[16] J.-P. Raymond, Boundary feedback stabilization of the two dimensional Navier-Stokes equations. SIAM J. Control Optim. 45 (2006) 790-828. | Zbl 1121.93064