no. 4
Nonlinear feedback stabilization of a two-dimensional Burgers equation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 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 : 10.1051/cocv/2009028
Classification : 93B52, 93C20, 93D15
Mots clés : Dirichlet control, Neumann control, feedback control, stabilization, Burgers equation, algebraic Riccati equation 
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     title = {Nonlinear feedback stabilization of a two-dimensional {Burgers} equation},
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Thevenet, Laetitia; Buchot, Jean-Marie; Raymond, Jean-Pierre. Nonlinear feedback stabilization of a two-dimensional Burgers equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 929-955. doi : 10.1051/cocv/2009028. http://archive.numdam.org/articles/10.1051/cocv/2009028/

[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).

[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

[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

[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

[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

[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.

[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

[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

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

[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

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

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

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

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

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