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.
Mots clés : 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}, pages = {929--955}, publisher = {EDP-Sciences}, volume = {16}, number = {4}, year = {2010}, doi = {10.1051/cocv/2009028}, mrnumber = {2744156}, zbl = {1202.93129}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2009028/} }
TY - JOUR AU - Thevenet, Laetitia AU - Buchot, Jean-Marie AU - Raymond, Jean-Pierre TI - Nonlinear feedback stabilization of a two-dimensional Burgers equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 929 EP - 955 VL - 16 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009028/ DO - 10.1051/cocv/2009028 LA - en ID - COCV_2010__16_4_929_0 ER -
%0 Journal Article %A Thevenet, Laetitia %A Buchot, Jean-Marie %A Raymond, Jean-Pierre %T Nonlinear feedback stabilization of a two-dimensional Burgers equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 929-955 %V 16 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2009028/ %R 10.1051/cocv/2009028 %G en %F 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, Tome 16 (2010) no. 4, pp. 929-955. doi : 10.1051/cocv/2009028. http://www.numdam.org/articles/10.1051/cocv/2009028/
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