We are interested in the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of perturbations. More precisely, we want to stabilize the laminar-to-turbulent transition location of a fluid flow over a flat plate. For that we study the Algebraic Riccati Equation (A.R.E.) of a control problem in which the state equation is a doubly degenerate linear parabolic equation. Because of the degenerate character of the state equation, the classical existence results in the literature of solutions to algebraic Riccati equations do not apply to this class of problems. Here taking advantage of the fact that the semigroup of the state equation is exponentially stable and that the observation operator is a Hilbert-Schmidt operator, we are able to prove the existence and uniqueness of solution to the A.R.E. satisfied by the kernel of the operator which associates the 'optimal adjoint state' with the 'optimal state'. In part 2 [Buchot and Raymond, Appl. Math. Res. eXpress (2010) doi:10.1093/amrx/abp007], we study problems in which the feedback law is determined by the solution to the A.R.E. and another nonhomogeneous term satisfying an evolution equation involving nonhomogeneous perturbations of the state equation, and a nonhomogeneous term in the cost functional.
Mots clés : feedback control law, Crocco equation, degenerate parabolic equations, Riccati equation, boundary layer equations, unbounded control operator
@article{COCV_2011__17_2_506_0,
author = {Buchot, Jean-Marie and Raymond, Jean-Pierre},
title = {Feedback stabilization of a boundary layer equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {506--551},
publisher = {EDP-Sciences},
volume = {17},
number = {2},
year = {2011},
doi = {10.1051/cocv/2010017},
mrnumber = {2801330},
zbl = {1243.93088},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2010017/}
}
TY - JOUR AU - Buchot, Jean-Marie AU - Raymond, Jean-Pierre TI - Feedback stabilization of a boundary layer equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 506 EP - 551 VL - 17 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010017/ DO - 10.1051/cocv/2010017 LA - en ID - COCV_2011__17_2_506_0 ER -
%0 Journal Article %A Buchot, Jean-Marie %A Raymond, Jean-Pierre %T Feedback stabilization of a boundary layer equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 506-551 %V 17 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010017/ %R 10.1051/cocv/2010017 %G en %F COCV_2011__17_2_506_0
Buchot, Jean-Marie; Raymond, Jean-Pierre. Feedback stabilization of a boundary layer equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 506-551. doi : 10.1051/cocv/2010017. http://www.numdam.org/articles/10.1051/cocv/2010017/
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