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Feedback stabilization of a boundary layer equation

Part 1: Homogeneous state equations*

Published online by Cambridge University Press:  23 April 2010

Jean-Marie Buchot  and
Jean-Pierre Raymond
Jean-Marie Buchot
Affiliation:
Université de Toulouse, UPS, Institut de Mathématiques, 31062 Toulouse Cedex 9, France. buchot@math.univ-toulouse.fr; raymond@math.univ-toulouse.fr
Jean-Pierre Raymond
Affiliation:
CNRS, Institut de Mathématiques, UMR 5219, 31062 Toulouse Cedex 9, France.
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Abstract

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.

Keywords

Feedback control lawCrocco equationdegenerate parabolic equationsRiccati equationboundary layer equationsunbounded control operator
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 2 , April 2011 , pp. 506 - 551
Copyright
© EDP Sciences, SMAI, 2010

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