Feedback stabilization of a boundary layer equation

Jean-Marie Buchot; Jean-Pierre Raymond

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 2, page 506-551
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Buchot, Jean-Marie, and Raymond, Jean-Pierre. "Feedback stabilization of a boundary layer equation." ESAIM: Control, Optimisation and Calculus of Variations 17.2 (2011): 506-551. <http://eudml.org/doc/276326>.

@article{Buchot2011,
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. },
author = {Buchot, Jean-Marie, Raymond, Jean-Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Feedback control law; Crocco equation; degenerate parabolic equations; Riccati equation; boundary layer equations; unbounded control operator; feedback control law},
language = {eng},
month = {5},
number = {2},
pages = {506-551},
publisher = {EDP Sciences},
title = {Feedback stabilization of a boundary layer equation},
url = {http://eudml.org/doc/276326},
volume = {17},
year = {2011},
}

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
DA - 2011/5//
PB - EDP Sciences
VL - 17
IS - 2
SP - 506
EP - 551
AB - 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.
LA - eng
KW - Feedback control law; Crocco equation; degenerate parabolic equations; Riccati equation; boundary layer equations; unbounded control operator; feedback control law
UR - http://eudml.org/doc/276326
ER -

References

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