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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  1. V. Barbu, I. Lasiecka and R. Triggiani, Extended algebraic Riccati equations in the abstract hyperbolic case. Nonlinear Anal.40 (2000) 105–129.  Zbl0961.49003
  2. A. Bensoussan, G. Da. Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Systems & Control: Fondations & Applications2. Boston, Birkhäuser (1993).  Zbl0790.93016
  3. J.-M. Buchot, Stabilization of the laminar turbulent transition location, in Proceedings MTNS 2000, El Jaï Ed. (2000).  
  4. J.-M. Buchot, Stabilisation et contrôle optimal des équations de Prandtl. Ph.D. Thesis, École supérieure d'Aéronautique et de l'Espace, Toulouse (2002).  
  5. J.-M. Buchot and J.-P. Raymond, A linearized model for boundary layer equations, in International Series of Numerical Mathematics139, Birkhäuser (2001) 31–42.  Zbl1029.35029
  6. J.-M. Buchot and J.-P. Raymond, A linearized Crocco equation. J. Math. Fluid Mech.8 (2006) 510–541.  Zbl1214.35035
  7. J.-M. Buchot and J.-P. Raymond, Feedback stabilization of a boundary layer equation – Part 2: Nonhomogeneous state equation and numerical experiments. Appl. Math. Res. eXpress (2010) doi:.  DOI10.1093/amrx/abp007
  8. R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique4. Masson, Paris (1988).  Zbl0664.47002
  9. F. Flandoli, Algebric Riccati Equations arising in boundary control problems. SIAM J. Control Optim.25 (1987) 612–636.  Zbl0617.49004
  10. F. Flandoli, I. Lasiecka and R. Triggiani, Algebraic Riccati Equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems. Ann. Math. Pura Appl.153 (1988) 307–382.  Zbl0674.49004
  11. I. Lasiecka and R. Triggiani, Control theory for partial differential equations I, Abstract parabolic systems. Cambridge University Press, Cambridge (2000).  Zbl0942.93001
  12. I. Lasiecka and R. Triggiani, Optimal Control and Algebraic Riccati Equations under Singular Estimates for eAtB in the Abscence of Analycity, Part I: The stable case, in Lecture Notes in Pure in Applied Mathematics225, Marcel Dekker (2002) 193–219.  Zbl1049.49023
  13. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes. Dunod, Paris (1968).  Zbl0165.10801
  14. O.A. Oleinik and V.N. Samokhin, Mathematical Models in Boundary Layer Theory,Applied Mathematics and Mathematical Computation15. Chapman & Hall/CRC, Boca Raton, London, New York (1999).  Zbl0928.76002
  15. A.J. Pritchard and D. Salamon, The linear quadratic control of problem for infinite dimensional systems with unbounded input and output operators. SIAM J. Control Optim.25 (1987) 121–144.  Zbl0615.93039
  16. H. Triebel, Interpolation theory, Functions spaces, Differential operators. North Holland (1978).  
  17. R. Triggiani, An optimal control problem with unbounded control operator and unbounded observation operator where Algebraic Riccati Equation is satisfied as a Lyapunov equation. Appl. Math. Letters10 (1997) 95–102.  Zbl0906.49001
  18. R. Triggiani, The Algebraic Riccati Equation with unbounded control operator: The abstract hyperbolic case revisited. Contemporary mathematics209 (1997) 315–338.  Zbl0908.49026
  19. G. Weiss and H. Zwart, An example in LQ optimal control. Syst. Control Lett.33 (1998) 339–349.  Zbl0901.49028
  20. Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system. Adv. Math.181 (2004) 88–133.  Zbl1052.35135
  21. J. Zabczyck, Mathematical Control Theory. Birkhäuser (1995).  

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