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/
[1] , and , Extended algebraic Riccati equations in the abstract hyperbolic case. Nonlinear Anal. 40 (2000) 105-129. | MR | Zbl
[2] , . Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Systems & Control: Fondations & Applications 2. Boston, Birkhäuser (1993). | MR | Zbl
[3] , Stabilization of the laminar turbulent transition location, in Proceedings MTNS 2000, El Jaï Ed. (2000).
[4] , Stabilisation et contrôle optimal des équations de Prandtl. Ph.D. Thesis, École supérieure d'Aéronautique et de l'Espace, Toulouse (2002).
[5] and , A linearized model for boundary layer equations, in International Series of Numerical Mathematics 139, Birkhäuser (2001) 31-42. | MR | Zbl
[6] and , A linearized Crocco equation. J. Math. Fluid Mech. 8 (2006) 510-541. | MR | Zbl
[7] and , Feedback stabilization of a boundary layer equation - Part 2: Nonhomogeneous state equation and numerical experiments. Appl. Math. Res. eXpress (2010) doi:10.1093/amrx/abp007. | Zbl
[8] and , Analyse mathématique et calcul numérique 4. Masson, Paris (1988).
[9] , Equations arising in boundary control problems. SIAM J. Control Optim. 25 (1987) 612-636. | MR | Zbl
[10] , and , Algebraic Riccati Equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems. Ann. Math. Pura Appl. 153 (1988) 307-382. | MR | Zbl
[11] and , Control theory for partial differential equations I, Abstract parabolic systems. Cambridge University Press, Cambridge (2000). | MR | Zbl
[12] and , 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 Mathematics 225, Marcel Dekker (2002) 193-219. | MR | Zbl
[13] and , Problèmes aux limites non homogènes. Dunod, Paris (1968). | Zbl
[14] and , Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation 15. Chapman & Hall/CRC, Boca Raton, London, New York (1999). | MR | Zbl
[15] and , The linear quadratic control of problem for infinite dimensional systems with unbounded input and output operators. SIAM J. Control Optim. 25 (1987) 121-144. | MR | Zbl
[16] , Interpolation theory, Functions spaces, Differential operators. North Holland (1978). | Zbl
[17] , An optimal control problem with unbounded control operator and unbounded observation operator where Algebraic Riccati Equation is satisfied as a Lyapunov equation. Appl. Math. Letters 10 (1997) 95-102. | MR | Zbl
[18] , The Algebraic Riccati Equation with unbounded control operator: The abstract hyperbolic case revisited. Contemporary mathematics 209 (1997) 315-338. | MR | Zbl
[19] and , An example in LQ optimal control. Syst. Control Lett. 33 (1998) 339-349. | MR | Zbl
[20] and , On the global existence of solutions to the Prandtl's system. Adv. Math. 181 (2004) 88-133. | MR | Zbl
[21] , Mathematical Control Theory. Birkhäuser (1995).
Cité par Sources :
