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Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system

Published online by Cambridge University Press:  21 October 2008

Mehdi Badra
Mehdi Badra*
Affiliation:
Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l'Adour, 64013 Pau Cedex, France. mehdi.badra@univ-pau.fr
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Abstract

We study the local exponential stabilization of the 2D and 3DNavier-Stokes equations in a bounded domain, around a givensteady-state flow, by means of a boundary control. We look for acontrol so that the solution to the Navier-Stokes equations be astrong solution. In the 3D case, such solutions may exist if theDirichlet control satisfies a compatibility condition with theinitial condition. In order to determine a feedback law satisfyingsuch a compatibility condition, we consider an extended systemcoupling the Navier-Stokes equations with an equation satisfied bythe control on the boundary of the domain. We determine a linearfeedback law by solving a linear quadratic control problem for thelinearized extended system. We show that this feedback law alsostabilizes the nonlinear extended system.

Keywords

Navier-Stokes equationfeedback stabilizationDirichlet controlRiccati equation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 4 , October 2009 , pp. 934 - 968
Copyright
© EDP Sciences, SMAI, 2008

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References

M. Badra, Feedback stabilization of 3-D Navier-Stokes equations based on an extended system, in Proceedings of the 22nd IFIP TC7 Conference (2005).
M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations. SIAM J. Contr. Opt. (to appear).
Barbu, V., Feedback stabilization of Navier-Stokes equations. ESAIM: COCV 9 (2003) 197–206 (electronic). CrossRef
V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoirs of the American Mathematical Society 181. AMS (2006).
Barbu, V. and Triggiani, R.L, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers. Indiana Univ. Math. J. 53 (2004) 1443–1494. CrossRef
A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite-dimensional systems 1, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, USA (1992).
P. Constantin and C. Foias, Navier-Stokes equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, USA (1988)
Coron, J.-M. and Fursikov, A.V., Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429–448.
Fernández-Cara, E., Guerrero, S., Imanuvilov, O.Yu. and Puel, J.-P., Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. CrossRef
Fujita, H. and Morimoto, H., On fractional powers of the Stokes operator. Proc. Japan Acad. 46 (1970) 1141–1143. CrossRef
Fursikov, A.V., Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control. J. Math. Fluid Mech. 3 (2001) 259–301. CrossRef
Fursikov, A.V., Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete Contin. Dyn. Syst. 10 (2004) 289–314. CrossRef
G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Vol. I, Linearized steady problems, Springer Tracts in Natural Philosophy, Vol. 38. Springer-Verlag, New York (1994).
G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Vol. II, Nonlinear steady problems, Springer Tracts in Natural Philosophy, Vol. 39. Springer-Verlag, New York (1994).
Grisvard, P., Caractérisation de quelques espaces d'interpolation. Arch. Rational Mech. Anal. 25 (1967) 40–63. CrossRef
P. Grisvard, Elliptic problems in nonsmooth domains, in Monographs and Studies in Mathematics, Vol. 24, Pitman (Advanced Publishing Program), Boston, MA, USA (1985).
E. Hille and R.S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31. American Mathematical Society, Providence, RI, USA, revised edition (1957).
I. Lasiecka and R. Triggiani, Control theory for partial differential equations: continuous and approximation theories. I. Abstract parabolic systems, in Encyclopedia of Mathematics and its Applications 74, Cambridge University Press, Cambridge (2000).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. I. Dunod, Paris (1968).
A. Pazy, Semigroups of linear operators and applications to partial differential equations, in Applied Mathematical Sciences 44, Springer-Verlag, New York (1983).
Raymond, J.-P., Feedback boundary stabilization of the two dimensional Navier-Stokes equations. SIAM J. Contr. Opt. 45 (2006) 790–828. CrossRef
Raymond, J.-P., Feedback boundary stabilization of the three dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl. 87 (2007) 627–669. CrossRef
J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions. Ann. Inst. H. Poincaré, Anal. Non Linéaire 24 (2007) 921–951.
M.E. Taylor, Partial differential equations. I. Basic theory, in Applied Mathematical Sciences 115, Springer-Verlag, New York (1996).
R. Temam, Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam, revised edition (1979). With an appendix by F. Thomasset.
H. Triebel, Interpolation theory, function spaces, differential operators. Johann Ambrosius Barth, Heidelberg, second edition (1995).
Yagi, A., Coïncidence entre des espaces d'interpolation et des domaines de puissances fractionnaires d'opérateurs. C. R. Acad. Sci. Paris Sér. I Math. 299 (1984) 173–176.