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Feedback stabilization of Navier–Stokes equations

Published online by Cambridge University Press:  15 September 2003

Viorel Barbu
Viorel Barbu*
Affiliation:
Department of Mathematics, “Al.I. Cuza" University, 6600 Iasi, Romania; [email protected].
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Abstract

One proves that the steady-state solutions to Navier–Stokes equations with internal controllers are locally exponentially stabilizable by linear feedback controllers provided by a LQ control problem associated with the linearized equation.

Keywords

Navier–Stokes systemRiccati equationlinearized systemsteady-state solutionweak solution.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 197 - 205
Copyright
© EDP Sciences, SMAI, 2003

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References

Abergel, F. and Temam, R., On some control problems in fluid mechanics. Theoret. Comput. Fluid Dynam. 1 (1990) 303-325. CrossRef
V. Barbu, Mathematical Methods in Optimization of Differential Systems. Kluwer, Dordrecht (1995).
Barbu, V., Local controllability of Navier-Stokes equations. Adv. Differential Equations 6 (2001) 1443-1462.
Barbu, V., The time optimal control of Navier-Stokes equations. Systems & Control Lett. 30 (1997) 93-100. CrossRef
Barbu, V. and Sritharan, S., $H^{\infty}$ -control theory of fluid dynamics. Proc. Roy. Soc. London 454 (1998) 3009-3033. CrossRef
Barbu, V. and Sritharan, S., Flow invariance preserving feedback controller for Navier-Stokes equations. J. Math. Anal. Appl. 255 (2001) 281-307. CrossRef
Th.R. Bewley, S. Liu, Optimal and robust control and estimation of linear path to transition. J. Fluid Mech. 365 (1998) 305-349. CrossRef
A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems. Birkhäuser, Boston, Bassel, Berlin (1992).
Cao, C., Kevrekidis, I.G. and Titi, E.S., Numerical criterion for the stabilization of steady states of the Navier-Stokes equations. Indiana Univ. Math. J. 50 (2001) 37-96. CrossRef
P. Constantin and C. Foias, Navier-Stokes Equations. University of Chicago Press, Chicago, London (1989).
Coron, J.M., On the controllability for the 2-D incompresssible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1996) 33-75.
Coron, J.M., On the null asymptotic stabilization of the 2-D incompressible Euler equations in a simple connected domain. SIAM J. Control Optim. 37 (1999) 1874-1896. CrossRef
Coron, J.M. and Fursikov, A., Global exact controllability of the 2-D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429-448.
Imanuvilov, O.A., Local controllability of Navier-Stokes equations. ESAIM: COCV 3 (1998) 97-131. CrossRef
Imanuvilov, O.A., On local controllability of Navier-Stokes equations. ESAIM: COCV 6 (2001) 49-97.
I. Lasiecka and R. Triggianni, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia of Mathematics and its Applications. Cambridge University Press (2000).
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis. SIAM Philadelphia (1983).