Feedback stabilization of Navier-Stokes equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 197-205

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.

DOI : 10.1051/cocv:2003009
Classification : 76D05, 76D55, 35B40, 35Q30
Keywords: Navier-Stokes system, Riccati equation, linearized system, steady-state solution, weak solution 
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     author = {Barbu, Viorel},
     title = {Feedback stabilization of {Navier-Stokes} equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {197--205},
     year = {2003},
     publisher = {EDP Sciences},
     volume = {9},
     doi = {10.1051/cocv:2003009},
     zbl = {1076.93037},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2003009/}
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Barbu, Viorel. Feedback stabilization of Navier-Stokes equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 197-205. doi: 10.1051/cocv:2003009

[1] F. Abergel and R. Temam, On some control problems in fluid mechanics. Theoret. Comput. Fluid Dynam. 1 (1990) 303-325. | Zbl

[2] V. Barbu, Mathematical Methods in Optimization of Differential Systems. Kluwer, Dordrecht (1995). | Zbl | MR

[3] V. Barbu, Local controllability of Navier-Stokes equations. Adv. Differential Equations 6 (2001) 1443-1462. | Zbl

[4] V. Barbu, The time optimal control of Navier-Stokes equations. Systems & Control Lett. 30 (1997) 93-100. | Zbl

[5] V. Barbu and S. Sritharan, H ∞ -control theory of fluid dynamics. Proc. Roy. Soc. London 454 (1998) 3009-3033. | Zbl | MR

[6] V. Barbu and S. Sritharan, Flow invariance preserving feedback controller for Navier-Stokes equations. J. Math. Anal. Appl. 255 (2001) 281-307. | Zbl

[7] Th.R. Bewley and S. Liu, Optimal and robust control and estimation of linear path to transition. J. Fluid Mech. 365 (1998) 305-349. | Zbl | MR

[8] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems. Birkhäuser, Boston, Bassel, Berlin (1992). | Zbl | MR

[9] C. Cao, I.G. Kevrekidis and E.S. Titi, Numerical criterion for the stabilization of steady states of the Navier-Stokes equations. Indiana Univ. Math. J. 50 (2001) 37-96. | Zbl

[10] P. Constantin and C. Foias, Navier-Stokes Equations. University of Chicago Press, Chicago, London (1989). | Zbl

[11] J.M. Coron, On the controllability for the 2-D incompresssible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1996) 33-75. | Zbl | Numdam

[12] J.M. Coron, 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. | Zbl | MR

[13] J.M. Coron and A. Fursikov, Global exact controllability of the 2-D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429-448. | Zbl

[14] O.A. Imanuvilov, Local controllability of Navier-Stokes equations. ESAIM: COCV 3 (1998) 97-131. | Zbl | Numdam

[15] O.A. Imanuvilov, On local controllability of Navier-Stokes equations. ESAIM: COCV 6 (2001) 49-97.

[16] 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). | Zbl

[17] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis. SIAM Philadelphia (1983). | Zbl

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