Far Eastern Mathematical Journal

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Finite-dimensional stabilization with given rate for the Navier – Stokes systems


A. Yu. Chebotarev

2010, issue 2, P. 199–204


Abstract
The stabilization for unstable stationary solution of operator equation with quadratic nonlinearity is studied. The bounded finite-dimensional feedback control exponentially stabilizing this solution is presented.

Keywords:
finite-dimensional control, Navier – Stokes equations, feedback stabilization

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References

[1] A. V. Fursikov, “Stabilizability of two-dimensional Navier – Stokes equations with help of a boundary feedback control”, J. Math. Fluid Mech, 3:3 (2001), 259–301.
[2] A. V. Fursikov, “Stabilizaciya s granicy reshenij sistemy Nav'e – Stoksa: razreshimost' i obosnovanie vozmozhnosti chislennogo modelirovaniya”, Dal'nevost. matem. zhurn., 4:1 (2003), 86–100.
[3] A. V. Fursikov, “Stabilization for the 3D Navier – Stokes system by feedback boundary control”, Discrete and Cont. Dyn. Syst., 10:1–2 (2004), 289–314.
[4] V. Barbu, “Feedback stabilization of Navier – Stokes equations”, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 197–205.
[5] V. Barbu, R. Triggiani, “Internal stabilization of Navier – Stokes equations with finite-dimensional controllers”, Indiana Univ. Math. J., 53:5 (2004), 1443–1494.
[6] V. Barbu, I. Lasiecka, R. Triggiani, “Abstract settings for tangential boundary stabilization of Navier – Stokes equations by high- and low-gain feedback controllers”, Nonlinear Analysis, 64:12 (2006), 2704–2746.
[7] J.-P. Raymond, “Feedback boundary stabilization of the three-dimensional incompressible Navier – Stokes equations”, J. Math. Pures Appl, 87:6 (2007), 627–669.
[8] S. S. Ravindran, “Stabilization of Navier – Stokes equations by boundary feedback”, Intern. J. of Num. Analysis and Modeling, 4:3–4 (2007), 608–624.
[9] R. Temam, Uravneniya Nav'e – Stoksa. Teoriya i chislennyj analiz, Mir, M., 1981.
[10] A. Yu. Chebotarev, “Obratnye zadachi dlya nelinejnyx e'volyucionnyx uravnenij tipa Nav'e – Stoksa”, Differencial'nye uravneniya, 31:3 (1995), 517–524.
[11] M. Sermange, R. Temam, “Some mathematical questions related to the MHD equations”, Comm. on Pure and Applied Math., 36 (1983), 635–664.
[12] A. Yu. Chebotarev, “Variacionnye neravenstva dlya operatora tipa Nav'e – Stoksa i odnostoronnie zadachi dlya uravnenij vyazkoj teploprovodnoj zhidkosti”, Matematicheskie zametki, 70:2 (2001), 296–307.

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