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• W2015445411 abstract "We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w , by means of a feedback boundary control. We first determine a feedback law for the linearized system around w . Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H 3 / 2 + ε , 3 / 4 + ε / 2 ( Q ) , with 0 < ε . In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t = 0 , is achieved by choosing a time varying control operator in a neighbourhood of t = 0 . Nous étudions la stabilisation locale des équations de Navier–Stokes en 3D au voisinage d'une solution stationnaire instable w , par un contrôle feedback frontière. Nous déterminons d'abord une loi de feedback pour le système linéarisé en w . Nous montrons que cette loi donne une stabilisation locale des équations de Navier–Stokes. Pour traiter le terme non linéaire, les solutions du système en boucle fermée doivent appartenir à H 3 / 2 + ε , 3 / 4 + ε / 2 ( Q ) , avec 0 < ε . Dans [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006) ; V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], cette régularité est obtenue avec un feedback déterminé par minimisation d'une fonctionnelle contenant une norme de la variable d'état suffisamment forte. Dans ce cas, le feedback ne peut pas être caractérisé par une équation de Riccati bien posée. Ici, nous choisissons une fonctionnelle contenant une norme très faible de la variable d'état. La condition de compatibilité entre la condition initiale et la loi de contrôle en t = 0 est obtenue en choisissant un opérateur de contrôle dépendant du temps dans un voisinage de t = 0 ." @default.
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• W2015445411 date "2007-06-01" @default.
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• W2015445411 title "Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations" @default.
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• W2015445411 doi "https://doi.org/10.1016/j.matpur.2007.04.002" @default.
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