Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers

Varga K. Kalantarov; Edriss S. Titi

doi:10.3934/dcdsb.2018153

Article Contents

2018, Volume 23, Issue 3: 1325-1345. Doi: 10.3934/dcdsb.2018153

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Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers

Varga K. Kalantarov^1,2, , and
Edriss S. Titi^3,4,

1.
Department of mathematics, Koç University, Rumelifeneri Yolu, Sariyer 34450, Istanbul, Turkey
2.
Institute of Matematics and Mechanics, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan
3.
Department of Mathematics, Texas AM University, 3368 TAMU, College Station, TX 77843-3368, USA
4.
Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

* Corresponding author: V. K. Kalantarov
* Corresponding author: V. K. Kalantarov

Received: April 30, 2017

Revised: June 30, 2017

Early access: February 2018

Published: June 2018

V.K.Kalantarov would like to thank the Weizmann Institute of Science for the generous hospitality during which this work was initiated. E.S.Titi would like to thank the ICERM, Brown University, for the warm and kind hospitality where this work was completed. The work of E.S.Titi was supported in part by the ONR grant N00014-15-1-2333

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In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped nonlinear wave equations and the nonlinear wave equation with nonlinear damping term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation. This algorithm capitalizes on the fact that such infinite-dimensional dissipative dynamical systems posses finite-dimensional long-time behavior which is represented by, for instance, the finitely many determining parameters of their long-time dynamics, such as determining Fourier modes, determining volume elements, determining nodes, etc..The algorithm utilizes these finite parameters in the form of feedback control to stabilize the relevant solutions. For the sake of clarity, and in order to fix ideas, we focus in this work on the case of low Fourier modes feedback controller, however, our results and tools are equally valid for using other feedback controllers employing other spatial coarse mesh interpolants.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 35Q35.

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