American Institute of Mathematical Sciences

Advanced
August  2010, 27(3): 1159-1187. doi: 10.3934/dcds.2010.27.1159

Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers

Jean-Pierre Raymond 1, and  Laetitia Thevenet 1,

1. 

Université de Toulouse & CNRS, Institut de Mathématiques, UMR 5219, 31062 Toulouse Cedex 9, France, France

Received  May 2009 Revised  February 2010 Published  March 2010

We study the boundary stabilization of the two-dimensional Navier-Stokes equations about an unstable stationary solution by controls of finite dimension in feedback form. The main novelty is that the linear feedback control law is determined by solving an optimal control problem of finite dimension. More precisely, we show that, to stabilize locally the Navier-Stokes equations, it is sufficient to look for a boundary feedback control of finite dimension, able to stabilize the projection of the linearized equation onto the unstable subspace of the linearized Navier-Stokes operator. The feedback operator is obtained by solving an algebraic Riccati equation in a space of finite dimension, that is to say a matrix Riccati equation.
Keywords: Oseen equations, Navier-Stokes equations, feedback control, finite dimensional controllers., Dirichlet control, Riccati equation, stabilization.
Mathematics Subject Classification: Primary: 93B52, 93C20, 93D15; Secondary: 35Q30, 76D55, 76D05, 76D0.
Citation: Jean-Pierre Raymond, Laetitia Thevenet. Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1159-1187. doi: 10.3934/dcds.2010.27.1159
[1]

Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169
[2]

Varga K. Kalantarov, Edriss S. Titi. Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1325-1345. doi: 10.3934/dcdsb.2018153
[3]

Tobias Breiten, Karl Kunisch. Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4197-4229. doi: 10.3934/dcds.2020178
[4]

Andrei Fursikov, Alexey V. Gorshkov. Certain questions of feedback stabilization for Navier-Stokes equations. Evolution Equations and Control Theory, 2012, 1 (1) : 109-140. doi: 10.3934/eect.2012.1.109
[5]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159
[6]

A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289
[7]

Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
[8]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations and Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89
[9]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations and Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147
[10]

Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085
[11]

Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control and Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029
[12]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[13]

Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109
[14]

Michael Herty, Lorenzo Pareschi, Sonja Steffensen. Mean--field control and Riccati equations. Networks and Heterogeneous Media, 2015, 10 (3) : 699-715. doi: 10.3934/nhm.2015.10.699
[15]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations and Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495
[16]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110
[17]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61
[18]

Cătălin-George Lefter, Elena-Alexandra Melnig. Feedback stabilization with one simultaneous control for systems of parabolic equations. Mathematical Control and Related Fields, 2018, 8 (3&4) : 777-787. doi: 10.3934/mcrf.2018034
[19]

Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations and Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1
[20]

Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (195)
  • HTML views (0)
  • Cited by (49)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy