# American Institute of Mathematical Sciences

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

 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.
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
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