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

Mehdi Badra

doi:10.3934/dcds.2012.32.1169

Article Contents

2012, Volume 32, Issue 4: 1169-1208. Doi: 10.3934/dcds.2012.32.1169

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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

Mehdi Badra^1,

1.
Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013 Pau Cedex

Received: September 30, 2010

Revised: April 30, 2011

Published: March 2012

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Abstract

Let $-A: \mathcal{D}(A)\to H$ be the generator of an analytic semigroup and $B : U \to [\mathcal{D}(A^*)]'$ a relatively bounded control operator such that $(A-\sigma,B)$ is stabilizable for some $\sigma>0$. In this paper, we consider the stabilization of the nonlinear system $y'+Ay+G(y,u)=Bu$ by means of a feedback or a dynamical control $u$. The control is obtained from the solution to a Riccati equation which is related to a low-gain optimal quadratic minimization problem. We provide a general abstract framework to define exponentially stable solutions which is based on the contruction of Lyapunov functions. We apply such a theory to stabilize, around an unstable stationary solution, the 2D or 3D Navier-Stokes equations with a Neumann control and the 2D or 3D Boussinesq equations with a Dirichlet control.

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Mathematics Subject Classification: Primary: 93D15, 93C20; Secondary: 76D05, 76D55, 35Q30, 35Q35.

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