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April 2012, 32(4): 1169-1208. doi: 10.3934/dcds.2012.32.1169
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,
Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l'Adour, 64013 Pau Cedex
Received October 2010 Revised May 2011 Published October 2011
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.
Keywords: Dirichlet control., Riccati equation, Navier-Stokes, Feedback stabilization, Neumann control, Boussinesq, Lyapunov function.
Mathematics Subject Classification: Primary: 93D15, 93C20; Secondary: 76D05, 76D55, 35Q30, 35Q3.
Citation: 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 & Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169
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Mehdi Badra