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
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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 To cite this version: Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccatibased strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete and Continuous Dynamical Systems Series A, American Institute of Mathematical Sciences, 2012, 252 (09), pp.5042–5075. . HAL Id: hal-00541563 https://hal.archives-ouvertes.fr/hal-00541563 Submitted on 30 Nov 2010 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. 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 1settings 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 Abstract. Let −A : D(A) → H be the generator of an analytic semigroup and B : U → [D(A)] a relatively bounded control operator such that (A − σ,B) is stabilizable for some σ > 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. Let −A : D(A) → H be the generator of an analytic semigroup and B : U → [D(A)] a relatively bounded control operator such that (A − σ,B) is stabilizable for some σ > 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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