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 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

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.

Mots clés: Feedback stabilization; Riccati equation; Lyapunov function; Navier-Stokes; Boussinesq; Neumann control; Dirichlet control; Feedback stabilization

Domaines: Equations aux dérivées partielles [math.AP]

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 - Series A, 2012, 252 (09), pp.5042--5075. ⟨10.3934/dcds.2012.32.1169⟩. ⟨hal-00541563⟩