Boundary Feedback Stabilization of the Boussinesq system with mixed boundary conditions

Mythily Ramaswamy (1) , Jean-Pierre Raymond (2) , Arnab Roy (1)

1 TIFR-CAM - Center for Applicable Mathematics [Bangalore]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We study the feedback stabilization of the Boussinesq system in a two dimensional domain , with mixed boundary conditions. After ascertaining the precise loss of regularity of the solution in such models, we prove first Green's formulas for functions belonging to weighted Sobolev spaces and then correctly define the underlying control system. This provides a rigorous mathematical framework for models studied in the engineering literature. We prove the stabilizability by extending to the linearized Boussinesq system a local Carleman estimate already established for the Oseen system. Then we determine a feedback control law able to stabilize the linearized system around the stationary solution, with any prescribed exponential decay rate, and able to stabilize locally the nonlinear system.

Mots clés: Boussinesq system; mixed boundary conditions; boundary Feedback Stabilization; finite dimensional controllers; fractional Sobolev spaces

Domaines: Mathématiques [math]

Mythily Ramaswamy, Jean-Pierre Raymond, Arnab Roy. Boundary Feedback Stabilization of the Boussinesq system with mixed boundary conditions. Journal of Differential Equations, In press. ⟨hal-01970455⟩