Boundary feedback stabilization of the Boussinesq system with mixed boundary conditions |
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Authors: | Mythily Ramaswamy Jean-Pierre Raymond Arnab Roy |
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Institution: | 1. T.I.F.R Centre for Applicable Mathematics, Post Bag No. 6503, GKVK Post Office, Bangalore-560065, India;2. Institut de Mathématiques de Toulouse, Université Paul Sabatier & CNRS, 31062 Toulouse Cedex, France |
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Abstract: | 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. |
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Keywords: | 93B52 93C20 93D15 76D05 76D55 26B20 Boussinesq system Mixed boundary conditions Boundary feedback stabilization Finite dimensional controllers Fractional Sobolev spaces |
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