Abstract
We study the numerical stabilization of the two dimensional Boussinesq equations. The stabilization is achieved by boundary feedback controls of finite dimension applied on the velocity and temperature, and localized on parts of the domain boundary. We consider a stationary solution of the Boussinesq system which is linearly unstable. The linearized system around this unstable stationary solution is used to construct a feedback law. For that, we determine the projected linearized system onto an invariant subspace (including the unstable subspace), and we determine a feedback control law stabilizing this projected system, by solving a Riccati equation, of small dimension, associated with this system. The degrees of stabilizability of the different real generalized eigenspaces of the control system are used to determine the invariant subspace on which the linearized system is projected. Next, the linear feedback law, stabilizing the projected system, is applied to the nonlinear model to study its ability to locally stabilize the flow and its temperature.
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