ECOSEA Project.

We are interested in the stabilization of the Navier- Stokes equations, locally in a neighborhood of unstable stationary solutions, by boundary controls of finite-dimension in the case of partial information. The new approaches that we have developed have been tested in two geometrical configurations in two dimensions : In the case of flow in a channel around a circular cylinder, and in the case of an open cavity. The continuous model is supposed to represent the physical model , it is of infinite dimension. The discrete model is the one used for calculating the feedback laws and the estimator, it is of finite size, but very large.

The methodology developed in this project is as follows.

– Gains and feedback filter are first determined for the continuous linearized model. Although the model is infinite dimensional, the gains are obtained by solving Riccati equations small.

– The approximate model is obtained by a mixed finite element method. The approximation of feedback and filtering gains requires a precise calculation of the stiffness and mass matrices of the system, and a precise calculation of twenty eigenmodes of the spectrum the linearized Navier- Stokes operator and of its adjoint.

– The size of the Riccati equations used for calculating the gain does not depend on the mesh selected for the approximation of the continuous model.

– The feedback laws depend on the choice of a finite dimensional space invariant and a chosen exponential decay rate. The control zone is chosen in an optimal way.