## Flow control of Navier-Stokes-Boussinesq model

In this work, we consider a flow control problem which is modeled by Navier-Stokes equations with Boussivesq approximation. The temperature is governed by the energy equation and it is coupled to the Navier-Stokes equation by the Boussivesq term. We take an unstable solution of this model and then try to stabilize it using three controls velocity and temperature at some part of the boundary and heat flow applied in another part. We have developed a finite element model and computed an unstable stationary solution. We have linearized the model around the stationary solution and obtained the feedback low. For this we have to deal with a descriptor system which is reduced by a Leray-type projection. The reduced system is further projected to the unstable subspace which is of small dimension. A Ricatti equation is solved to find the feedback which causes the unstable eigenvalues to be shifted to the stable side. Numerical codes have been developed to perform all these tasks.

At the Reynolds number considered there are two unstable eigenvalues. A feedback law based on two dimensional unstable subspace does not stabilize the nonlinear model. Additional stable eigenvalues close to the origin are included in the unstable subspace by adding a shift which helps to obtain a feedback law that stabilizes the nonlinear model also. Moreover, the best control location for the velocity and temperature control is identified through numerical experiments.

**People**: Mythily Ramaswamy, Praveen C, Jean-Pierre Raymond

**Status**: Ongoing work