A Mathematical Analysis of an Optimal Control Problem for a Generalized Boussinesq Model for Viscous Incompressible Flows

We consider an optimal control problem governed by a systems of nonlinear partial differential equations modeling viscous incompressible flows submitted to variations of temperature, using a generalized Boussinesq approximation. We obtain existence for the optimal control as well as first order optimality conditions of Pontriagyn type by using the formalism due to Dubovitskii and Milyutin.