Optimal Boundary Control of the Boussinesq Approximation for Polymeric Fluids

We consider an optimal control problem for non-isothermal steady flows of low-concentrated aqueous polymer solutions in a bounded 3D domain. In this problem, the state functions are the flow velocity and the temperature, while the control function is the heat flux through a given part of the boundary of the flow domain. We obtain sufficient conditions for the existence of weak solutions that minimize a cost functional under a given bounded set of admissible controls. It is shown that the marginal function of the considered control system is lower semi-continuous and the optimal states operator generates a continuous branch in a suitable function space.