The optimal start control problem for 2D Boussinesq equations

We consider the problem of the optimal start control for two-dimensional Boussinesq equations describing non-isothermal flows of a viscous fluid in a bounded domain. Using the study of the properties of admissible tuples and of the evolution operator, we prove the solubility of the optimization problem under natural assumptions about the model data. We derive a variational inequality which is satisfied by the optimal control provided that the objective functional is determined by the final observation. We also obtain sufficient conditions for the uniqueness of an optimal control.