  Motivated by an energy efficient building application, we want to optimize a
quadratic cost functional subject to the Boussinesq approximation of the
Navier-Stokes equations and to bilateral state and control constraints. Since
the computation of such an optimal solution is numerically costly, we design an
efficient strategy to compute a sub-optimal (but applicationally acceptable)
solution with significantly reduced computational effort. We employ an economic
Model Predictive Control (MPC) strategy to obtain a feedback control. The MPC
sub-problems are based on a linear-quadratic optimal control problem subjected
to mixed control and state constraints and a convection-diffusion equation,
reduced with proper orthogonal decomposition. To solve each sub-problem, we
apply a primal-dual active set strategy. The method can be fully parallelized,
which enables the solution of large problems with real-world parameters.
