Approximation methods for boundary control of the Boussinesq equations

In this paper we discuss an approximation method for dealing with Dirichlet boundary control of thermal-fluid systems. The physics of displacement ventilation and buoyancy-driven flows are described by the Boussinesq equations. We first develop a computational algorithm for solving the corresponding LQR control problem for the Boussinesq equations with general Robin boundary conditions. This scheme is combined with a finite element method that generalizes Nitsche's perturbation theory for approximating Dirichlet boundary conditions. Using this approach we are able to avoid imposing the “compatibility condition” that is required for Dirichlet boundary control in 3D problems. Numerical examples are presented to illustrate the computational algorithm.