A boundary control problem associated to the Rayleigh-B\'enard-Marangoni system

In this paper, we study a boundary control problem associated to the stationary Rayleigh-B\'enard-Marangoni (RBM) system in presence of controls for the velocity and the temperature on parts of the boundary. We analyze the existence, uniqueness and regularity of weak solutions for the stationary RBM system in polyhedral domains of $\mathbb{R}^3,$ and then, we prove the existence of the optimal solution. By using the Theorem of Lagrange multipliers, we derive an optimality system. We also give a second-order sufficient optimality condition and we establish a result of uniqueness of the optimal solution.