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

TL;DR

This paper investigates a boundary control problem for the stationary Rayleigh-Bénard-Marangoni system, establishing existence, uniqueness, regularity of solutions, and deriving optimality conditions for control strategies in three-dimensional polyhedral domains.

Contribution

It provides the first comprehensive analysis of boundary control for the RBM system, including existence, uniqueness, regularity, and optimality conditions in 3D polyhedral domains.

Findings

Existence and uniqueness of weak solutions established.

Optimal control solutions exist and are unique under certain conditions.

Derived second-order optimality conditions and an optimality system.

Abstract

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.