Optimal Control of Stationary Doubly Diffusive Flows on Lipschitz Domains

TL;DR

This paper develops a rigorous mathematical framework for controlling stationary doubly diffusive flows within Lipschitz domains, establishing existence, differentiability, and optimality conditions for the control problem.

Contribution

It provides the first comprehensive analysis of optimal control for stationary doubly diffusive flows, including existence, differentiability, and second-order optimality conditions.

Findings

Existence of an optimal control with quadratic cost

Fréchet differentiability of the control-to-state map

First and second-order optimality conditions established

Abstract

In this work, we study the control constrained distributed optimal control of a stationary doubly diffusive flow model. For the control problem, we use a well-posedness analysis based on minimal assumptions on data and domain. We show the existence of an optimal control with quadratic type cost functional, study the Fr\'echet differentiability properties of the control-to-state map and establish the first-order necessary optimality conditions corresponding to the optimal control problem. Expanding on this we prove the local optimality of a reference control using second-order sufficient optimality condition for the control problem.