Optimal control problem associated with three-dimensional critical convective Brinkman-Forchheimer equations

TL;DR

This paper investigates an optimal control problem for 3D critical convective Brinkman-Forchheimer equations, focusing on deriving necessary conditions for optimality despite challenges in differentiability and regularity.

Contribution

It establishes first-order necessary optimality conditions for the control problem, addressing differentiability issues of the control-to-state mapping in 3D critical equations.

Findings

Derived intermediate optimality conditions.

Established limit processes for necessary conditions.

Addressed regularity challenges in control-to-state mapping.

Abstract

In this article, we are concerned about the velocity tracking optimal control problem for 3D critical convective Brinkman-Forchheimer equations defined on a simply connected bounded domain $\mathbb{D}\subset\mathbb{R}^3$ with $\mathrm{C}^2$-boundary $\partial\mathbb{D}$. The control is introduced through an external force. The objective is to optimally minimize a velocity tracking cost functional, for which the velocity vector field is oriented towards a target velocity. Most importantly, we are concerned about the first-order necessary optimality conditions for above-mentioned optimal control problem which is the main challenging task of this article. To overcome the difficulties related to the differentiability of the control-to-state mapping, consequence of the lack of regularity of the state variable on bounded domains, we first establish some intermediate optimality conditions and then pass to the limit.