Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations

TL;DR

This paper studies optimal control problems for 2D convective Brinkman-Forchheimer equations, establishing existence, optimality conditions, and applications to data assimilation in meteorology.

Contribution

It provides the first analysis of optimal control problems for 2D CBF equations, including existence, first and second order optimality conditions, and a data assimilation application.

Findings

Existence of optimal solutions for the control problems.

First order necessary conditions derived via Euler-Lagrange system.

Second order conditions established for the case r=3.

Abstract

The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. In this work, we consider some distributed optimal control problems like total energy minimization, minimization of enstrophy, etc governed by the two dimensional CBF equations with the absorption exponent $r=1,2$ and $3$. We show the existence of an optimal solution and the first order necessary conditions of optimality for such optimal control problems in terms of the Euler-Lagrange system. Furthermore, for the case $r=3$, we show the second order necessary and sufficient conditions of optimality. We also investigate an another control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is 2D CBF equations, using optimal control techniques.