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

Kush Kinra, Fernanda Cipriano

TL;DR

This work establishes a rigorous velocity-tracking optimal control framework for the $3$D critical convective Brinkman–Forchheimer equations on a bounded domain, with the control entering as a distributed force. The authors address the key difficulty of 3D regularity by deriving intermediate optimality conditions via a difference-quotient approach and regularizing an intermediate adjoint system, then passing to the limit. They prove the existence of an optimal state–control pair and derive the first-order necessary optimality conditions through a backward adjoint equation and a variational inequality that characterizes optimal forces. The results extend known 2D torus analyses to 3D bounded domains, providing a robust mathematical basis for velocity-tracking control in porous-media flows with potential industrial applications.

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.

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

TL;DR

This work establishes a rigorous velocity-tracking optimal control framework for the D critical convective Brinkman–Forchheimer equations on a bounded domain, with the control entering as a distributed force. The authors address the key difficulty of 3D regularity by deriving intermediate optimality conditions via a difference-quotient approach and regularizing an intermediate adjoint system, then passing to the limit. They prove the existence of an optimal state–control pair and derive the first-order necessary optimality conditions through a backward adjoint equation and a variational inequality that characterizes optimal forces. The results extend known 2D torus analyses to 3D bounded domains, providing a robust mathematical basis for velocity-tracking control in porous-media flows with potential industrial applications.

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 with -boundary . 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.
Paper Structure (14 sections, 8 theorems, 95 equations)

This paper contains 14 sections, 8 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Functional setting
  4. Linear operator
  5. Trilinear form and bilinear operator
  6. Nonlinear operator
  7. State equation
  8. Abstract formulation and weak solution
  9. Well-posedness and Lipschitz estimates
  10. Intermediate adjoint system
  11. Regularization of the system \ref{['eqn-adjoint']}
  12. Passing $\delta$ to zero
  13. Existence of an optimal pair and first-order necessary optimality conditions
  14. Proof of Theorem \ref{['thm-Main-result']}

Key Result

Theorem 1.2

Assume that $\mathfrak{m}_0 \in \mathbb{L}^2(\mathfrak{D})$. Then the control problem eqn-control-problem admits, at least, one optimal solution where $\widetilde{\mathfrak{m}}$ is the unique solution of 1 with $\boldsymbol{f}$ replaced by $\widetilde{\boldsymbol{f}}$. In addition, let the Hypothesis Para-Hypo be satisfied, then there exists a weak solution $\widetilde{\mathfrak{q}}$ (in the sens

Theorems & Definitions (19)

  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Theorem 3.1: Gautam+Mohan_2025
  • Proposition 3.2
  • proof
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.1
  • ...and 9 more