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Optimal velocity control of a Brinkman-Cahn-Hilliard system with curvature effects

Pierluigi Colli, Gianni Gilardi, Andrea Signori, Jürgen Sprekels

TL;DR

The paper analyzes an optimal control problem for a Brinkman–Cahn–Hilliard system with curvature effects, where a distributed velocity field drives the flow and phase-field dynamics. It establishes a rigorous variational framework: existence of optimal controls, Fréchet differentiability of the control-to-state map via a linearized system, and first-order optimality conditions derived from an adjoint system, including sparsity considerations. The adjoint-based condition yields explicit projection formulas for the optimal control under box constraints and a sparsity-promoting term. These results provide a foundation for rigorous analysis and numerical design in microfluidic and soft-matter applications where curvature energy governs pattern formation. Overall, the work advances the mathematical theory of high-order diffuse-interface models coupled with Brinkman flows and offers a pathway to sparsity-driven control in complex multiphase systems.

Abstract

We address an optimal control problem governed by a system coupling a Brinkman-type momentum equation for the velocity field with a sixth-order Cahn-Hilliard equation for the phase variable, incorporating curvature effects in the free energy. The control acts as a distributed velocity control, allowing for the manipulation of the flow field and, consequently, the phase separation dynamics. We establish the existence of optimal controls, prove the Fréchet differentiability of the control-to-state operator, and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. We also discuss the aspect of sparsity. Beyond its analytical novelty, this work provides a rigorous control framework for Brinkman-Cahn-Hilliard systems incorporating a curvature regularization, offering a foundation for applications in microfluidic design and controlled pattern formation.

Optimal velocity control of a Brinkman-Cahn-Hilliard system with curvature effects

TL;DR

The paper analyzes an optimal control problem for a Brinkman–Cahn–Hilliard system with curvature effects, where a distributed velocity field drives the flow and phase-field dynamics. It establishes a rigorous variational framework: existence of optimal controls, Fréchet differentiability of the control-to-state map via a linearized system, and first-order optimality conditions derived from an adjoint system, including sparsity considerations. The adjoint-based condition yields explicit projection formulas for the optimal control under box constraints and a sparsity-promoting term. These results provide a foundation for rigorous analysis and numerical design in microfluidic and soft-matter applications where curvature energy governs pattern formation. Overall, the work advances the mathematical theory of high-order diffuse-interface models coupled with Brinkman flows and offers a pathway to sparsity-driven control in complex multiphase systems.

Abstract

We address an optimal control problem governed by a system coupling a Brinkman-type momentum equation for the velocity field with a sixth-order Cahn-Hilliard equation for the phase variable, incorporating curvature effects in the free energy. The control acts as a distributed velocity control, allowing for the manipulation of the flow field and, consequently, the phase separation dynamics. We establish the existence of optimal controls, prove the Fréchet differentiability of the control-to-state operator, and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. We also discuss the aspect of sparsity. Beyond its analytical novelty, this work provides a rigorous control framework for Brinkman-Cahn-Hilliard systems incorporating a curvature regularization, offering a foundation for applications in microfluidic design and controlled pattern formation.
Paper Structure (6 sections, 10 theorems, 180 equations)

This paper contains 6 sections, 10 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Notation, assumptions and main results
  3. The existence of an optimal control
  4. The control-to-state mapping
  5. Necessary conditions for optimality
  6. Remarks on sparsity

Key Result

Theorem 2.1

Assume --h on the structure, and suppose that the data satisfy --r. Then there exists a unique quadruplet $({\boldsymbol v},\varphi,\mu,w)$ with the regularity --r that solves Problem --p. Moreover, this solution satisfies the estimate with a constant $K_1$ that depends only on the structure of the system, $\Omega$, $T$ and an upper bound for the norms of the data related to --r.

Theorems & Definitions (20)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 10 more