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On the optimal control of viscous Cahn-Hilliard systems with hyperbolic relaxation of the chemical potential

Pierluigi Colli, Jürgen Sprekels

TL;DR

This work analyzes an optimal control problem for a viscous Cahn--Hilliard system augmented by a hyperbolic relaxation term in the chemical potential. It establishes the Fréchet differentiability of the control-to-state map, derives a well-posed adjoint system, and obtains first-order optimality conditions, including a sparsity result via an $L^1$ penalty. The paper proves existence of optimal controls for fixed hyperbolic relaxation parameter $\\alpha>0$ and conducts a detailed asymptotic analysis as $\\alpha\to0$, showing convergence of state, adjoint, and control to the CP$_0$ limit. Under additional smoothness assumptions, convergence of adjoint variables is also proved, linking the relaxed and nonrelaxed problems through a rigorous limit passage.

Abstract

In this paper, we study an optimal control problem for a viscous Cahn--Hilliard system with zero Neumann boundary conditions in which a hyperbolic relaxation term involving the second time derivative of the chemical potential has been added to the first equation of the system. For the initial-boundary value problem of this system, results concerning well-posedness, continuous dependence and regularity are known. We show Fréchet differentiability of the associated control-to-state operator, study the associated adjoint state system, and derive first-order necessary optimality conditions. Concerning the nonlinearities driving the system, we can include the case of logarithmic potentials. In addition, we perform an asymptotic analysis of the optimal control problem as the relaxation coefficient approaches zero.

On the optimal control of viscous Cahn-Hilliard systems with hyperbolic relaxation of the chemical potential

TL;DR

This work analyzes an optimal control problem for a viscous Cahn--Hilliard system augmented by a hyperbolic relaxation term in the chemical potential. It establishes the Fréchet differentiability of the control-to-state map, derives a well-posed adjoint system, and obtains first-order optimality conditions, including a sparsity result via an penalty. The paper proves existence of optimal controls for fixed hyperbolic relaxation parameter and conducts a detailed asymptotic analysis as , showing convergence of state, adjoint, and control to the CP limit. Under additional smoothness assumptions, convergence of adjoint variables is also proved, linking the relaxed and nonrelaxed problems through a rigorous limit passage.

Abstract

In this paper, we study an optimal control problem for a viscous Cahn--Hilliard system with zero Neumann boundary conditions in which a hyperbolic relaxation term involving the second time derivative of the chemical potential has been added to the first equation of the system. For the initial-boundary value problem of this system, results concerning well-posedness, continuous dependence and regularity are known. We show Fréchet differentiability of the associated control-to-state operator, study the associated adjoint state system, and derive first-order necessary optimality conditions. Concerning the nonlinearities driving the system, we can include the case of logarithmic potentials. In addition, we perform an asymptotic analysis of the optimal control problem as the relaxation coefficient approaches zero.

Paper Structure

This paper contains 11 sections, 12 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. General assumptions and properties of the state system
  3. Differentiability of the solution operator
  4. The optimal control problem
  5. Existence
  6. Necessary optimality conditions
  7. Sparsity of controls
  8. Asymptotic analysis
  9. Convergence of the state variables
  10. Convergence of optimal controls
  11. Convergence of the adjoint variables

Key Result

Theorem 2.1

Suppose that (A1)-- (A5) are satisfied. Then the state system --s has for any $u\in {\cal U}_R$ and every $\alpha\in (0,1]$ a unique solution triple $(\mu^\alpha,\varphi^\alpha,w^\alpha)$ satisfying the regularity requirements \Regsoluz -> - regmu regw Moreover, there is a constant $K_1>0$, which depends only on the data of the state system and $R$ and not on $\alpha\in (0,1]$, such that In add

Theorems & Definitions (26)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 4.1
  • ...and 16 more