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

Pierluigi Colli; Jürgen Sprekels; Pierluigi Colli; Jürgen Sprekels

doi:10.3934/cam.2025027

Communications in Analysis and Mechanics

2025, Volume 17, Issue 3: 683-706. doi: 10.3934/cam.2025027

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

Pierluigi Colli ^{1
,
,},
Jürgen Sprekels ²

1.
Dipartimento di Matematica "F. Casorati", Università di Pavia and Research Associate at the IMATI – C.N.R. Pavia, via Ferrata 5, 27100 Pavia, Italy
2.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany

Received: 19 March 2025 Revised: 23 June 2025 Accepted: 02 July 2025 Published: 09 July 2025
35M33, 49K20, 49N90, 93C20, 37D35

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.
- Optimal control,
- Cahn–Hilliard system,
- hyperbolic relaxation,
- first-order optimality conditions,
- asymptotic analysis
Citation: Pierluigi Colli, Jürgen Sprekels. On the optimal control of viscous Cahn–Hilliard systems with hyperbolic relaxation of the chemical potential[J]. Communications in Analysis and Mechanics, 2025, 17(3): 683-706. doi: 10.3934/cam.2025027

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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.

References

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