Optimal distributed control of nonlinear Cahn–Hilliard systems with computational realization

In this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a three-dimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective Cahn-Hilliard system, which consists of two nonlinearly coupled second-order partial differential equations for the unknown quantities, the chemical potential and an order parameter representing the scaled density of one of the phases. In contrast to other contributions, in which zero Neumann boundary conditions were are assumed for both the chemical potential and the order parameter, we consider the case of dynamic boundary conditions, which model the situation when another phase transition takes place on the boundary. The phase transition processes in the bulk and on the boundary are in the unknowns ρ, the order parameter, and µ, the chemical potential. In the above equations, τ Ω is a nonnegative constant, f ′ is the derivative of a double-well potential f , and u is a given velocity field.Typical and significant examples of f are the so-called classical regular potential, the logarithmic double-well potential , and the double obstacle potential , which are given bywhere the constants in (1.3) and (1.4) satisfy c 1 > 1 and c 2 > 0, so that that f log and f 2obs are nonconvex. In cases like (1.4), one has to split f into a nondifferentiable convex part (the indicator function of [−1, 1] in the present example) and a smooth perturbation. Accordingly, one has to replace the derivative of the convex part by the subdifferential and interpret the second identity in (1.1) as a differential inclusion.As far as the conditions on the boundary Γ := ∂Ω are concerned, instead of the classical homogeneous Neumann boundary conditions, the dynamic boundary condition for both µ and ρ are considered, namely,

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“…There also exist articles addressing some discretized versions of general Cahn-Hilliard systems, cf. [31,41].…”

Section: Introductionmentioning

confidence: 99%

On a Cahn–Hilliard system with convection and dynamic boundary conditions

Colli

Gilardi

Sprekels

2018

Annali di Matematica

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“…There also exist contributions addressing some discretized versions of general Cahn-Hilliard systems, cf. [16,22]. However, about the optimal control of viscous or non-viscous Cahn-Hilliard systems with dynamic boundary conditions of the form (1.4), we only know of the papers [10] and [6] dealing with the viscous case; to the best of our knowledge, the present contribution is the first paper treating the optimal control of the pure Cahn-Hilliard system with dynamic boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions

Colli¹,

Gilardi²,

Sprekels³

2015

Preprint

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“…Recent investigations have been also addressed to the optimal control problem for Cahn-Hilliard systems: let us mention [8,9,[15][16][17]27]. We also refer to [43,44] which deals with the convective Cahn-Hilliard equation, and to [28,41], where some discretized versions of the general Cahn-Hilliard systems are studied.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Cahn-Hilliard systems have been rather investigated from the viewpoint of optimal control (see, e.g., [22]). We also refer to [37,38] which deals with the convective Cahn-Hilliard equation, and to [23,35], where some discretized versions of the general Cahn-Hilliard systems are studied.…”

Section: Introductionmentioning

confidence: 99%

On a Class of Conserved Phase Field Systems with a Maximal Monotone Perturbation

Colturato

2017

Appl Math Optim

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We prove existence and regularity for the solutions to a Cahn-Hilliard system describing the phenomenon of phase separation for a material contained in a bounded and regular domain. Since the first equation of the system is perturbed by the presence of an additional maximal monotone operator, we show our results using suitable regularization of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments. Next, under further assumptions, we deduce a continuous dependence estimate whence the uniqueness property is also achieved. Then, we consider the related sliding mode control (SMC) problem and show that the chosen SMC law forces a suitable linear combination of the temperature and the phase to reach a given (spacedependent) value within finite time.

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Optimal distributed control of nonlinear Cahn–Hilliard systems with computational realization

Cited by 7 publications

References 35 publications

On a Cahn–Hilliard system with convection and dynamic boundary conditions

On a Cahn–Hilliard system with convection and dynamic boundary conditions

A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions

On a Class of Conserved Phase Field Systems with a Maximal Monotone Perturbation

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