Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials

TL;DR

This paper extends the analysis of Cahn-Hilliard systems with fractional operators and double obstacle potentials, focusing on optimal control problems and introducing the deep quench approximation to handle nondifferentiable nonlinearities.

Contribution

It develops a deep quench approximation method for optimal control of Cahn-Hilliard systems with non-differentiable double obstacle potentials, providing convergence analysis and optimality conditions.

Findings

Established convergence of the deep quench approximation.

Derived first-order optimality conditions using variational inequalities.

Demonstrated the method's effectiveness for nondifferentiable potentials.

Abstract

The paper arXiv:1804.11290 contains well-posedness and regularity results for a system of evolutionary operator equations having the structure of a Cahn-Hilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated double-well potentials driving the phase separation process modeled by the Cahn-Hilliard system could be of a very general type that includes standard physically meaningful cases such as polynomial, logarithmic, and double obstacle nonlinearities. In the subsequent paper arXiv:1807.03218, an analysis of distributed optimal control problems was performed for such evolutionary systems, where only the differentiable case of certain polynomial and logarithmic double-well potentials could be admitted. Results concerning existence of optimizers and first-order necessary optimality conditions were derived. In the present paper, we complement these results by studying a distributed control problem for such evolutionary systems in the case of nondifferentiable nonlinearities of double obstacle type. For such nonlinearities, it is well known that the standard constraint qualifications cannot be applied to construct appropriate Lagrange multipliers. To overcome this difficulty, we follow here the so-called "deep quench" method. We first give a general convergence analysis of the deep quench approximation that includes an error estimate and then demonstrate that its use leads in the double obstacle case to appropriate first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint state system.

Full text

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Deep quench approximation and optimal control

of general Cahn–Hilliard systems with fractional

operators and double obstacle potentials

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centerPierluigi Colli*(1)*

e-mail: pierluigi.colli@unipv.it

Gianni Gilardi*(1)*

e-mail: gianni.gilardi@unipv.it

Jürgen Sprekels*(2)*

e-mail: sprekels@wias-berlin.de

(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) Department of Mathematics

Humboldt-Universität zu Berlin

Unter den Linden 6, 10099 Berlin, Germany

and

Weierstrass Institute for Applied Analysis and Stochastics

Mohrenstrasse 39, 10117 Berlin, Germany