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The paper arXiv:1804.11290 contains well-posedness and regularity results fora system of evolutionary operator equations having the structure of aCahn-Hilliard system. The operators appearing in the system equations werefractional versions in the spectral sense of general linear operators A and Bhaving compact resolvents and are densely defined, unbounded, selfadjoint, andmonotone in a Hilbert space of functions defined in a smooth domain. Theassociated double-well potentials driving the phase separation process modeledby the Cahn-Hilliard system could be of a very general type that includesstandard physically meaningful cases such as polynomial, logarithmic, anddouble obstacle nonlinearities. In the subsequent paper arXiv:1807.03218, ananalysis of distributed optimal control problems was performed for suchevolutionary systems, where only the differentiable case of certain polynomialand logarithmic double-well potentials could be admitted. Results concerningexistence of optimizers and first-order necessary optimality conditions werederived. In the present paper, we complement these results by studying adistributed control problem for such evolutionary systems in the case ofnondifferentiable nonlinearities of double obstacle type. For suchnonlinearities, it is well known that the standard constraint qualificationscannot be applied to construct appropriate Lagrange multipliers. To overcomethis difficulty, we follow here the so-called "deep quench" method. We firstgive a general convergence analysis of the deep quench approximation thatincludes an error estimate and then demonstrate that its use leads in thedouble obstacle case to appropriate first-order necessary optimality conditionsin terms of a variational inequality and the associated adjoint state system.

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