Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials. (English) Zbl 1458.35244 Discrete Contin. Dyn. Syst., Ser. S 14, No. 1, 243-271 (2021). Summary: Recently, the authors derived well-posedness and regularity results for general evolutionary operator equations having the structure of a Cahn-Hilliard system. The involved operators were fractional versions in the spectral sense of general linear operators that have compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions. The class of admissible double-well potentials driving the phase separation process modeled by the Cahn-Hilliard system included polynomial, logarithmic, and double obstacle nonlinearities. In a subsequent paper, distributed optimal control problems for such systems were investigated, where only differentiable polynomial and logarithmic potentials were admitted. Existence of optimizers and first-order optimality conditions were derived. In this paper, these results are complemented for nondifferentiable double obstacle nonlinearities. It is well known that for such nonlinearities standard constraint qualifications to construct Lagrange multipliers cannot be applied. To overcome this difficulty, we follow the so-called “deep quench” method, which has proved to be a powerful tool in optimal control problems with double obstacle potentials. We give a general convergence analysis of the deep quench approximation, including an error estimate, and demonstrate that its use leads to meaningful first-order necessary optimality conditions. Cited in 4 Documents MSC: 35K90 Abstract parabolic equations 35K45 Initial value problems for second-order parabolic systems 49K20 Optimality conditions for problems involving partial differential equations 49K27 Optimality conditions for problems in abstract spaces 35R11 Fractional partial differential equations Keywords:fractional operators; optimal control; double obstacles; necessary optimality conditions PDFBibTeX XMLCite \textit{P. Colli} et al., Discrete Contin. Dyn. Syst., Ser. S 14, No. 1, 243--271 (2021; Zbl 1458.35244) Full Text: DOI arXiv References: [1] M. Ainsworth; Z. Mao, Analysis and approximation of a fractional Cahn-Hilliard equation, SIAM J. Numer. Anal., 55, 1689-1718 (2017) · Zbl 1369.65124 [2] G. Akagi; G. Schimperna; A. Segatti, Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations, J. Differential Equations, 261, 2935-2985 (2016) · Zbl 1342.35429 [3] H. Antil, R. Khatri and M. 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