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Recent results on well-posedness and optimal control for a class of generalized fractional Cahn–Hilliard systems

Colli Pierluigi, Gilardi Gianni, Sprekels Jurgen

Control and Cybernetics

2019

Vol. 48, No. 1

153--197

In this paper, we give an overview of results for Cahn–Hilliard systems involving fractional operators that have recently been established by the authors of this note. We address problems concerning existence, uniqueness, and regularity of the solutions to the system equations, and we study optimal control problems for the systems. The well-posedness results are valid for a wide class of fractional operators of spectral type and for the typical double-well nonlinearities appearing in the Cahn–Hilliard system equations, namely the classical diﬀerentiable, the logarithmic, and the nondiﬀerentiable double obstacle potentials. While this also applies to the existence of optimal controls in the related optimal control problems, the establishment of ﬁrst-order necessary optimality conditions requires imposing much stronger assumptions on the admissible class of fractional operators. One main reason for this is the necessity of deriving suitable diﬀerentiability properties for the associated control-to-state mapping. Nevertheless, it turns out that also in the singular case of logarithmic potentials, the ﬁrst-order necessary optimality conditions can be established under suitable assumptions, and a “deep quench” approximation, based on the results derived for logarithmic nonlinearities makes even the case of double obstacle potentials accessible.