Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density (Q2884601)

The authors study the distributed optimal control for the Cahn-Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the double-obstacle potential. The authors show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its Moreau-Yosida approximation. Corresponding first-order optimality conditions for the mollified problems are given. For this purpose a new result on the continuous Fréchet differentiability of superposition operators with values in Sobolev spaces is established. Besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem, the authors also derive first-order optimality conditions for the original problem by a limit process. The newly derived stationarity system corresponds to a function space version of \(C\)-stationarity.