This talk concerns a sixth-order Cahn–Hilliard system, a higher-order extension of the classical model that includes a source term where the control variable acts as a distributed mass regulator. The additional spatial derivatives account for curvature effects, enabling a more accurate description of isothermal phase separation in complex materials.
We discuss well-posedness under smooth double-well potentials and address the associated optimal control problem, proving existence of optimal controls and deriving first-order necessary conditions via an adjoint-based variational inequality. These results were obtained in collaboration with G. Gilardi (University of Pavia), A. Signori (Politecnico di Milano), and J. Sprekels (WIAS Berlin). Some extension to couplings with Brinkman flows will be possibly mentioned.