Optimal boundary control of the Bulk-surface Coupled Cahn-Hilliard-diffusion system for lipid raft formation

This paper investigates the optimal boundary control problem for the bulk-surface coupled Cahn-Hilliard-diffusion system, modeling lipid-phase separation, lipid-cholesterol interactions and cytosol-membrane cholesterol exchange on cell membranes. The system consists of a bulk diffusion equation for cytosolic cholesterol relative concentration, a surface Cahn-Hilliard equation for relative concentration of saturated lipids and a surface diffusion-reaction equation for the relative concentration of membrane-bound cholesterol on the membrane. In cancer therapy, the boundary control $ h $ can be interpreted as medications, while the standard tracking type cost function aims to regulate saturated lipids levels in the membrane, thereby restoring membrane endocytic function, promoting tumor cell apoptosis and inhibiting tumor growth. First, we address the strong well-posedness of the state system. Subsequently, the existence of optimal controls is established. Thereafter, the Fréchet differentiability of the control-to-state operator $ \mathcal S $ is characterized. Finally, the first-order necessary optimality conditions are derived by formulating the backward adjoint system.