Optimal distributed control of the viscous Cahn-Hilliard-Oono system with chemotaxis (Q6615612)

The system under study is a simplified, fluid-free version of the thermodynamically consistent diffuse interface model for a two-phase incompressible fluid mixture. This system includes chemotaxis, active transport, and nonlocal interaction. The control problem consists in approaching a given difference of volume fractions of the two components by minimizing a cost functional. To obtain optimality conditions for the control, first well-posedness of this system is established, and then differentiability of the control to state operator is studied. The analysis of the backward adjoint system leads to necessary optimality conditions.