Optimal distributed control of the viscous Cahn-Hilliard-Oono system with chemotaxis.

In this paper, we investigated a distributed optimal control problem for the viscous Cahn-Hilliard-Oono system. In this system, chemotaxis, active transport, and nonlocal interaction of Oono's type were also taken into account. The system consisted of a viscous Cahn-Hilliard equation for the difference in volume fractions of the two components $ \varphi $ and a diffusion-reaction equation for the nutrient concentration $ \sigma $. Our aim was to achieve a desired goal of the difference in volume fractions of the two components by minimizing the cost functional of standard tracking type. To begin, the strong well-posedness of the state system was addressed. Then, we establish the existence of optimal controls. Afterward, we presented the Fréchet differentiability of the control-to-state operator $ \mathcal S $. Lastly, we derived the first-order necessary optimality conditions by considering the backward adjoint system. [ABSTRACT FROM AUTHOR]