Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential

In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the Navier--Stokes equations, nonlinearly coupled with a convective nonlocal Cahn--Hilliard equation. The system rules the evolution of the volume-averaged velocity $\uvec$ of the mixture and the (relative) concentration difference $\varphi$ of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the control-to-state map $\vvec \mapsto [\uvec,\varphi]$, and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with E.~Rocca in [19]. There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and C.\,G.~Gal in [14].