Optimal boundary control of a simplified Ericksen-Leslie system for nematic liquid crystal flows in 2D

In this paper, we investigate an optimal boundary control problem for a two dimensional simplified Ericksen-Leslie system modelling the incompressible nematic liquid crystal flows. The hydrodynamic system consists of the Navier-Stokes equations for the fluid velocity coupled with a convective Ginzburg-Landau type equation for the averaged molecular orientation. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the molecular orientation is subject to a time-dependent Dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals. We first establish the existence of optimal boundary controls. Then we show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.