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In this paper, we investigate an optimal boundary control problem for a twodimensional simplified Ericksen--Leslie system modelling the incompressiblenematic liquid crystal flows. The hydrodynamic system consists of theNavier--Stokes equations for the fluid velocity coupled with a convectiveGinzburg--Landau type equation for the averaged molecular orientation. Thefluid velocity is assumed to satisfy a no-slip boundary condition, while themolecular orientation is subject to a time-dependent Dirichlet boundarycondition that corresponds to the strong anchoring condition for liquidcrystals. We first establish the existence of optimal boundary controls. Thenwe show that the control-to-state operator is Fr\'echet differentiable betweenappropriate Banach spaces and derive first-order necessary optimalityconditions in terms of a variational inequality involving the adjoint statevariables.

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