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

TL;DR

This paper studies an optimal boundary control problem for a 2D simplified Ericksen--Leslie system modeling nematic liquid crystal flows, establishing existence, differentiability, and optimality conditions for controls.

Contribution

It introduces the first analysis of optimal boundary controls for a simplified Ericksen--Leslie system, including existence and optimality conditions.

Findings

Existence of optimal boundary controls proven.

Control-to-state operator shown to be Fréchet differentiable.

First-order necessary optimality conditions derived.

Abstract

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 Fr\'echet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.