We present an analysis and numerical study of an optimal control problem for the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a crucial component in modern technology. They exhibit long range orientational order in their nematic phase, which is represented by a tensor-valued (spatial) order parameter $Q = Q(x)$. Equilibrium LC states correspond to $Q$ functions that (locally) minimize an LdG energy functional. Thus, we consider an $L^2$-gradient flow of the LdG energy that allows for finding local minimizers and leads to a semi-linear parabolic PDE, for which we develop an optimal control framework. We then derive several a priori estimates for the forward problem, including continuity in space-time, that allow us to prove existence of optimal boundary and external ``force'' controls and to derive optimality conditions through the use of an adjoint equation. Next, we present a simple finite element scheme for the LdG model and a straightforward optimization algorithm. We illustrate optimization of LC states through numerical experiments in two and three dimensions that seek to place LC defects (where $Q(x) = 0$) in desired locations, which is desirable in applications.

翻译：本文针对向列液晶（LC）的Landau-de Gennes（LdG）模型的最优控制问题进行了分析与数值研究，该模型在现代技术中具有关键作用。液晶在其向列相中表现出长程取向序，该序由张量值（空间）序参数$Q = Q(x)$描述。平衡液晶态对应于（局部）最小化LdG能量泛函的$Q$函数。因此，我们考虑LdG能量的一种$L^2$梯度流，该梯度流可用于寻找局部极小值，并导出一个半线性抛物型偏微分方程，我们为其建立了最优控制框架。随后，我们推导了正问题的一些先验估计，包括时空连续性，这些估计使我们能够证明最优边界和外部“力”控制的存在性，并通过伴随方程推导最优性条件。接着，我们提出了LdG模型的一个简单有限元方案和一种直接的优化算法。通过二维和三维数值实验，我们展示了液晶态的优化过程，旨在将液晶缺陷（其中$Q(x) = 0$）置于期望位置，这在应用中具有需求。