Optimal Control of the Landau-de Gennes Model of Nematic Liquid Crystals

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.