@article{17310,
  author = {Thomas Surowiec and Shawn Walker},
  title = {Optimal control of the Landau-de Gennes model of nematic liquid crystals},
  abstract = {We present an analysis and numerical study of an optimal control problem for the Landau{\textendash}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 semilinear 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 {\textquotedblleft}force{\textquotedblright} 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.},
  year = {2023},
  journal = {SIAM Journal on Control and Optimization},
  volume = {61},
  number = {4},
  pages = {2546-2570},
  publisher = {Society for Industrial and Applied Mathematics},
  url = {https://doi.org/10.1137/22M1506158},
}