Controlling defects in the Landau--de Gennes model of nematic liquid crystals

This talk presents an optimal control framework for the time-dependent, Landau--de Gennes (LdG) model of nematic liquid crystals.  We develop parabolic, optimal control techniques for controlling the L^2 gradient flow of the LdG energy.  The controls are through the boundary conditions (by weak anchoring) and a body force term.  We seek to find optimal controls that drive the LdG Q-tensor variable toward a desired "texture" state, including defect positions.  The objective functional we minimize is of tracking type with additional regularization terms for the controls.  To the best of our knowledge, this is the first time PDE-based optimal control has been developed for the LdG model. 

Existence of a minimizer for the control problem is established.  Moreover, with various regularity estimates, we prove first order Freche't differentiability results for the control objective.  This enables gradient based optimization methods through an adjoint PDE.  In the talk, we highlight the analytical issues that arise, especially those due to the gradient flow being a parabolic *system*.  We then describe a finite element discretization of the full control problem and present numerical simulations in two and three dimensions demonstrating that point and line defect positions can be controlled.

This is joint work with Thomas M. Surowiec ( thomasms@simula.no, Simula Research Laboratory, Oslo, Norway)