Feedback control of flow alignment in sheared liquid crystals

Based on a continuum theory, we investigate the manipulation of the nonequilibrium behavior of a sheared liquid crystal via closed-loop feedback control. Our goal is to stabilize a specific dynamical state, that is, the stationary flow alignment, under conditions where the uncontrolled system displays oscillatory director dynamics with in-plane symmetry. To this end we employ time-delayed feedback control (TDFC), where the equation of motion for the ith component q(i) (t) of the order parameter tensor is supplemented by a control term involving the difference q(i) (t) -q(i) (t - tau). In this diagonal scheme, tau is the delay time. We demonstrate that the TDFC method successfully stabilizes flow alignment for suitable values of the control strength K and tau; these values are determined by solving an exact eigenvalue equation. Moreover, our results show that only small values of K are needed when the system is sheared from an isotropic equilibrium state, contrary to the case where the equilibrium state is nematic.