Feedback control of flow alignment in sheared liquid crystals

Abstract

Based on a continuum theory, we investigate the manipulation of the non-equilibrium 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-τ). In this diagonal scheme, τis the delay time. We demonstrate that the TDFC method successfully stabilizes flow alignment for suitable values of the control strength, K, and τ; 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.

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