Optimal control for phase locking of synchronized oscillator populations via dynamical reduction techniques

We present a framework for controlling the collective phase of a system of coupled oscillators described by the Kuramoto model under the influence of a periodic external input by combining the methods of dynamical reduction and optimal control. We employ the Ott-Antonsen ansatz and phase-amplitude reduction theory to derive a pair of one-dimensional equations for the collective phase and amplitude of mutually synchronized oscillators. We then use optimal control theory to derive the optimal input for controlling the collective phase based on the phase equation and evaluate the effect of the control input on the degree of mutual synchrony using the amplitude equation. We setup an optimal control problem for the system to quickly resynchronize with the periodic input after a sudden phase shift in the periodic input, a situation similar to jet lag, and demonstrate the validity of the framework through numerical simulations.