Control of Oscillator Networks with Mean-Field Measurement: A Hybrid Open/Closed-Loop Approach

Controlling large populations of limit-cycle oscillators with a broadcast input is a challenging problem with applications spanning from engineering to biology. The sheer scale of these networks restricts measurements at the population level, typically in the form of the population mean, thereby presenting further challenges for control design. In this work, we introduce a hybrid open/closed-loop approach to construct desired dynamic patterns in oscillator populations using only their mean measurement. Specifically, we design a minimum-power global signal that acts as an open-loop input for each oscillation period and is subsequently updated in the next oscillation cycle using the feedback from the Fourier coefficients of the population mean. The input parameters are determined by solving a constrained quadratic convex program, which is formulated utilizing the phase model description of oscillators. Our findings indicate that this approach is able to produce a variety of synchronization patterns and remains robust against Gaussian measurement noise, as it relies only on the population mean Fourier coefficients for control design. We validate the efficacy of our method through numerical simulations, demonstrating its capability to construct synchronized clusters within neuronal networks.