  Motivated from a wide range of applications, various methods to control
synchronization in coupled oscillators have been proposed. Previous studies
have demonstrated that global feedback typically induces three macroscopic
behaviors: synchronization, desynchronization, and oscillation quenching.
However, analyzing all of these transitions within a single theoretical
framework is difficult, and thus the feedback effect is only partially
understood in each framework. Herein, we analyze a model of globally coupled
phase oscillators exposed to global feedback, which shows all of the typical
macroscopic dynamical states. Analytical tractability of the model enables us
to obtain detailed phase diagrams where transitions and bistabilities between
different macroscopic states are identified. Additionally, we propose
strategies to steer the oscillators into targeted states with minimal feedback
strength. Our study provides a useful overview of the effect of global feedback
and is expected to serve as a benchmark when more sophisticated feedback needs
to be designed.
