Feedback control drives synchronization of self-propelled particles moving in unbounded space

The fascinating spatio-temporal patterns emerging from collective motion constitute one of nature’s most prominent features. Typically, self-propelled particles in Vicsek model are confined within periodic or reflective boundary conditions. When these limitations are broken, it becomes challenging for them to self-organize into a cohesively moving swarm. Inspired by the feedback mechanism adopted in the classic control domain, in this work, we explore the effect of the proportional–integral–derivative (PID) control strategy in the collective dynamics of the particles moving in open space. The results reveal various collective motion patterns elicited by the control parameters, featuring the dot-like, ring-, arc- and umbrella-shaped configurations. The system demonstrates a complete synchronized, clustering or disorderly state under the different adjustments of the control strategy. Despite experiencing time delay and disturbance, the PID regulation effectively suppresses particle diffusion and fosters compact clustering, showing robustness in complex and noisy environments. Particularly, the system perturbed from one ordered state to another disorderly one is also analyzed, finding that the PID model exhibits a superior dynamic response capability relative to the original Vicsek model during transient processes.