Optimal collision avoidance in swarms of active Brownian particles

The effectiveness of collective navigation of biological or artificial agents requires to accommodate for contrasting requirements, such as staying in a group while avoiding close encounters and at the same time limiting the energy expenditure for maneuvering. Here, we address this problem by considering a system of active Brownian particles in a finite two-dimensional domain and ask what is the control that realizes the optimal tradeoff between collision avoidance and control expenditure. We couch this problem in the language of optimal stochastic control theory and by means of a mean-field game approach we derive an analytic mean-field solution, characterized by a second-order phase transition in the alignment order parameter. We find that a mean-field version of a classical model for collective motion based on alignment interactions (Vicsek model) performs remarkably close to the optimal control. Our results substantiate the view that observed group behaviors may be explained as the result of optimizing multiple objectives and offer a theoretical ground for biomimetic algorithms used for artificial agents.