Controlling collective motions of self-propelled particles by mean field couplings defined by topology

Weakly interacting particle systems have been important model systems for exploring the behavior of flocks of birds, of swarms of fish, and of human crowds and traffic patterns. Models from the microscopic level of the individual have focused on finding parameters that when time-evolved show stable states. Recent work on the D'Orsogna model showed that topological analysis can be used more effectively than order parameters to define the stable states that are determined by collective motions. Using ideas from Mean Field Games and Mean Field Controls we show that a coupling to the individual behaviors can be added to control collective states defined previously by analysis of trajectory data. This leads us to define controls for interacting systems where the macroscopic couplings are defined by topological descriptors. Our approach extends work on traffic flows that used one-dimensional models and density estimates to couple microscopic and macroscopic behavior. This coupling between microscopic and macroscopic behavior defined by topological analysis may be fruitful for control of continuous dynamical systems, often described by non-linear coupled differential equations, where predefined sets of finite states are often difficult to describe in advance. We thus suggest that controls defined in this manner can be a form of dual control, where collective motions are defined and controlled without a need to understand the entire event space from microscopic details.