Steering the Distribution of Agents in Mean-Field Games

In this work we pose and solve the problem to guide a collection of weakly interacting dynamical systems e.g., agents, to a specified target distribution. The problem is formulated using the mean-field game theory where each agent seeks to minimize its own cost. The underlying dynamics is assumed to be linear and the cost is assumed to be quadratic. In our framework a terminal cost is added as an incentive term to accomplish the task; we establish that the map between terminal costs and terminal probability distributions is onto. By adding a proper terminal cost/incentive, the agents will reach any desired terminal distribution providing they are adopting the Nash equilibrium strategy. A similar problem is considered in the cooperative game setting where the agents work together to minimize a total cost. Our approach relies on and extends the theory of optimal mass transport and its generalizations.