  We consider the problem of controlling the group behavior of a large number
of dynamic systems that are constantly interacting with each other. These
systems are assumed to have identical dynamics (e.g., birds flock, robot swarm)
and their group behavior can be modeled by a distribution. Thus, this problem
can be viewed as an optimal control problem over the space of distributions. We
propose a novel algorithm to compute a feedback control strategy so that, when
adopted by the agents, the distribution of them would be transformed from an
initial one to a target one over a finite time window. Our method is built on
optimal transport theory but differs significantly from existing work in this
area in that our method models the interactions among agents explicitly. From
an algorithmic point of view, our algorithm is based on a generalized version
of the proximal gradient descent algorithm and has a convergence guarantee with
a sublinear rate. We further extend our framework to account for the scenarios
where the agents are from multiple species. In the linear quadratic setting,
the solution is characterized by coupled Riccati equations which can be solved
in closed-form. Finally, several numerical examples are presented to illustrate
our framework.
