  We study discrete-time finite-horizon optimal control problems in probability
spaces, whereby the state of the system is a probability measure. We show that,
in many instances, the solution of dynamic programming in probability spaces
results from two ingredients: (i) the solution of dynamic programming in the
"ground space" (i.e., the space on which the probability measures live) and
(ii) the solution of an optimal transport problem. From a multi-agent control
perspective, a separation principle holds: The "low-level control of the agents
of the fleet" (how does one reach the destination?) and "fleet-level control"
(who goes where?) are decoupled.
