  We consider the problem of the transport of a density of states from an
initial state distribution to a desired final state distribution through a
dynamical system with actuation. In particular, we consider the case where the
control signal is a function of time, but not space; that is, the same
actuation is applied at every point in the state space. This is motivated by
several problems in fluid mechanics, such as mixing and manipulation of a
collection of particles by a global control input such as a uniform magnetic
field, as well as by more general control problems where a density function
describes an uncertainty distribution or a distribution of agents in a
multi-agent system. We formulate this problem using the generators of the
Perron-Frobenius operator associated with the drift and control vector fields
of the system. By considering finite-dimensional approximations of these
operators, the density transport problem can be expressed as a control problem
for a bilinear system in a high-dimensional, lifted state. With this system, we
frame the density control problem as a problem of driving moments of the
density function to the moments of a desired density function, where the
moments of the density can be expressed as an output which is linear in the
lifted state. This output tracking problem for the lifted bilinear system is
then solved using differential dynamic programming, an iterative trajectory
optimization scheme.
