  This paper considers the relaxed version of the transport problem for general
nonlinear control systems, where the objective is to design time-varying
feedback laws that transport a given initial probability measure to a target
probability measure under the action of the closed-loop system. To make the
problem analytically tractable, we consider control laws that are stochastic,
i.e., the control laws are maps from the state space of the control system to
the space of probability measures on the set of admissible control inputs.
Under some controllability assumptions on the control system as defined on the
state space, we show that the transport problem, considered as a
controllability problem for the lifted control system on the space of
probability measures, is well-posed for a large class of initial and target
measures. We use this to prove the well-posedness of a fixed-endpoint optimal
control problem defined on the space of probability measures, where along with
the terminal constraints, the goal is to optimize an objective functional along
the trajectory of the control system. This optimization problem can be posed as
an infinite-dimensional linear programming problem. This formulation
facilitates numerical solutions of the transport problem for low-dimensional
control systems, as we show in two numerical examples.
