  We consider the problem of steering the joint state probability density
function of a static feedback linearizable control system over finite time
horizon. Potential applications include controlling neuronal populations, swarm
guidance, and probabilistic motion planning. Our theoretical developments
reveal the structure of the minimum energy controller for the same, and can be
viewed as a generalization of the Benamou-Brenier theory for dynamic optimal
transport. Further analytical results are derived for solving the feasibility
problem, i.e., for finding feedback that steers a given joint density function
to another in fixed time, subject to the controlled nonlinear dynamics. An
algorithm based on the Schr\"{o}dinger bridge is proposed to approximate a
feasible controller; a numerical example is worked out to illustrate the same.
