  In this paper, we study the feedback synthesis problem for steering the joint
state density or ensemble subject to multi-input state feedback linearizable
dynamics. This problem is of interest to many practical applications including
that of dynamically shaping a robotic swarm. Our results here show that it is
possible to exploit the structural nonlinearities to derive the feedback
controllers steering the joint density from a prescribed shape to another while
minimizing the expected control effort to do so. The developments herein build
on our previous work, and extend the theory of the Schr\"{o}dinger bridge
problem subject to feedback linearizable dynamics.
