  We consider a controlled-diffusion process pertaining to a chain of
distributed systems with random perturbations that satisfies a weak H\"ormander
type condition. In particular, we consider a stochastic control problem with
the following objectives that we would like to achieve. The first one being of
a reachability-type that consists of determining a set of attainable
distributions at a given time starting from an initial distribution, while the
second one involves minimizing the relative entropy subject to the initial and
desired final attainable distributions. Using the logarithmic transformations
approach from Fleming, we provide a sufficient condition on the existence of an
optimal admissible control for such a stochastic control problem which is
amounted to changing the drift by a certain perturbation suggested by Jamison
in the context of reciprocal processes. Moreover, such a perturbation coincides
with a minimum energy control among all admissible controls forcing the
controlled-diffusion process to the desired final attainable distribution
starting from the initial distribution. Finally, we briefly remark on the
invariance property of the path-space measure for such a controlled-diffusion
process pertaining to the chain of distributed systems.
