  Many problems in dynamic data driven modeling deals with distributed rather
than lumped observations. In this paper, we show that the Monge-Kantorovich
optimal transport theory provides a unifying framework to tackle such problems
in the systems-control parlance. Specifically, given distributional
measurements at arbitrary instances of measurement availability, we show how to
derive dynamical systems that interpolate the observed distributions along the
geodesics. We demonstrate the framework in the context of three specific
problems: (i) \emph{finding a feedback control} to track observed ensembles
over finite-horizon, (ii) \emph{finding a model} whose prediction matches the
observed distributional data, and (iii) \emph{refining a baseline model} that
results a distribution-level prediction-observation mismatch. We emphasize how
the three problems can be posed as variants of the optimal transport problem,
but lead to different types of numerical methods depending on the problem
context. Several examples are given to elucidate the ideas.
