Optimal Transport in Systems and Control

Optimal transport began as the problem to efficiently redistribute goods between production and consumers, and evolved into a far reaching geo- metric variational framework for studying flows of distributions on metric spaces. This theory interests in enabling ways a class of stochastic control problems, to regulate dynamical systems so as to limit uncertainty to within specifications. Representative control examples include the landing of a spacecraft aimed probabilistically towards a target, and the suppression of undesirable effects of thermal noise on resonators; in either, the goal is to regulate the flow of the distribution of the random state. Thence, a most unlikely link turned up between transport of probability distributions and a “maximum entropy” inference problem of E. Schroedinger, where the latter is seen as an entropy-regularized version of the former. These intertwined topics, of optimal transport, stochastic control, and inference, are the subject of the current review; it aims to highlight connections, insights, and computational tools, while at the same time making contact with quadratic regulator theory and probabilistic flows on discrete spaces/networks.