Optimal Transport in Systems and Control

Abstract

Optimal transport began as the problem of how to efficiently redistribute goods between production and consumers and evolved into a far-reaching geometric variational framework for studying flows of distributions on metric spaces. This theory enables a class of stochastic control problems to regulate dynamical systems so as to limit uncertainty to within specified limits. Representative control examples include the landing of a spacecraft aimed probabilistically toward a target and the suppression of undesirable effects of thermal noise on resonators; in both of these examples, the goal is to regulate the flow of the distribution of the random state. A most unlikely link turned up between transport of probability distributions and a maximum entropy inference problem posed by Erwin Schrödinger, 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 this review, which aims to highlight connections, insights, and computational tools while touching on quadratic regulator theory and probabilistic flows in discrete spaces and networks. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 4 is May 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

The most likely evolution of diffusing and vanishing particles: Schrodinger Bridges with unbalanced marginals

“…Next we briefly discuss the solution to SBP. For an in depth exposition see [25] and the review articles [36], [27].…”

Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure

“…More recently however Schödinger bridges and entropy-regularised OT are being studied for their own sake, finding applications in control, computational statistics and machine learning, see e.g. Bernton et al (2019); Chen et al (2021); Corenflos et al (2021); De Bortoli et al (2021); Huang et al (2021); Vargas et al (2021). In these applications, the entropy regularisation may be a desirable feature rather than an approximation, and the main source of error is the fact that the marginal distributions are typically intractable and often approximated by empirical versions.…”

Covariance Steering for Nonlinear Control-affine Systems

“…1 2 ut 2 that only depends on the control and V (Xt) that only depends on the state. The problem (10) has been investigated in the study of distribution control [26], [27] for general distributions ρ0, ρ1, which links control theory and optimal transport theory [26], [28].…”

“…For instance, one can set P u 0 to be the process dXt = √ ǫB(t)dWt, which means A0(t) ≡ 0 and a0(t) ≡ 0. A different option is a linearization of the prior process (28). More precisely, set z 0 t to be the solution to…”