Controllability and Tracking of Ensembles: An Optimal Transport Theory Viewpoint

TL;DR

This paper introduces a novel optimal transport-based framework for analyzing and controlling ensembles of systems, enabling efficient state tracking with limited observations and establishing new controllability concepts for nonlinear ensembles.

Contribution

It develops a new approach linking ensemble control to optimal transport theory, providing computational methods and theoretical insights for nonlinear and Gaussian output ensembles.

Findings

Reformulates ensemble control as a linear program using Kantorovich's optimal transport.

Establishes notions of controllability and observability for nonlinear ensembles.

Demonstrates the effectiveness of the approach through numerical examples.

Abstract

This paper explores the controllability and state tracking of ensembles from the perspective of optimal transport theory. Ensembles, characterized as collections of systems evolving under the same dynamics but with varying initial conditions, are a fundamental concept in control theory and applications. By leveraging optimal transport, we provide a novel framework for analyzing and solving the state tracking problem of ensembles, particularly when state observations are limited and only accessible at discrete time points. This study establishes connections between the ensemble dynamics and finite-horizon optimal control problems, demonstrating that the problem can be reformulated as a computationally efficient linear program using Kantorovich's formulation of optimal transport. We raise notions of observability and controllability for nonlinear ensembles, and propose methods for state tracking in Gaussian output distributions settings. Numerical examples and theoretical insights are provided to validate the approach, highlighting the utility of optimal transport in ensemble control problems.