Discrete-Time Linear-Quadratic Regulation via Optimal Transport

TL;DR

This paper links optimal transport with discrete-time stochastic control, providing a closed-form solution for linear-quadratic regulation with probabilistic initial and target states, and demonstrates its practical use through numerical examples.

Contribution

It introduces a novel connection between optimal transport and stochastic control, deriving a closed-form solution for linear-quadratic problems with probabilistic states.

Findings

Closed-form solution for optimal transport map in linear systems

Algorithm for computing the optimal transport map

Numerical examples demonstrating practical applicability

Abstract

In this paper, we consider a discrete-time stochastic control problem with uncertain initial and target states. We first discuss the connection between optimal transport and stochastic control problems of this form. Next, we formulate a linear-quadratic regulator problem where the initial and terminal states are distributed according to specified probability densities. A closed-form solution for the optimal transport map in the case of linear-time varying systems is derived, along with an algorithm for computing the optimal map. Two numerical examples pertaining to swarm deployment demonstrate the practical applicability of the model, and performance of the numerical method.