# Dynamic Programming in Probability Spaces via Optimal Transport

## TL;DR

This paper develops a framework for solving discrete-time optimal control problems in probability spaces by combining dynamic programming on the ground space with optimal transport, enabling decoupled multi-agent control strategies.

## Contribution

It introduces a novel approach that links dynamic programming in probability spaces with optimal transport, providing a separation principle for multi-agent control.

## Key findings

- Solution of dynamic programming in probability spaces via ground space and optimal transport
- Decoupling of low-level agent control and fleet-level control
- Applicable to multi-agent systems with probabilistic states

## Abstract

We study discrete-time finite-horizon optimal control problems in probability spaces, whereby the state of the system is a probability measure. We show that, in many instances, the solution of dynamic programming in probability spaces results from two ingredients: (i) the solution of dynamic programming in the "ground space" (i.e., the space on which the probability measures live) and (ii) the solution of an optimal transport problem. From a multi-agent control perspective, a separation principle holds: The "low-level control of the agents of the fleet" (how does one reach the destination?) and "fleet-level control" (who goes where?) are decoupled.