# Mean-field optimal control and optimality conditions in the space of probability measures

## TL;DR

This paper develops a calculus on the space of probability measures to derive optimality conditions for mean-field control problems, connecting particle systems, PDEs, and numerical methods.

## Contribution

It introduces a new calculus on probability measures for optimal control, linking particle-based and mean-field formulations, and provides convergence rates for controls as particle number increases.

## Key findings

- Derived first-order optimality system on probability measures
- Established relations between particle and mean-field optimal controls
- Proved convergence rate of particle controls to mean-field controls

## Abstract

We derive a framework to compute optimal controls for problems with states in the space of probability measures. Since many optimal control problems constrained by a system of ordinary differential equations (ODE) modelling interacting particles converge to optimal control problems constrained by a partial differential equation (PDE) in the mean-field limit, it is interesting to have a calculus directly on the mesoscopic level of probability measures which allows us to derive the corresponding first-order optimality system. In addition to this new calculus, we provide relations for the resulting system to the first-order optimality system derived on the particle level, and the first-order optimality system based on $L^2$-calculus under additional regularity assumptions. We further justify the use of the $L^2$-adjoint in numerical simulations by establishing a link between the adjoint in the space of probability measures and the adjoint corresponding to $L^2$-calculus. Moreover, we prove a convergence rate for the convergence of the optimal controls corresponding to the particle formulation to the optimal controls of the mean-field problem as the number of particles tends to infinity.