# Mean field control hierarchy

arXiv: https://arxiv.org/abs/1608.01728 · HTML: https://mlsanity.com/paper/1608.01728/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Aug 2016.

## Built on by (most cited first, 3 of 3)

- [Data/moment-driven approaches for fast predictive control of collective dynamics](https://mlsanity.com/paper/2402.15611/) (2024)
- [Optimal Control of Agent-Based Dynamics under Deep Galerkin Feedback Laws](https://mlsanity.com/paper/2406.09141/) (2024)
- [A Schr"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control](https://mlsanity.com/paper/2603.23173/) (2026)

## Abstract (arXiv)

In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrainted by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population. We show the existence of mean field optimal controls both in the stochastic and deterministic setting. We derive rigorously the first order optimality conditions useful for numerical computation of mean field optimal controls. We introduce a novel approximating hierarchy of sub-optimal controls based on a Boltzmann approach, whose computation requires a very moderate numerical complexity with respect to the one of the optimal control. We provide numerical experiments for models in opinion formation comparing the behavior of the control hierarchy.