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

arXiv: https://arxiv.org/abs/1902.05339 · HTML: https://mlsanity.com/paper/1902.05339/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2019.

## Built on by (most cited first, 2 of 2)

- [Global Variational Inference Enhanced Robust Domain Adaptation](https://mlsanity.com/paper/2507.03291/) (2025)
- [Taming Epilepsy: Mean Field Control of Whole-Brain Dynamics](https://mlsanity.com/paper/2603.18035/) (2026)

## Abstract (arXiv)

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 L2-calculus under additional regularity assumptions. We further justify the use of the L2-adjoint in numerical simulations by establishing a link between the adjoint in the space of probability measures and the adjoint corresponding to L2-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.