# Optimal Control of ensembles of dynamical systems

arXiv: https://arxiv.org/abs/2203.16485 · HTML: https://mlsanity.com/paper/2203.16485/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Mar 2022.

## Built on by (most cited first, 5 of 5)

- [Variational formulations of ODE-Net as a mean-field optimal control problem and existence results](https://mlsanity.com/paper/2303.05924/) (2023)
- [From NeurODEs to AutoencODEs: a mean-field control framework for width-varying Neural Networks](https://mlsanity.com/paper/2307.02279/) (2023)
- [Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs](https://mlsanity.com/paper/2311.01404/) (2023)
- [Optimal Control of Probabilistic Dynamics Models via Mean Hamiltonian Minimization](https://mlsanity.com/paper/2504.02543/) (2025)
- [State-space models through the lens of ensemble control](https://mlsanity.com/paper/2603.13587/) (2026)

## Abstract (arXiv)

In this paper we consider the problem of the optimal control of an ensemble of affine-control systems. After proving the well-posedness of the minimization problem under examination, we establish a Gamma-convergence result that allows us to substitute the original (and usually infinite) ensemble with a sequence of finite increasing-in-size sub-ensembles. The solutions of the optimal control problems involving these sub-ensembles provide approximations in the L2-strong topology of the minimizers of the original problem. Using again a Gamma-convergence argument, we manage to derive a Maximum Principle for ensemble optimal control problems with end-point cost. Moreover, in the case of finite sub-ensembles, we can address the minimization of the related cost through numerical schemes. In particular, we propose an algorithm that consists of a subspace projection of the gradient field induced on the space of admissible controls by the approximating cost functional. In addition, we consider an iterative method based on the Pontryagin Maximum Principle. Finally, we test the algorithms on an ensemble of linear systems in ℝ2.