PreprintCC BY-NC-ND 4.0HAL
Optimal control of the Dyson equation
Abstract
We provide a complete study of the optimal control problem associated to the so-called Dyson equation, which describes the mean field evolution of the spectrum of large random matrices. We treat both the case of regular and singular terminal data. We construct admissible controls and prove the existence of optimal ones. We then proceed to give a thorough mathematical analysis of the associated Hamilton-Jacobi-Bellman equation by means of the theory of viscosity solutions in the space of measures. In particular, we characterize the value function of the optimal control problem as the unique viscosity solution to this infinite dimensional HJB equation.
Related
- Solution to a fully coupled McKean-Vlasov forward-backward stochastic difference equation and applications to optimal control with law-delay
- Time optimal control for the heat equation with inverse-square potentials
- Iterative Method for Stochastic Optimal Control Problem Constrained by Parabolic Partial Differential Equation
- On the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data
- Finiteness of null subvarieties and optimal destabilizers for the J-equation
BibTeX
@misc{bertucci2026optimal,
title = {{Optimal control of the Dyson equation}},
author = {Charles Bertucci and Pierre-Louis Lions and Panagiotis E. Souganidis},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06460},
eprint = {2610.06460},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05775958v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Optimal control of the Dyson equation
AU - Charles Bertucci
AU - Pierre-Louis Lions
AU - Panagiotis E. Souganidis
PY - 2026
DA - 2026/10/02
PB - HAL
DO - 10.48550/arxiv.2610.06460
UR - https://hal.science/hal-05775958v1
AB - We provide a complete study of the optimal control problem associated to the so-called Dyson equation, which describes the mean field evolution of the spectrum of large random matrices. We treat both the case of regular and singular terminal data. We construct admissible controls and prove the existence of optimal ones. We then proceed to give a thorough mathematical analysis of the associated Hamilton-Jacobi-Bellman equation by means of the theory of viscosity solutions in the space of measures. In particular, we characterize the value function of the optimal control problem as the unique viscosity solution to this infinite dimensional HJB equation.
ER -