Variance Optimization and Control Regularity for Mean-Field Dynamics

TL;DR

This paper investigates optimal control problems balancing control effort and variance in multi-agent systems, revealing conditions for the existence and regularity of solutions in both finite and infinite agent limits.

Contribution

It provides a detailed analysis of the existence and regularity of optimal controls in mean-field dynamics, highlighting the impact of time horizon and penalization parameters.

Findings

Explicit solutions for finite-agent problems

Existence of Lipschitz controls for certain conditions

Non-existence of regular controls for large time horizons

Abstract

We study a family of optimal control problems in which one aims at minimizing a cost that mixes a quadratic control penalization and the variance of the system, both for finitely many agents and for the mean-field dynamics as their number goes to infinity. While solutions of the discrete problem always exist in a unique and explicit form, the behavior of their macroscopic counterparts is very sensitive to the magnitude of the time horizon and penalization parameter. When one minimizes the final variance, there always exists a Lipschitz-in-space optimal controls for the infinite dimensional problem, which can be obtained as a suitable extension of the optimal controls for the finite-dimensional problems. The same holds true for variance maximizations whenever the time horizon is sufficiently small. On the contrary, for large final times (or equivalently for small penalizations of the control cost), it can be proven that there does not exist Lipschitz-regular optimal controls for the macroscopic problem.