Sparse stabilization of mean-field agent dynamics through a three-operator splitting method

TL;DR

This paper introduces a three-operator splitting method for sparse control of mean-field multi-agent systems, enabling scalable and efficient consensus with minimal control effort through a particle-based discretization.

Contribution

It develops a novel three-operator splitting approach for nonsmooth mean-field optimal control problems with sparsity, demonstrated on the Cucker-Smale model.

Findings

Effective consensus achieved with sparse control actions

Scalable particle-based Monte Carlo discretization preserves mean-field structure

Method demonstrates robustness and efficiency in numerical experiments

Abstract

We study the sparse stabilization of nonlinear multi-agent systems within a mean-field optimal control framework. The goal is to drive large populations of interacting agents toward consensus with minimal control effort. In the mean-field limit, the dynamics are described by a Vlasov-type kinetic equation, and sparsity is enforced through an l1-l2 penalization in the cost functional. The resulting nonsmooth optimization problem is solved via a three-operator splitting (TOS) method that separately handles smooth, nonsmooth, and constraint components through gradient, shrinkage, and projection steps. A particle-based Monte Carlo discretization with random batch interactions enables scalable computation while preserving the mean-field structure. Numerical experiments on the Cucker-Smale model demonstrate effective consensus formation with sparse, localized control actions, confirming the efficiency and robustness of the proposed approach.