# Indefinite Linear Quadratic Mean Field Social Control Problems with Multiplicative Noise

arXiv: https://arxiv.org/abs/2009.09864 · HTML: https://mlsanity.com/paper/2009.09864/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2020.

## Built on by (most cited first, 3 of 3)

- [Mean Field LQG Social Optimization: A Reinforcement Learning Approach](https://mlsanity.com/paper/2410.15119/) (2024)
- [Robust Mean Field Social Control: A Unified Reinforcement Learning Framework](https://mlsanity.com/paper/2502.20029/) (2025)
- [Inverse reinforcement learning for indefinite mean-field social optimization with multiplicative noise](https://mlsanity.com/paper/2608.07252/) (2026)

## Abstract (arXiv)

This paper studies uniform stabilization and social optimality for linear quadratic (LQ) mean field control problems with multiplicative noise, where agents are coupled via dynamics and individual costs. The state and control weights in cost functionals are not limited to be positive semi-definite. This leads to an indefinite LQ mean field control problem, which may still be well-posed due to deep nature of multiplicative noise. We first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback control by decoupling the FBSDEs. By using solutions to two Riccati equations, we design a set of decentralized control laws, which is further shown to be asymptotically social optimal. Some equivalent conditions are given for uniform stabilization of the systems with the help of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed control laws.