# Mean Field Linear Quadratic Control: FBSDE and Riccati Equation Approaches

## TL;DR

This paper develops a comprehensive framework for mean field linear quadratic control problems, deriving decentralized control laws via FBSDE and Riccati equations, and establishing their social optimality and Nash equilibrium properties.

## Contribution

It introduces a novel approach combining FBSDE and Riccati equations to design decentralized controls for mean field LQ control and game problems, linking open-loop and feedback solutions.

## Key findings

- Decentralized controls are asymptotically social optimal.
- Decentralized controls form an asymptotic Nash equilibrium.
- Proposed controls are equivalent to previous feedback strategies.

## Abstract

This paper studies social optima and Nash games for mean field linear quadratic control systems, where subsystems are coupled via dynamics and individual costs. For the social control problem, we first obtain a set of forward-backward stochastic differential equations (FBSDE) from variational analysis, and construct a feedback-type control by decoupling the FBSDE. By using solutions of two Riccati equations, we design a set of decentralized control laws, which is further proved to be asymptotically social optimal. Two equivalent conditions are given for uniform stabilization of the systems in different cases. For the game problem, we first design a set of decentralized control from variational analysis, and then show that such set of decentralized control constitute an asymptotic Nash equilibrium by exploiting the stabilizing solution of a nonsymmetric Riccati equation. It is verified that the proposed decentralized control laws are equivalent to the feedback strategies of mean field control in previous works. This may illustrate the relationship between open-loop and feedback solutions of mean field control (games).