Mean Field Linear Quadratic Control: Uniform Stabilization and Social Optimality

TL;DR

This paper develops decentralized control strategies for mean field linear quadratic systems, ensuring uniform stabilization and social optimality without requiring positive definiteness of the state weight, applicable to both finite and infinite horizons.

Contribution

It introduces a novel approach using FBSDEs and Riccati equations to achieve decentralized control with stability and optimality guarantees in complex mean field systems.

Findings

Decentralized controls are asymptotically social optimal.

Conditions for uniform stabilization are established.

Control laws are compared favorably to previous strategies.

Abstract

This paper is concerned with uniform stabilization and social optimality for general mean field linear quadratic control systems, where subsystems are coupled via individual dynamics and costs, and the state weight is not assumed with the definiteness condition. For the finite-horizon problem, we first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback-type control by decoupling the FBSDEs. For the infinite-horizon problem, by using solutions to two Riccati equations, we design a set of decentralized control laws, which is further proved to be asymptotically social optimal. Some equivalent conditions are given for uniform stabilization of the systems in different cases, respectively. Finally, the proposed decentralized controls are compared to the asymptotic optimal strategies in previous works.