Indefinite Linear Quadratic Mean Field Social Control Problems with Multiplicative Noise

TL;DR

This paper develops decentralized control laws for indefinite linear quadratic mean field control problems with multiplicative noise, ensuring stability and social optimality through Riccati equations and FBSDEs.

Contribution

It introduces a novel approach to indefinite LQ mean field control with multiplicative noise, including new stabilization conditions and decentralized control design.

Findings

Decentralized control laws achieve asymptotic social optimality.

Stability conditions are characterized by linear matrix inequalities.

Numerical example demonstrates effectiveness of the proposed methods.

Abstract

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.