Linear-Quadratic Mean Field Control: The Hamiltonian Matrix and Invariant Subspace Method

TL;DR

This paper introduces a Hamiltonian matrix and invariant subspace method to analyze and solve linear quadratic mean field control and game problems, improving both theoretical understanding and computational efficiency.

Contribution

It develops a novel subspace decomposition approach based on Hamiltonian matrix structure for solving LQ mean field control and game problems.

Findings

Effective for existence and uniqueness analysis

Enables efficient numerical solutions

Extends to mean field games

Abstract

This paper studies the existence and uniqueness of a solution to linear quadratic (LQ) mean field social optimization problems with uniform agents. We exploit a Hamiltonian matrix structure of the associated ordinary differential equation (ODE) system and apply a subspace decomposition method to find the solution. This approach is effective for both the existence analysis and numerical computations. We further extend the decomposition method to LQ mean field games.