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

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.