This note studies linear quadratic (LQ)mean field social optimization problems with non-uniform agents and indefinite state weight, and analyzes the resulting linear system of ordinary differential equations (ODEs)subject to a partial initial condition and a growth condition. We apply a subspace decomposition method for existence analysis and computation. This is accomplished by carefully choosing the unspecified initial condition in the ODE system such that the modes exceeding the admissible growth rate are not activated. We further extend this method to LQ mean field games.

Algebraic Riccati equation, Hamiltonian matrix, Linear quadratic control, Mean field game, Social optimization
School of Mathematics and Statistics

Chen, X. (Xiang), & Huang, M. (2019). Linear-quadratic mean field control: The invariant subspace method. Automatica. doi:10.1016/j.automatica.2019.04.021