Model-free method for LQ mean-field social control problems with one-dimensional state space

This paper presents a novel model-free method to solve linear quadratic (LQ) mean-field control problems with one-dimensional state space and multiplicative noise. The focus is on the infinite horizon LQ setting, where the conditions for solution either stabilization or optimization can be formulated as two algebraic Riccati equations (AREs). The proposed approach leverages the integral reinforcement learning technique to iteratively solve the drift-coefficient-dependent stochastic ARE (SARE) and other indefinite ARE, without requiring knowledge of the system dynamics. A numerical example is given to demonstrate the effectiveness of the proposed algorithm.