Inverse Problems for Costs and Controls in LQG MFGs via Mean Field Trajectories

Abstract

This paper investigates inverse problems for Linear-Quadratic-Gaussian (LQG) Mean Field Games (MFGs) based entirely on the observation of mean-covariance trajectories. We address three sequential challenges: identifying the optimal control for observed initializations, determining the control for arbitrary initializations, and recovering consistent cost parameters. After establishing the existence and uniqueness of the forward Nash equilibrium under mild hypotheses, we analyze the injectivity of the parameter-to-trajectory mapping, demonstrating that it is inherently non-injective and providing sufficient conditions for parameter equivalence. We prove that while the optimal control is locally identifiable for observed initializations under minimal assumptions, global identifiability requires a deeper structural recovery of the game's costs. To bridge this gap, we propose a constructive semidefinite programming method to infer cost parameters that are strictly consistent with the observed population dynamics. Numerical experiments illustrate this method.