Discrete-time decentralized control using the risk-sensitive performance criterion in the large population regime: A mean field approach

This paper considers a discrete-time decentralized control problem using the risk-sensitive cost function when there is a large number of agents. We solve this problem via mean field control theory. We first obtain an individual robust decentralized controller that is a function of the local state information and a bias term that is related to the mean field term. We then construct an auxiliary system that characterizes the best approximation to the mean field term in the mean-square sense when the number of agents, say N, goes to infinity. We prove that the set of individual decentralized controllers is an ε-Nash equilibrium, where ε can be made arbitrarily close to zero when N → ∞. Finally, we show that in view of the relationship with risk-sensitive, H^∞, and LQG control, the equilibrium features robustness, and converges to that of the LQG mean field game when the risk-sensitivity parameter goes to infinity.