Social optimal control in mixed mean field models with control input constraints

In this paper, we study a class of mixed mean field social optimal control with one major player and many minor players. The control input for each player in the system is restricted to a given closed convex set. Firstly, we transform the cooperative mean field social optimal control 问题 into non-cooperative mean field games by the person-by-person optimality principle;secondly, using the Nash certainty equivalence principle, stochastic maximum principle and the law of conditional large numbers, we deduce the forward and backward stochastic differential equation for the estimation of the state average term of minor players; thirdly, we obtain the existence and uniqueness of the 解 of the forward-backward stochastic differential equation by compression mapping principle, and design a sequence of decentralized strategies based on individual local information for the system by the estimation of the state average term of minor players; finally, we prove that the estimation of the state average term is consistent with the true value for the systems, and the sequence of strategies designed is a decentralized asymptotic social optimal sequence.