Risk-Sensitive Optimal Control And Mean Field Games With Major And Minor Players

Due to their wide background in applications ranging from engineering,economics and biology,mean field games have become hot topics in the operational research and cybernetics.Using the stochastic maximum principle,Nash certainly equivalence principle,and the law of conditional large numbers,and combining the method of state aggregation,this thesis studies the problems of optimal controls with quadratic-tracking risk-sensitive cost functionals,mixed mean field games with risk sensitivity cost functionals,and mixed mean field social optimal control with control input constraints.In detail,the contents include the following aspects:1.The optimal control problems with quadratic-tracking type risk-sensitive cost functionals are considered.For this kind of problems,firstly,a new verification theorem that does not need to assume that the admissible control laws are Markov strategies is proposed.Then the optimal control strategy is designed for the system by the new verification theorem.2.A class of non-cooperative mean field games containing a large number of major and minor players is considered.Each player minimizes a quadratic-tracking type risk-sensitive cost functional,where the reference signal is a function of the state average term of the major and minor players.To reduce the complexity of solving the problem,we design a sequence of decentralized strategies by the Nash certainty equivalence principle and the new verification theorem obtained in the first part.Firstly,we apply the two-layer state aggregation method and the new verification theorem to construct the fixed-point equations for the estimations of the state average terms of major and minor players,give the conditions for the existence and uniqueness of the fixed points,and design a sequence of decentralized strategies by the estimations of the state average terms based on local information.Secondly,it is proved that the estimations of the state average terms are consistent with the true values for the closed-loop systems,and the sequence of strategies designed is a decentralized asymptotic Nash equilibrium.3.A class of social optimal control containing one major player and a large number of minor players is considered.The admissible control set of each player is a closed convex set,and the diffusion terms contain state and control variables.For the system,firstly,we transform the cooperative mean field social optimal control problem into a series of non-cooperative optimal control problems by applying the person-by-person optimality principle in team decision theory;Secondly,using Nash certainly equivalence principle,the stochastic maximum principle and the law of conditional large numbers,the forward and backward stochastic differential equation for the estimation of the state average term of minor players is constructed;Thirdly,the existence and uniqueness of the solution of the forward-backward stochastic differential equation is obtained by the compression mapping principle,and a sequence of decentralized strategies is designed for the system by the estimation of the state average for minor players;Finally,it is proved 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.