Stochastic Maximum Principle For Time-delayed Mean-field Doubly Stochastic Control Systems

As a main branch of modern control theory,optimal control theory focuses on the conditions and methods to optimize the performance indicators of the control system.With the development of stochastic differential equations and its related theories,it is found that using stochastic differential equation to describe the control system can better explain some practical phenomena.Therefore,a control system described by a stochastic differential equation is called stochastic control system,and the theory which is used to optimize the performance of stochastic control systems is called stochastic optimal control theory.Subsequently,the theory of stochastic optimal control was developed rapidly,and obtained rich research results.On the basis of the previous,the stochastic maximum principle of the time-delayed meanfield doubly stochastic control system is proposed in this paper.The stochastic maximum principle is a set of necessary conditions that must be satisfied by any optimal solution which the state equation of the control system is described by time-delayed mean-field doubly stochastic differential equations.To begin with,the time-delayed mean-field doubly stochastic differential equation and anticipated mean-field backward doubly stochastic differential equation are introduced,under the assumption of Lipschitz condition,the existence and uniqueness of solutions for these two equations by using the martingale representation theorem and contraction mapping principle is obtained.The next,under the assumption that the control domain U is a convex set,the stochastic maximum principle of the time-delayed mean-field doubly stochastic control system is deduced by utilizing the classical variational theory.After that,by using the duality relation between the state equation and the corresponding adjoint equation,the sufficient conditions for the optimal control of the system are given.Eventually,the stochastic maximum principle is applied to the research of time-delayed mean-field doubly stochastic linear quadratic optimal control system,and the explicit expression of the optimal control is obtained,which verifies the correctness of the conclusion.