【作者】 娄延俊；

【导师】 李娟；

【作者基本信息】 山东大学 ， 运筹学与控制论， 2013， 硕士

【摘要】 本文主要基于正倒向随机微分方程、平均场正倒向随机微分方程和最优控制理论,研究了一类特殊的初始条件耦合的平均场正倒向随机微分方程,然后研究了该类方程的线性二次最优控制以及非零和微分对策问题,最后我们进一步研究了平均场线性正倒向随机系统的近似最优控制问题。首先我们主要研究了下列初始条件耦合的平均场正倒向随机微分方程在单调性假设条件下解的存在性和唯一性：然后我们给出了上述方程组中正向方程的伴随方程,利用相应的平均场正倒向随机微分方程的解得到了线性二次最优控制问题的最优控制的显式解,也得到了非零和微分对策问题的纳什均衡点的显式解,并且分别证明了它们是唯一的。其次,我们研究了平均场正倒向线性随机控制系统的近似最优控制问题。结合Zhou [25]我们给出了近似最优控制的定义,证明了近似最优控制的充分性条件和必要性条件。更多还原

【Abstract】 Based on the theory of forward-backward stochastic differential equations (FBSDEs), mean-field forward-backward stochastic differential equations (mean-field FBSDEs) and the optimal control, we study a new type of initial coupled mean-field FBSDEs. Then, we study mean-field linear quadratic optimal control problems and the nonzero-sum differential games of such equations. In the end we study the near-optimal control problems of mean-field linear forward-backw-ard stochastic systems.In this paper, firstly, we prove that there exists a unique solution of initial coupled mean-field FBSDEs under the monotonic conditions: Then we give the adjoint equation corresponding to the forward stochastic equa-tion. With the help of the solutions of mean-field FBSDEs, we get the explicit form of the optimal control for the linear quadratic optimal control problems and the open-loop Nash equilibrium point of nonzero-sum differential games.Secondly, we study the near-optimal control problems of mean-field linear forward-backward stochastic systems. Inspired by Zhou [25] we give the definit-ion of the near-optimality, and establish the sufficient condition and the necess--ary condition of the near-optimality in the form of Pontryagin stochastic maxi--mum principle.更多还原