时滞正倒向随机方程的可解性及相关控制问题研究

Forward and backward stochastic equations (FBSEs) are the foundation of the stochastic optimal control and have wide applications in areas such as financial mathematics. However, due to the facts that time delays lead to the infinite dimension and the separation principle does not hold for the systems with multiplicative noise, it remains challenging for the solvability and control problems of the delayed FBSEs. Based on our previous work to solve the stochastic control problem with delay in terms of Riccati-ZXL equation, this project will study the solvability, optimal control and stabilization problems for the delayed FBSEs. Firstly, the necessary and sufficient condition and the explicit solution will be proposed for delayed FBSEs by establishing a non-homogeneous relationship between the forward and backward stochastic processes; By applying the stochastic maximum principle, the optimal control for delayed forward and backward stochastic systems (FBSSs) will be reduced to the solvability of delayed FBSEs. Combining with the solvability of FBSEs, the equivalent conditions for the existence of the optimal control will be proposed and the explicit solution will be given; In addition, a predictor controller will be proposed and the stabilization condition will also be analyzed. These works will further enrich the theory of delayed systems and stochastic systems, and have important values for the development of the stochastic control.

正倒向随机方程是随机最优控制的基础，在金融数学等领域具有广泛应用背景。而考虑到时滞导致无限维及乘性噪声系统分离原理不成立，时滞正倒向随机方程的求解及相关控制问题仍面临巨大挑战。在我们前期得到的利用Riccati-ZXL方程求解时滞随机控制问题基础上，本项目拟研究时滞正倒向随机方程的可解性及相关最优控制和镇定问题。首先，通过建立正倒向随机过程之间的非齐次关系，提出时滞正倒向随机方程可解的充要条件及显式解；进而利用随机极大值原理，将时滞正倒向随机系统的最优控制问题转化为时滞正倒向随机方程的求解问题，结合可解性提出最优控制存在的充要条件及显式解；此外，提出预报反馈形式的控制器，分析时滞正倒向随机系统的镇定条件。本项目的研究将丰富时滞随机系统理论，对随机控制理论的发展具有重要意义。

随机系统的最优控制在网络控制系统、金融投资等实际问题中具有重要应用。基于随机极大值原理，最优控制器的设计转化为在平衡条件约束下的正倒向随机方程求解问题。此外，时滞现象广泛存在于众多实际系统中，如交通控制系统、流体控制系统、热传导系统等。因此，时滞正倒向随机方程求解的研究具有重要的理论意义和实际应用价值，是时滞随机系统最优控制的基础。然而，时间延迟的存在导致无限维，使得问题求解更具挑战，也导致相关的时滞系统最优控制问题有待解决。针对上述问题，项目组提出了时滞正倒向随机差分方程的可解条件及解析解的显式表达；提出了时滞随机领导者-跟随者博弈控制问题的可解条件及显式最优控制器；提出了一类正倒向随机微分方程的可解条件及相关线性二次最优控制问题的显式最优控制器；提出了具有非一致丢包的时滞多智能体系统的可趋同条件。在顺利完成原研究计划的基础上，项目负责人及其研究团队进一步研究提出了理性预期模型的Stackelberg解；提出了反馈Nash策略的强化学习算法。项目执行期间项目负责人与参与人员发表的学术论文中，有37篇受到了该基金的资助，其中SCI检索论文28篇，其中控制领域国际顶级期刊IEEE Transactions on Automatic Control、Automatica论文7篇，其他IEEE汇刊5篇。