扩展MF-AFBSDDEswCN及线性系统的随机最优控制
Extended MF-AFBSDDEswCN and stochastic optimal controls of linear system
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- School of Mathematics and Statistics, South-Central Minzu University（中南民族大学数学与统计学院）
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中文总结 AI 辅助
本文通过扩展支配-单调性条件至非线性情形，建立了扩展平均场前瞻-倒向随机延迟微分方程的适定性，并据此解决了两类带路径依赖初始值的随机最优控制问题，给出了最优控制的显式闭式解。
中文摘要 AI 辅助
本研究通过扩展支配-单调性条件，研究了带共同噪声的扩展平均场前瞻-倒向随机延迟微分方程（扩展MF-AFBSDDEswCN）。我们将条件从线性情形推广到非线性情形，引入了非线性伴随函数，这些函数是保证扩展MF-AFBSDDEswCN适定性的关键。利用这一更广泛的扩展MF-AFBSDDEswCN适定性框架，结合其他精细的分析工具，我们研究了两类随机最优控制问题。这些问题包括一个线性凸问题和一个带输入约束的线性二次问题，其中输入约束既可以是时变的也可以是随机的。此外，关于初始值的控制元素是路径依赖的，这将导致一些本质性的困难。对于每个问题，我们建立了最优控制的存在性和唯一性，并给出了其显式的闭式表示。
英文摘要
The research studies the extended mean-field anticipated forward-backward stochastic delayed differential equations with common noise (extended MF-AFBSDDEswCN) by expanding the domination-monotonicity conditions. We generalize the conditions from a linear setting to a nonlinear one by including nonlinear adjoint functions which are key to guaranteeing the well-posedness of extended MF-AFBSDDEswCN. Utilizing this broader well-posedness framework for extended MF-AFBSDDEswCN, combined with other refined analytical tools, we examine two classes of stochastic optimal control problems. These include a linear-convex problem and a linear-quadratic problem with input constraints that can be both time-dependent and random. Moreover, the control elements regarding the initial values are path-dependent, which will cause some essential difficulties. For each problem, we establish the existence and uniqueness of optimal controls and provide their explicit, closed-form representations.
