Scaling Limits for some Stochastic Control Problems with Applications to Stochastic Networks

本研究项目考虑了大流量条件下可控随机处理网络的尺度限制。布朗控制问题（BCP）已被提出作为一种形式的扩散近似，适用于广泛的控制网络。目前，在网络控制问题及其相关的BCP之间建立严格联系的一般理论严重缺乏。本研究的第一个目标是建立对于广泛的控制形式、网络结构和优化准则，适当缩放的控制网络模型的值函数收敛于相应扩散控制问题的值函数。第二个目标是研究由上述渐近分析引起的扩散控制问题的定性性质。这些问题对应于一类非光滑域上具有状态约束的奇异控制问题。这类控制问题的Hamilton-Jacobi-Bellman （HJB）方程是一类具有梯度约束和非标准边界条件的退化椭圆型非线性偏微分方程。针对一系列此类控制问题和成本准则，将发展HJB方程的存在性、唯一性和正则性理论。这类方程的适定性是获得近最优控制的数值格式发展的一个中心因素。此外，规律性是读出最优控制形式的有用定性信息的关键。这类信息的一个例子是将一个最优控制过程描述为在一个由合适的自由边界问题确定的区域上的反射扩散。这些奇异控制问题的表征结果是该领域中最优雅和最有用的结果之一，它们对状态约束问题的研究将是本研究的重点。激励这项工作的动态控制问题出现在广泛的应用领域，如电信，制造，服务工程，计算等。这种网络中的控制可以采取多种形式，例如，调度、排序、作业的路由和接收，以及输入和处理速率控制。感兴趣的网络通常相当复杂，因此人们寻求易于处理的近似模型。本研究的总体主题是利用扩散近似的数学理论，为这类随机处理系统的一般族获得良好的调度策略的技术发展。这项工作将有助于改进复杂制造、通信和计算机系统的设计、稳定性和管理。研究项目将支持两名研究生的培训，并与非博士教师开展国际合作。授予机构。

This research project considers scaling limits of certain controlled stochastic processing networks in heavy traffic. Brownian control problems(BCP) have been proposed as formal diffusion approximations for a broad range of controlled networks.Currently there is a critical lack of general theory that establishes rigorous connections between a network control problem and its associated BCP. The first goal of this research is to establish that for a wide range of control forms, network structures and optimization criteria, the value functions of suitably scaled controlled network models converge to that of the corresponding diffusion control problem. The second goal is the study of qualitative properties of diffusion control problems arising from the above asymptotic analysis. These problems correspond to a family of singular control problems with state constraints in non-smooth domains. The Hamilton-Jacobi-Bellman (HJB) equations for such control problems are a challenging class of degenerate elliptic nonlinear partial differential equations with gradient constraints and somewhat non-standard boundary conditions. Existence, uniqueness and regularity theory for HJB equations for a range of such control problems and cost criteria will be developed. Wellposedness of such equations is a central ingredient in development of numerical schemes for obtaining near optimal controls. Additionally, regularity is key in reading off useful qualitative information on the form of an optimal control. An example of such information is the characterization of an optimally controlled process as a reflected diffusion over a domain determined in terms of a suitable free boundary problem. Such characterization results for singular control problems are some of the most elegant and useful results in the field and their study for problems with state constraints will be a focus of this research.Dynamic control problems that motivate this work arise from a wide range of application areas, such as, telecommunications, manufacturing, service engineering, computing, etc. Control in such networks can take a variety of forms, examples include, scheduling, sequencing, routing and admissions of jobs, and input and processing rate controls. Networks of interest are in general quite complex and thus one seeks tractable approximate models. The overall theme of this research is the development of techniques for obtaining good scheduling policies for general families of such stochastic processing systems using the mathematical theory of diffusion approximations. The work will lead to improved design, stability and regulation of complex manufacturing, communication and computer systems. Research project will support the training of two graduate students and develop international collaborations and with faculty from non-Ph.D. granting institutions.