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

本研究计划考虑在繁忙的交通中，某些受控随机处理网络的标度限制。 布朗控制问题（BCP）作为一种形式化的扩散近似被提出来，但目前还缺乏一种能将网络控制问题与BCP紧密联系起来的通用理论。本研究的第一个目标是建立一个广泛的控制形式，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到相应的扩散控制问题。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。 这些问题对应于一族在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。存在性，唯一性和正则性理论的HJB方程的一系列这样的控制问题和成本标准将被开发。这种方程的适定性是一个核心成分，在发展的数值计划，以获得接近最优控制。 此外，规律性是阅读关于最优控制形式的有用定性信息的关键。 这样的信息的一个例子是一个最佳控制过程的表征为一个合适的自由边界问题确定的域上的反射扩散。奇异控制问题的这种特征结果是该领域中最优雅和最有用的结果之一，并且它们对具有状态约束问题的研究将是本研究的重点。激发这项工作的动态控制问题来自广泛的应用领域，例如，电信，制造，服务工程，计算等。在这样的网络中的控制可以采取各种形式，实例包括作业的调度、排序、路由和许可，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发展技术，以获得良好的调度政策，一般家庭这样的随机处理系统，使用扩散近似的数学理论。 这项工作将改善复杂制造、通信和计算机系统的设计、稳定性和监管。研究项目将支持两名研究生的培训，并与非博士的教师开展国际合作。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.