Collaborative Research: Infinite horizon risk-sensitive control of diffusions with applications in stochastic networks

这一研究将推进随机控制中的数学分析，并对应用概率和随机网络做出重要贡献。该研究还将对大型数据中心、制造业、电信、医疗保健、库存和服务系统中的实际应用产生影响，提供有效管理这些应用的技能和工具。这样的系统通常可以建模为随机网络，具有多个作业和许多服务器，以及复杂的网络拓扑。这些复杂的网络系统的运行和管理在各种随机环境下受到许多风险因素的影响。该研究将开发先进的方法和算法，以提供解决方案，减轻大规模网络模型系统中的潜在运营风险。该模型系统大致描述了大规模并行服务器网络中的系统动力学。该研究将提供近似的最佳调度和其他运营策略。风险敏感控制具有在存在干扰和不确定性的情况下仍能获得良好性能的优点。它还限制了大的波动，因为它惩罚了运行成本的较高时刻。研究人员将把他们的发现纳入现有的随机网络和控制研究生课程，并通过相关研究课题的研讨会传播。该研究涉及一个跨学科研究人员团队，包括来自代表性不足的少数群体的研究人员，并为具有新数学技能的研究生提供培训机会。本文的研究目标是：（1）建立一个研究椭圆型系统和积分微分算子特征值的理论框架，以解决无限时域风险敏感控制（IHRS）中的状态转换和跳跃扩散问题。(2)发展建立渐近最优性所需的技术，并研究相关的随机微分对策和大偏差特征。(3)研究滚动时域控制和强化学习方法的基础--大时间渐近行为和相对值迭代算法。这项研究将大大推进积分微分算子和椭圆系统的特征值理论，并为扩散（带跳跃）和状态切换扩散的风险敏感控制提供开创性的方法。在分析方面，这项研究将大大有助于目前的努力，在文献中有关的非线性特征值问题的无界域。在非对称算子的变分特征、最大偏差原理和大偏差原理以及相应的Feynman-Kac半群等方面，我们期望得到大量的结果。 本文的另一个重要贡献是分析了系统的大时间渐近行为，包括相对值迭代算法和滚动时域控制的研究。该研究还将推进对大规模并行服务器网络的风险敏感的渐近最优调度策略的理解，包括那些在随机环境中引起跳跃扩散和状态切换扩散限制的策略。新的方法涉及到等价的随机微分对策和空间截断技术将被开发来证明渐近最优性的上下界。最后但并非最不重要的是，这项研究旨在缩小概率和分析方法之间的差距，旨在改善两个社区之间的互动。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

This research will advance mathematical analysis in stochastic control and make important contributions to applied probability and stochastic networks. The research will also have an impact on real-world applications in large-scale data centers, manufacturing, telecommunications, healthcare, inventory, and service systems, providing skills and tools to manage them effectively. Such systems can often be modeled as a stochastic network, with multiple jobs and many servers, and complex network topology. The operations and management of these sophisticated networked systems are subject to many risk factors under various random environments. This research will develop advanced methods and algorithms to provide solutions that mitigate the potential operational risks in a large-scale network model system. The model system roughly describes the system dynamics in large-scale parallel server networks. The research will provide approximate optimal scheduling and other operational policies. Risk-sensitive control has the advantage of achieving good performance in the presence of disturbances and uncertainty. It also limits large fluctuations since it penalizes higher moments of the running cost. The investigators will incorporate their findings into the existing graduate courses in stochastic networks and control, and disseminate them through seminars on relevant research topics. The research involves a team of interdisciplinary researchers, including those from underrepresented minority groups, and provides training opportunities for graduate students with new mathematical skills. The objectives of the research are: (1) To develop a comprehensive theoretical framework for the study of eigenvalues of elliptic systems and integro-differential operators to address the associated problems in infinite-horizon risk sensitive control (IHRS) of regime-switching and jump diffusions. (2) To develop the techniques required to establish asymptotic optimality and study the associated stochastic differential games and large deviation characterizations. (3) To study the large-time asymptotic behavior and relative value iteration algorithms, which form the basis of rolling horizon control and reinforcement learning methods. This research will greatly advance the theory of eigenvalues of integro-differential operators and elliptic systems and produce ground-breaking methodologies for risk-sensitive control of diffusions (with jumps) and regime-switching diffusions. On the analytical side, this research will greatly contribute to the current efforts in the literature concerning nonlinear eigenvalue problems in unbounded domains. A wealth of results on variational characterizations, maximum and large deviation principles, and the associated Feynman-Kac semigroup for nonsymmetric operators are expected to be obtained. Another important contribution of the proposed research is analyzing large-time asymptotic behavior, which includes the study of relative value iteration algorithms and rolling horizon control. The research will also advance the understanding of the risk-sensitive asymptotically optimal scheduling policies for large-scale parallel server networks, including those in random environments that give rise to jump-diffusion and regime-switching diffusion limits. New methods involving the equivalent stochastic differential game and spatial truncation techniques will be developed to prove lower and upper bounds for asymptotic optimality. Last, but not least, this research aims to close the gap between probabilistic and analytical methods, aiming to improve the interaction between the two communities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.