Analysis, Algorithm Design, and Computation for Stochastic Systems and Optimization

在新兴应用的推动下，这项建议涵盖了系统论和随机优化方法的几个研究主题。(1)发展新的随机逼近算法(具有时延、分布式处理器和代表随机环境的切换过程)。这些算法的渐近性质和相关的极限结果将被建立。所得结果可应用于共识控制等问题。(2)研究具有随机时滞(可能是由于通信延迟)的系统的稳定性。得到了非线性系统稳定的充分条件和具有随机时滞的泛函微分系统的判据。预期的结果将为处理与延迟相关的系统的稳定性提供更多线索。(3)对于使用常规观测和量化观测的系统辨识，将获得大偏差形式的误差估计。通过考虑量化方面的空间复杂性和相对于数据窗口大小的时间复杂性，本研究侧重于更好地理解概率误差与代表算法中的数据大小、分析中的样本大小和通信中的信道带宽的资源之间的基本关系。(4)为了逼近扩散的第一个退出时间，将发展马尔可夫链近似方法，并得到它们的收敛速度。为了处理具有连续状态依赖切换的随机微分方程组的数值解，将使用与原始系统在更大的概率空间中具有相同分布的重新嵌入的数值解序列来确定路径收敛速度。该项目旨在将系统理论、随机优化方法和应用联系起来。提出的研究课题包括利用并行处理机开发考虑随机环境的迭代算法，研究随机时滞系统的稳定性，获得不同观测模式下系统辨识的估计误差界，以及设计和分析含有随机不确定性的微分方程问题的数值解。所要研究的问题是从实际应用中提炼出来的，或者是受到实际应用的启发的。预计本研究的结果将在网络系统、无线通信、金融工程、常规观测和量化观测的系统辨识以及某些涉及随机微分方程的问题的数值解中应用。将要建立的模型、系统的内在特性以及将要开发的数值方法和算法将潜在地将随机系统理论和优化方法的进展转移到上述应用中。建议的研究将包括研究生的参与；它还将包括本科生的研究项目。通过将拟议的研究与教学和学生培养相结合，计划的工作将有助于数学系统理论的进一步发展和数学教育的改进。

Motivated by emerging applications, this proposal encompasses several research topics in systems theory and stochastic optimization methods. (1) It aims to develop new stochastic approximation algorithms (with delays, distributed processors, and a switching process representing the random environment). Asymptotic properties of these algorithms and related limit results will be established. The results can be applied to consensus control problems among others. (2) Stability of systems with random delays (possibly due to communication latency) will be investigated. Sufficient conditions for stability of nonlinear systems and criteria for functional differential systems with random delays will be obtained. The expected results will shed more lights on treating stability of systems that are delay dependent. (3) Error estimates in the form of large deviations for system identification using regular and quantized observations will be obtained. By considering both space complexity in terms of quantization and time complexity with respect to data window sizes, this study focuses on providing a better understanding to the fundamental relationship between probabilistic errors and resources that represent data sizes in algorithms, sample sizes in analysis, and channel bandwidths in communications. (4) To approximate the first exit time for diffusions, Markov chain approximation methods will be developed and their rates of convergence will be obtained. To treat numerical solutions to stochastic differential equations with continuous-state-dependent switching, pathwise rates of convergence will be ascertained using a sequence of re-embedded numerical solutions having the same distribution as the original systems in an enlarged probability space.This project aims to bridge systems theory, stochastic optimization methods, and applications. The research topics proposed include developing iterative algorithms using parallel processors and taking random environment into consideration, investigating stability of systems involving random delays, obtaining lower and upper estimation error bounds for system identification under different observation patterns, and designing and analyzing numerical solutions of problem involving certain differential equations with random uncertainty. The problems to be studied have been extracted from or motivated by real applications. It is anticipated that the results of this research will be useful for applications in networked systems, wireless communication, financial engineering, system identification with regular and quantized observations, and numerical solutions of certain problems involving random differential equations. The models to be constructed, the intrinsic properties of the systems, and the numerical methods and algorithms to be developed will lead to potential transfer of advances in stochastic systems theory and optimization methods to the aforementioned applications. Theproposed research will involve participation of graduate students; it will also include undergraduate student research projects. By integrating the proposed research with teaching and student training, the planned work will contribute to the further development of mathematical systems theory and the improvement of mathematics education.