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

在新兴应用的激励下，本提案涵盖了系统理论和随机优化方法的几个研究课题。(1)旨在开发新的随机逼近算法（具有延迟、分布式处理器和代表随机环境的切换过程）。建立了这些算法的渐近性质和相关的极限结果。结果可以应用于共识控制等问题。(2)研究随机延迟（可能是由于通信延迟）系统的稳定性。得到了非线性系统稳定的充分条件和随机时滞泛函微分系统的判据。预期的结果将为处理依赖于延迟的系统的稳定性提供更多的启示。(3)以大偏差形式的误差估计，用于使用规则和量化的观测进行系统识别。通过考虑量化方面的空间复杂性和数据窗口大小方面的时间复杂性，本研究的重点是更好地理解概率误差与资源之间的基本关系，这些资源表示算法中的数据大小、分析中的样本大小和通信中的信道带宽。(4)为了近似扩散的第一次退出时间，将发展马尔可夫链近似方法并获得其收敛速率。为了处理具有连续状态依赖切换的随机微分方程的数值解，将使用在扩大的概率空间中具有与原始系统相同分布的重新嵌入数值解序列来确定路径收敛率。本项目旨在衔接系统理论、随机优化方法和应用。提出的研究课题包括：利用并行处理器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下界和上界；设计和分析具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取出来或受到实际应用的启发。预计本研究结果将在网络系统、无线通信、金融工程、正则和量化观测的系统识别以及涉及随机微分方程的某些问题的数值解等方面得到应用。要构建的模型、系统的内在特性以及要开发的数值方法和算法将导致随机系统理论和优化方法的进步潜在地转移到上述应用中。拟议的研究将涉及研究生的参与；它还将包括本科生的研究项目。通过将研究与教学和学生训练相结合，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.