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.
