Stochastic Systems and Control

本研究的主要焦点是分数布朗运动的随机微积分在随机系统中的应用，特别是在识别和控制问题上。分数布朗运动在随机模型中的重要性已经在各种各样的应用中得到展示，例如，水文学，经济学和电信。本研究的目标之一是进一步发展分数阶布朗运动的随机微积分，类似于布朗运动的随机微积分的发展方式，为解决分数阶布朗运动的随机系统问题提供工具。为了解决具有分数布朗运动的随机系统的辨识、滤波和控制问题，一些计划开发的随机微积分工具是必要的。所研究的分数布朗运动包括有限维空间和无限维空间中具有值的过程。研究了具有分数阶布朗运动的随机微分方程和随机偏微分方程解的存在性和唯一性，以及由分数阶布朗运动形成的鞅。另一类计划研究的随机模型是隐马尔可夫模型。研究了分数阶布朗运动和隐马尔可夫模型描述的系统的自适应控制问题。本文主要研究分数阶布朗运动的随机微积分。分数布朗运动是一种随机过程，在水文、经济和电信等物理系统的建模中起着重要作用。随机微积分为解决包含分数布朗运动的随机模型问题提供了基本工具。分数阶布朗运动具有自相似性质，许多运动具有在物理现象中经常出现的长距离依赖性质。计划研究包含分数布朗运动和隐马尔可夫模型的部分已知系统的控制。这种控制问题在应用中自然会出现。

0204669DuncanThe major focus of this research is the application of a stochastic calculus for a fractional Brownian motion to stochastic systems, especially for the problems of identification and control. The importance of a fractional Brownian motion in stochastic models has been exhibited in a wide variety of applications, such as, hydrology, economics and telecommunications. One goal of the research is to develop further a stochastic calculus for a fractional Brownian motion similar to the way that the stochastic calculus for Brownian motion was developed to provide the tools for solving problems of stochastic systems with a fractional Brownian motion. Some tools of stochastic calculus that are planned for development are necessary for solving the problems of identification, filtering and control of a stochastic system with a fractional Brownian motion. The fractional Brownian motions for study include processes with values in both finite and infinite dimensional spaces. The study includes the existence and the uniqueness of solutions of stochastic differential equations and stochastic partial differential equations with a fractional Brownian motion and martingales that are formed from a fractional Brownian motion. Another family of stochastic models that are planned for study is hidden Markov models. An investigation of adaptive control for systems described by both fractional Brownian motions and hidden Markov models is planned.This research focuses on a stochastic calculus for a fractional Brownian motion. A fractional Brownian motion is a stochastic process that has been important in modeling physical systems such as those arising in hydrology, economics and telecommunications. Stochastic calculus provides the basic tool for solving problems of stochastic models that contain a fractional Brownian motion. Fractional Brownian motions have a self similarity property and many have a long range dependence property that often occur in physical phenomena. It is planned to study the control of partially known systems that contain fractional Brownian motion and hidden Markov models. Such control problems naturally arise in applications.