Dynamic Optimization: Time Scales and Nonsmooth Analysis

动态优化：时间尺度和非光滑分析

• 批准号: 0707789
• 负责人: Vera Zeidan
• 金额: $ 18万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707789&HistoricalAwards=false
• 关键词: Dynamic; Optimization; Time; Scales; Nonsmooth

This research program lies at the interface between six subjects, namely, optimization theory; nonsmooth analysis; discrete systems; continuous systems; systems over time scales; and impulsive-systems. These subjects govern a broad spectrum of applications arising from industry, engineering, economics, biology, and many others fields. In the last decade, more applications pointed toward combining both continuous-time and discrete-time systems which resulted in the birth of two different fields: Systems over time scales and impulsive hybrid systems. Nowadays they are two of the most rapidly growing areas of current research in dynamical systems theory. Results for either type of system are challenging, due to the discrepancies between the continuous and discrete settings and to their combined complexities. The objective of this investigation is to address key challenges in four topics of applied mathematics: optimal controls over time scales or over impulsive systems, nonsmooth analysis, and infinite dimensional optimization. More specifically, the aim of the proposal is four-fold. (i) To launch the field of optimal controls over time scales in which constraints are allowed. (ii) To build a bridge between optimal controls over time scales and that over impulsive systems.(iii) To introduce derivative- and Hessian- like objects for infinite dimensional "nonsmooth" functions that approximate such functions by linear and bilinear operators and that actually possess all the properties required from such sets; as was the case with Clarke's generalized Jacobian and Hessian in finite dimension. (iv) To develop in terms of those objects, optimality criteria for infinite dimensional nonsmooth optimization problems with constraints. In order execute these projects, a mixed bag of new and known techniques is needed such as nonsmooth and variational analysis techniques, optimization methods, Dubovitskii-Milyutin-type approach, dynamic programming- type techniques, time scales tools etc.Optimal control is a field whose mere existence is the product of applications arising from fields such as: aerospace, mechanical and electrical engineering, automatics, robotics, automotive electronics, economics, biology, and more. Therefore, important systematic developments of this research program have a significant impact on those fields. The outcome of this project will be a fundamental breakthrough leading to a new methodology in approaching some hybrid systems, and a new incentive to further advance optimal controls over time scales. The investigator integrates her research into education by incorporating her findings into her two graduate courses, which are well attended by engineering students, and by crafting a capstone course at the undergraduate level in which the applications of optimal control theory to the above disciplines are enhanced. Such an activity contributes greatly in training K-12 science and math teachers who graduate from MSU.

本研究项目处于六个学科的交叉点，即优化理论；非光滑分析;离散系统;连续系统;时间尺度上的系统；和impulsive-systems。这些学科管理着工业、工程、经济学、生物学和许多其他领域的广泛应用。在过去的十年中，更多的应用指向连续时间和离散时间系统的结合，这导致了两个不同领域的诞生：时间尺度系统和脉冲混合系统。现在它们是动力系统理论中发展最快的两个领域。由于连续和离散设置之间的差异以及它们的综合复杂性，这两种类型的系统的结果都具有挑战性。本研究的目的是解决应用数学的四个主题中的关键挑战：时间尺度或脉冲系统的最优控制，非光滑分析和无限维优化。更具体地说，这项提议的目的有四个方面。(i)在允许有限制的情况下开展时间尺度上的最优控制领域。在时间尺度上的最优控制和脉冲系统的最优控制之间建立一座桥梁。（iii）为无限维“非光滑”函数引入导数和类Hessian对象，这些函数用线性和双线性算子近似这些函数，并且实际上具有这些集合所需的所有性质；如有限维的Clarke广义雅可比矩阵和Hessian矩阵。（iv）根据这些对象，发展具有约束的无限维非光滑优化问题的最优性准则。为了执行这些项目，需要混合新的和已知的技术，如非光滑和变分分析技术，优化方法，dubovitskii - milyutin类型的方法，动态规划类型的技术，时间尺度工具等。最优控制是一个领域，其存在仅仅是应用领域的产物，如：航空航天，机械和电气工程，自动化，机器人，汽车电子，经济学，生物学等等。因此，该研究计划的重要系统发展对这些领域产生了重大影响。该项目的成果将是一项根本性的突破，为研究一些混合系统提供了一种新的方法，并为进一步推进时间尺度上的最佳控制提供了新的动力。研究者将她的研究与教育相结合，将她的发现纳入她的两门研究生课程，这两门课程有很多工程专业的学生参加，并在本科阶段精心制作了一门顶点课程，其中最优控制理论在上述学科中的应用得到加强。这样的活动有助于培养从密歇根州立大学毕业的K-12科学和数学教师。