Dynamic Optimization: Time Scales and Nonsmooth Analysis

这项研究计划位于六门学科的交界处，即最优化理论、非光滑分析、离散系统、连续系统、时间尺度系统和脉冲系统。这些学科管理着工业、工程、经济、生物和许多其他领域的广泛应用。在过去的十年中，更多的应用指向将连续时间系统和离散时间系统相结合，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。如今，它们是当前动力系统理论研究中发展最快的两个领域。由于连续和离散设置之间的差异及其组合的复杂性，这两种类型的系统的结果都是具有挑战性的。这项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，该提案的目的有四个方面。(1)在允许限制的时间尺度上开展最优控制领域。(Ii)在时间尺度上的最优控制和脉冲系统上的最优控制之间架起一座桥梁。(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逼近此类函数，并且实际上具有此类集合所需的所有性质；例如Clarke的有限维广义Jacobian和Hessian对象。(4)根据这些目标，发展了无限维带约束非光滑优化问题的最优性准则。为了执行这些项目，需要一系列新的和已知的技术，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，它的存在本身就是一个应用领域的产物，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，生物等。因此，这一研究计划的重大系统性发展对这些领域产生了重大影响。该项目的结果将是一个根本性的突破，导致在接近一些混合系统方面的新方法，以及进一步推进时间尺度上的最优控制的新动机。这位研究人员将她的研究成果融入到她的两门研究生课程中，这两门课程都很受工科学生的欢迎，并在本科水平上精心设计了一门顶峰课程，在这门课程中，最优控制理论在上述学科中的应用得到了加强。这样的活动对培养密歇根州立大学毕业的K-12科学和数学教师有很大贡献。

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.