Mathematical Sciences: Nonsmooth Analysis and Control Theory

摘要：提案：DMS-9623406 PI: Wolenski本提案的主要重点是利用非光滑分析的思想和最新结果来研究所谓的微分包含。微分包涵是一个常微分方程的推广，可以看作是一个多向流动。该公式的实际应用是由控制理论中的问题所激发的，一段时间以来，人们已经知道，控制系统可以被重新表述为微分包含，而不会丢失有关其轨迹的信息。但与任何重新制定的模型一样，人们必须仔细分析假设从一个模型到另一个模型的精确转移。本提案中的部分研究旨在弥合在不同假设集下已知的不同内含物和控制系统之间的差距。Wolenski的论点是，控制理论中的许多问题将无法被正确理解，直到隐含期望行为的基本特征在相关的差异包含中被揭示出来。Wolenski计划解决的具体主题包括最小时间问题中的Hamilton-Jacobi型分析，可达集边界的详细分析，Hopf表示公式的扩展，半完整性在分析和优化中的应用的调查，以及遗传控制系统理论的扩展。控制理论的应用在应用科学中非常普遍，它对我们社会的影响无疑将继续扩大。从制造业到医学到工程到经济学，控制系统越来越多地被引入作为描述复杂系统行为的模型。最近控制理论中的许多数学进展正在进入实践，这表明需要对控制现象进行进一步的数学理解。非光滑分析是近二十年来发展迅速的一门数学学科，主要是由于它在优化和最优控制理论中的许多应用。它现在包含了一个令人惊讶的完整的结果体，无论是在方法上还是在内容上都扩展了大多数经典微分学。本提案旨在将一些最新进展应用于控制理论的主题。由于控制理论在应用科学中的影响很难被高估，并且随着越来越多的数学进展在整个社会中得到应用，人们需要对控制系统的特性有更深入的了解。这项研究的成功完成将导致对控制系统演变的更大的数学理解，并且可以通过允许设计师专注于设计的关键元素来提高性能，从而在建模过程中产生重大影响。

ABSTRACT Proposal: DMS-9623406 PI: Wolenski The main emphasis of this proposal is aimed at studying a so-called differential inclusion by using ideas and recent results from nonsmooth analysis. A differential inclusion is a generalization of an ordinary differential equation, and can be viewed as a multidirectional flow. The practical uses of the formulation are motivated by issues in control theory, where it has been known for some time that a control system can be reformulated as a differential inclusion without losing information regarding its trajectories. But as with any reformulated model, one must take care in analysing the precise transfer of assumptions from one to the other. Part of the research in this proposal is aimed at bridging this gap between what is known in differential inclusions and control systems under different sets of assumptions. It is the contention of the Wolenski that many issues in control theory will not be properly understood until the fundamental traits implying the desired behavior are brought out in the associated differential inclusion. Specific topics Wolenski plans to address include a Hamilton-Jacobi type analysis in the minimal time problem, a detailed analysis of the boundary to a reachable set, an extension of the Hopf representation formula, a survey of the uses of semiconvexity in analysis and optimization, and a broadening of the theory of hereditary control systems. Applications of control theory are prevalent in applied science, and its influence on our society will undoubtedly continue to expand. From manufacturing to medicine to engineering to economics, control systems are increasingly being introduced as models for describing complicated system behavior. The many recent mathematical advances in control theory are finding their way into practice, and this exposes the need for further mathematical understanding of control phenomena. Nonsmooth analysis is a mathematical discipline that has developed rapidly in the last twen ty years, mainly motivated by its many applications to optimization and optimal control theory. It now contains a surprisingly complete body of results that both in method and substance extend most of the classical differential calculus. The present proposal is directed toward applying some of the more recent advances to topics in control theory. Since the influence of control theory in applied science can hardly be overestimated, and as more of the recent mathematical advances are finding applications throughout society, a need has arisen for a greater understanding of the properties of control systems. The successful completion of this research will lead to a greater mathematical understanding of the evolution of control systems, and could have significant consequences in the modelling process by allowing the designer to focus on crucial elements of the design that would lead to an improved performance.