Mathematical Sciences: Nonsmooth Analysis and Control Theory

摘要提案：DMS-9623406 PI：Wolenski 这个建议的主要重点是研究一个所谓的微分包含使用的想法和最近的结果从非光滑分析。 微分包含是常微分方程的推广，可以看作是一个多向流。 该公式的实际应用是由控制理论中的问题所激发的，在控制理论中，一段时间以来人们都知道，控制系统可以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，必须小心分析假设从一个到另一个的精确转移。 本提案中的部分研究旨在弥合微分包含和控制系统在不同假设下的已知差距。 Wolenski的论点是，控制理论中的许多问题将无法得到正确的理解，直到相关的微分包含中包含了暗示所需行为的基本特征。 具体议题Wolenski计划解决包括一个汉密尔顿-雅可比型分析的最小时间问题，一个详细的分析边界到一个可达集，一个扩展的霍普夫表示公式，调查使用的分析和优化，并扩大了遗传控制系统的理论。 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。 从制造业到医药工程到经济学， 控制系统越来越多地被引入作为用于描述复杂系统行为的模型。 控制理论中的许多最新数学进展正在走向实践， 这暴露了对控制现象的进一步数学理解的需要。 非光滑分析是近二十年来迅速发展的一门数学学科，主要是出于以下动机 它在最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机构的结果，无论是在方法和实质上扩展了大多数的经典微分 微积分 目前的建议是针对应用一些最新的进展，在控制理论的主题。 由于控制理论在应用科学中的影响是不可低估的，并且随着越来越多的数学进步在整个社会中得到应用，人们需要更好地理解控制系统的性质。 这项研究的成功完成将导致更好的数学理解控制系统的演变，并可能在建模过程中产生重大影响，使设计师专注于设计的关键要素，从而提高性能。

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.