Investigation of Stability and Approximation Issues via Renorming for Distributed Parameter Systems

通过分布式参数系统重整研究稳定性和逼近问题

• 批准号: 0105253
• 负责人: Richard Fabiano
• 金额: $ 7.33万
• 依托单位: University of North Carolina Greensboro
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0105253&HistoricalAwards=false
• 关键词: Investigation; Stability; Approximation; Issues; via

The project is concerned with issues related to exponential and asymptotic stability of solutions to linear and nonlinear distributed parameter systems. Particular attention is focused on the investigation of how stability properties are preserved by finite dimensional semidiscrete approximation schemes and how stability properties are affected by disturbances or by system parameters. The PI studies most issues by making use of an appropriate renorming of the underlying state space. The idea of the renorming method is that while every mathematical model is associated with a natural norm (which measures an important quantity in the model, such as energy), it is often the case that a judicious choice of a new norm can be especially useful for insight into a specific issue such as exponential stability. Frequently a new norm also allows construction of improved Galerkin approximation schemes. For linear models, the PI considers applications including hybrid systems (i.e. coupled dynamics, such as models of thermoelasticity) and systems of delay equations, as well as questions such as the implications of renorming in feedback control problems. For nonlinear models, the renorming is an issue for investigation at the stages of linearization and approximation. Today's scientists and engineers are using increasingly complex and sophisticated mathematical models, and they often require quite accurate answers to difficult and delicate questions. A particularly important issue for many models, and the main focus of this project, is exponential stability. Typical of questions related to exponential stability would be - do vibrations of a mechanical structure (robot arm, smart material actuator, satellite antenna, digital reading device, ...) dissipate and at what rate; does the flow of a fluid (around a wing or rudder, in a mixing process, ...) stay smooth or become turbulent; do computer simulations preserve the stability behavior of the original model; etc. These and other issues will be investigated using a mathematical method known as renorming, which also often leads to the construction of improved computer simulation methods (approximation schemes). Undergraduate students will be actively involved in some lines of research, giving them an exposure to topical and scientifically relevant mathematical models, training in the use of the latest scientific computing algorithms and software, and an appreciation of applied mathematics.

本课题研究线性和非线性分布参数系统解的指数稳定性和渐近稳定性问题。特别关注的是如何通过有限维半离散近似方案来保持稳定性，以及稳定性如何受到干扰或系统参数的影响。PI通过对底层状态空间进行适当的改造来研究大多数问题。重整方法的思想是，虽然每个数学模型都与自然规范（衡量模型中的重要数量，如能量）相关联，但通常情况下，明智地选择新规范对于洞察特定问题（如指数稳定性）特别有用。通常新的范数也允许构造改进的伽辽金近似格式。对于线性模型，PI考虑的应用包括混合系统（即耦合动力学，如热弹性模型）和延迟方程系统，以及诸如在反馈控制问题中改造的含义等问题。对于非线性模型，在线性化和逼近阶段的改造是一个有待研究的问题。今天的科学家和工程师正在使用越来越复杂和精密的数学模型，他们经常需要对困难和微妙的问题给出相当准确的答案。对于许多模型来说，一个特别重要的问题，也是这个项目的主要焦点，是指数稳定性。与指数稳定性相关的典型问题是——机械结构（机器人手臂、智能材料致动器、卫星天线、数字阅读设备等）的振动是否消散，消散的速度如何；流体的流动（在机翼或方向舵周围，在混合过程中……）是保持平稳还是变得湍流？计算机模拟是否能保持原始模型的稳定性；等。这些和其他问题将使用称为重整的数学方法进行调查，这也经常导致构建改进的计算机模拟方法（近似方案）。本科学生将积极参与一些研究，让他们接触到主题和科学相关的数学模型，训练他们使用最新的科学计算算法和软件，并了解应用数学。