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

该项目关注线性和非线性分布参数系统解的指数和渐近稳定性问题。 特别注意的是集中在调查的稳定性如何保持有限维半离散近似计划和稳定性如何受到干扰或系统参数的影响。 PI研究的大多数问题，利用一个适当的底层状态空间的重赋范。 重赋范方法的思想是，虽然每个数学模型都与一个自然范数（它测量模型中的一个重要量，如能量）相关联，但通常情况下，明智地选择一个新的范数对于洞察特定问题（如指数稳定性）特别有用。 通常一个新的规范也允许建设改进的Galerkin近似计划。 对于线性模型，PI考虑应用程序，包括混合动力系统（即耦合动力学，如模型的热弹性）和系统的延迟方程，以及问题，如反馈控制问题中的重赋范的影响。 对于非线性模型，在线性化和近似化阶段，重赋范是一个需要研究的问题。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，他们经常需要非常准确的答案来解决困难和微妙的问题。 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.）的振动。消散和以什么速度;流体的流动（围绕机翼或方向舵，在混合过程中，.）保持平滑或变得湍流;计算机模拟是否保持了原始模型的稳定性行为;等等。这些和其他问题将使用称为重定范的数学方法进行研究，这也常常导致构建改进的计算机模拟方法（近似方案）。 本科生将积极参与一些研究领域，让他们接触到专题和科学相关的数学模型，培训使用最新的科学计算算法和软件，并欣赏应用数学。

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.