The exponential map for flows and its application in geometric control theory

流动指数图及其在几何控制理论中的应用

• 批准号: RGPIN-2019-04554
• 负责人: Lewis, Andrew
• 金额: $ 1.53万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759036
• 关键词: exponential; map; flows; its; application

This research is connected with the area of control theory, a field with many areas of application, ranging from robotics, to chemical processes, to scheduling of medical treatments.  The primary focus of the work is on a sub-discipline of control theory known as geometric control theory.  The tools of geometric control theory are most useful for studying control systems described by ordinary differential equation models that are not linear.  For nonlinear ordinary differential equation models, many of the basic problems of control theory remain unanswered.  Three such important basic problems are (1) controllability, (2) stabilisability, and (3) optimality.  The problem of controllability is, in rough terms, that of using control to steer a system from a given initial state to a desired final state.  A simple example is the steering of a mobile robot from one configuration to another.  The problem of stabilisability is to use control to render a (possibly unstable) behaviour stable.  A simple example is the balancing of a pendulum in its upright configuration.  The problem of optimality is that of determining whether a given behaviour minimises a prescribed cost.  A simple example is the mobile robot steering problem mentioned above, but now requiring that the reconfiguration be done while using the least possible energy. All three of these problems have a component of their resolution that relies on a detailed understanding of the local structure of a system.  It is the aim of this work to understand this local structure with the objective of coming to a more complete understanding of the basic problems described above.  While existing work has contributed greatly to our understanding of this structure, it is fair to say that a comprehensive understanding remains elusive.  To make advances on this understanding, the proposed research uses a new modelling framework for geometric control systems, one founded in the mathematics of advanced functional analysis, sheaf theory, and pseudogroups.  Functional analysis is used to topologise spaces of vector fields, and recent work by the proposer's group has made significant advances by understanding this topology for real analytic vector fields, an especially important case for geometric control theory.  With these functional analysis tools, it becomes possible to employ methods of sheaf theory (for spaces of vector fields) and the theory of pseudogroups (for spaces of flows) to express the subtle behaviour exhibited by control systems.  These tools have hitherto not been widely used in control theory, and so provide unexplored avenues for fruitful future work. The proposal is concerned with a long-term program of basic research.  The intention of the work, however, is to shed light on applied problems, and eventually produce methodologies based on the deep understanding revealed by new results and techniques.  An important facet of the proposal is to utilise and train highly qualified personnel.

这项研究与控制理论领域相关，该领域有许多应用领域，从机器人技术到化学过程，再到医疗调度。​工作的主要重点是控制的子学科。几何控制理论的工具对于研究非线性常微分方程模型描述的控制系统最有用。对于非线性常微分方程模型，控制理论的许多基本问题仍然没有得到解答。这些重要的基本问题是（1）可控性，（2）稳定性，以及（3）最优性，粗略地说，可控性问题是使用控制将系统从给定的初始状态引导到所需的最终状态。一个简单的例子是转向。移动机器人从一种配置到另一种配置的稳定性问题是使用控制来使（可能不稳定的）行为稳定。一个简单的例子是摆在其直立配置中的平衡。一个简单的例子是上面提到的移动机器人转向问题，但现在要求在使用尽可能少的能量的情况下完成重新配置，这三个问题都有其解决方案的一部分。依赖于对系统局部结构的详细理解，这项工作的目的是理解这种局部结构，以更全面地理解上述基本问题。可以公平地说，这对我们对这种结构的理解有很大影响为了在这种理解上取得进展，所提出的研究使用了一种新的几何控制系统建模框架，该框架建立在高级泛函分析、层理论和伪群的数学基础上，用于拓扑空间。向量场，并且提出者小组最近的工作通过理解实解析向量场的这种拓扑取得了重大进展，这是几何控制理论的一个特别重要的例子，有了这些泛函分析工具，就可以采用层理论的方法（对于几何控制理论）。向量场空间）和伪群理论（用于流空间）来表达控制系统表现出的微妙行为，这些工具迄今为止尚未在控制理论中得到广泛应用，因此为未来富有成效的工作提供了尚未探索的途径。然而，这项工作的目的是阐明应用问题，并最终根据新结果和技术揭示的深刻理解产生方法论。该提案是利用和培养高素质人才。