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有关，这是一个具有许多应用领域的领域，从机器人技术到化学过程，工作的主要重点是在一个子-几何控制理论是控制理论的一门学科，几何控制理论的工具对于研究由常微分方程模型描述的控制系统是最有用的，线性。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。三个这样重要的基本问题是（1）可控性，（2）稳定性，和（3）最优性。可控性问题，粗略地说，利用控制将系统从给定的初始状态转向期望的最终状态。一个简单的例子是将移动的机器人从一种配置转向另一种配置。稳定性问题是使用控制来使一个（可能不稳定的）行为稳定。一个简单的例子是平衡一个摆在它的直立配置。最优性问题是确定一个给定的行为是否最小化规定的成本。一个简单的例子是上面提到的移动的机器人转向问题，但现在要求在使用尽可能少的能量的情况下进行重新配置。所有这三个问题的解决都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，以便更全面地理解上述基本问题。虽然现有的工作极大地促进了我们对这种结构的理解，公平地说，全面的理解仍然是难以捉摸的。为了在这一理解上取得进展，拟议的研究使用了一个新的几何控制系统建模框架，一个建立在高级泛函分析数学，层理论，和伪群。泛函分析被用来拓扑空间的向量场，最近的工作由提案人的小组取得了重大进展，通过理解这种拓扑结构的真实的分析向量场，几何控制理论的一个特别重要的案例。有了这些功能分析工具，就有可能运用层理论的方法（向量场空间）和伪群理论（对于流的空间）来表达控制系统所表现出的微妙行为。这些工具迄今为止还没有被广泛用于控制理论，从而为未来富有成效的工作提供了未经探索的途径。这是一项长期的基础研究计划，但其目的是揭示应用问题，并最终在新成果和新技术的深刻理解的基础上产生方法论。这一计划的一个重要方面是使用和培养高素质的人才。