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

这项研究与控制理论领域有关，该领域有许多应用领域，从机器人到化学过程，再到医疗计划。这项工作的主要重点是控制理论的一个分支学科，即几何控制理论。几何控制理论的工具对于研究由非线性常微分方程模型描述的控制系统是最有用的。对于非线性常微分方程模型，控制理论的许多基本问题仍未得到解答。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。可控性的问题，粗略地说，就是使用控制来引导系统从给定的初始状态到期望的最终状态。一个简单的例子是移动机器人从一种配置转向另一种配置。稳定性的问题是使用控制使（可能不稳定的）行为稳定。一个简单的例子是使摆摆保持直立状态的平衡。最优性问题是确定给定行为是否使规定成本最小化的问题。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。所有这三个问题的解决方案都依赖于对系统局部结构的详细理解。这项工作的目的是了解这种局部结构，目的是对上述基本问题有更全面的了解。虽然现有的工作对我们对这种结构的理解做出了很大的贡献，但公平地说，全面的理解仍然是难以捉摸的。为了在这种理解上取得进展，提出的研究使用了一种新的几何控制系统建模框架，该框架建立在高级功能分析、束理论和伪群的数学基础上。泛函分析被用于向量场空间的拓扑化，最近由提议者小组的工作通过理解实际解析向量场的拓扑结构取得了重大进展，这是几何控制理论的一个特别重要的案例。有了这些泛函分析工具，就可以使用束理论（用于向量场的空间）和伪群理论（用于流的空间）的方法来表达控制系统所表现出的微妙行为。这些工具迄今尚未在控制理论中广泛使用，因此为卓有成效的未来工作提供了未经探索的途径。该提案涉及一项长期的基础研究计划。然而，这项工作的目的是阐明应用问题，并最终产生基于新结果和新技术所揭示的深刻理解的方法。该建议的一个重要方面是利用和培训高素质的人员。

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.