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

这项研究是与控制理论领域相联系的，控制理论是一个应用领域很广的领域，从机器人到化学过程，再到医疗调度。工作的主要焦点是控制理论的一个子学科，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答案。这三个重要的基本问题是(1)可控性，(2)稳定性，和(3)最优性。粗略地说，就是使用控制将系统从给定的初始状态引导到期望的最终状态。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来使(可能不稳定的)行为稳定。一个简单的例子是直立摆的平衡。最优的问题是确定给定的行为是否使规定的成本最小化。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了解。这项工作的目的是了解这种局部结构，目的是更全面地理解上述基本问题。虽然现有的工作对我们对这种结构的理解做出了很大贡献，但公平地说，全面的理解仍然难以实现。为了在这种理解上取得进展，拟议的研究使用了一种新的几何控制系统的建模框架，该框架建立在高级泛函分析、鞘理论和伪群的数学基础上。泛函分析用于拓扑矢量场的空间，作者的团队最近的工作取得了重大进展，通过理解实解析向量场的这种拓扑学，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，就有可能使用层理论(对于向量场空间)和伪群理论(对于流空间)的方法来表达控制系统所表现出的微妙行为。这些工具到目前为止还没有被广泛应用于控制理论，因此为富有成效的未来工作提供了尚未探索的途径。该提案涉及一个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，并最终基于新成果和技术所揭示的深刻理解而产生方法。该提案的一个重要方面是使用和培训高素质的人才。

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.