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，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，再到医疗调度。 工作的主要重点是控制理论的一个子学科，称为几何控制理论。 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。 可控性的问题，粗略地说，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。 一个简单的例子是一个移动的机器人从一种配置转向到另一种配置。 可稳定性的问题是使用控制来使（可能不稳定的）行为稳定。 一个简单的例子是平衡一个摆在其直立配置。 最优性的问题是确定一个给定的行为是否最小化了一个规定的成本。 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完成重新配置。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。 这是这项工作的目的，了解这种局部结构的目的是来一个更完整的理解上述基本问题。 虽然现有的工作极大地促进了我们对这一结构的理解，但公平地说，全面理解仍然是可望而不可及的。 为了在这一认识上取得进展，拟议的研究使用了一个新的几何控制系统建模框架，一个建立在数学的高级泛函分析，层理论和伪群。 泛函分析是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，了解这种拓扑结构的真实的分析向量场，一个特别重要的情况下，几何控制理论。 有了这些功能分析工具，就有可能采用层理论（向量场空间）和伪群理论（流空间）的方法来表达控制系统所表现出的微妙行为。 这些工具迄今尚未被广泛应用于控制理论，因此提供了富有成效的未来工作的未探索的途径。这项建议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.