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

点击翻译按钮获取中文摘要

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.