Sub-Riemannian geometry and nonlinear control theory

在许多实际控制应用中，感兴趣的系统是非线性和欠驱动的。在大多数情况下，这些系统都是非完整的。这样的系统出现在广泛不同的环境中，从经典的机械系统，如汽车，到核磁共振成像中使用的量子力学系统。非完整控制的关键问题之一是各种应用的显式最优控制的计算。这是一个具有挑战性的问题，除了在一些非常特殊的情况下，还没有完全解决。因此，理解这些最优控制特性的基础研究，以及计算或近似它们的方法，是相当重要的。本课题的核心思想是通过共同研究非完整系统所对应的亚黎曼几何及其所定义的亚椭圆扩散来研究非完整系统。具体来说，我们想研究局部次黎曼不变量与最优控制之间的关系。作为起点，我们将研究亚黎曼接触结构的局部几何不变量。我们还将研究相应的亚椭圆扩散的性质，我们将从计算这些扩散的热核或参数开始，因为我们知道它们与最优控制问题的值函数密切相关。我们期望这项工作将有助于更深入地理解非完整系统和亚黎曼几何，特别是它们之间的关系。

In many practical control applications, the systems of interest are non-linear and under-actuated. In most cases of interest, these systems are furthermore non-holonomic. Such systems appear in widely distinct contexts, ranging from classical mechanical systems, such as cars, to the quantum mechanical systems used in Nuclear Magnetic Resonance Imaging. One of the key problems of non-holonomic control is the computation of explicit optimal controls for various applications. This is a challenging problem which has not been entirely settled except in some very special cases. Fundamental research into understanding the properties of these optimal controls, as well as ways to compute or approximate them, is then of considerable importance. The core idea of this research proposal is to study non-holonomic systems by jointly studying the corresponding sub-Riemannian geometry and sub-elliptic diffusion they define. Specifically, we want to investigate the relations between the local sub-Riemannian invariants and the optimal controls. As a starting point, we will study the local geometric invariants of sub-Riemannian contact structures. We will also investigate the properties of the corresponding sub-elliptic diffusions, and we will begin by computing heat kernels or parametrices for these diffusions, since we know they are intimately related to the value function of the optimal control problem. We expect this work will contribute to a deeper understanding of non-holonomic systems and sub-Riemannian geometry, and, in particular, of their relation.