Geometric, topological, and stochastic approaches in nonlinear control theory

非线性控制理论是控制理论的一个重要分支，在电网、航空航天系统、生物医学系统、纳米工程、过程控制、化学工程等领域有着广泛的应用。它的突出之处在于它的问题的丰富性和它所连接的数学领域的多样性。在后者中值得注意的是次黎曼几何，它在我们对非线性控制系统的理解中继续发挥着关键作用。由于线性控制理论将自己局限于一类狭隘且行为良好的控制系统，它拥有丰富而成熟的理论工具和技术，主要来自于线性泛函分析。相比之下，非线性控制理论远远没有达到这种成熟的程度，人们在设计新的工具和方法方面付出了巨大的努力。从第一个现代控制系统的出现到今天，控制理论的两个关键问题仍然是非常广泛的控制和镇定。虽然这些概念和它们之间的关系对于线性控制是很好的理解的，但对于非线性控制系统来说，情况还远远不是这样。我们研究计划的长期目标是，在非常广泛的条件下，理解非线性控制系统中能控性和可镇定性的深层本质以及它们之间的关系，就像目前线性控制理论中的情况一样。*在短期内，我们计划集中解决以下四个问题：*(A)定义与非线性控制系统相关的适当几何结构，计算该几何的局部不变量，并将这些不变量与控制系统的局部能控性和镇定性联系起来。*(B)通过研究与控制系统规范相关的拓扑结构来改进现有的非线性控制系统局部镇定的拓扑障碍。*(C)将李群上左不变次黎曼结构的亚椭圆热算子与这些几何的共轭结构联系起来。*(D)将Stein的经典概率逼近方法推广到亚椭圆热核短时间渐近的计算。*将为这项研究计划做出贡献的研究生和本科生将深入研究非常丰富的数学理论，从而获得出色的数学研究培训，以及为进入数学科学职业做好准备。**

Nonlinear control theory is an important branch of control theory, with applications in areas and industries as diverse as power grids, aerospace systems, biomedical systems, nano-engineering, process control, chemical engineering, and many others. It stands out by the richness of its problems and the diversity of mathematical areas that it connects to. Worthy of note among the latter is sub-Riemannian geometry, which continues to play a key role in our understanding of nonlinear control systems. By virtue of restricting itself to a narrow and well-behaved class of control systems, linear control theory enjoys a rich and well-established body of theoretical tools and techniques, drawn principally from linear functional analysis. By contrast, nonlinear control theory is far from enjoying this level of maturity, and intensive efforts have been deployed in devising new tools and approaches.***From the advent of the first modern control systems to the present day, the two key problems of control theory continue to be, very broadly, Control and Stabilization. Whereas these notions and the relations between them are very well understood for linear control, this is still far from being the case for nonlinear control systems. The long term objective of our research program is, in very broad terms, to understand the deep nature of and the relations between controllability and stabilizability in nonlinear control systems, to the same extent as is currently the case in linear control theory. ***In the shorter term, we plan to concentrate on the following four problems:***(a) Define the appropriate geometrical structure associated to a nonlinear control system, compute the local invariants of this geometry, and relate these invariants to local controllability and stabilizability properties of the control system.****(b) Refine the existing topological obstructions to the local stabilization of a nonlinear control system through the study of topological structures canonically associated to a control system.***(c) Relate the hypoelliptic heat operator of left-invariant sub-Riemannian structures on Lie groups to the conjugate structure of those geometries.***(d) Extend Stein's classical method of probabilistic approximation to the computation of hypoelliptic heat kernel short-time asymptotics. ***The graduate and undergraduate students who will contribute to this research program will delve into very rich mathematical theories, thereby acquiring an excellent training in mathematics research, as well as an excellent preparation for a career in the mathematical sciences.**