Geometric, topological, and stochastic approaches in nonlinear control theory

点击翻译按钮获取中文摘要

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.