Ugo Boscain

Sub-Riemannian geometry is the geometry of a world with nonholonomic constraints. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other. In the last two decades sub-Riemannian geometry has emerged as an independent research domain impacting on several areas of pure and applied mathematics, with applications to many areas such as quantum control, Hamiltonian dynamics, robotics and Lie theory. This comprehensive introduction proceeds from classical topics to cutting-edge theory and applications, assuming only standard knowledge of calculus, linear algebra and differential equations. The book may serve as a basis for an introductory course in Riemannian geometry or an advanced course in sub-Riemannian geometry, covering elements of Hamiltonian dynamics, integrable systems and Lie theory. It will also be a valuable reference source for researchers in various disciplines.

This monograph is the first devoted to optimal syntheses in control theory and focuses on minimum time on 2-D manifolds. The first chapter outlines book results and examples of applicability. Then a quick introduction to geometric methods in control theory is provided. The following chapters contain a deep analysis of single input systems on 2-D manifolds including classifications of optimal syntheses and feedbacks, their singularities, extremals projection and minimum time singularities. Various extensions and applications are also illustrated. The volume is suitable for a graduate level one-semester course on optimal syntheses theory for mathematicians or engineers with a solid mathematical background.