Geometric Control Theory [Paperback]

A modern version of the calculus of variations, encompassing geometric mechanics, differential geometry, and optimal control.Geometric control theory concerns the differential equations described by non-commuting vector fields. It builds ideas from the theory of differential systems and the calculus of variations into a cohesive mathematical framework applicable to a wide range of problems from differential geometry, applied mathematics, physics and engineering.Geometric control theory concerns the differential equations described by non-commuting vector fields. It builds ideas from the theory of differential systems and the calculus of variations into a cohesive mathematical framework applicable to a wide range of problems from differential geometry, applied mathematics, physics and engineering.This book describes the mathematical theory inspired by the irreversible nature of time-evolving events. The first part of the book deals with the ability to steer a system from any point of departure to any desired destination. The second part deals with optimal control--the problem of finding the best possible course. The author demonstrates an overlap with mathematical physics using the maximum principle, a fundamental concept of optimality arising from geometric control, which is applied to time-evolving systems governed by physics as well as to man-made systems governed by controls. He draws applications from geometry, mechanics, and control of dynamical systems. The geometric language in which the author expresses the results allows clear visual interpretations and makes the book accessible to physicists and engineers as well as to mathematicians.Introduction; Acknowledgments; Part I. Reachable Sets and Controllability: 1. Basic formalism and typical problems; 2. Orbits of families of vector fields; 3. Reachable sets of Lie-determined systems; 4. Control affine systems; 5. Linear and polynomial control systems; 6. Systems on Lie groups and homogenous spaces; Part Ilc¿