Geometric Control Theory

A modern version of the calculus of variations, encompassing geometric mechanics, differential geometry, and optimal control.

Velimir Jurdjevic (Author)

9780521495028, Cambridge University Press

Hardback, published 28 December 1996

512 pages, 81 b/w illus.
22.9 x 15.2 x 3.3 cm, 0.92 kg

Review of the hardback: '… an important reference for graduate students and mathematicians … well written, almost self-contained, and easy to read.' M. F. Silva Leite, Zentralblatt MATH

Geometric control theory is concerned with the evolution of systems subject to physical laws but having some degree of freedom through which motion is to be controlled. This book describes the mathematical theory inspired by the irreversible nature of time evolving events. The first part of the book deals with the issue of being able to steer the system from any point of departure to any desired destination. The second part deals with optimal control, the question of finding the best possible course. An overlap with mathematical physics is demonstrated by the Maximum principle, a fundamental principle 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. Applications are drawn from geometry, mechanics, and control of dynamical systems. The geometric language in which the results are expressed 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 II. Optimal Control Theory: 7. Linear systems with quadratic costs
8. The Riccati equation and quadratic systems
9. Singular linear quadratic problems
10. Time-optimal problems and Fuller's phenomenon
11. The maximum principle
12. Optimal problems on Lie groups
13. Symmetry, integrability and the Hamilton-Jacobi theory
14. Integrable Hamiltonian systems on Lie groups: the elastic problem, its non-Euclidean analogues and the rolling-sphere problem
References
Index.

Subject Areas: Optimization [PBU]