Contemporary Trends in Nonlinear Geometric Control Theory and Its Applications

by Anzaldo-Meneses, A.; Monroy-Perez, F.; Bonnard, B.; Gauthier, J. P.

• ISBN13: 9789810248413
• ISBN10: 9810248415
• Format: Hardcover
• Copyright: 2002-03-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Concerns contemporary trends in nonlinear geometric control theory and its applications.

Table of Contents

Foreword	xiii
Part I Invited Survey Chapters	1	(166)
Variational Problems on Lie Groups and their Homogeneous Spaces: Elastic Curves, Tops, and Constrained Geodesic Problems	3	(50)
V. Jurdjevic
F. Monroy-Perez
Introduction	3	(3)
Space forms and their frame bundles	6	(13)
Hamiltonians and the extremal curves	19	(34)
Controllability of Lie Systems	53	(24)
J. D. Lawson
D. Mittenhuber
Introduction	53	(1)
Control systems on Lie groups	54	(3)
Groups irrelevant for transitivity	57	(2)
Exploiting compactness and irrelevancy	59	(2)
Irrelevant groups and algebras	61	(3)
Irrelevant groups and algebras: the solvable case	64	(11)
Irrelevant groups and algebras: the semisimple case	75	(2)
Canonical Contact Systems for Curves: A Survey	77	(36)
W. Respondek
W. Pasillas-Lepine
Introduction	77	(3)
The canonical contact system for curves	80	(3)
History	83	(5)
Involutive subdistributions of corank one	88	(4)
Contact systems, characteristic distributions and involutive subdistributions	92	(10)
Flatness of contact systems	102	(3)
An example	105	(2)
Singular points and extended Kumpera-Ruiz normal forms	107	(6)
The Brachistochrone Problem and Modern Control Theory	113	(54)
H.J. Sussmann
J. C. Willems
Introduction	113	(5)
Johann Bernoulli and the brachistochrone problem	118	(5)
The standard formulation and Johann Bernoulli's solution	123	(5)
Spurious solutions and the calculus of variations approach	128	(3)
The optimal control approach	131	(5)
The differential-geometric connection	136	(14)
Five modern variations on the theme of the brachistochrone	150	(17)
Part II Contributed Chapters	167	(306)
Symplectic Methods for Strong Local Optimality in the Bangbang Case	169	(14)
A. A. Agrachev
G. Stefani
P. L. Zezza
Introduction	169	(2)
Main results	171	(5)
Sketch of the proof	176	(7)
Charges in Magnetic Fields and Sub-Riemannian Geodesics	183	(20)
A. Anzaldo-Meneses
F. Monroy-Perez
Introduction	183	(1)
Sub-Riemannian geometry and classical particles	184	(4)
Polynomial magnetic fields	188	(7)
Linear magnetic fields, and Cartan's five dimensional case	195	(8)
Topological Versus Smooth Linearization of Control Systems	203	(14)
L. Baratchart
J. B. Pomet
M. Chyba
Introduction	203	(3)
Preliminaries on equivalence of control systems	206	(2)
Main result on topological linearization	208	(3)
An open question	211	(2)
Implications in control theory	213	(4)
Local Approximation of the Reachable Set of Control Processes	217	(16)
R. M. Bianchini
Introduction	217	(1)
Tangent cones	218	(7)
Examples of g-variations	225	(3)
Applications	228	(2)
Open Problems	230	(3)
Geometric Optimal Control of the Atmospheric Arc for a Space Shuttle	233	(24)
B. Bonnard
E. Busvelle
G. Launay
Introduction	233	(2)
The model	235	(3)
The control problem	238	(2)
The minimal principle without state constraints-extremal curves	240	(9)
Optimal control with state constraints	249	(8)
High-Gain and Non-High-Gain Observers for Nonlinear Systems	257	(30)
E. Busvelle
J. P. Gauthier
Introduction, systems under consideration	257	(4)
Justification of the assumptions and observability	261	(3)
Statement and proof of the theoretical result	264	(9)
Application: observation of a binary distillation column	273	(10)
Appendix: Technical lemmas	283	(4)
Lie Systems in Control Theory	287	(18)
J.F. Carinena
A. Ramos
Introduction	287	(1)
Systems of differential equations admitting a superposition rule	288	(2)
Control and controllability of systems on Lie groups	290	(1)
The Wei-Norman method	291	(1)
Illustrative examples	292	(13)
From the Geometry to the Algebra of Nonlinear Observability	305	(42)
S. Diop
Introduction	305	(3)
The differential algebraic geometric approach	308	(11)
Computing	319	(28)
Appendix: Basic differential algebraic geometry	328	(6)
Appendix: Characteristic sets	334	(13)
Existence Theorems in Nonlinear Realization Theory and a Cauchy-Kowalewski Type Theorem	347	(12)
B. Jakubczyk
Introduction	347	(1)
Existence of analytic realizations	348	(3)
Convergence along vector fields and their commutators	351	(3)
Existence of analytic solutions of PDE's	354	(5)
Normality, Local Controllability and NOC for Multiobjective Optimal Control Problems	359	(22)
A. Jourani
Introduction	359	(4)
Background	363	(5)
Normality implies seminormality	368	(4)
Hamiltonian normality implies seminormality	372	(1)
Normality implies seminormality for systems of Mayer type	373	(2)
NOC for multiobjective optimal control problems	375	(6)
Controllability and Coordinates of the First Kind	381	(24)
M. Kawski
Introduction	381	(2)
Integral manifolds and controllability	383	(4)
Separating invariant flows from time-varying functionals	387	(4)
Coordinates of the 1st kind	391	(4)
Supporting hyperplanes	395	(10)
Variational Equations of Lagrangian Systems and Hamilton's Principle	405	(18)
H. N. Nunez-Yepez
J. Delgado
A. L. Salas-Brito
Introduction	405	(1)
The variational principle	406	(3)
Symmetries and constants of motion	409	(3)
Intrinsic formulation	412	(5)
Examples	417	(6)
Control of the Hovercraft Vessel: A Flatness plus Second Order Sliding Mode control Approach	423	(18)
H. Sira-Ramirez
Introduction	423	(1)
The hovercraft model	424	(5)
Trajectory tracking for the hovercraft system	429	(4)
Simulation results	433	(8)
Optimality of Singular Trajectories and Asymptotics of Accessibility Sets under Generic Assumptions	441	(18)
E. Trelat
Introduction	441	(2)
Optimality of singular trajectories	443	(8)
Asymptotics of the accessibility sets	451	(8)
Control Theory and Holomorphic Diffeomorphisms	459	(14)
D. Varolin
Introduction	459	(1)
Complex analytic considerations	460	(2)
Holomorphic vector fields	462	(4)
Jets	466	(7)
Index	473