Geometric control theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
1997
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Cambridge studies in advanced mathematics
51 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 492 S. |
| ISBN: | 0521495024 |
Internformat
MARC
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| 100 | 1 | |a Jurdjevic, Velimir |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Geometric control theory |c Velimir Jurdjevic |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1997 | |
| 300 | |a XVIII, 492 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Cambridge studies in advanced mathematics |v 51 | |
| 515 | |a Fälschlich als Bd. 51 bezeichnet | ||
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Datensatz im Suchindex
| _version_ | 1804125842767347712 |
|---|---|
| adam_text | Contents
Introduction page xiii
Acknowledgments xvii
Part one: Reachable sets and controllability 1
1 Basic formalism and typical problems 3
1 Differentiable manifolds 3
1.1 Differentiable mappings 6
1.2 The tangent space 7
1.3 The cotangent space 8
2 Vector fields, flows, and differential forms 11
2.1 Derivations 11
2.2 The tangent bundle and cotangent bundle 12
2.3 Vector fields and differential forms 14
3 Control systems 20
3.1 Families of vector fields and control systems 27
Notes and sources 31
2 Orbits of families of vector fields 32
1 The orbit theorem 32
1.1 Submanifolds 33
1.2 Integral submanifolds 35
1.3 Proof of the orbit theorem 36
2 Lie brackets of vector fields and involutivity 40
2.1 Lie brackets of vector fields 40
2.2 Lie algebras 42
2.3 Involutivity and integral manifolds 43
3 Analytic vector fields and their orbit properties 48
3.1 Lie groups 50
vii
viii Contents
3.2 Group translations and invariant vector fields 51
3.3 Orbits of invariant vector fields 52
3.4 GLn(R) and its subgroups 54
3.5 Homogeneous spaces 55
4 Zero time orbits of families of vector fields 56
4.1 Zero time orbits of analytic vector fields 59
Notes and sources 63
3 Reachable sets of Lie determined systems 64
1 Topological properties of reachable sets 65
1.1 Reachable sets of the form A?(x, T) 65
1.2 Reachable sets of the form AAx, T) 70
2 The closure of the reachable sets and its invariants 75
2.1 Closure and convexification of families of vector fields 76
2.2 Time scaling and normalizers 82
3 The Lie saturate and controllability 86
4 Exact time controllability 89
Notes and sources 94
4 Control affine systems 95
1 Kinematic equations of a rolling sphere 96
2 Linear systems 100
3 Control of a rigid body by means of jet torques 103
4 Reachability by piecewise constant controls 105
5 Reachability by smooth controls 110
6 Recurrent drifts and control of a rigid body 112
7 Compact constraints and the closure of the reachable sets 117
8 Non holonomic aspects of control theory 121
Notes and sources 124
5 Linear and polynomial control systems 125
1 Feedback, controllability, and the structure of linear systems 126
1.1 Controllability indices and the feedback decomposition
theorem 129
1.2 Controllability and the spectrum 133
2 Bounded controls and the bang bang principle 134
3 Controllability of linear systems with bounded controls 137
4 Polynomial drifts 141
4.1 Homogeneous polynomial vector fields and their Lie
algebras 143
4.2 Controllability 147
Notes and sources 148
Contents ix
6 Systems on Lie groups and homogeneous spaces 150
1 Families of right invariant vector fields on a Lie group 153
1.1 Compact Lie groups 154
1.2 Orthogonal and symplectic groups and the unitary group 159
1.3 Stiff Serret Frenet frames 162
1.4 The Grassmann manifolds 163
1.5 Motions of a sphere rolling on another sphere 165
1.6 Quaternions and rotations 169
2 Semidirect products of Lie groups 177
3 Controllability properties of affine systems 182
4 Controllability on semisimple Lie groups 185
Notes and sources 193
Part two: Optimal control theory 195
7 Linear systems with quadratic costs 199
1 Assumptions and their consequences 200
1.1 Optimality and the boundaries of the reachable sets 202
2 The maximum principle 204
2.1 Canonical coordinates and Hamiltonian vector fields 204
2.2 Necessary and sufficient conditions of optimality 207
2.3 The Euler Lagrange equation 214
3 Conjugate points for the regular problem 216
4 Applications: Wirtinger s inequality and Hardy Littlewood
systems 219
Notes and sources 226
8 The Riccati equation and quadratic systems 228
1 Symplectic vector spaces 230
1.1 The geometry of linear Lagrangians 232
2 Lagrangians and the Riccati equation 234
3 The algebraic Riccati equation 241
4 Infinite horizon optimal problems 244
5 Hardy Littlewood inequalities 251
Notes and sources 258
9 Singular linear quadratic problems 259
1 The structure of the strong Lie saturate 261
1.1 The structure of jump fields 267
1.2 The saturated linear quadratic system 272
2 The maximum principle and its consequences 275
3 The reduction procedure 283
x Contents
4 The optimal synthesis 290
Notes and sources 299
10 Time optimal problems and Fuller s phenomenon 300
1 Linear time optimal problems: the maximum principle 301
1.1 Time optimal control of linear mechanical systems 307
2 The brachistochrone problem and Zermelo s navigation
problem 311
3 Linear quadratic problems with constraints, and Fuller s
phenomenon 319
Notes and sources 331
11 The maximum principle 332
1 The maximum principle in W 333
1.1 Background 333
1.2 The basic optimal problem and the maximum principle 335
1.3 The maximum principle and the classic necessary
conditions for optimality 338
1.4 The minimal surface of revolution 345
2 Extensions to differentiable manifolds 348
2.1 The symplectic structure of the cotangent bundle 349
2.2 Variational problems on manifolds and the maximum
principle 354
2.3 Euler s elastic problem and the problem of Dubins 360
Notes and sources 367
12 Optimal problems on Lie groups 368
1 Hamiltonian vector fields 369
1.1 Realization of the cotangent bundle as the product GxC* 369
1.2 The symplectic form 370
2 The rigid body and the equations for the heavy top 372
3 Left invariant control systems and co adjoint orbits 379
4 The elastic problem in M3 and the kinetic analogue of Kirchhoff 3 84
5 Casimir functions and the conservation laws 394
5.1 Left invariant optimal problems on the group of motions
of a plane 398
5.2 Left invariant optimal problems on SO^(R) and SO(2,
1) 401
Notes and sources 406
13 Symmetry, integrability, and the Hamilton
Jacobi theory 407
1 Symmetry, Noether s theorem, and the maximum principle 409
Contents xi
2 The geometry of Lagrangian manifolds and the Hamilton
Jacobi theory 414
3 Integrability 424
3.1 Integrable systems on the Heisenberg group 424
3.2 Integrable systems on the group of motions of a plane 430
3.3 Integrability on SO3(K) 436
Notes and sources 443
14 Integrable Hamiltonian systems on Lie groups: the elastic
problem, its non Euclidean analogues, and the rolling sphere
problem 444
1 The symmetric elastic problem in K3 446
1.1 Euler angles and elastic curves 453
2 Non Euclidean symmetric elastic problems 456
2.1 Algebraic preliminaries 456
2.2 The structure of extremal curves 458
2.3 The Kowalewski elastic problem 465
3 Rolling sphere problems 467
3.1 The extremals for the rolling sphere in E2 468
3.2 Noninfiectional solutions 477
3.3 Inflectional solutions 479
Notes and sources 481
References 483
Index 489
|
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515.64 |
| dewey-search | 515.64 |
| dewey-sort | 3515.64 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| physical | XVIII, 492 S. |
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| spelling | Jurdjevic, Velimir Verfasser aut Geometric control theory Velimir Jurdjevic 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1997 XVIII, 492 S. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 51 Fälschlich als Bd. 51 bezeichnet Kontrolltheorie (DE-588)4032317-1 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf 1\p (DE-588)1071861417 Konferenzschrift gnd-content Differentialgeometrie (DE-588)4012248-7 s Kontrolltheorie (DE-588)4032317-1 s DE-604 Cambridge studies in advanced mathematics 51 (DE-604)BV000003678 51 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007615478&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Jurdjevic, Velimir Geometric control theory Cambridge studies in advanced mathematics Kontrolltheorie (DE-588)4032317-1 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4032317-1 (DE-588)4012248-7 (DE-588)1071861417 |
| title | Geometric control theory |
| title_auth | Geometric control theory |
| title_exact_search | Geometric control theory |
| title_full | Geometric control theory Velimir Jurdjevic |
| title_fullStr | Geometric control theory Velimir Jurdjevic |
| title_full_unstemmed | Geometric control theory Velimir Jurdjevic |
| title_short | Geometric control theory |
| title_sort | geometric control theory |
| topic | Kontrolltheorie (DE-588)4032317-1 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Kontrolltheorie Differentialgeometrie Konferenzschrift |
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