Holdings: Geometric control theory

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Bibliographic Details
Main Author:	Jurdjevic, Velimir 1940- (Author)
Format:	Book
Language:	English
Published:	Cambridge [u.a.] Cambridge Univ. Press 2008
Edition:	Paperback re-issue
Series:	Cambridge studies in advanced mathematics 52
Subjects:	Control theory Geometry, Differential Kontrolltheorie Differentialgeometrie Konferenzschrift
Online Access:	Inhaltsverzeichnis
Physical Description:	XVIII, 492 S. Ill., graph. Darst.
ISBN:	9780521058247 9780521495028

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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 1 1 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 і .3 Proof of the orbit theorem 36 2 Lie brackets of vector fields and involutivity 40 2.1 Lie brackets of vector fields 40 ?.?- Lie algebras 42 2.3 Involutivity and integral manifolds 43 3 Analytic vector fields and their orbit properties 48 3.1 Lie groups 50 vn viii Contents 3.2 Group translations and invariant vector fields 51 3.3 Orbits of invariant vector fields 52 3.4 GL„ (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, <F) 65 1.2 Reachable sets of the form A?(x , 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 14 1 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. і 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 231 Notes and sources 258 9 Singular linear quadratic problems 2559 Î The structure of the strong Lie saturate 261 1.1 The structure of jump fields 267 і .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 IR 333 1.1 Background 333 1.2 The basic optimal problem and the maximum principie 335 1.3 The maximum principle and the classic necessary conditions for optimality 338 1.4 The minimal surface of revolution 34S 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 G x C* 369 1.2 The sympîectic 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 Ћ? and the kinetic analogue of Kirchhoff 384 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 S Оъ(Ш) 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 Ш3 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 Noninflectional solutions 477 3.3 Inflectional solutions 479 Notes and sources 481 References 483 Index 489
adam_txt	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 1 1 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 і .3 Proof of the orbit theorem 36 2 Lie brackets of vector fields and involutivity 40 2.1 Lie brackets of vector fields 40 ?.?- Lie algebras 42 2.3 Involutivity and integral manifolds 43 3 Analytic vector fields and their orbit properties 48 3.1 Lie groups 50 vn viii Contents 3.2 Group translations and invariant vector fields 51 3.3 Orbits of invariant vector fields 52 3.4 GL„ (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, <F) 65 1.2 Reachable sets of the form A?(x , 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 14 1 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. і 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 231 Notes and sources 258 9 Singular linear quadratic problems 2559 Î The structure of the strong Lie saturate 261 1.1 The structure of jump fields 267 і .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 IR" 333 1.1 Background 333 1.2 The basic optimal problem and the maximum principie 335 1.3 The maximum principle and the classic necessary conditions for optimality 338 1.4 The minimal surface of revolution 34S 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 G x C* 369 1.2 The sympîectic 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 Ћ? and the kinetic analogue of Kirchhoff 384 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 S Оъ(Ш) 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 Ш3 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 Noninflectional solutions 477 3.3 Inflectional solutions 479 Notes and sources 481 References 483 Index 489
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spelling	Jurdjevic, Velimir 1940- (DE-588)137500033 aut Geometric control theory Velimir Jurdjevic Paperback re-issue Cambridge [u.a.] Cambridge Univ. Press 2008 XVIII, 492 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 52 Control theory Geometry, Differential 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 52 (DE-604)BV000003678 52 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016509891&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 1940- Geometric control theory Cambridge studies in advanced mathematics Control theory Geometry, Differential 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_exact_search_txtP	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	Control theory Geometry, Differential Kontrolltheorie (DE-588)4032317-1 gnd Differentialgeometrie (DE-588)4012248-7 gnd
topic_facet	Control theory Geometry, Differential Kontrolltheorie Differentialgeometrie Konferenzschrift
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