Singular trajectories and their role in control theory:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Paris [u.a.]
Springer
2003
|
| Series: | Mathématiques & applications
40 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVI, 357 S. graph. Darst. |
| ISBN: | 3540008381 |
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| 245 | 1 | 0 | |a Singular trajectories and their role in control theory |c Bernard Bonnard ; Monique Chyba |
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| 490 | 1 | |a Mathématiques & applications |v 40 | |
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| adam_text | IMAGE 1
BERNARD BONNARD
MONIQUE CHYBA
SINGULAR TRAJECTORIES
AND THEIR ROLE IN CONTROL THEORY
SPRINGER
IMAGE 2
TABLE OF CONTENTS
1 LINEAR SYSTEMS AND THE TIME OPTIMAL CONTROL PROBLEM 1
1.1 LINEAR SYSTEMS 1
1.2 ACCESSIBILITY SET AND CONTROLLABILITY 1
1.3 CONTROLLABILITY AND FEEDBACK CLASSIFICATION IN THE AUTONOMOUS CASE 2
1.3.1 CONTROLLABILITY 2
1.3.2 LINEAR CLASSIFICATION 3
1.3.3 FEEDBACK CLASSIFICATION 5
1.3.4 STABILIZATION 6
1.4 CONTROLLABILITY IN THE NONAUTONOMOUS CASE 7
1.4.1 APPLICATION 8
1.5 TIME OPTIMAL CONTROL FOR LINEAR SYSTEMS 9
1.5.1 NOTATIONS 9
1.6 FEEDBACK EQUIVALENCE TO A LINEAR SYSTEM 15
1.6.1 PROBLEM STATEMENT 15
1.7 TIME MINIMAL SYNTHESIS 17
1.7.1 NOTION OF SYNTHESIS 17
1.7.2 THE GENERAL ALGORITHM IN THE PLANE 21
EXERCISES 24
2 OPTIMAL CONTROL FOR NONLINEAR SYSTEMS 27
2.1 A SHORT VISIT INTO THE CLASSICAL CALCULUS OF VARIATIONS 27
2.1.1 STATEMENT OF THE PROBLEM IN THE HOLONOMIC CASE 27
2.1.2 HAMILTONIAN EQUATIONS 29
2.1.3 HAMILTON-JACOBI-BELLMAN EQUATION 30
2.1.4 EULER-LAGRANGE EQUATIONS AND CHARACTERISTICS OF THE HJB EQUATION
31
2.1.5 SECOND ORDER CONDITIONS 31
2.1.6 THE ACCESSORY PROBLEM AND THE JACOBI EQUATION 32
2.1.7 CONJUGATE POINT AND LOCAL MORSE THEORY 33
2.1.8 SCALAR RICCATI EQUATION 34
2.1.9 LOCAL C MINIMIZER - EXTREMAL FIELD - HUBERT INVARI ANT INTEGRAL
35
2.2 OPTIMAL CONTROL AND THE CALCULUS OF VARIATIONS 36
IMAGE 3
X TABLE OF CONTENTS
2.2.1 PROBLEM STATEMENT 36
2.2.2 THE AUGMENTED SYSTEM 37
2.2.3 RELATED PROBLEMS 38
2.2.4 OPTIMAL CONTROL AND THE CLASSICAL CALCULUS OF VARIATIONS 38 2.2.5
SINGULAR TRAJECTORIES AND THE WEAK MAXIMUM PRINCIPLE 39 2.2.6 FIRST AND
SECOND VARIATIONS OF E X T 39
2.2.7 GEOMETRIE INTERPRETATION OF THE ADJOINT VECTOR 42
2.2.8 THE WEAK MAXIMUM PRINCIPLE 42
2.2.9 ABNORMALITY 43
2.2.10 THE WEAK MAXIMUM PRINCIPLE AND EULER-LAGRANGE EQUATION 43
2.2.11 COMPARISON WITH THE CALCULUS OF VARIATIONS 44
2.2.12 LQ-CONTROL AND THE WEAK MAXIMUM PRINCIPLE 44
2.3 PONTRYAGIN S MAXIMUM PRINCIPLE (PMP) 45
2.4 FILIPPOV EXISTENCE THEOREM 52
2.4.1 COMMENTS ABOUT THE EXISTENCE THEOREM 52
2.5 DYNAMIC PROGRAMMING AND THE MAXIMUM PRINCIPLE 53
EXERCISES 56
3 GEOMETRIE OPTIMAL CONTROL 65
3.1 INTRODUCTION TO SYMPLECTIC GEOMETRY 65
3.2 GEOMETRIE CLASSIFICATION OF EXTREMAIS IN ONE DIMENSIONAL PROBLEMS OF
CALCULUS OF VARIATIONS 69
3.2.1 PROBLEM STATEMENT AND PRELIMINARIES 69
3.2.2 SINGULARITY ANALYSIS 71
3.2.3 GENERIC CLASSIFICATION NEAR .EI 72
3.2.4 CLASSIFICATION AND EXISTENCE OF OPTIMAL SOLUTIONS 75
3.3 TIME MINIMUM CONTROL PROBLEM 76
3.4 DETERMINATION OF THE SINGULAR EXTREMAIS 77
3.4.1 HAMILTONIAN FORMALISM AND SINGULAR EXTREMAIS 78
3.5 GEOMETRIE CLASSIFICATION OF EXTREMAIS NEAR EI 79
3.5.1 NORMAL SWITCHING POINTS 79
3.5.2 THE FOLD CASE 80
3.6 COMPARAISON BETWEEN THE CALCULUS OF VARIATIONS AND THE TIME MINIMUM
PROBLEM FOR AFFINE SYSTEMS. THE ROLE OF SIN GULAR EXTREMAIS 82
3.7 THE FUELLER PHENOMENON 82
3.7.1 FUELLER EXAMPLE 82
3.8 GEOMETRY OF THE TIME OPTIMAL CONTROL IN THE PLANE 84
3.8.1 PRELIMINARIES 84
3.8.2 GENERIC POINTS 85
3.8.3 SINGULAR ARC 86
3.8.4 ELLIPTIC CASE - THE CONCEPT OF CONJUGATE POINT 88
EXERCISES 94
IMAGE 4
TABLE OF CONTENTS XI
SINGULAR TRAJECTORIES AND FEEDBACK CLASSIFICATION 97
4.1 CLASSIFICATION OF AFFINE SYSTEMS 99
4.1.1 COMPUTATIONS OF SINGULAR CONTROLS 99
4.1.2 SINGULAR TRAJECTORIES AND FEEDBACK CLASSIFICATION 102
4.1.3 TIME OPTIMALITY AND FEEDBACK CLASSIFICATION 105
4.2 SINGULAR TRAJECTORIES AND THE PROBLEM OF CLASSIFICATION OF
DISTRIBUTIONS 108
4.2.1 PRELIMINARIES 108
4.2.2 LOCAL CLASSIFICATION IN DIMENSION 3, WITH RANK D = 2 . 109 4.3
FEEDBACK CLASSIFICATION AND ANALYTIC GEOMETRY 111
4.3.1 PRELIMINARIES 111
4.3.2 CRITICAL HAMILTONIANS AND SYMBOLS 112
EXERCISES 113
CONTROUABILITY - HIGHER ORDER MAXIMUM PRINCIPLE - LEGENDRECLEBSCH AND
GOH NECESSARY OPTIMALITY CONDITIONS 117
5.1 SOME NOTATIONS AND FORMULAS FROM DIFFERENTIAL GEOMETRY . .. 117
5.1.1 CONTROUABILITY WITH PIECEWISE CONSTANT CONTROLS 119
5.2 INTEGRATING DISTRIBUTIONS 120
5.2.1 PRELIMINARIES 120
5.2.2 FROBENIUS THEOREM 120
5.2.3 NAGANO-SUSSMANN THEOREM 121
5.2.4 C-COUNTER EXAMPLE 122
5.3 NONLINEAR CONTROUABILITY AND CHOW THEOREM 122
5.4 POISSON STABILITY AND CONTROUABILITY 124
5.4.1 APPLICATION 125
5.5 CONTROUABILITY AND ENLARGEMENT TECHNIQUE 125
5.5.1 PROBLEM STATEMENT 125
5.6 EVALUATION OF THE ACCESSIBILITY SET 127
5.6.1 LEGENDRE-CLEBSCH CONDITION 129
5.7 THE MULTI-INPUTS CASE: GOH CONDITION 131
5.8 THE CONCEPT OF RIGIDITY - STRONG LEGENDRE CLEBSCH AND GOH CONDITIONS
AS NECESSARY CONDITIONS FOR RIGIDITY 132
5.8.1 INTRINSIC SECOND VARIATION - MORSE INDEX 133
EXERCISES 134
THE CONCEPT OF CONJUGATE POINTS IN THE TIME MINIMAL CONTROL PROBLEM FOR
SINGULAR TRAJECTORIES, C-OPTIMALITY . . .. 143 6.1 SINGLE-INPUT CASE
143
6.1.1 PRELIMINARIES 143
6.1.2 FEEDBACK SEMI-NORMAL FORMS IN THE HYPERBOLIC AND ELLIPTIC CASES
146
6.1.3 LQ-MODEL 147
6.1.4 APPROXIMATION OF THE END-POINT MAPPING USING THE LQ-MODEL IN THE
ELLIPTIC CASE 148
IMAGE 5
TABLE OF CONTENTS
6.1.5 CONCLUSION 153
6.1.6 THE EXCEPTIONAL CASE 153
6.1.7 CONCLUSION 155
6.1.8 COMPARISON OF THE ELLIPTIC-HYPERBOLIC CASE AND THE EXCEPTIONAL
CASE 156
6.2 APPLICATIONS 157
6.2.1 TIME OPTIMAL SYNTHESIS FOR PLANAR SYSTEM 157
6.2.2 EXAMPLE IN DIMENSION 3 AND CONNECTION WITH THE
HAMILTON-JACOBI-BELLMAN EQUATION 157
6.3 THE CASE IN R 3 . INTRINSIC COMPUTATIONS OF CONJUGATE POINTS -
CONNECTION WITH THE TIME MINIMAL SYNTHESIS PROBLEM. THE CONCEPT OF
CURVATURE 160
6.3.1 EULER-LAGRANGE EQUATION 160
6.3.2 GEOMETRIE INTERPRETATION 161
6.3.3 INTRINSIC COMPUTATION 162
6.3.4 THE CONCEPT OF CURVATURE 162
6.3.5 CONJUGATE POINTS AND TIME MINIMAL SYNTHESIS 163
6.4 CONNECTION WITH THE LIU-SUSSMANN EXAMPLE 164
6.4.1 PRELIMINARIES 164
6.4.2 CONCLUSION 166
6.4.3 STATEMENT AND PROOF OF LIU-SUSSMANN RESULT 166
6.4.4 CONCLUSION 167
6.4.5 CONNECTION WITH THE SUB-RIEMANNIAN GEOMETRY 168
TIME MINIMAL CONTROL OF CHEMICAL BATCH REACTORS AND SIN GULAR
TRAJECTORIES 171
7.1 INTRODUCTION 171
7.2 MATHEMATICAL MODEL OF CHEMICAL BATCH REACTORS AND DESCRIPTION OF THE
CONTROL PROBLEM 172
7.2.1 CHEMICAL KINETICS 172
7.2.2 CONTROL DEVICE 174
7.2.3 THE OPTIMAL CONTROL PROBLEM 175
7.2.4 PROJECTED PROBLEM 175
7.3 SINGULAR EXTREMAIS - CURVATURE - CONJUGATE POINTS 176
7.3.1 PROJECTED SYSTEM 177
7.4 TIME MINIMAL SYNTHESIS FOR PLANAR SYSTEMS IN THE NEIGHBORHOOD OF A
TERMINAL MANIFOLD OF CODIMENSION ONE 180
7.4.1 PROBLEM STATEMENT 180
7.4.2 ASSUMPTION CL 181
7.4.3 THE GENERIC CL CASE ,181
7.4.4 THE GENERIC CL FIAT CASE 181
7.4.5 THE CASE CL OF CODIMENSION ONE 182
7.4.6 GENERIC HYPERBOLIC CASES 185
7.4.7 GENERIC EXCEPTIONAL CASE 188
7.4.8 GENERIC FIAT EXCEPTIONAL CASE 189
IMAGE 6
TABLE OF CONTENTS XIII
7.5 GLOBAL TIME MINIMAL SYNTHESIS 191
7.5.1 PRELIMINARIES 191
7.5.2 SINGULAR ARE 191
7.5.3 REGULAER ARES 192
7.5.4 OPTIMAL SYNTHESIS IN THE NEIGHBORHOOD OF AT 193
7.5.5 SWITCHING RULES 193
7.5.6 OPTIMAL SYNTHESIS 194
7.6 STATE CONSTRAINTS DUE TO THE TEMPERATURE 195
7.6.1 PRELIMINARIES 195
7.6.2 OPTIMAL SYNTHESIS 198
7.7 THE PROBLEM IN DIMENSION 3 199
7.7.1 PRELIMINARIES 199
7.7.2 STRATIFICATION OF N BY THE OPTIMAL FEEDBACK SYNTHESIS 199 7.7.3
ORIENTATION PRINCIPLE 200
7.7.4 SWITCHING RULES 201
7.7.5 LOCAL CLASSIFICATION NEAR THE TARGET 203
7.7.6 FOCAL POINTS 206
8 GENERIC PROPERTIES OF SINGULAR TRAJECTORIES 209
8.1 INTRODUCTION AND NOTATIONS 209
8.2 DETERMINATION OF THE SINGULAR EXTREMAIS WITH MINIMAL ORDER 210 8.3
STATEMENT OF THE FIRST GENERIC PROPERTY 211
8.4 GEOMETRIE INTERPRETATION AND THE GENERAL CONCEPT OF ORDER . 2 11 8.5
PROOF OF THEOREM 25 213
8.5.1 PARTIALLY ALGEBRAIC AND SEMI-ALGEBRAIC FIBER BUNDLES. 217 8.5.2
COORDINATE SYSTEMS ON P(D, N) 218
8.5.3 EVALUATION OF CODIMENSION OF THE F{N) 218
8.5.4 END OF THE PROOF OF THEOREM 25 222
8.6 GENERICITY OF CODIMENSION ONE SINGULARITY 222
8.7 PROOF OF THEOREM 26 222
8.7.1 THE BAD SET FOR THEOREM 26 223
8.7.2 EVALUATION OF THE CODIMENSION OF B C (N, Q) 224
8.8 SINGULARITIES OF THE SINGULAR FLOW OF MINIMAL ORDER 226
8.8.1 PRELIMINARIES (SEE SECT. 4.1.3) 226
8.8.2 LOCAL CLASSIFICATION NEAR C 227
8.8.3 LOCAL CLASSIFICATION NEAR S C 228
8.8.4 THE QUADRATIC CASE 229
EXERCISES 231
9 SINGULAR TRAJECTORIES IN SUB-RIEMANNIAN GEOMETRY 233
9.1 INTRODUCTION 233
9.2 GENERALITIES ABOUT SR-GEOMETRY 234
9.2.1 DEFINITION 234
9.2.2 OPTIMAL CONTROL THEORY FORMULATION 234
IMAGE 7
XIV TABLE OF CONTENTS
9.2.3 COMPUTATIONS OF THE EXTREMAIS AND EXPONENTIAL MAPPING 235
9.3 RESEARCH PROGRAM IN SR-GEOMETRY 239
9.3.1 CLASSIFICATION 239
9.3.2 SINGULARITY THEORY OF THE EXPONENTIAL MAPPING AND OF THE DISTANCE
FUNCTION 239
9.4 PRIVILEGED COORDINATES AND GRADED NORMAL FORMS 240
9.4.1 REGULAER AND SINGULAR POINTS 240
9.4.2 ADAPTED AND PRIVILEGED COORDINATES 241
9.4.3 NILPOTENT APPROXIMATION 242
9.4.4 GRADED APPROXIMATION 242
9.5 THE CONTACT CASE OF ORDER -1 OR THE HEISENBERG-BROCKETT EXAMPLE 242
9.5.1 THE CONTACT CASE IN K 3 242
9.5.2 SYMMETRY GROUP IN THE HEISENBERG CASE 243
9.5.3 HEISENBERG SR-GEOMETRY AND THE DIDO PROBLEM 243
9.5.4 GEODESICS 244
9.5.5 CONJUGATE POINTS 245
9.5.6 SPHERE AND WAVE FRONT 245
9.5.7 CONCLUSION ABOUT THE SR-HEISENBERG GEOMETRY 246
9.6 THE GENERIC CONTACT CASE 247
9.6.1 NORMAL FORMS IN THE CONTACT CASE 247
9.6.2 GENERIC CONJUGATE LOCUS 248
9.7 THE MARTINET CASE 249
9.7.1 PRELIMINARIES 249
9.7.2 NORMAL FORM 250
9.7.3 ORTHONORMAL FRAME 253
9.7.4 GRADED NORMAL FORM 253
9.7.5 GEODESICS 254
9.7.6 RIEMANNIAN METRIE ON THE PLANE (X, Y) 255
9.7.7 ASYMPTOTIC FOLIATION ASSOCIATED TO THE NORMAL FORM AT ORDER 0 256
9.7.8 PROPERTIES OF THE ASYMPTOTIC FOLIATIONS 257
9.7.9 INTEGRABLE CASE OF ORDER 0 AND ELLIPTIC INTEGRALS 258
9.7.10 THE MARTINET FIAT CASE 262
9.8 ESTIMATES OF THE SPHERE AND OF THE WAVE FRONT NEAR THE AB NORMAL
DIRECTION IN THE FIAT CASE AND EXP -IN CATEGORY . . .. 266 9.8.1
INTERSECTION OF 5(0, R) WITH THE CUT LOCUS 266
9.9 CONCLUSION DEDUCED FROM THE MARTINET SR-FLAT GEOMETRY CONCERNING THE
ROLE OF ABNORMAL GEODESICS IN SR-GEOMETRY . 268 9.9.1 BEHAVIORS OF THE
NORMAL GEODESICS NEAR THE ABNORMAL DIRECTION - GEODESIC C-RIGIDITY 268
9.9.2 NONPROPERNESS OF THE FIRST RETURN MAPPING R AND GEOMETRIE
CONSEQUENCE 269
IMAGE 8
TABLE OF CONTENTS XV
9.10 CUT LOCUS IN MARTINET SR-GEOMETRY 271
EXERCISES 273
10 MICRO-LOCAL RESOLUTION OF THE SINGULARITY NEAR A SINGULAR TRAJECTORY
- LAGRANGIAN MANIFOLDS AND SYMPLECTIC STRATIFICATIONS 281
10.1 INTRODUCTION 281
10.2 LAGRANGIAN MANIFOLDS 281
10.3 APPLICATION TO CLASSICAL CALCULUS OF VARIATIONS 282
10.4 SINGULARITY THEORY OF THE GENERATING FUNCTION - GENERATING FAMILY
283
10.5 APPLICATION TO OPTIMAL CONTROL THEORY AND TO SR-GEOMETRY 285 10.5.1
THE SR-NORMAL CASE 285
10.5.2 JACOBI FIELDS 287
10.5.3 REDUCTION ALGORITHM IN THE SR-CASE 288
10.6 THE SINGULAR CASE 289
10.6.1 A GEOMETRIE REMARK 289
10.6.2 HEISENBERG CASE 289
10.6.3 MARTINET FIAT CASE 289
10.7 RESOLUTION OF THE SINGULARITY IN THE HYPERBOLIC CASE 292
10.7.1 NORMAL FORM 292
10.7.2 INTRINSIC SECOND-ORDER DERIVATIVE 293
10.7.3 GOH TRANSFORMATION 294
10.8 LAGRANGIAN MANIFOLDS IN THE HYPERBOLIC CASE 294
10.9 EXAMPLES 296
10.10 RESOLUTION OF THE SINGULARITY IN THE EXCEPTIONAL CASE 302
10.10.1 NORMAL FORM - INTRINSIC DERIVATIVE 302
10.10.2 THE OPERATORS 303
10.10.3 ISOTROPIE MANIFOLDS IN THE EXCEPTIONAL CASE 303
10.11 MICRO-LOCAL ANALYSIS OF SR-MARTINET BALL OF SMALL RADIUS
LAGRANGIAN STRATIFICATION OF THE MARTINET SECTOR 306
10.11.1 PRELIMINARIES 306
10.11.2 THE SMOOTH ABNORMAL SECTOR 307
10.11.3 THE SMOOTHNESS OF THE SPHERE IN THE ABNORMAL DI RECTUM 307
10.11.4 THE MARTINET SECTOR 309
10.11.5 SYMPLECTIC STRATIFICATIONS 312
10.11.6 TRANSCENDENCE OF THE SECTOR 312
10.11.7 THE MICRO-LOCAL ANALYSIS OF THE WHOLE MARTINET SPHERE313
EXERCISES 314
11 NUMERICAL COMPUTATIONS 321
11.1 INTRODUCTION 321
11.2 NUMERICAL ALGORITHM 321
11.3 APPLICATIONS 324
IMAGE 9
XVI TABLE OF CONTENTS
11.3.1 CONTACT CASE IN DIMENSION 3 324
11.3.2 THE MARTINET FIAT CASE 324
11.4 THE TANGENTIAL CASE 328
11.4.1 THE ELLIPTIC CASE 329
11.4.2 THE HYPERBOLIC CASE 330
12 CONCLUSION AND PERSPECTIVES 335
12.1 THE CATEGORY OF THE DISTANCE FUNCTION IN SR-GEOMETRY 335
12.2 SR-DISTANCES AND CONTROL OF THE OSCILLATIONS OF SOLUTIONS OF
ORDINARY DIFFERENTIAL EQUATIONS 335
12.3 OPTIMAL CONTROL PROBLEMS WITH BOUNDED STATE VARIABLES . . .. 338
12.3.1 MAXIMUM PRINCIPLE WITH STATE CONSTRAINTS 338
12.4 SINGULAR TRAJECTORIES AND REGULARITY OF STABILIZING FEEDBACK . 339
12.5 COMPUTATIONS OF SINGULAR TRAJECTORIES 340
12.6 COMPUTATIONS OF CONJUGATE POINTS 340
EXERCISES 341
REFERENCES 343
INDEX 349
|
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| id | DE-604.BV017038567 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:13:03Z |
| institution | BVB |
| isbn | 3540008381 |
| language | English |
| lccn | 2003045543 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010282373 |
| oclc_num | 51886430 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-29T DE-91G DE-BY-TUM DE-703 DE-384 DE-634 DE-11 DE-20 |
| owner_facet | DE-355 DE-BY-UBR DE-29T DE-91G DE-BY-TUM DE-703 DE-384 DE-634 DE-11 DE-20 |
| physical | XVI, 357 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Mathématiques & applications |
| series2 | Mathématiques & applications |
| spelling | Bonnard, Bernard Verfasser aut Singular trajectories and their role in control theory Bernard Bonnard ; Monique Chyba Paris [u.a.] Springer 2003 XVI, 357 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathématiques & applications 40 Includes bibliographical references and index Calculus of variations Control theory Observers (Control theory) Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Geometrische Methode (DE-588)4156715-8 gnd rswk-swf Kontrolltheorie (DE-588)4032317-1 gnd rswk-swf Trajektorie Mathematik (DE-588)4406251-5 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 s Trajektorie Mathematik (DE-588)4406251-5 s Kontrolltheorie (DE-588)4032317-1 s DE-604 Geometrische Methode (DE-588)4156715-8 s Chyba, Monique Sonstige oth Mathématiques & applications 40 (DE-604)BV006642035 40 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010282373&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bonnard, Bernard Singular trajectories and their role in control theory Mathématiques & applications Calculus of variations Control theory Observers (Control theory) Singularität Mathematik (DE-588)4077459-4 gnd Geometrische Methode (DE-588)4156715-8 gnd Kontrolltheorie (DE-588)4032317-1 gnd Trajektorie Mathematik (DE-588)4406251-5 gnd |
| subject_GND | (DE-588)4077459-4 (DE-588)4156715-8 (DE-588)4032317-1 (DE-588)4406251-5 |
| title | Singular trajectories and their role in control theory |
| title_auth | Singular trajectories and their role in control theory |
| title_exact_search | Singular trajectories and their role in control theory |
| title_full | Singular trajectories and their role in control theory Bernard Bonnard ; Monique Chyba |
| title_fullStr | Singular trajectories and their role in control theory Bernard Bonnard ; Monique Chyba |
| title_full_unstemmed | Singular trajectories and their role in control theory Bernard Bonnard ; Monique Chyba |
| title_short | Singular trajectories and their role in control theory |
| title_sort | singular trajectories and their role in control theory |
| topic | Calculus of variations Control theory Observers (Control theory) Singularität Mathematik (DE-588)4077459-4 gnd Geometrische Methode (DE-588)4156715-8 gnd Kontrolltheorie (DE-588)4032317-1 gnd Trajektorie Mathematik (DE-588)4406251-5 gnd |
| topic_facet | Calculus of variations Control theory Observers (Control theory) Singularität Mathematik Geometrische Methode Kontrolltheorie Trajektorie Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010282373&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV006642035 |
| work_keys_str_mv | AT bonnardbernard singulartrajectoriesandtheirroleincontroltheory AT chybamonique singulartrajectoriesandtheirroleincontroltheory |