Viability theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston u.a.
Birkhäuser
1991
|
| Series: | Systems & control: foundations & applications
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 497 - 537 |
| Physical Description: | XXV, 543 S. graph. Darst. |
| ISBN: | 0817635718 3764335718 |
Staff View
MARC
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| 100 | 1 | |a Aubin, Jean-Pierre |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Viability theory |c Jean-Pierre Aubin |
| 264 | 1 | |a Boston u.a. |b Birkhäuser |c 1991 | |
| 300 | |a XXV, 543 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Systems & control: foundations & applications | |
| 500 | |a Literaturverz. S. 497 - 537 | ||
| 650 | 7 | |a Applications mesurables à valeurs convexes compactes |2 ram | |
| 650 | 7 | |a Inclusions différentielles |2 ram | |
| 650 | 7 | |a Réaction, systèmes à |2 ram | |
| 650 | 4 | |a Differential inclusions | |
| 650 | 4 | |a Feedback control systems | |
| 650 | 4 | |a Set-valued maps | |
| 650 | 0 | 7 | |a Differentialinklusion |0 (DE-588)4149777-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Evolution |0 (DE-588)4071050-6 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804119700364328960 |
|---|---|
| adam_text | Contents
Introduction 1
Outline of the Book 11
1 Viability Theorems for Ordinary and Stochastic Dif¬
ferential Equations 19
1.1 Viability Invariance Properties 23
1.2 Nagumo Theorem 26
1.3 Numerical Schemes 32
1.4 Replicator Systems 34
1.5 Stochastic Viability and Invariance 44
1.5.1 Stochastic Tangent Sets 44
1.5.2 Stochastic Viability 44
1.5.3 Necessary Conditions 46
1.5.4 Sufficient Conditions for Stochastic Invariance . 47
2 Set Valued Maps 53
2.1 Semicontinuous Set Valued Maps 54
2.1.1 Definitions 54
2.1.2 Continuity Concepts 56
2.2 Closed Set Valued Maps 60
2.2.1 Upper Semicontinuity of Closed Maps 60
2.2.2 Marchaud Maps 61
2.3 Support Functions 62
2.4 Convergence Theorem 65
2.5 Closed Convex Processes 70
xvii
xviii
3 Viability Theorems for Differential Inclusions 77
3.1 Solution Class 79
3.2 Viability Domains 83
3.2.1 Definition of Viability Domains 84
3.2.2 Subnormal Cones 85
3.2.3 Dual Characterization of Viability Domains . . 85
3.3 Statement of Viability Theorems 88
3.4 Proofs of the Viability Theorems 92
3.5 Solution Map 101
3.5.1 Upper Semicontinuity of Solution Maps .... 101
3.5.2 Closure of a Viability Domain 103
3.5.3 Reachable Map 104
3.5.4 Representation Property 105
3.6 Stability of Viability Domains 106
3.7 w Limit Sets and Equilibria 109
3.7.1 w Limit Sets 109
3.7.2 Cesaro means of the velocities 112
3.7.3 Viability implies Stationarity 113
3.8 Chaotic Solutions to Differential Inclusions 117
4 Viability Kernels and Exit Tubes 119
4.1 Viability Kernels 121
4.1.1 Existence of the Viability Kernel 121
4.1.2 Permanence and Fluctuation 125
4.1.3 Viability Envelopes 130
4.2 Hitting and Exit Tubes 131
4.2.1 Hitting and Exit Functional 132
4.2.2 Hitting and Exit Functions 134
4.2.3 Exit Tubes 137
4.3 Anatomy of a Set 140
4.3.1 Contingent Cone to the Boundary 140
4.3.2 Strict Invariance 142
4.3.3 Inward and Outward Areas 145
4.3.4 Boundary of Viability Kernels 146
4.4 Viability Domain Algorithms 147
4.4.1 Viability Kernel Algorithm 147
4.4.2 Global Contingent Sets 148
4.4.3 Viability Domain Algorithm 151
xix
4.4.4 Fast Viability Kernel Algorithm 152
4.5 Finite Difference Approximation of Viability Kernels . 153
4.5.1 Viable Subsets under a Discrete System .... 153
4.5.2 Finite Difference Approximations 154
5 Invariance Theorems for Differential Inclusions 157
5.1 External Contingent Cones 158
5.1.1 External Contingent Cones 158
5.1.2 Sleek Subsets 160
5.1.3 Tangent Cones to Convex Sets 162
5.1.4 Calculus of Contingent Cones 163
5.1.5 Inequality Constraints 166
5.2 Invariance Domains 167
5.3 Invariance Theorem 170
5.3.1 Filippov s Theorem 170
5.3.2 Characterization of Local Invariance 173
5.3.3 Graphical Lower Limits of Solution Maps . . . 174
5.3.4 Accessibility Map 176
5.3.5 Proof of Convergence of the Fast Viability Ker¬
nel Algorithm 178
5.4 Invariance Kernels 182
5.4.1 Existence of the Invariance Kernel 182
5.4.2 Complement of the Invariance Kernel 183
5.4.3 Stability of Invariance Domains 184
5.4.4 Global Exit and Hitting Functions 185
5.4.5 Invariance Envelopes 186
5.5 Boundaries of Viability and Invariance Kernels .... 188
5.5.1 Semipermeability of the Boundary of the Via¬
bility Kernel 188
5.5.2 Viability of the Boundary of the Invariance
Kernel 190
5.6 Defeat and Victory domains of a Target and its Barrier 190
5.7 Linear Differential Inclusions 193
5.7.1 Viability Cones 193
5.7.2 Projection on the sphere 195
5.7.3 Projection on a compact sole 196
5.7.4 Duality between Viability and Invariance . . . 198
XX
6 Regulation of Control Systems 199
6.1 Regulation Map 201
6.2 A Simple Economic Example 204
6.3 Lower Semicontinuity Criteria 211
6.4 Lower Semicontinuity of the Regulation Map 216
6.4.1 General State Constraints 216
6.4.2 Output Dependent Constraints 217
6.4.3 Output Regulation Map 219
6.4.4 Duality Criterion 220
6.4.5 Decoupling the Regulation Map 221
6.5 Selection Theorems 223
6.5.1 Minimal Selection 223
6.5.2 Selection Procedures 224
6.5.3 Michael s Selection Theorem 228
6.6 Closed Loop Controls and Slow Solutions 228
6.6.1 Continuous Closed Loop Controls 228
6.6.2 Slow Viable Solutions 229
6.6.3 Other Selections of Viable Solutions 231
6.6.4 Examples of Selection of Viable Solutions . . . 233
7 Smooth and Heavy Viable Solutions 235
7.1 Contingent Derivatives 240
7.2 Smooth Viable Solutions 242
7.2.1 Regularity Theorem 242
7.2.2 Subregulation and Metaregulation Maps .... 247
7.3 Second Order Differential Inclusions 253
7.4 Metaregulation Map of High Order 256
7.5 Punctuated Equilibria, Ramp Controls and Polyno¬
mial Open Loop Controls 259
7.6 Heavy Viable Solutions 261
7.6.1 Dynamical Closed Loops 261
7.6.2 Heavy Viable Solutions 263
7.6.3 Heavy Viable Solutions under Equality Con¬
straints 265
7.6.4 Heavy Viable Solutions of High Order 267
xxi
8 Partial Differential Inclusions of Tracking Problems 275
8.1 The Tracking Property 282
8.1.1 Characterization of the Tracking Property . . . 283
8.1.2 Construction of trackers 286
8.1.3 The Observation Problem 287
8.1.4 Construction of Observers 291
8.2 The Tracking Problem 292
8.2.1 Tracking Control Systems 292
8.2.2 Decentralization of a control system 293
8.2.3 Hierarchical Decomposition Property 295
8.3 Partial Differential Inclusions 297
8.3.1 Decomposable Case 297
8.3.2 Existence of a Lipschitz Contingent Solution . 305
8.3.3 Comparison Results 307
8.4 The Variational Principle 308
8.4.1 Definition of the Functional 308
8.4.2 Convergence Properties of the Codifferentials .311
8.5 Feedback Controls Regulating Smooth Evolutions ... 313
9 Lyapunov Functions 315
9.1 Contingent Epiderivatives 319
9.1.1 Extended Functions and their Epigraphs .... 319
9.1.2 Contingent Epiderivatives 320
9.1.3 Epidifferential Calculus 325
9.2 Lyapunov Functions 327
9.2.1 The Characterization Theorem 327
9.2.2 Stability Theorems 329
9.2.3 W Monotone Set Valued Maps 330
9.2.4 Attractors 331
9.2.5 Universal Lyapunov Functions 332
9.3 Optimal Lyapunov Functions 333
9.3.1 Smallest Lyapunov Functions 333
9.3.2 Smallest Universal Lyapunov Functions .... 336
9.4 Other Monotonicity Properties 337
9.4.1 Monotone Solutions 337
9.4.2 LaSalle s Theorem 340
9.4.3 Example: Gradient Inclusions 341
9.4.4 Feedbacks Regulating Monotone Solutions . . . 342
xxii
9.5 Lyapunov Preorders 344
9.5.1 Monotone solutions with respect to a preorder 345
9.5.2 Comparison of solutions 346
9.6 Asymptotic Observability of Differential Inclusions . . 348
10 Miscellaneous Viability Issues 351
10.1 Variational Differential Inequalities 353
10.1.1 The Equivalence Theorem 354
10.1.2 Slow Solutions 357
10.1.3 Projected differential inclusions onto smooth
subsets 359
10.2 Fuzzy Viability 361
10.2.1 Fuzzy Sets 361
10.2.2 Fuzzy Differential Inclusions 362
10.2.3 Fuzzy Viability Domains 364
10.3 Finite Difference Schemes 367
10.3.1 Implicit Finite Difference Scheme 367
10.3.2 Explicit Finite Difference Scheme 368
10.3.3 Approximation of an Equilibrium 370
10.4 Newton s Method 371
11 Viability Tubes 377
11.1 Viability Tubes 381
11.2 Cauchy Problem for Viability Tubes 385
11.3 Asymptotic Target 386
11.4 Examples of Viability Tubes 387
11.5 An Abstract K C Problem 391
11.6 Invariant Tubes 393
11.7 Measurable Time Dependence 395
12 Functional Viability 401
12.1 Definitions and Examples 402
12.2 Functional Viability Theorem 405
12.3 History dependent Viability Constraints 413
12.3.1 Viability constraints with delays 418
12.3.2 Volterra Viability constraints 419
12.4 Functional Viability Kernel 419
12.5 Functional Viability Tubes 421
xxiii
13 Viability Theorems for Partial Differential Inclusions425
13.1 Unbounded operators 427
13.2 Operational Differential Inclusions 430
13.3 Elliptic Parabolic Inclusions 438
13.4 Distributed Control Systems 446
13.5 Lyapunov Functions of Parabolic Inclusions 448
14 Differential Games 451
14.1 Contingent Isaacs Equations 454
14.2 Playable Differential Games 462
14.3 Feedback Solutions 465
14.4 Discriminating and Leading Feedbacks 470
14.5 Closed Loop Decision Rules 474
Bibliographical Comments 485
Bibliography 497
Index 539
|
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| id | DE-604.BV005485731 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:30:21Z |
| institution | BVB |
| isbn | 0817635718 3764335718 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003434469 |
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| physical | XXV, 543 S. graph. Darst. |
| publishDate | 1991 |
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| publisher | Birkhäuser |
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| spelling | Aubin, Jean-Pierre Verfasser aut Viability theory Jean-Pierre Aubin Boston u.a. Birkhäuser 1991 XXV, 543 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Systems & control: foundations & applications Literaturverz. S. 497 - 537 Applications mesurables à valeurs convexes compactes ram Inclusions différentielles ram Réaction, systèmes à ram Differential inclusions Feedback control systems Set-valued maps Differentialinklusion (DE-588)4149777-6 gnd rswk-swf Evolution (DE-588)4071050-6 gnd rswk-swf Unsicherheit (DE-588)4186957-6 gnd rswk-swf Regelungssystem (DE-588)4134712-2 gnd rswk-swf Mengenwertige Abbildung (DE-588)4270772-9 gnd rswk-swf Regelungssystem (DE-588)4134712-2 s Evolution (DE-588)4071050-6 s Unsicherheit (DE-588)4186957-6 s Differentialinklusion (DE-588)4149777-6 s Mengenwertige Abbildung (DE-588)4270772-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003434469&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Aubin, Jean-Pierre Viability theory Applications mesurables à valeurs convexes compactes ram Inclusions différentielles ram Réaction, systèmes à ram Differential inclusions Feedback control systems Set-valued maps Differentialinklusion (DE-588)4149777-6 gnd Evolution (DE-588)4071050-6 gnd Unsicherheit (DE-588)4186957-6 gnd Regelungssystem (DE-588)4134712-2 gnd Mengenwertige Abbildung (DE-588)4270772-9 gnd |
| subject_GND | (DE-588)4149777-6 (DE-588)4071050-6 (DE-588)4186957-6 (DE-588)4134712-2 (DE-588)4270772-9 |
| title | Viability theory |
| title_auth | Viability theory |
| title_exact_search | Viability theory |
| title_full | Viability theory Jean-Pierre Aubin |
| title_fullStr | Viability theory Jean-Pierre Aubin |
| title_full_unstemmed | Viability theory Jean-Pierre Aubin |
| title_short | Viability theory |
| title_sort | viability theory |
| topic | Applications mesurables à valeurs convexes compactes ram Inclusions différentielles ram Réaction, systèmes à ram Differential inclusions Feedback control systems Set-valued maps Differentialinklusion (DE-588)4149777-6 gnd Evolution (DE-588)4071050-6 gnd Unsicherheit (DE-588)4186957-6 gnd Regelungssystem (DE-588)4134712-2 gnd Mengenwertige Abbildung (DE-588)4270772-9 gnd |
| topic_facet | Applications mesurables à valeurs convexes compactes Inclusions différentielles Réaction, systèmes à Differential inclusions Feedback control systems Set-valued maps Differentialinklusion Evolution Unsicherheit Regelungssystem Mengenwertige Abbildung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003434469&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT aubinjeanpierre viabilitytheory |