Hamilton-Jacobi equations: approximations, numerical analysis and applications ; Cetraro, Italy 2011
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2013
|
| Schriftenreihe: | Lecture notes in mathematics
2074 : CIME Foundation subseries |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XV, 301 S. graph. Darst. |
| ISBN: | 9783642364327 |
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Datensatz im Suchindex
| _version_ | 1806325091064086528 |
|---|---|
| adam_text |
FINITE DIFFERENCE METHODS FOR MEAN FIELD GAMES 1
YVES ACHDOU
1 INTRODUCTION 1
2 FINITE DIFFERENCE SCHEMES 4
2.1 DESCRIPTION OF THE SCHEMES 4
2.2 EXISTENCE AND A PRIORI BOUNDS 9
2.3 A FUNDAMENTAL IDENTITY 14
2.4 UNIQUENESS 16
2.5 A PRIORI ESTIMATES FOR (21)*(22) WITH LOCAL OPERATORS T 16
3 EXAMPLES OF CONVERGENCE RESULTS 19
4 ALGORITHMS FOR SOLVING THE DISCRETE LINEAR SYSTEMS 24
4.1 NEWTON METHODS FOR SOLVING (21 )-(22) 24
4.2 ITERATIVE STRATEGIES FOR SOLVING (54) BASED
ON ELIMINATING % 26
5 SOME SIMULATIONS 31
6 THE PLANNING PROBLEM 35
6.1 DESCRIPTION OF THE PLANNING PROBLEM 35
6.2 THE FINITE DIFFERENCE SCHEME AND AN OPTIMAL
CONTROL FORMULATION 36
6.3 UNIQUENESS 44
6.4 A PENALTY METHOD 45
REFERENCES 45
AN INTRODUCTION TO THE THEORY OF VISCOSITY SOLUTIONS
FOR FIRST-ORDER HAMILTON-JACOBI EQUATIONS AND APPLICATIONS 49
GUY BARLES
1 INTRODUCTION 49
2 PRELIMINARIES: A RUNNING EXAMPLE 51
HTTP://D-NB.INFO/103008422X
XII CONTENTS
3 THE NOTION OF CONTINUOUS VISCOSITY SOLUTIONS: DEFINITION(S)
AND FIRST PROPERTIES 53
3.1 WHY A "GOOD" NOTION OF WEAK SOLUTION IS NEEDED? 53
3.2 CONTINUOUS VISCOSITY SOLUTIONS 54
3.3 BACK TO THE RUNNING EXAMPLE (1): THE VALUE FUNCTION
U IS A VISCOSITY SOLUTION OF (7) 56
3.4 AN EQUIVALENT DEFINITION AND ITS CONSEQUENCES 58
4 THE FIRST STABILITY RESULT FOR VISCOSITY SOLUTIONS 60
5 UNIQUENESS: THE BASIC ARGUMENTS AND ADDITIONAL RECIPES 64
5.1 A FIRST BASIC RESULT 64
5.2 SEVERAL VARIATIONS 70
5.3 FINITE SPEED OF PROPAGATION 72
6 DISCONTINUOUS VISCOSITY SOLUTIONS, DISCONTINUOUS
NONLINEARITIES AND THE "HALF-RELAXED LIMITS" METHOD 74
6.1 DISCONTINUOUS VISCOSITY SOLUTIONS 74
6.2 BACK TO THE RUNNING EXAMPLE (II): THE DIRICHLET
BOUNDARY CONDITION FOR THE VALUE-FUNCTION 76
6.3 THE HALF-RELAXED LIMIT METHOD 77
6.4 STRONG COMPARISON RESULTS 81
7 EXISTENCE OF VISCOSITY SOLUTIONS: PERRON'S METHOD 82
8 REGULARITY RESULTS 86
9 CONVEX HAMILTONIANS, BARRON-JENSEN SOLUTIONS 89
10 LARGE TIME BEHAVIOR OF SOLUTIONS OF HAMILTON-JACOBI EQUATIONS 92
10.1 INTRODUCTION 92
10.2 EXISTENCE AND REGULARITY OF THE SOLUTION 93
10.3 ERGODIC BEHAVIOR 94
10.4 ASYMPTOTIC BEHAVIOR OF U(X,
T) * CT 97
10.5 THE NAMAH-ROQUEJOFFRE FRAMEWORK 98
10.6 THE "STRICTLY CONVEX" FRAMEWORK 100
10.7 CONCLUDING REMARKS 106
REFERENCES 107
A SHORT INTRODUCTION TO VISCOSITY SOLUTIONS AND THE LARGE
TIME BEHAVIOR OF SOLUTIONS OF HAMILTON-JACOBI EQUATIONS ILL
HITOSHI ISHII
1 INTRODUCTION TO VISCOSITY SOLUTIONS 114
1.1 HAMILTON-JACOBI EQUATIONS 114
1.2 AN OPTIMAL CONTROL PROBLEM 115
1.3 CHARACTERIZATION OF THE VALUE FUNCTION 121
1.4 SEMICONTINUOUS VISCOSITY SOLUTIONS AND THE PERRON METHOD 128
1.5 AN EXAMPLE 139
1.6 SUP-CONVOLUTIONS 141
2 NEUMANN BOUNDARY VALUE PROBLEMS '43
3 INITIAL-BOUNDARY VALUE PROBLEM FOR HAMILTON-JACOBI EQUATIONS ^8
3.1 INITIAL-BOUNDARY VALUE PROBLEMS '^8
CONTENTS
XIII
3.2 ADDITIVE EIGENVALUE PROBLEMS 152
3.3 PROOF OF COMPARISON THEOREM 155
4 STATIONARY PROBLEM: WEAK KAM ASPECTS 165
4.1 AUBRY SETS AND REPRESENTATION OF SOLUTIONS 166
4.2 PROOF OF THEOREM 4.2 174
5 OPTIMAL CONTROL PROBLEM ASSOCIATED WITH (ENP)-(ID) 185
5.1 SKOROKHOD PROBLEM 185
5.2 VALUE FUNCTION 1 191
5.3 BASIC LEMMAS 194
5.4 VALUE FUNCTION II 202
5.5 DISTANCE-LIKE FUNCTION
D
208
6 LARGE-TIME ASYMPTOTIC SOLUTIONS 211
6.1 PRELIMINARIES TO ASYMPTOTIC SOLUTIONS 214
6.2 PROOF OF CONVERGENCE 219
6.3 REPRESENTATION OF THE ASYMPTOTIC SOLUTION U
00
222
6.4 LOCALIZATION OF CONDITIONS (A9) 226
A.L LOCAL MAXIMA TO GLOBAL MAXIMA 229
A.2 A QUICK REVIEW OF CONVEX ANALYSIS 230
A.3 GLOBAL LIPSCHITZ REGULARITY 235
A.4 LOCALIZED VERSIONS OF LEMMA 4.2 238
A.5 A PROOF OF LEMMA 5.4 242
A.6 RADEMACHER'S THEOREM 245
REFERENCES 247
IDEMPOTENT/TROPICAL ANALYSIS, THE HAMILTON-JACOBI
AND BELLMAN EQUATIONS 251
GRIGORY L. LITVINOV
1 INTRODUCTION 251
2 THE MASLOV DEQUANTIZATION 253
3 SEMIRINGS AND SEMIFIELDS: THE IDEMPOTENT CORRESPONDENCE
PRINCIPLE 254
4 IDEMPOTENT ANALYSIS 255
5 THE SUPERPOSITION PRINCIPLE AND LINEAR EQUATIONS 256
5.1 HEURISTICS 256
5.2 THE CAUCHY PROBLEM FOR THE HAMILTON-JACOBI EQUATIONS 259
6 CONVOLUTION AND THE FOURIER-LEGENDRE TRANSFORM 260
7 IDEMPOTENT FUNCTIONAL ANALYSIS 261
7.1 IDEMPOTENT SEMIMODULES AND IDEMPOTENT LINEAR SPACES 262
7.2 BASIC RESULTS 265
7.3 IDEMPOTENT ^-SEMIALGEBRAS 266
7.4 LINEAR OPERATOR, ^-SEMIMODULES
AND SUBSEMIMODULES 267
7.5 FUNCTIONAL SEMIMODULES 268
7.6 INTEGRAL REPRESENTATIONS OF LINEAR OPERATORS
IN FUNCTIONAL SEMIMODULES 270
\
XIV CONTENTS
7.7 NUCLEAR OPERATORS AND THEIR INTEGRAL REPRESENTATIONS 272
7.8 THE ^-APPROXIMATION PROPERTY AND 6-NUCLEAR
SEMIMODULES AND SPACES 272
7.9 KERNEL THEOREMS FOR FUNCTIONAL FE-SEMIMODULES 273
7.10 INTEGRAL REPRESENTATIONS OF OPERATORS IN ABSTRACT
IDEMPOTENT SEMIMODULES 273
8 THE DEQUANTIZATION TRANSFORM, CONVEX GEOMETRY
AND THE NEWTON POLYTOPES 275
8.1 DEQUANTIZATION TRANSFORM: ALGEBRAIC PROPERTIES 276
8.2 GENERALIZED POLYNOMIALS AND SIMPLE FUNCTIONS 277
8.3 SUBDIFFERENTIALS OF SUBLINEAR FUNCTIONS 278
8.4 NEWTON SETS FOR SIMPLE FUNCTIONS 279
9 DEQUANTIZATION OF SET FUNCTIONS AND MEASURES ON METRIC SPACES 280
10 DEQUANTIZATION OF GEOMETRY 281
11 SOME SEMIRING CONSTRUCTIONS AND THE MATRIX BELLMAN EQUATION 282
11.1 COMPLETE IDEMPOTENT SEMIRINGS AND EXAMPLES 282
11.2 CLOSURE OPERATIONS 282
11.3 MATRICES OVER SEMIRINGS 283
11.4 DISCRETE STATIONARY BELLMAN EQUATIONS 284
11.5 WEIGHTED DIRECTED GRAPHS AND MATRICES OVER SEMIRINGS 284
12 UNIVERSAL ALGORITHMS 287
13 UNIVERSAL ALGORITHMS OF LINEAR ALGEBRA OVER SEMIRINGS 288
14 THE CORRESPONDENCE PRINCIPLE FOR COMPUTATIONS 293
15 THE CORRESPONDENCE PRINCIPLE FOR HARDWARE DESIGN 293
16 THE CORRESPONDENCE PRINCIPLE FOR SOFTWARE DESIGN 295
17 INTERVAL ANALYSIS IN IDEMPOTENT MATHEMATICS 296
REFERENCES 297 |
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| spelling | Hamilton-Jacobi equations approximations, numerical analysis and applications ; Cetraro, Italy 2011 Yves Achdou ... Ed.: Paola Loreti ... Berlin [u.a.] Springer 2013 XV, 301 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 2074 : CIME Foundation subseries Hamilton-Jacobi-Differentialgleichung (DE-588)4158954-3 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2011 Cetraro gnd-content Hamilton-Jacobi-Differentialgleichung (DE-588)4158954-3 s DE-604 Achdou, Yves Sonstige (DE-588)140964363 oth Loreti, Paola Sonstige (DE-588)1036676641 oth Centro Internazionale Matematico Estivo Sonstige (DE-588)1025933-8 oth Erscheint auch als Online-Ausgabe 978-3-642-36433-4 Lecture notes in mathematics 2074 : CIME Foundation subseries (DE-604)BV000676446 2074 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4240598&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026069530&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hamilton-Jacobi equations approximations, numerical analysis and applications ; Cetraro, Italy 2011 Lecture notes in mathematics Hamilton-Jacobi-Differentialgleichung (DE-588)4158954-3 gnd |
| subject_GND | (DE-588)4158954-3 (DE-588)1071861417 |
| title | Hamilton-Jacobi equations approximations, numerical analysis and applications ; Cetraro, Italy 2011 |
| title_auth | Hamilton-Jacobi equations approximations, numerical analysis and applications ; Cetraro, Italy 2011 |
| title_exact_search | Hamilton-Jacobi equations approximations, numerical analysis and applications ; Cetraro, Italy 2011 |
| title_full | Hamilton-Jacobi equations approximations, numerical analysis and applications ; Cetraro, Italy 2011 Yves Achdou ... Ed.: Paola Loreti ... |
| title_fullStr | Hamilton-Jacobi equations approximations, numerical analysis and applications ; Cetraro, Italy 2011 Yves Achdou ... Ed.: Paola Loreti ... |
| title_full_unstemmed | Hamilton-Jacobi equations approximations, numerical analysis and applications ; Cetraro, Italy 2011 Yves Achdou ... Ed.: Paola Loreti ... |
| title_short | Hamilton-Jacobi equations |
| title_sort | hamilton jacobi equations approximations numerical analysis and applications cetraro italy 2011 |
| title_sub | approximations, numerical analysis and applications ; Cetraro, Italy 2011 |
| topic | Hamilton-Jacobi-Differentialgleichung (DE-588)4158954-3 gnd |
| topic_facet | Hamilton-Jacobi-Differentialgleichung Konferenzschrift 2011 Cetraro |
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