Generalized characteristics of first order PDEs: applications in optimal control and differential games
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
1998
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 301 - 307 |
| Physical Description: | XIV, 310 S. graph. Darst. |
| ISBN: | 3764339845 0817639845 |
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| 100 | 1 | |a Melikyan, Arik A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Generalized characteristics of first order PDEs |b applications in optimal control and differential games |c A. A. Melikyan |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 1998 | |
| 300 | |a XIV, 310 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 500 | |a Literaturverz. S. 301 - 307 | ||
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| 650 | 4 | |a Differential equations, Partial | |
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Record in the Search Index
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| adam_text | Contents
Preface xi
Introduction 1
1 Method of Characteristics in Smooth Problems 7
1.1 Classical Cauchy problem for first order PDE 7
1.1.1 Problem statement 7
1.1.2 Characteristic equations 8
1.1.3 Construction of the initial conditions 9
1.1.4 A note on the approach in small 11
1.1.5 Construction of twice differentiable solution 11
1.1.6 Irregular characteristic problem 13
1.1.7 A 2D example; problem formulation 17
1.1.8 Construction of the solution 18
1.2 Cauchy problem for integral surfaces 19
1.2.1 Geometrical formulation of the Problem 1.1 19
1.2.2 Generalized Cauchy problem 21
1.2.3 Characteristic field on a manifold 24
1.2.4 Construction of the reference solution 27
1.2.5 Explicit expressions for A for small m 28
1.2.6 Sufficient conditions for the Problems 1.3, 1.4 .... 29
1.2.7 The geometry of the characteristic field 31
1.2.8 Characteristic points of the manifold W 32
1.2.9 Some particular characteristic systems for m = 1 . . 34
1.3 Cauchy problem with movable boundary 35
1.3.1 Regular problem with movable boundary 35
1.3.2 Irregular problem 38
1.3.3 Jacobi brackets of different levels 38
1.3.4 A sufficiency condition 39
1.3.5 Classical irregular non characteristic problem .... 41
1.3.6 Illustrative example 52
Exercises 54
2 Generalized Solutions and Singular Characteristics of First
Order PDEs 55
2.1 Viscosity solutions and their singular manifolds 55
2.1.1 Definition of viscosity solution 55
vi Contents
2.1.2 Regular and singular points of a solution; simplest
singularity 57
2.1.3 Necessary conditions for a simplest singularity ... 60
2.1.4 Singular characteristics, definition and classification 62
2.1.5 Some properties of IVP and TVP 64
2.2 Dispersal surface 64
2.2.1 General conditions 64
2.2.2 Linear and nonlinear Hamiltonians 66
2.3 Singular characteristics for equivocal surface 68
2.3.1 Four types of surfaces, necessary conditions 68
2.3.2 Equations of singular characteristics 71
2.3.3 Some properties of characteristic system 72
2.4 Singular characteristics for focal surface 74
2.4.1 Six types of surfaces, necessary conditions 74
2.4.2 Focal surface — hyperplane 77
2.4.3 Non symmetric surface, collinear fields 80
2.4.4 Degenerate surfaces 84
2.4.5 Initial conditions and identification of singular surfaces 84
2.4.6 Modifications for TVP 85
2.5 An IVP example 86
2.5.1 Problem formulation 86
2.5.2 The case 1), a b 88
2.5.3 The case 2), a = b 90
2.5.4 The case 3), a b 91
2.5.5 Some modifications for non symmetric case 93
2.5.6 Concluding remarks 96
Exercises 97
3 First Order PDEs in Variation Calculus, Optimal Control
and Differential Games 99
3.1 Hamilton Jacobi equation in Variation Calculus 99
3.1.1 First variation formula 99
3.1.2 The case of non homogeneous Lagrangian 102
3.1.3 Variational problem on geodesic line 103
3.1.4 General homogeneous Lagrangian 105
3.2 Bellman equation in Optimal Control 106
3.2.1 Fixed time problem 106
3.2.2 Time optimal problem 109
3.2.3 Feedback controls Ill
3.3 The Isaacs equation in Differential Games 112
3.3.1 Fixed time game. Value function 112
3.3.2 Pursuit evasion games 115
3.4 Generalized solutions of the HJBI equation 116
3.4.1 Classical and viscosity solutions 116
3.4.2 Generalized main equation, A.I.Subbotin s inequalitiesll8
Contents vii
3.5 Singular paths and singular characteristics 120
3.5.1 Singular surfaces and paths: definition and classifica¬
tion 120
3.5.2 Theory of the equivocal surface 123
3.5.3 Singular paths and characteristics 126
3.6 A linear pursuit evasion game with elliptical vectograms . . 127
3.6.1 Problem formulation 127
3.6.2 Dispersal surface 130
3.6.3 Focal surface 131
3.6.4 Boundary of the indifferent zone 132
Exercises 135
4 Differential Games with Simple Motions on the Manifolds 137
4.1 Problem statement 137
4.1.1 Games with simple motion 137
4.1.2 Dynamic equations 138
4.1.3 Cost functions for two games 139
4.2 Primary solution 140
4.2.1 Properties of the geodesic length 140
4.2.2 Primary and secondary domains 141
4.3 Necessary optimality conditions 142
4.3.1 Generalized main equation, regular paths 142
4.3.2 Singular surface in primary domain 145
4.3.3 Analysis of the surface Fo using viscosity conditions 148
4.4 Two branches of the equivocal surface 153
4.4.1 Identification of the equivocal surfaces 153
4.4.2 Main result 155
4.4.3 Construction algorithm 158
4.5 Game of pursuit in the presence of an obstacle 159
4.5.1 Problem formulation 159
4.5.2 Planar problem 162
4.5.3 Examples 164
Exercises 168
5 Games of Simple Pursuit and Approach on Two Dimensional
Cone 169
5.1 Game formulation in different coordinate systems 169
5.1.1 Dynamics in Cartesian and relative variables .... 169
5.1.2 Self similar variables, complex coordinates 173
5.1.3 Primary solutions 175
5.2 Analysis of the primary domain 176
5.2.1 Necessary optimality conditions 176
5.2.2 Construction of the set B, parametric analysis . . . 178
5.2.3 Construction of the equivocal surface 182
5.3 Analysis of the secondary domain 184
viii Contents
5.3.1 Game of pursuit 184
5.3.2 The critical cone, v = 1 sine* 188
5.3.3 Game of approach 190
5.3.4 The case u = 1 194
5.3.5 On the algorithm of synthesis and computer simulation. 195
Exercises 198
6 Smooth Solutions of a PDE with Nonsmooth Hamiltonianl99
6.1 Open loop and feedback analysis of singular paths in Opti¬
mal Control 199
6.1.1 Introduction 199
6.1.2 Singular arc in Optimal Control problem, open loop
approach 201
6.1.3 Linear problem 201
6.1.4 Two sets of variables 203
6.1.5 Necessary conditions in invariant form 204
6.1.6 Singular universal surface in general problem .... 205
6.2 First order PDEs with nonsmooth Hamiltonian 208
6.2.1 Necessary conditions for singular hyperplane .... 208
6.2.2 An auxiliary theorem 210
6.2.3 Necessary conditions in invariant form 212
6.2.4 Singular characteristics for the universal surface . . 213
6.2.5 Applications to the control problem 214
6.2.6 Example 217
6.3 Second order singularity 220
6.3.1 Two optimal phase portraits; Kopp Moyer condition 220
6.3.2 Invariant form of the second order conditions .... 222
6.3.3 Singular characteristics for the synthesis 52 223
Exercises 225
7 Shock Waves Related to First Order PDEs 227
7.1 Singular characteristics in two dimensional problems .... 227
7.1.1 Two dimensional problem 227
7.1.2 Equations for a focal line 228
7.1.3 Equations for equivocal line 231
7.1.4 Singular characteristics of two dimensional
Hamilton Jacobi equation 231
7.2 Shock waves generated by the boundary conditions 234
7.2.1 Initial conditions 234
7.2.2 Convexification of the function g(p) 236
7.2.3 Analysis of the second derivative 241
7.3 Main results on the number of waves 243
7.3.1 Simplified expressions for Jacobi brackets 243
7.3.2 The case of simple segments 245
7.3.3 Secondary waves 250
Contents ix
7.3.4 A result concerning non simple segment 250
7.3.5 S.N.Kruzhkov s theorem 252
7.3.6 Example 253
7.3.7 Some generalizations for multidimensional case . . . 255
7.4 Other applications of the MSC 257
7.4.1 Singular characteristics in conservation laws 257
7.4.2 On a class of systems of first order PDEs 258
Exercises 261
8 Singular Surfaces of Nonsmooth Solutions to Multiple In¬
tegral Variational Problems 263
8.1 Multiple integral Variational Problem 263
8.1.1 Nonsmooth solution of second order PDE 263
8.1.2 First variation formula 264
8.1.3 Necessary conditions for singular surface 267
8.2 Construction of singular surface 271
8.2.1 Equations of singular characteristics 271
8.2.2 Initial conditions 272
8.3 Quadratic Lagrangian 276
8.3.1 Degenerate necessary conditions 276
8.3.2 Singular characteristics 277
8.3.3 The perturbed problem 280
8.3.4 Initial conditions 281
8.4 Example 282
8.4.1 Problem formulation 282
8.4.2 Taylor expansions 284
8.4.3 Particular cases 287
Exercises 289
9 Appendix 291
9.1 Implicit function theorem 291
9.2 Jacobi brackets 292
9.3 Invariance of Jacobi brackets 293
9.4 Field straightening 296
9.5 Reduction to the simple problem 298
Bibliography 301
Subject Index 308
|
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| isbn | 3764339845 0817639845 |
| language | English |
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| spelling | Melikyan, Arik A. Verfasser aut Generalized characteristics of first order PDEs applications in optimal control and differential games A. A. Melikyan Boston [u.a.] Birkhäuser 1998 XIV, 310 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 301 - 307 Control theory Differential equations, Partial Differential games Differentialspiel (DE-588)4012253-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Optimale Kontrolle (DE-588)4121428-6 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Optimale Kontrolle (DE-588)4121428-6 s Differentialspiel (DE-588)4012253-0 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008208882&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Melikyan, Arik A. Generalized characteristics of first order PDEs applications in optimal control and differential games Control theory Differential equations, Partial Differential games Differentialspiel (DE-588)4012253-0 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Optimale Kontrolle (DE-588)4121428-6 gnd |
| subject_GND | (DE-588)4012253-0 (DE-588)4044779-0 (DE-588)4121428-6 |
| title | Generalized characteristics of first order PDEs applications in optimal control and differential games |
| title_auth | Generalized characteristics of first order PDEs applications in optimal control and differential games |
| title_exact_search | Generalized characteristics of first order PDEs applications in optimal control and differential games |
| title_full | Generalized characteristics of first order PDEs applications in optimal control and differential games A. A. Melikyan |
| title_fullStr | Generalized characteristics of first order PDEs applications in optimal control and differential games A. A. Melikyan |
| title_full_unstemmed | Generalized characteristics of first order PDEs applications in optimal control and differential games A. A. Melikyan |
| title_short | Generalized characteristics of first order PDEs |
| title_sort | generalized characteristics of first order pdes applications in optimal control and differential games |
| title_sub | applications in optimal control and differential games |
| topic | Control theory Differential equations, Partial Differential games Differentialspiel (DE-588)4012253-0 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Optimale Kontrolle (DE-588)4121428-6 gnd |
| topic_facet | Control theory Differential equations, Partial Differential games Differentialspiel Partielle Differentialgleichung Optimale Kontrolle |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008208882&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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