Lectures on the calculus of variations and optimal control theory

Title......Page 1
Foreword......Page 3
Preface......Page 5
Contents......Page 7
Volume 1- Lectures on the Calculus of Variations......Page 12
1. Introduction......Page 13
2. The Place of the Calculus of Variations in Relation to the Rest of Mathematics and to Space Science......Page 14
3. Statement of the Simplest Problem and Some Cognate Matters......Page 15
4. Extremals in Some Classical Problems......Page 18
5. Solutions of the Preceding Problems (a), (b), (c)......Page 20
6. The Euler-Lagrange Lemma and Schwartz Distributions......Page 27
7. Alternative Forms of the Lemma......Page 29
8. Proof of the Main Form of the Lemma......Page 30
9. First Variation, Euler Equation, Transversality......Page 32
10. Perron\'s Paradox......Page 33
11. Introduction......Page 35
12. The Variational Algorithm ofHuygens......Page 36
13. A Link with Elementary Convexity......Page 38
14. Reappearance of the Euler Equation......Page 41
15. The Theorem of Malus......Page 43
16. Sufficient Conditions for Independence of the Hilbert Integral......Page 45
17. lnvariance Properties and an Envelope Theorem......Page 46
18. General Comments and the Applications to Plane Problems......Page 49
19. Background on Fix-Points and on Existence Theorems for Differential Equations and Implicit Functions......Page 51
21. The Legendre Transformation......Page 57
22. The Hamiltonian and its Properties......Page 58
23. Cauchy Characteristics......Page 60
24. Duality and the Standard Hamiltonian in the Parametric Case......Page 61
25. Other Admissible Parametric Hamiltonians......Page 64
26. Local Passage from Parametric to Nonparametric Case......Page 66
27. The Embedding of Small Extremals in Small Tubes......Page 67
28. Local Existence Theory for Nonparametric Variational Problems and for Ordinary Second Order Differential Equations......Page 69
29. Local Parametric Existence Theory for the Elliptic Case......Page 75
30. Introduction......Page 81
31. First and Second Variations and Transversality......Page 82
32. The Second Variation Fallacy......Page 84
33. The Secondary Hamiltonian......Page 85
34. Geometrical Interpretation of Exactness......Page 87
35. Distinguished Families......Page 89
36. Canonical Embeddings and Focal Points......Page 92
37. The Jacobi Theory of Conjugate Points......Page 94
38. The Index of Stability of an Extremal......Page 99
39. The Second Stage of the Morse Theory......Page 103
40. Introduction......Page 105
41. Center of Gravity and Dispersal Zone......Page 106
42. Convexity and the Hahn-Banach Theorem......Page 109
43. The Conceptual Heritage of Georg Cantor......Page 112
44. Duality of Convex Figures......Page 116
45. Duality of Convex Functions......Page 119
46. Hamiltonians in the Large and Reformulated Variational Theory......Page 121
47. Remarks on Classical Inequalities......Page 123
48. The Dual Unit Ball of a Functional Space......Page 124
49. The Riesz Representation......Page 129
50. Introduction......Page 133
51. The Hilbert Construction and Some of its Consequences in the Standard Parametric Case......Page 134
52. The Parametric Theory of Conjugate Points and the Parametric Jacobi Condition......Page 139
53. The Tonelli-Caratheodory Unicity Theorem......Page 144
54. Absolute and Homotopic Minima on B · · i-Compact Domains and Manifolds......Page 154
55. Toward an Automatic Existence Theory......Page 158
56. First Stage of an Abstract Approach: Semicontinuity in a B · · i- Compact Set......Page 162
57, 58, 59......Page 165
60. Introduction......Page 166
61. Intuitive Background......Page 167
62. A Question of Semantics......Page 171
63. Parametric Curves in the Calculus of Variations......Page 172
64. Admissible Curves as Elements of a Dual Space......Page 174
65. A Human Analogy......Page 176
66. Generalized Curves and Flows, and Their Boundaries......Page 177
67. Parametric Representation of Generalized Curves......Page 182
68. Existence of a Minimum......Page 189
69. The Nature of the Generalized Solutions......Page 190
71. Separation Theorem for a Convex Cone in \'1&\'0 (A)......Page 195
72. The Lemma of the Insufficient Radius......Page 196
73. The Dual Separation Theorem......Page 198
74. A Localization Lemma for a B · · i-Compact Set......Page 199
75. Riesz Measures......Page 200
77. An Elementary Norm Estimate......Page 201
78. Vector Integration......Page 202
79. Closure of a Convex Hull......Page 203
80. Introduction......Page 205
81. Polygonal Flows......Page 206
83. The Variational Convexity Principle in its Elementary Form......Page 208
84. A First Extension......Page 209
85. The Enlargement Principle and the First Closure Theorem for Generalized Flows......Page 210
86. The Extension to Consistent Flows and Boundaries......Page 211
87. Preliminary Information on Mixtures and on the Lagrange Representation......Page 213
88. Further Comments on Measures, Mixtures, and Consistent Flows......Page 215
89. The Lagrange Representation of a Consistent Flow......Page 220
Volume II - Optimal Control Theory......Page 224
1. Introduction......Page 225
2. The Multiplier Rule......Page 226
3. Optimal Control and the Lagrange Problem......Page 228
4. The Sad Facts of Life......Page 229
5. A First Revision of the Euler Equation and of the Multiplier Rule......Page 231
6. The Weierstrass Condition, Transversality, Hamiltonians and a Strong Revised Euler Recipe......Page 233
7. The Classical Constrained Hamiltonians......Page 235
8. Controls and the Maximum Principle......Page 239
9. The Maximum Principle and Its Special Cases as Definitions......Page 242
10. Solutions of Two Elementary Time-Optimal Problems......Page 244
11. Introduction......Page 254
12. Discrete Time and Programming......Page 255
13. Some Basic Remarks on Linear Differential Equations......Page 258
14. Suspected Solutions of the Simplest Time-Optimal Problems......Page 261
15. Unicity and Optimality......Page 262
16. Two Dimensional Problems: Switching Times and Basic Constructions......Page 264
17. Discussion of Case (a) ......Page 267
18. Discussion of Case (b1)......Page 268
19. Discussion of Case (b2)......Page 270
20. Introduction......Page 272
21. Trajectories and Lines of Flight......Page 274
22. The Synchronization Condition and the Notions of Standard Projection and Descriptive Map......Page 277
23. The Notion of a Spray of Flights......Page 278
24. The Hilbert Independence Integral......Page 280
25. Preliminary Lemmas......Page 283
26. The Theorem of Malus......Page 285
27. Chains of Flights......Page 286
28. Piecing Together Fragments of Curves......Page 287
29. The Fundamental Theorem and Its Consequences......Page 290
30. Introduction......Page 293
31. The Preproblem......Page 296
32. More Semantics......Page 298
33. Conventional and Chattering Controls in Differential Equations......Page 300
34. The Halfway Principle and the Filippov Lemma......Page 303
35. Unicity and a Key Lemma for Approximations......Page 308
36. Control Measures......Page 311
37. A Proper Setting for Optimal Control Problems......Page 315
38. Hilbert\'s Principle of Minimum......Page 318
39. Pontrjagin\'s Maximum Principle......Page 319
39A. The Perturbation......Page 323
39B. Reduction to a Separation Theorem......Page 327
39C. An Equivalent Form of the Separation......Page 329
39D. Proof of the Maximum Principle......Page 330
39E. Epilogue......Page 332
REFERENCES......Page 336
Additional references......Page 338
Index......Page 344