Calculus of variations and optimal control theory

Series ......Page 1
Title page ......Page 2
Copyright ......Page 3
Dedication ......Page 4
Preface ......Page 6
Contents ......Page 8
1. Introduction ......Page 12
2. Euclidean $n$-space and linear spaces ......Page 13
3. Continuous and lower-semicontinuous functions ......Page 16
4. Positively homogeneous functions ......Page 19
5. Elementary properties of linear functionals ......Page 22
6. Functions of class $C^{(m)}$, local minima of functions ......Page 27
7. Implicit function theorems ......Page 33
8. Tangent cones ......Page 36
9. Minima of functions subject to equality constraints ......Page 40
10. Minima of functions subject to inequality constraints ......Page 45
11. An alternate definition of directional derivatives ......Page 51
12. On positively homogeneous functions ......Page 54
13. Convex functions ......Page 56
14. Existence and embedding theorems for differential equations ......Page 59
15. Fundamental lemmas ......Page 61
1. Introduction ......Page 64
2. Terminology and initial results ......Page 68
3. Examples ......Page 72
4. Proof of the necessary condition of Weierstrass ......Page 77
5. The first and second variation of $I$ ......Page 79
6. Variations as differentials ......Page 83
7. Alternate form of the first necessary condition ......Page 86
8. Parametric problems ......Page 89
9. Isoperimetric problems ......Page 94
10. Variable endpoint problems and the transversality condition ......Page 98
11. Boundary arcs as minimizing arcs ......Page 104
12. Proofs of results given in Section 11 ......Page 109
13. Minima of integrals involving higher derivatives ......Page 114
14. Quadratic integrals ......Page 119
15. Proof of Lemma 14.2 ......Page 125
1. Introduction ......Page 129
2. Extremals and canonical variables ......Page 130
3. The condition of Jacobi ......Page 133
4. Determination of conjugate points ......Page 136
5. Positiveness of the second variation ......Page 139
6. Sufficient conditions for relative minima ......Page 143
7. Mayer fields and a fundamental sufficiency theorem ......Page 147
8. Proof of Theorem 6.1 ......Page 148
9. Construction of fields ......Page 150
10. Transversal surfaces and Hamiltonian-Jacobi theory ......Page 154
11. Optimal fields ......Page 156
12. A property of the Weierstrass $E$-function ......Page 162
13. Auxiliary lemmas ......Page 163
14. An indirect sufficiency proof ......Page 170
1. Introduction ......Page 178
2. Tangent cones, derived cones, and derived sets ......Page 181
3. A general multiplier rule ......Page 187
4. The linear case ......Page 190
5. Functions of $n$-variables ......Page 192
6. A formal application of Theorem 3.2 ......Page 194
7. Auxiliary theorems ......Page 197
8. Proof of Theorem 3.1 ......Page 203
9. An alternate proof of a minimum principle ......Page 207
1. Introduction ......Page 213
2. A simple integral problem ......Page 214
3. Auxiliary lemmas ......Page 217
4. An extension of Theorem 2.2 ......Page 219
5. Simple integral problems with integral side conditions ......Page 223
6. Proof of Theorem 5.1 ......Page 231
7. An optimal control problem ......Page 235
8. An extension of Theorem 5.1 ......Page 240
9. Further applications ......Page 245
10. Integral problems involving parameters ......Page 248
11. Optimal control problems linear in the state variables ......Page 251
12. An extension of Theorem 10.1 ......Page 256
13. An approximation lemma ......Page 259
1. Introduction ......Page 261
2. A maximum principle ......Page 264
3. A control problem of Lagrange with equality constraints ......Page 267
4. Control problems of Lagrange with inequality constraints ......Page 271
5. The isoperimetric problem of Lagrange ......Page 273
6. Proof of Theorem 2.1 ......Page 277
7. Normality and the closed derived cone $K^{\ast\ast}$ ......Page 281
8. Normality and abnormality on subintervals ......Page 290
9. Second-order necessary conditions for a minimum 283 10 Optimal fields ......Page 297
1. Introduction ......Page 307
2. Hypotheses and preliminary results ......Page 312
3. A maximum principle for boundary arcs ......Page 319
4. A maximum principle for a minimum problem ......Page 325
5. Transfer functions for an arc $x_0$ ......Page 328
6. Derived sets and boundary arcs ......Page 332
7. Proof of Lemma 6.3 ......Page 339
8. An extension of Theorem 4.1 ......Page 343
9. A general control problem of Mayer ......Page 348
10. A general control problem involving integrals ......Page 354
11. A general control problem of Bolza ......Page 357
1. Introduction ......Page 363
2. The first necessary condition for a minimum ......Page 364
3. Remarks on conclusion (iii) of Theorem 2.1 ......Page 368
4. An alternate form of Theorem 2.1 ......Page 372
5. Auxiliary lemmas ......Page 376
6. Proof of Theorem 4.1 ......Page 381
1. Introduction ......Page 386
2. Hypotheses and initial results ......Page 387
3. An existence theorem ......Page 392
4. An embedding theorem ......Page 395
5. Linear differential equations ......Page 398
6. A differentiability theorem ......Page 401
7. An extension of Theorem 4.1 ......Page 406
8. Differential equations of class $C^{(m)}$ ......Page 407
9. An application of Theorem 4.1 ......Page 409
Bibliography ......Page 412
Index ......Page 414