Theory of Extremal Problems (eBook)

Front Cover 1
Theory of Extremal Problems 4
Copyright Page 5
CONTENTS 12
Preface 6
Basic notation 9
CHAPTER 0 INTRODUCTION. BACKGROUND MATERIAL 14
0.1 Functional analysis 25
0.2 Differential calculus 35
0.3 Convex analysis 57
0.4 Differential equations 63
CHAPTER 1 NECESSARY CONDITIONS FOR AN EXTREMUM 78
1.1 Statements of the problems and formulations of basic theorems 78
1.2 Smooth problems. The Lagrange multiplier rule 90
1.3 Convex problems. Proof of the Kuhn–Tucker theorem 93
1.4 Mixed problems. Proof of the extremal principle 97
CHAPTER 2 NECESSARY CONDITIONS FOR AN EXTREMUM IN THE CLASSICAL PROBLEMS OF THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL 106
2.1 Statements of the problems 106
2.2 Elementary derivation of necessary conditions for an extremum in simplest problems of the classical calculus of variations 114
2.3 The Lagrange problem. The Euler–Lagrange equation 137
2.4 The Pontrjagin maximum principle. Formulation and discussion 145
2.5 Proof of the maximum principle 160
CHAPTER 3 ELEMENTS OF CONVEX ANALYSIS 174
3.1 Convex sets and separation theorems 174
3.2 Convex functions 180
3.3 Conjugate functions. The Fenche1–Moreau theorem 184
3.4 Duality theorems 191
3.5 Convex analysis in finite-dimensional spaces 197
CHAPTER 4 LOCAL CONVEX ANALYSIS 204
4.1 Homogeneous functions and directional derivatives 204
4.2 Subdifferentials. Basic theorems 209
4.3 Cones of supporting functionals 218
4.4 Locally convex functions 221
4.5 The subdifferentials of certain functions 229
CHAPTER 5 LOCALLY CONVEX PROBLEMS AND THE MAXIMUM PRINCIPLE FOR PROBLEMS WITH PHASE CONSTRAINTS 237
5.1 Locally convex problems 237
5.2 Optimal control problems with phase constraints 246
5.3 Proof of the maximum principle for problems with phase constraints 254
CHAPTER 6 SPECIAL PROBLEMS 268
6.1 Linear programming 268
6.2 The theory of quadratic forms in Hilbert space 271
6.3 Quadratic functionals in the classical caculus of variations 279
6.4 Discrete optimal control problems 290
CHAPTER 7 SUFFICIENT CONDITIONS FOR AN EXTREMUM 297
7.1 The perturbation method 297
7.2 Smooth problems 303
7.3 Convex problems 312
7.4 Sufficient conditions for an extremum in the classical calculus of variations 316
CHAPTER 8 MEASURABLE MULTIMAPPINGS AND CONVEX ANALYSIS OF INTEGRAL FUNCTIONALS 334
8.1 Multimappings and measurability 334
8.2 Integration of multimappings 347
8.3 Integral functionals 353
CHAPTER 9 EXISTENCE OF SOLUTIONS IN PROBLEMS OF THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL 367
9.1 Semicontinuity of the functionals in the calculus of variations and the compactness of their level sets 367
9.2 Theorems on the existence of solutions 382
9.3 The convolution integral and linear problems 398
CHAPTER 10 APPLICATION OF THE THEORY TO SPECIFIC PROBLEMS 417
10.1 Problems in geometric optics 417
10.2 Young's inequality and Helly's theorem 427
10.3 Optimal excitation of an oscillator 431
Problems 436
Bibliography 456
Subject Index 470