Variational Calculus and Optimal Control: Optimization with Elementary Convexity

✍ Scribed by John L. Troutman

• Publisher: Springer
• Year: 1995
• Tongue: English
• Leaves: 479
• Series: Undergraduate Texts in Mathematics
• Edition: 2nd
• Category: Library

✦ Synopsis

I had read/studied most of this book when I was a graduate student in chemical engineering at Syracuse University (in 1987-88). I also took two courses on the subject from Professor Troutman. I strongly recommend this book to any "newcomer" to the subject. The author is a mathematician, and a large fraction of the book consists of theorems, lemmas, propositions, corollaries (and their rigorous proofs). The book also contains, however, a good number of illustrative examples and exercises which make it useful to engineers and scientists as well as to students of mathematics who want to learn more about applications of mathematics to physical sciences.

✦ Table of Contents

Cover......Page 1
Series: Undergraduate Texts in Mathematics......Page 2
Variational Calculus and Optimal Control: Optimization with Elementary Convexity, Second edition......Page 4
Copyright - ISBN: 0387945113......Page 5
Preface......Page 8
Acknowledgments......Page 10
Contents......Page 12
CHAPTER 0. Review of Optimization in IR^d......Page 18
Problems......Page 24
PART ONE. BASIC THEORY......Page 28
1.1. Geodesic Problems......Page 30
(a) Geodesies in IR^d......Page 31
(b) Geodesies on a Sphere......Page 32
(a) The Brachistochrone......Page 34
(b) Steering and Control Problems......Page 37
1.3. Isoperimetric Problems......Page 38
(a) Minimal Surface of Revolution......Page 41
(b) Minimal Area Problem......Page 42
1.5. Summary: Plan of the Text......Page 43
Notation: Uses and Abuses......Page 46
Problems......Page 48
2.1. Real Linear Spaces......Page 53
2.2. Functions from Linear Spaces......Page 55
2.3. Fundamentals of Optimization......Page 56
Constraints......Page 58
Application: Rotating Fluid Column......Page 59
2.4. The Gâteaux Variations......Page 62
Problems......Page 67
CHAPTER 3. Minimization of Convex Functions......Page 70
3.1. Convex Functions......Page 71
3.2. Convex Integral Functions......Page 73
Free End-Point Problems......Page 77
3.3. [Strongly] Convex Functions......Page 78
(a) Geodesies on a Cylinder......Page 82
(b) A Brachistochrone......Page 83
(c) A Profile of Minimum Drag......Page 86
(d) An Economics Problem......Page 89
(e) Minimal Area Problem......Page 91
3.5. Minimization with Convex Constraints......Page 93
The Hanging Cable......Page 95
Optimal Performance......Page 98
3.6. Summary: Minimizing Procedures......Page 100
Problems......Page 101
CHAPTER 4. The Lemmas of Lagrange and Du Bois-Reymond......Page 114
Problems......Page 118
5.1. Norms for Linear Spaces......Page 120
5.2. Normed Linear Spaces: Convergence and Compactness......Page 123
5.3. Continuity......Page 125
5.4. (Local) Extremal Points......Page 131
5.5. Necessary Conditions: Admissible Directions......Page 132
5.6. Affine Approximation: The Frechet Derivative......Page 137
Tangency......Page 144
5.7. Extrema with Constraints: Lagrangian Multipliers......Page 146
Problems......Page 156
CHAPTER 6. The Euler-Lagrange Equations......Page 162
6.1. The First Equation: Stationary Functions......Page 164
6.2. Special Cases of the First Equation......Page 165
(b) When f = f(x, z)......Page 166
(c) When f = f(y, z)......Page 167
6.3. The Second Equation......Page 170
Application: Jakob Bernoulli's Brachistochrone......Page 173
Transversal Conditions......Page 174
6.5. Integral Constraints: Lagrangian Multipliers......Page 177
6.6. Integrals Involving Higher Derivatives......Page 179
Buckling of a Column under Compressive Load......Page 181
6.7. Vector Valued Stationary Functions......Page 186
Application 1: The Isoperimetric Problem......Page 188
Lagrangian Constraints......Page 190
Application 2: Geodesies on a Surface......Page 194
6.8. Invariance of Stationarity......Page 195
6.9. Multidimensional Integrals......Page 198
Application: Minimal Area Problem......Page 201
Natural Boundary Conditions......Page 202
Problems......Page 203
PART TWO. ADVANCED TOPICS......Page 212
CHAPTER 7. Piecewise C^1 Extremal Functions......Page 214
7.1. Piecewise C^1 Functions......Page 215
(a) Smoothing......Page 216
(b) Norms for \hat{C}^1......Page 218
7.2. Integral Functions on \hat{C}^1......Page 219
7.3. Extremals in \hat{C}^1[a, b]:The Weierstrass-Erdmann Corner Conditions......Page 221
Application: A Sturm-Liouville Problem......Page 226
7.4. Minimization Through Convexity......Page 228
Internal Constraints......Page 229
7.5. Piecewise C^1 Vector-Valued Extremals......Page 232
Application: Minimal Surface of Revolution......Page 234
Hilbert's Differentiability Criterion......Page 237
7.6. Conditions Necessary for a Local Minimum......Page 238
(a) The Weierstrass Condition......Page 239
(b) The Legendre Condition......Page 241
Bolza's Problem......Page 242
Problems......Page 244
CHAPTER 8. Variational Principles in Mechanics......Page 251
8.1. The Action Integral......Page 252
8.2. Hamilton's Principle: Generalized Coordinates......Page 253
Bernoulli's Principle of Static Equilibrium......Page 256
8.3. The Total Energy......Page 257
Application: Spring-Mass-Pendulum System......Page 258
8.4. The Canonical Equations......Page 260
8.5. Integrals of Motion in Special Cases......Page 264
Jacobi's Principle of Least Action......Page 265
8.6. Parametric Equations of Motion......Page 267
8.7 The Hamilton-Jacobi Equation......Page 268
8.8. Saddle Functions and Convexity; Complementary Inequalities......Page 271
Example 1. The Cycloid Is the Brachistochrone......Page 274
Example 2 .Dido's Problem......Page 275
(a) Taut String......Page 277
The Nonuniform String......Page 281
(b) Stretched Membrane......Page 283
Static Equilibrium of (Nonplanar) Membrane......Page 286
Problems......Page 287
CHAPTER 9. Sufficient Conditions for a Minimum......Page 299
9.1. The Weierstrass Method......Page 300
9.2. [Strict] Convexity of f(\underline{x}, \underline{Y}, Z)......Page 303
9.3. Fields......Page 305
Exact Fields and the Hamilton-Jacobi Equation......Page 310
9.4. Hilbert's Invariant Integral......Page 311
Application: The Brachistochrone......Page 313
Variable End-Point Problems......Page 314
9.5. Minimization with Constraints......Page 317
The Wirtinger Inequality......Page 321
9.6. Central Fields......Page 325
Smooth Minimal Surface of Revolution......Page 329
9.7. Construction of Central Fields with Given Trajectory: The Jacobi Condition......Page 331
9.8. Sufficient Conditions for a Local Minimum......Page 336
Application: Hamilton's Principle......Page 337
(b) Trajectory Results......Page 338
9.9. Necessity of the Jacobi Condition......Page 339
9.10. Concluding Remarks......Page 344
Problems......Page 346
PART THREE. OPTIMAL CONTROL......Page 356
CHAPTER 10. Control Problems and Sufficiency Considerations......Page 358
10.1. Mathematical Formulation and Terminology......Page 359
10.2. Sample Problems......Page 361
(a) Some Easy Problems......Page 362
(b) A Bolza Problem......Page 364
(c) Optimal Time of Transit......Page 365
(d) A Rocket Propulsion Problem......Page 367
(e) A Resource Allocation Problem......Page 369
(f) Excitation of an Oscillator......Page 372
(g) Time-Optimal Solution by Steepest Descent......Page 374
10.3. Sufficient Conditions Through Convexity......Page 376
Linear State-Quadratic Performance Problem......Page 378
10.4. Separate Convexity and the Minimum Principle......Page 382
Problems......Page 389
11.1. Necessity of the Minimum Principle......Page 395
(a) Effects of Control Variations......Page 397
(b) Autonomous Fixed Interval Problems......Page 401
(c) General Control Problems......Page 408
11.2. Linear Time-Optimal Problems......Page 414
Problem Statement......Page 415
A Free Space Docking Problem......Page 418
11.3. General Lagrangian Constraints......Page 421
(a) Control Sets Described by Lagrangian Inequalities......Page 422
(b) Variational Problems with Lagrangian Constraints......Page 423
(c) Extensions......Page 427
Problems......Page 430
A.0. Compact Sets in IR^d......Page 436
A.1. The Intermediate and Mean Value Theorems......Page 438
A.2. The Fundamental Theorem of Calculus......Page 440
A.3. Partial Integrals: Leibniz' Formula......Page 442
A.4. An Open Mapping Theorem......Page 444
A.5. Families of Solutions to a System of Differential Equations......Page 446
A.6. The Rayleigh Ratio......Page 452
A.7*. Linear Functional and Tangent Cones in IR^d......Page 458
Bibliography......Page 462
Historical References......Page 467
Answers to Selected Problems......Page 469
Index......Page 474