Introduction to the Calculus of Variations and Control with Modern Applications

564 Pages·2014·4.93 MB·English

by  John A. Burns

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Introduction to
The Calculus of
Variations
and
Control
with Modern Applications
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CHAPMAN & HALL/CRC APPLIED MATHEMATICS
AND NONLINEAR SCIENCE SERIES
Introduction to
The Calculus of
Variations
and
Control
with Modern Applications
John A. Burns
Virginia Tech
Blacksburg, Virginia, USA
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CRC Press
Taylor & Francis Group
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Contents
Preface xi
Acknowledgments xvii
I Calculus of Variations 1
1 Historical Notes on the Calculus of Variations 3
1.1 Some Typical Problems . . . . . . . . . . . . . . . . 6
1.1.1 Queen Dido’s Problem . . . . . . . . . . . . 6
1.1.2 The Brachistochrone Problem . . . . . . . . 7
1.1.3 Shape Optimization . . . . . . . . . . . . . . 8
1.2 Some Important Dates and People . . . . . . . . . . 11
2 Introduction and Preliminaries 17
2.1 Motivating Problems . . . . . . . . . . . . . . . . . 17
2.1.1 Problem 1: The Brachistochrone Problem . . 17
2.1.2 Problem 2: The River Crossing Problem . . 18
2.1.3 Problem 3: The Double Pendulum . . . . . 20
2.1.4 Problem 4: The Rocket Sled Problem . . . 21
2.1.5 Problem 5: Optimal Control in the Life Sci-
ences . . . . . . . . . . . . . . . . . . . . . . 22
2.1.6 Problem 6: Numerical Solutions of Bound-
ary Value Problems . . . . . . . . . . . . . . 24
2.2 Mathematical Background . . . . . . . . . . . . . . 26
2.2.1 A Short Review and Some Notation . . . . . 26
2.2.2 A Review of One Dimensional Optimization 35
2.2.3 Lagrange Multiplier Theorems . . . . . . . . 42
v
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vi
2.3 Function Spaces . . . . . . . . . . . . . . . . . . . 57
2.3.1 Distances between Functions . . . . . . . . 64
2.3.2 An Introduction to the First Variation . . . 68
2.4 Mathematical Formulation of Problems . . . . . . . 69
2.4.1 The Brachistochrone Problem . . . . . . . . 69
2.4.2 The Minimal Surface of Revolution Problem 72
2.4.3 The River Crossing Problem . . . . . . . . . 73
2.4.4 The Rocket Sled Problem . . . . . . . . . . 74
2.4.5 The Finite Element Method . . . . . . . . . 76
2.5 Problem Set for Chapter 2 . . . . . . . . . . . . . . 86
3 The Simplest Problem in the Calculus of Variations
91
3.1 The Mathematical Formulation of the SPCV . . . . 91
3.2 The Fundamental Lemma of the Calculus of Varia-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3 The First Necessary Condition for a Global Mini-
mizer . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3.1 Examples . . . . . . . . . . . . . . . . . . . 112
3.4 Implications and Applications of the FLCV . . . . 117
3.4.1 Weak and Generalized Derivatives . . . . . . 118
3.4.2 Weak Solutions to Differential Equations . 124
3.5 Problem Set for Chapter 3 . . . . . . . . . . . . . 125
4 Necessary Conditions for Local Minima 131
4.1 Weak and Strong Local Minimizers . . . . . . . . . 132
4.2 The Euler Necessary Condition - (I) . . . . . . . . . 135
4.3 The Legendre Necessary Condition - (III) . . . . . . 139
4.4 Jacobi Necessary Condition - (IV) . . . . . . . . . . 146
4.4.1 Proof of the Jacobi Necessary Condition . . 152
4.5 Weierstrass Necessary Condition - (II) . . . . . . . 155
4.5.1 Proof of the Weierstrass Necessary Condition159
4.5.2 Weierstrass Necessary Condition for a Weak
Local Minimum . . . . . . . . . . . . . . . . 171
4.5.3 A Proof of Legendre’s Necessary Condition . 176
4.6 Applying the Four Necessary Conditions . . . . . . 178
4.7 Problem Set for Chapter 4 . . . . . . . . . . . . . 180
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vii
5 Sufficient Conditions for the Simplest Problem 185
5.1 A Field of Extremals . . . . . . . . . . . . . . . . . 186
5.2 The Hilbert Integral . . . . . . . . . . . . . . . . . 190
5.3 Fundamental Sufficient Results . . . . . . . . . . . 192
5.4 Problem Set for Chapter 5 . . . . . . . . . . . . . . 197
6 Summary for the Simplest Problem 203
7 Extensions and Generalizations 213
7.1 Properties of the First Variation . . . . . . . . . . . 213
7.2 The Free Endpoint Problem . . . . . . . . . . . . . 215
7.2.1 The Euler Necessary Condition . . . . . . . 218
7.2.2 Examples of Free Endpoint Problems . . . . 221
7.3 The Simplest Point to Curve Problem . . . . . . . . 224
7.4 Vector Formulations and Higher Order Problems . . 238
7.4.1 Extensions of Some Basic Lemmas . . . . . 242
7.4.2 The Simplest Problem in Vector Form . . . 249
7.4.3 The Simplest Problem in Higher Order Form 252
7.5 Problems with Constraints: Isoperimetric Problem . 255
7.5.1 Proof of the Lagrange Multiplier Theorem . 259
7.6 Problems with Constraints: Finite Constraints . . . 263
7.7 An Introduction to Abstract Optimization Prob-
lems . . . . . . . . . . . . . . . . . . . . . . . . . . 265
7.7.1 The General Optimization Problem . . . . 265
7.7.2 General Necessary Conditions . . . . . . . . 267
7.7.3 Abstract Variations . . . . . . . . . . . . . . 271
7.7.4 Application to the SPCV . . . . . . . . . . . 273
7.7.5 Variational Approach to Linear Quadratic
Optimal Control . . . . . . . . . . . . . . . 274
7.7.6 An Abstract Sufficient Condition . . . . . . 275
7.8 Problem Set for Chapter 7 . . . . . . . . . . . . . . 278
8 Applications 283
8.1 Solution of the Brachistochrone Problem . . . . . . 283
8.2 Classical Mechanics and Hamilton’s Principle . . . 287
8.2.1 Conservation of Energy . . . . . . . . . . . . 292
8.3 A Finite Element Method for the Heat Equation . . 295
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viii
8.4 Problem Set for Chapter 8 . . . . . . . . . . . . . . 303
II Optimal Control 307
9 Optimal Control Problems 309
9.1 An Introduction to Optimal Control Problems . . . 309
9.2 The Rocket Sled Problem . . . . . . . . . . . . . . 313
9.3 Problems in the Calculus of Variations . . . . . . . 315
9.3.1 The Simplest Problem in the Calculus of
Variations . . . . . . . . . . . . . . . . . . . 315
9.3.2 Free End-Point Problem . . . . . . . . . . . 318
9.4 Time Optimal Control . . . . . . . . . . . . . . . . 319
9.4.1 Time Optimal Control for the Rocket Sled
Problem . . . . . . . . . . . . . . . . . . . . 319
9.4.2 The Bushaw Problem . . . . . . . . . . . . . 333
9.5 Problem Set for Chapter 9 . . . . . . . . . . . . . . 338
10 Simplest Problem in Optimal Control 341
10.1 SPOC: Problem Formulation . . . . . . . . . . . . . 341
10.2 The Fundamental Maximum Principle . . . . . . . 343
10.3 Application of the Maximum Principle to Some
Simple Problems . . . . . . . . . . . . . . . . . . . 351
10.3.1 The Bushaw Problem . . . . . . . . . . . . 351
10.3.2 The Bushaw Problem: Special Case γ = 0
and κ = 1 . . . . . . . . . . . . . . . . . . . 358
10.3.3 A Simple Scalar Optimal Control Problem . 362
10.4 Problem Set for Chapter 10 . . . . . . . . . . . . . 367
11 Extensions of the Maximum Principle 373
11.1 A Fixed-Time Optimal Control Problem . . . . . . 373
11.1.1 The Maximum Principle for Fixed t . . . . 375
1
11.2 Application to Problems in the Calculus of Varia-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . 377
11.2.1 The Simplest Problem in the Calculus of
Variations . . . . . . . . . . . . . . . . . . . 377
11.2.2 Free End-Point Problems . . . . . . . . . . 384
11.2.3 Point-to-Curve Problems . . . . . . . . . . 385
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ix
11.3 Application to the Farmer’s Allocation Problem . . 393
11.4 Application to a Forced Oscillator Control Problem 400
11.5 Application to the Linear Quadratic Control
Problem . . . . . . . . . . . . . . . . . . . . . . . . 404
11.5.1 Examples of LQ Optimal Control Problems 410
11.5.2 The Time Independent Riccati Differential
Equation . . . . . . . . . . . . . . . . . . . 419
11.6 The Maximum Principle for a Problem of Bolza . . 429
11.7 The Maximum Principle for Nonautonomous
Systems . . . . . . . . . . . . . . . . . . . . . . . . 436
11.8 Application to the Nonautonomous LQ Control
Problem . . . . . . . . . . . . . . . . . . . . . . . . 446
11.9 Problem Set for Chapter 11 . . . . . . . . . . . . . 453
12 Linear Control Systems 459
12.1 Introduction to Linear Control Systems . . . . . . . 459
12.2 Linear Control Systems Arising from Nonlinear
Problems . . . . . . . . . . . . . . . . . . . . . . . 473
12.2.1 Linearized Systems . . . . . . . . . . . . . . 474
12.2.2 Sensitivity Systems . . . . . . . . . . . . . 475
12.3 Linear Quadratic Optimal Control . . . . . . . . . . 478
12.4 The Riccati Differential Equation for a Problem of
Bolza . . . . . . . . . . . . . . . . . . . . . . . . . . 480
12.5 Estimation and Observers . . . . . . . . . . . . . . 490
12.5.1 The Luenberger Observer . . . . . . . . . . 494
12.5.2 An Optimal Observer: The Kalman Filter . 498
12.6 The Time Invariant Infinite Interval Problem . . . 506
12.7 The Time Invariant Min-Max Controller . . . . . . 509
12.8 Problem Set for Chapter 12 . . . . . . . . . . . . . 511
Bibliography 519
Index 539
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