This is a book for those who want to understand the main ideas in the theory of optimal problems. It provides a good introduction to classical topics (under the heading of "the calculus of variations") and more modern topics (under the heading of "optimal control"). It employs the language and terminology of functional analysis to discuss and justify the setup of problems that are of great importance in applications. The book is concise and self-contained, and should be suitable for readers with a standard undergraduate background in engineering mathematics.
Author(s): L.P. Lebedev, Michael J. Cloud
Series: Series on Stability, Vibration and Control of Systems, Series A - Vol. 12
Publisher: World Scientific Publishing Company
Year: 2003
Language: English
Pages: 435
Tags: Автоматизация;Теория автоматического управления (ТАУ);Книги на иностранных языках;
Contents......Page 12
Foreword......Page 6
Preface......Page 10
1.1 Introduction......Page 15
1.2 Euler's Equation for the Simplest Problem......Page 28
1.3 Some Properties of Extremals of the Simplest Functional......Page 33
1.4 Ritz's Method......Page 36
1.5 Natural Boundary Conditions......Page 44
1.6 Some Extensions to More General Functionals......Page 47
1.7 Functionals Depending on Functions in Many Variables......Page 57
1.8 A Functional with Integrand Depending on Partial Derivatives of Higher Order......Page 62
1.9 The First Variation......Page 68
1.10 Isoperimetric Problems......Page 80
1.11 General Form of the First Variation......Page 87
1.12 Movable Ends of Extremals......Page 92
1.13 Weierstrass-Erdmann Conditions and Related Problems......Page 96
1.14 Sufficient Conditions for Minimum......Page 102
1.15 Exercises......Page 111
2.1 A Variational Problem as a Problem of Optimal Control......Page 113
2.2 General Problem of Optimal Control......Page 115
2.3 Simplest Problem of Optimal Control......Page 118
2.4 Fundamental Solution of a Linear Ordinary Differential Equation......Page 125
2.5 The Simplest Problem Continued......Page 126
2.6 Pontryagin's Maximum Principle for the Simplest Problem......Page 127
2.7 Some Mathematical Preliminaries......Page 132
2.8 General Terminal Control Problem......Page 145
2.9 Pontryagin's Maximum Principle for the Terminal Optimal Problem......Page 151
2.10 Generalization of the Terminal Control Problem......Page 154
2.11 Small Variations of Control Function for Terminal Control Problem......Page 159
2.12 A Discrete Version of Small Variations of Control Function for Generalized Terminal Control Problem......Page 161
2.13 Optimal Time Control Problems......Page 165
2.14 Final Remarks on Control Problems......Page 169
2.15 Exercises......Page 171
3. Functional Analysis......Page 173
3.1 A Normed Space as a Metric Space......Page 174
3.2 Dimension of a Linear Space and Separability......Page 179
3.3 Cauchy Sequences and Banach Spaces......Page 183
3.4 The Completion Theorem......Page 194
3.5 Contraction Mapping Principle......Page 198
3.6 Lp Spaces and the Lebesgue Integral......Page 206
3.7 Sobolev Spaces......Page 213
3.8 Compactness......Page 219
3.9 Inner Product Spaces Hilbert Spaces......Page 229
3.10 Some Energy Spaces in Mechanics......Page 234
3.11 Operators and Functional......Page 254
3.12 Some Approximation Theory......Page 259
3.13 Orthogonal Decomposition of a Hilbert Space and the Riesz Representation Theorem......Page 263
3.14 Basis Gram-Schmidt Procedure Fourier Series in Hilbert Space......Page 267
3.15 Weak Convergence......Page 273
3.16 Adjoint and Self-adjoint Operators......Page 281
3.17 Compact Operators......Page 287
3.18 Closed Operators......Page 295
3.19 Introduction to Spectral Concepts......Page 299
3.20 The Fredholm Theory in Hilbert Spaces......Page 304
3.21 Exercises......Page 315
4.1 Some Problems of Mechanics from the Viewpoint of the Calculus of Variations; the Virtual Work Principle......Page 321
4.2 Equilibrium Problem for a Clamped Membrane and its Generalized Solution......Page 327
4.3 Equilibrium of a Free Membrane......Page 329
4.4 Some Other Problems of Equilibrium of Linear Mechanics......Page 331
4.5 The Ritz and Bubnov-Galerkin Methods......Page 339
4.6 The Hamilton-Ostrogradskij Principle and the Generalized Setup of Dynamical Problems of Classical Mechanics......Page 342
4.7 Generalized Setup of Dynamic Problems for a Membrane......Page 344
4.8 Other Dynamic Problems of Linear Mechanics......Page 359
4.9 The Fourier Method......Page 360
4.10 An Eigenfrequency Boundary Value Problem Arising in Linear Mechanics......Page 362
4.11 The Spectral Theorem......Page 366
4.12 The Fourier Method Continued......Page 372
4.13 Equilibrium of a von Karman Plate......Page 377
4.14 A Unilateral Problem......Page 387
4.15 Exercises......Page 394
Appendix A Hints for Selected Exercises......Page 397
References......Page 429
Index......Page 431
