Contents

Preface                                                                                                                 ix 

1 Basic Calculus of Variations                                                                                1

  1.1 Introduction                                                                                                   1

  1.2 Euler's Equation for the Simplest Problem                                                        14

  1.3 Some Properties of Extremals of the Simplest Functional                                  19

  1.4 Ritz's Method                                                                                                 22

  1.5 Natural Boundary Conditions                                                                          30

  1.6 Some Extensions to More General Functionals                                                 33

  1.7 Functionals Depending on Functions in Many Variables                                    43

  1.8 A Functional with Integrand Depending on Partial Derivatives of Higher Order   48

  1.9 The First Variation  .                                                                                      54

  1.10 Isoperimetric Problems                                                                                 66

  1.11 General Form of the First Variation                                                               73

  1.12 Movable Ends of Extremals                                                                          78

  1.13 Weierstrass-Erdmann Conditions and Related Problems                                 82

  1.14 Sufficient Conditions for Minimum                                                                 88

  1.15 Exercises                                                                                                     97

2 Elements of Optimal Control Theory                                                                    99

  2.1 A Variational Problem as a Problem of Optimal Control                                   99

  2.2 General Problem of Optimal Control                                                                101

  2.3 Simplest Problem of Optimal Control                                                               104

  2.4 Fundamental Solution of a Linear Ordinary Differential Equation                       111

  2.5 The Simplest Problem, Continued                                                                    112

  2.6 Pontryagin's Maximum Principle for the Simplest Problem                                113

  2.7 Some Mathematical Preliminaries                                                                    118

  2.8 General Terminal Control Problem                                                                  131

  2.9 Pontryagin's Maximum Principle for the Terminal Optimal Problem                   137

  2.10 Generalization of the Terminal Control Problem                                              140

  2.11 Small Variations of Control Function for Terminal Control Problem                  145

  2.12 A Discrete Version of Small Variations of Control Function for
         Generalized Terminal Control Problem                                                           147

  2.13 Optimal Time Control Problems                                                                     151

  2.14 Final Remarks on Control Problems                                                               155

  2.15 Exercises                                                                                                     157

3 Functional Analysis                                                                                             159

  3.1 A Normed Space as a Metric Space                                                               160

  3.2 Dimension of a Linear Space and Separability                                                  165

  3.3 Cauchy Sequences and Banach Spaces                                                           169

  3.4 The Completion Theorem                                                                               180

  3.5 Contraction Mapping Principle                                                                        184

  3.6Lp Spaces and the Lebesgue Integral                                                               192

  3.7 Sobolev Spaces                                                                                              199

  3.8 Compactness                                                                                                 205

  3.9 Inner Product Spaces, Hilbert Spaces                                                              215

  3.10 Some Energy Spaces in Mechanics                                                               220

  3.11 Operators and Functionals                                                                             240

  3.12 Some Approximation Theory                                                                         245

  3.13 Orthogonal Decomposition of a Hilbert Space and
         the Riesz Representation Theorem                                                                249

  3.14 Basis, Gram–Schmidt Procedure, Fourier Series in Hilbert Space                    253

  3.15 Weak Convergence                                                                                      259

  3.16 Adjoint and Self-adjoint Operators                                                                 267

  3.17 Compact Operators                                                                                      273

  3.18 Closed Operators                                                                                         281

  3.19 Introduction to Spectral Concepts                                                                  285

  3.20 The Fredholm Theory in Hilbert Spaces                                                         290

  3.21 Exercises                                                                                                     301

4 Some Applications in Mechanics                                                                          307

  4.1 Some Problems of Mechanics from the Viewpoint of
       the Calculus of Variations; the Virtual Work Principle                                       307

  4.2 Equilibrium Problem for a Clamped Membrane and its Generalized Solution        313

  4.3 Equilibrium of a Free Membrane                                                                     315

  4.4 Some Other Problems of Equilibrium of Linear Mechanics                                317

  4.5 The Ritz and Bubnov–Galerkin Methods                                                          325

  4.6 The Hamilton–Ostrogradskij Principle and the Generalized
        Setup of Dynamical Problems of Classical Mechanics                                     328

  4.7 Generalized Setup of Dynamic Problems for a Membrane                                330

  4.8 Other Dynamic Problems of Linear Mechanics                                                345

  4.9 The Fourier Method                                                                                       346

  4.10 An Eigenfrequency Boundary Value Problem Arising in Linear Mechanics       348

  4.11 The Spectral Theorem                                                                                  352

  4.12 The Fourier Method, Continued                                                                     358

  4.13 Equilibrium of a von Karman Plate                                                                363

  4.14 A Unilateral Problem                                                                                    373

  4.15 Exercises                                                                                                     380

Appendix A Hints for Selected Exercises                                                                383

References                                                                                                           415