Contents -- Foreword -- Preface -- 1. Basic Calculus of Variations -- 1.1 Introduction -- 1.2 Euler's Equation for the Simplest Problem -- 1.3 Some Properties of Extremals of the Simplest Functional -- 1.4 Ritz's Method -- 1.5 Natural Boundary Conditions -- 1.6 Some Extensions to More General Functionals -- 1.7 Functionals Depending on Functions in Many Variables -- 1.8 A Functional with Integrand Depending on Partial Derivatives of Higher Order -- 1.9 The First Variation -- 1.10 Isoperimetric Problems -- 1.11 General Form of the First Variation -- 1.12 Movable Ends of Extremals -- 1.13 Weierstrass-Erdmann Conditions and Related Problems -- 1.14 Sufficient Conditions for Minimum -- 1.15 Exercises -- 2. Elements of Optimal Control Theory -- 2.1 A Variational Problem as a Problem of Optimal Control -- 2.2 General Problem of Optimal Control -- 2.3 Simplest Problem of Optimal Control -- 2.4 Fundamental Solution of a Linear Ordinary Differential Equation -- 2.5 The Simplest Problem Continued -- 2.6 Pontryagin's Maximum Principle for the Simplest Problem -- 2.7 Some Mathematical Preliminaries -- 2.8 General Terminal Control Problem -- 2.9 Pontryagin's Maximum Principle for the Terminal Optimal Problem -- 2.10 Generalization of the Terminal Control Problem -- 2.11 Small Variations of Control Function for Terminal Control Problem -- 2.12 A Discrete Version of Small Variations of Control Function for Generalized Terminal Control Problem -- 2.13 Optimal Time Control Problems -- 2.14 Final Remarks on Control Problems -- 2.15 Exercises -- 3. Functional Analysis -- 3.1 A Normed Space as a Metric Space -- 3.2 Dimension of a Linear Space and Separability -- 3.3 Cauchy Sequences and Banach Spaces -- 3.4 The Completion Theorem -- 3.5 Contraction Mapping Principle -- 3.6 Lp Spaces and the Lebesgue Integral -- 3.7 Sobolev Spaces.

3.8 Compactness -- 3.9 Inner Product Spaces Hilbert Spaces -- 3.10 Some Energy Spaces in Mechanics -- 3.11 Operators and Functional -- 3.12 Some Approximation Theory -- 3.13 Orthogonal Decomposition of a Hilbert Space and the Riesz Representation Theorem -- 3.14 Basis Gram-Schmidt Procedure Fourier Series in Hilbert Space -- 3.15 Weak Convergence -- 3.16 Adjoint and Self-adjoint Operators -- 3.17 Compact Operators -- 3.18 Closed Operators -- 3.19 Introduction to Spectral Concepts -- 3.20 The Fredholm Theory in Hilbert Spaces -- 3.21 Exercises -- 4. Some Applications in Mechanics -- 4.1 Some Problems of Mechanics from the Viewpoint of the Calculus of Variations -- the Virtual Work Principle -- 4.2 Equilibrium Problem for a Clamped Membrane and its Generalized Solution -- 4.3 Equilibrium of a Free Membrane -- 4.4 Some Other Problems of Equilibrium of Linear Mechanics -- 4.5 The Ritz and Bubnov-Galerkin Methods -- 4.6 The Hamilton-Ostrogradskij Principle and the Generalized Setup of Dynamical Problems of Classical Mechanics -- 4.7 Generalized Setup of Dynamic Problems for a Membrane -- 4.8 Other Dynamic Problems of Linear Mechanics -- 4.9 The Fourier Method -- 4.10 An Eigenfrequency Boundary Value Problem Arising in Linear Mechanics -- 4.11 The Spectral Theorem -- 4.12 The Fourier Method Continued -- 4.13 Equilibrium of a von Karman Plate -- 4.14 A Unilateral Problem -- 4.15 Exercises -- Appendix A Hints for Selected Exercises -- References -- Index.