Table of Contents

Preface v

1 The Classical Calculus of Variations 1

1.1 Problem Statement 1

1.2 The Euler-Lagrange Equations 3

1.3 Problems with Isoperimetric Constraints 5

1.4 Sufficient Optimality Conditions via Joint Convexity 6

1.5 Noether's Theorem 7

2 Fractional Calculus of Variations via Riemann-Liouville Operators 11

2.1 Riemann-Liouville Fractional Integrals and Derivatives 11

2.2 Fundamental Problem of the Calculus of Variations 17

2.2.1 Problems with Classical and Fractional Derivatives 17

2.2.2 Problems with Fractional Derivatives and Integrals 19

2.3 Fractional Isoperimetric Problems 21

2.3.1 Problems with Fractional Derivatives 22

2.3.2 Problems with Classical and Fractional Derivatives 26

2.3.3 Problems with Fractional Derivatives and Integrals 29

2.3.4 Some Extensions 29

2.4 Sufficient Conditions for Optimality 35

2.5 Fractional Noether's Theorem 37

2.6 Multidimensional Fractional Noether's Theorem 45

2.6.1 Multidimensional Euler-Lagrange Equations 46

2.6.2 Fractional Noether-Type Theorem 49

2.7 Modified Optimal Energy and Initial Memory 51

2.7.1 Linear Systems and the Controllability Gramian 52

2.7.2 Steering Laws 56

2.8 Fractional Conservation Laws in Optimal Control 58

2.9 Fractional Version of Leitmann's Direct Method 65

3 Fractional Calculus of Variations via Caputo Operators 73

3.1 Caputo Fractional Derivatives 74

3.2 Fundamental Problem of the Calculus of Variations 80

3.3 Variable End-points 82

3.3.1 Generalized Natural Boundary Conditions 82

3.3.2 Transversality Conditions I 87

3.3.3 Transversality Conditions II 91

3.3.4 Infinite Horizon Fractional Variational Problems 93

3.4 The Fractional Isoperimetric Problem 96

3.5 Sufficient Conditions for Optimality 99

3.6 Minimal Modified Energy Control 102

3.6.1 Fractional Linear Control Systems and the Gramian 102

3.6.2 Rank Conditions and Steering Controls 106

3.7 A Noether Theorem for Fractional Optimal Control 107

3.8 Fractional Riesz-Caputo Derivatives 116

3.8.1 Riesz-Caputo Conservation of Momentum 116

3.8.2 The Noether Theorem in the Riesz-Caputo Sense 120

3.8.3 Optimal Control of Riesz-Caputo Systems 122

3.9 Multidimensional Lagrangians 126

3.9.1 Multidimensional Euler-Lagrange Equations 127

3.9.2 Natural Boundary Conditions 129

3.9.3 Sufficient Condition for Optimality 130

3.9.4 Examples 132

3.9.5 Fractional Noether-Type Theorem 133

3.10 On the Second Noether Theorem 136

3.10.1 Single Integral Case 136

3.10.2 Multiple Integral Case 141

3.10.3 Example 144

3.11 Further Generalizations 145

3.11.1 Calculus of Variations via CDα,βγ 147

3.11.2 Multiobjective Fractional Optimization 160

3.11.3 Problems Depending on Indefinite Integrals 166

4 Other Approaches to the Fractional Calculus of Variations 191

4.1 Fractional Action-Like Variational Approach (FALVA) 191

4.1.1 Fractional Action-Like Noether's Theorem 192

4.1.2 Higher-Order Euler-Lagrange Equations 195

4.1.3 Higher-Order DuBois-Reymond Condition 198

4.1.4 Stationary Conditions for Optimal Control 201

4.1.5 Higher-Order Noether's Theorem 203

4.2 Cresson's Approach 208

4.2.1 The Fractional Derivative of Cresson 209

4.2.2 Fractional Euler-Lagrange Equations 209

4.2.3 The Fractional Hamiltonian Formalism 211

4.2.4 Double-Weighted Fractional Variational Principles 213

4.2.5 N-Weighted Fractional Variational Principles 215

4.3 Jumarie's Approach 216

4.3.1 Jumarie's Riemann-Liouville Derivative 217

4.3.2 The Euler-Lagrange Equations 220

4.3.3 Natural Boundary Conditions 222

4.3.4 The Isoperimetric Problem 223

4.3.5 Holonomic Constraints 224

4.3.6 Lagrangians Depending on the Free End-Points 225

4.3.7 Composition Functionals 228

4.3.8 Fractional Theorems of Green, Gauss, and Stokes 234

4.3.9 Fractional Variational Calculus with Multiple Integrals 239

4.3.10 Applications and Possible Extensions 249

5 Towards a Combined Fractional Mechanics and Quantization 253

5.1 Preliminaries 253

5.2 Hamiltonian Formulation of the Combined Euler-Lagrange Equations 254

5.3 Fractional Constants of Motion 256

5.4 Canonical Fractional Transformations of the First Kind 256

5.5 Canonical Fractional Transformations of the Second Kind 258

5.6 Fractional Hamilton-Jacobi Equation 259

5.7 Conclusion 259

Bibliography 261

Index 271