Advances in Mathematical Modeling, Optimization and Optimal Control (eBook)

Contents 6
Introduction 7
Bregman Distances in Inverse Problems and Partial Differential Equations 9
1 Introduction 9
2 Bregman Distances and Their Basic Properties 10
2.1 Examples of Bregman Distances 12
2.2 Bregman Distances and Duality 13
2.3 Bregman Distances and Fenchel Duality 14
2.4 Bregman Distances for One-Homogeneous Functionals 15
3 Applications in Inverse Problems and Imaging 16
3.1 Error Estimates 17
3.2 Asymptotics 19
3.3 Bregman Iterations and Inverse Scale Space Methods 20
4 Applications in Partial Differential Equations 23
4.1 Entropy Dissipation Methods for Gradient Systems 23
4.2 Lyapunov Functionals for Gradient Systems Out of Equilibrium 26
4.3 Doubly Nonlinear Evolution Equations 28
4.4 Error Estimates for Nonlinear Elliptic Problems 30
5 Further Developments 32
5.1 Uncertainty Quantification in Inverse Problems 32
5.2 Bregman Distances and Optimal Transport 35
5.3 Infimal Convolution of Bregman Distances 36
References 36
On Global Attractor for Parabolic Partial Differential Inclusion and Its Time Semidiscretization 40
1 Introduction 40
2 Definitions 42
3 Problem Definition 45
4 Pseudomonotonicity of Nemytskii Operator for A 47
5 Convergence of Semi-Discrete Scheme 48
6 Global Attractor for Time Continuous Problem 57
7 Global Attractor for the Time Discrete Problem 61
8 Upper-Semicontinuous Convergence of Attractors 62
9 Examples 65
References 67
Passive Control of Singularities by Topological Optimization: The Second-Order Mixed Shape Derivatives of Energy Functionals for Variational Inequalities 70
1 Introduction 71
1.1 Asymptotic Approximation for Variational Inequalities 72
2 Applications of Steklov–Poincaré Operators in Asymptotic Analysis 75
3 Asymptotic Analysis by Domain Decomposition Method 77
4 Asymptotic Analysis of Boundary Value Problems in Rings or Spherical Shells 78
4.1 Elasticity Boundary Value Problems 78
4.2 Explicit form of the Operator B for the Laplacian in Two Spatial Dimensions 81
4.3 Explicit form of the Operator B for the Laplacian in Three Spatial Dimensions 82
4.4 Explicit form of the Operator B for Elasticity in Two Spatial Dimensions 82
4.5 Explicit form of the Operator B for Elasticity in Three Spatial Dimensions 83
4.6 Laplace Spherical Polynomials 85
5 Asymptotic Analysis of Steklov–Poincaré Operators in Reinforced Rings in Two Spatial Dimensions 86
5.1 Model Problem 87
5.2 Steklov–Poincaré Operator 89
5.3 Asymptotic Expansion 90
5.4 Extension to Linear Elasticity 92
6 Asymptotic Expansions of the Steklov–Poincaré Operators and Perturbations of Bilinear Forms in Particular Cases 93
7 Directional Differentiability of the Metric Projection onto Positive Cone in Fractional Sobolev Spaces 94
7.1 Metric Projection onto Positive Cone in H1/200(c) 99
8 Rectilinear Crack in Two Spatial Dimensions 101
8.1 Green Formulae and Steklov–Poincaré Operators 103
9 Shape and Topological Derivatives of Elastic Energy in Two Spatial Dimensions for an Inclusion 105
9.1 Shape and Topological Derivatives of the Energy Functional in ?R with Respect to the Inclusion ? 105
References 106
Optimal Control for Applications in Medical and Rehabilitation Technology: Challenges and Solutions 108
1 Introduction 108
2 Mathematical Models for Motion Studies in Medical and Rehabilitation Engineering 111
2.1 Whole-Body Models of Humans 112
2.2 Including Neuromuscular Models in the Human Model 114
2.3 Modeling the Rehabilitation Device and Establishing a Combined Model 118
3 Mathematical Stability Criteria for Human Movement 120
3.1 ZMP Related Criteria 120
3.2 Lyapunov's First Method 121
3.3 Lyapunov's Second Method 123
3.4 Capture Point Stability Criteria 124
3.5 Angular Momentum 125
4 Formulation and Solution of Optimal Control Problems for Motion Generation 125
4.1 Multi-Phase Hybrid Optimal Control Problems with Standard Criteria 126
4.2 Multi-Phase Hybrid Optimal Control Problems with Non-standard Criteria Related to Stability 127
4.3 Numerical Solution of Optimal Control Problems 129
5 Formulation and Solution of Inverse Optimal Control Problems for Analysis of Motions in Medical Applications 129
6 Model-Based Optimization for Physical Assistive Devices for Geriatric Patients 132
7 Optimization and Analysis of Motions with Lower Limb Prostheses 135
8 Model-Based Optimization for the Design of Lower Limb Exoskeletons 137
9 Optimization of Functional Electrical Stimulation for Walking Motions of Hemiplegic Patients 139
10 Stability Studies of Human Walking 142
11 Conclusion and Perspectives 144
References 146
Second-Order Optimality Conditions for Broken Extremals and Bang-Bang Controls: Theory and Applications 151
1 Introduction 152
2 Second-Order Optimality Conditions for Broken Extremals in the Simplest Problem in the Calculus of Variations 153
2.1 The Simplest Problem in the Calculus of Variations 153
2.2 Second-Order Optimality Conditions for Broken Extremals 154
3 Second-Order Optimality Conditions for Discontinuous Controls in the General Problem of the Calculus of Variations on a Fixed Time Interval 156
3.1 The General Problem in the Calculus of Variations on a Fixed Time Interval 156
3.2 First-Order Necessary Conditions 157
3.3 Second-Order Necessary Conditions 159
3.4 Second-Order Sufficient Conditions 161
4 The General Problem in the Calculus of Variations on a Variable Time Interval 165
4.1 Statement of the Problem 165
4.2 First-Order Necessary Conditions 166
4.3 Second-Order Necessary Conditions 168
4.4 Second-Order Sufficient Conditions 169
5 Second-Order Optimality Conditions for Bang-Bang Controls 170
5.1 Optimal Control Problems with Control Appearing Linearly 170
5.2 Necessary Optimality Conditions: The Minimum Principle of Pontryagin et al. 171
5.3 Second-Order Necessary Optimality Conditions 174
5.4 Second-Order Sufficient Optimality Conditions (SSC) 176
6 Induced Optimization Problem for Bang-Bang Controls and the Verification of SSC 177
6.1 Formulation of the Induced Optimization Problem and Necessary Optimality Conditions 177
6.2 Second-Order Optimality Conditions for Bang-Bang Controls in Terms of the Induced Optimization Problem 179
6.3 Numerical Methods for Solving the Induced Optimization Problem 180
7 Numerical Example with Fixed Final Time: Optimal Control of the Chemotherapy of HIV 182
8 Numerical Example with Free Final Time: Time-Optimal Control of Two-Link Robots 188
9 Optimal Control Problems with Mixed Control-State Constraints and Control Appearing Linearly 197
9.1 Statement of the Problem and Transformed Control Problem 197
9.2 Numerical Example: Optimal Control of the Rayleigh Equation 198
10 Conclusion 202
References 203