Stabilization, Optimal and Robust Control (eBook)

Preface 7
Contents 11
Notation and Symbols 18
1 General Introduction 21
1.1 Motivations and Objectives 22
1.2 General Process of the Robust Control Theory 26
1.3 Applications to Biological and Physical Sciences 27
Part I Convex Analysis and Duality Principles 31
2 Convexity and Topology 32
2.1 Convex Sets 32
2.2 Convex Functions 38
2.3 G-Regularization and Continuous Affine Functions 58
3 A Brief Overview of Sobolev Spaces 61
3.1 Tools and Definitions 61
3.2 Some Properties of Sobolev Spaces 67
4 Legendre–Fenchel Transformation and Duality 74
4.1 Fenchel Conjugate Functions 74
4.2 Subdifferentials and Superdifferentials of Extended-value Functions 79
4.3 Applications of the Duality 94
5 Lagrange Duality Theory 116
5.1 Frenchel–Rockafellar Duality in Optimization 116
5.2 Lagrange Duality 125
5.3 Minimax Duality 143
5.4 Duality and Parametric Variational Problems 164
Part II General Results and Concepts on Robust and Optimal Control Theory for Evolutive Systems 177
6 Studied Systems and General Results 178
6.1 Hypotheses and Properties 178
6.2 Evolution Problems, Existence and Stability Results 181
6.3 Regularity Results 186
6.4 Examples of Operators and Spaces 192
7 Optimal Control Problems 198
7.1 Introduction 198
7.2 Basic Framework 199
7.3 Linear Control Problems 202
7.4 Examples of Controls and Observations 208
7.5 Parameter Estimations and Bilinear Control Problems 217
7.6 Non-linear Control for Non-linear Evolutive PDE Problems 223
8 Stabilization and Robust Control Problem 241
8.1 Motivation and Objectives 241
8.2 Basic Framework 243
8.3 Linear Robust Control Problems 246
8.4 Examples of Controls, Disturbances and Observations 254
8.5 Bilinear-type Robust Control Problems 267
8.6 Non-linear Robust Control for Non-linear Evolutive Problems 280
8.7 Non-linear Time-varying Delay Systems 310
9 Remarks on Numerical Techniques 332
9.1 Introduction and Studied Problem 332
9.2 Continuous Case 334
9.3 Discrete Problem 341
Part III Applications in the Biological and Physical Sciences: Modeling and Stabilization 348
10 Vortex Dynamics in Superconductors and Ginzburg–Landau-type Models 351
10.1 Introduction 351
10.2 Existence and Uniqueness of the Solution of the MTDGL Model 357
10.3 The Perturbation Problem 358
10.4 Differentiability of the Operator Solution 360
10.5 Robust Control Problems 362
11 Multi-scale Modeling of Alloy Solidification and Phase-field Model 381
11.1 Introduction 382
11.2 Existence, Uniqueness and a Maximum Principle 388
11.3 The Perturbation Problem 390
11.4 Differentiability of the Operator Solution 392
11.5 Robust Control Problems 394
12 Large-scale Ocean in the Climate System 406
12.1 Introduction and Formulation of the Problem 406
12.2 The Perturbation Problem 411
12.3 Robust Control Problems 421
12.4 Primitive Ocean Equations with Vertical Viscosity 429
13 Heat Transfer Laws on Temperature Distribution in Biological Tissues 437
13.1 Introduction 437
13.2 The State System 442
13.3 The Perturbation Problem 447
13.4 Robust Control Problems 449
13.5 Other Situations 455
14 Lotka–Volterra-type Systems with Logistic Time-varying Delays 460
14.1 Introduction and Mathematical Setting 460
14.2 Existence and Uniqueness of the Solution 463
14.3 The Perturbation Problem 468
14.4 Robust Control Problems 469
14.5 Other Situations 477
15 Other Systems 482
15.1 Micropolar Fluids and Blood Pressure 482
15.2 Semiconductor Melt Flow in Crystal Growth 487
References 491
Index 506