Stabilization, optimal and robust control : theory and applications in biological and physical sciences

Stabilization, Optimal and Robust Control develops robust control of infinite-dimensional dynamical systems derived from time-dependent coupled PDEs associated with boundary-value problems. Rigorous analysis takes into account nonlinear system dynamics, evolutionary and coupled PDE behaviour and the selection of function spaces in terms of solvability and model quality.

Mathematical foundations are provided so that the book remains accessible to the non-control-specialist. Following chapters giving a general view of convex analysis and optimization and robust and optimal control, problems arising in fluid mechanical, biological and materials scientific systems are laid out in detail.

The combination of mathematical fundamentals with application of current interest will make this book of much interest to researchers and graduate students looking at complex problems in mathematics, physics and biology as well as to control theorists.

Notation and Symbols	p. xix
1 General Introduction	p. 1
1.1 Motivations and Objectives	p. 2
1.2 General Process of the Robust Control Theory	p. 6
1.3 Applications to Biological and Physical Sciences	p. 7
1.3.1 Material Sciences	p. 8
1.3.2 Fluid Mechanics	p. 9
1.3.3 Biological Models	p. 9
1.3.4 Other Systems	p. 10
Part I Convex Analysis and Duality Principles
2 Convexity and Topology	p. 13
2.1 Convex Sets	p. 13
2.1.1 Definitions	p. 13
2.1.2 Topological Spaces and Properties	p. 14
2.1.3 Hahn-Banach and Separation Between Convex Sets	p. 17
2.2 Convex Functions	p. 19
2.2.1 Definitions	p. 19
2.2.2 Closure and Semi-continuous Functions	p. 22
2.2.3 Weak Topologies and Dual Spaces	p. 24
2.2.4 Separable Spaces	p. 28
2.2.5 Dual of Banach Spaces and Reflexivity	p. 32
2.2.6 Closure and Continuity of Convex Functions	p. 37
2.3 [Gamma]-Regularization and Continuous Affine Functions	p. 39
3 A Brief Overview of Sobolev Spaces	p. 43
3.1 Tools and Definitions	p. 43
3.1.1 Definitions and Notations	p. 43
3.1.2 Some Fundamental Inequalities and Convergence Criteria	p. 45
3.1.3 Definition of Sobolev Spaces	p. 47
3.2 Some Properties of Sobolev Spaces	p. 49
3.2.1 Density Results	p. 49
3.2.2 Embedding Results	p. 49
3.2.3 Compactness Results	p. 50
3.2.4 Trace Results and Green's Formula	p. 50
3.2.5 Truncation Operations	p. 53
3.2.6 Interpolation Theory	p. 54
4 Legendre-Fenchel Transformation and Duality	p. 57
4.1 Fenchel Conjugate Functions	p. 57
4.1.1 Definitions and Properties	p. 57
4.1.2 Examples	p. 61
4.2 Subdifferentials and Superdifferentials of Extended-value Functions	p. 62
4.2.1 Definition and Characterization	p. 62
4.2.2 General Case	p. 66
4.2.3 Calculus Rules with Subdifferentials	p. 68
4.2.4 Connection with Directional Derivative	p. 70
4.3 Applications of the Duality	p. 77
4.3.1 Fundamental Equations	p. 78
4.3.2 Duality Mapping in Banach Spaces	p. 79
4.3.3 Duality and Fundamental Equations	p. 82
4.3.4 Euler-Lagrange Equation and the Non-linear Operator	p. 86
4.3.5 Minimization of Convex Functions	p. 93
4.3.6 General Boundary Value Problems	p. 95
5 Lagrange Duality Theory	p. 99
5.1 Frenchel-Rockafellar Duality in Optimization	p. 99
5.1.1 Primal and Dual Problems	p. 100
5.1.2 Normal and Stability Problems	p. 103
5.1.3 Optimality Conditions and Existence	p. 106
5.1.4 Bidual Problem and Duality in Variational Inequalities	p. 107
5.2 Lagrange Duality	p. 108
5.2.1 Definitions and Critical Points of Lagrangians	p. 108
5.2.2 Lagrangian Duality and Saddle Points	p. 113
5.2.3 Application and Boundary-value Problems	p. 116
5.3 Minimax Duality	p. 126
5.3.1 Motivation	p. 126
5.3.2 Saddle Point and Properties	p. 127
5.3.3 Banach Spaces and Saddle Points	p. 131
5.3.4 Connection with Duality and Application	p. 140
5.3.5 Ky Fan's Minimax Inequality and Non-potential Operators	p. 142
5.4 Duality and Parametric Variational Problems	p. 147
5.4.1 Abstract Framework	p. 147
5.4.2 Geometrically Non-linear Lagrangian Representation	p. 151
Part II General Results and Concepts on Robust and Optimal Control Theory for Evolutive Systems
6 Studied Systems and General Results	p. 163
6.1 Hypotheses and Properties	p. 163
6.2 Evolution Problems, Existence and Stability Results	p. 166
6.3 Regularity Results	p. 171
6.4 Examples of Operators and Spaces	p. 177
6.4.1 Dirichlet Boundary Condition	p. 177
6.4.2 Neumann Boundary Condition	p. 178
6.4.3 Robin Boundary Condition	p. 179
6.4.4 Non-homogeneous Neumann and Dirichlet Boundary Conditions	p. 180
7 Optimal Control Problems	p. 183
7.1 Introduction	p. 183
7.2 Basic Framework	p. 184
7.3 Linear Control Problems	p. 187
7.3.1 Position of the Problem, Existence and Uniqueness of the Optimal Solution	p. 187
7.3.2 Optimality Conditions and Identification of the Gradients	p. 188
7.4 Examples of Controls and Observations	p. 193
7.4.1 Boundary Control	p. 194
7.4.2 Pointwise Observations	p. 195
7.4.3 Pointwise Controls	p. 198
7.4.4 Boundary Controls and Boundary Observations	p. 199
7.4.5 Data Assimilation Problem and Initial Condition Control	p. 201
7.5 Parameter Estimations and Bilinear Control Problems	p. 202
7.5.1 State Problem	p. 202
7.5.2 Existence of Optimal Solutions	p. 203
7.5.3 First-order Optimality Conditions	p. 204
7.6 Non-linear Control for Non-linear Evolutive PDE Problems	p. 208
7.6.1 State Problem and Assumptions	p. 208
7.6.2 Existence and Uniqueness of the Solution	p. 210
7.6.3 The Control Framework	p. 211
7.6.4 Initial Condition Control	p. 219
7.6.5 Example	p. 224
8 Stabilization and Robust Control Problem	p. 227
8.1 Motivation and Objectives	p. 227
8.2 Basic Framework	p. 229
8.3 Linear Robust Control Problems	p. 232
8.3.1 Position of the Problem, and the Existence and Uniqueness of the Optimal Solution	p. 232
8.3.2 Optimality Conditions and Identification of the Gradients	p. 234
8.4 Examples of Controls, Disturbances and Observations	p. 240
8.4.1 Boundary Disturbance	p. 242
8.4.2 Pointwise Observations	p. 243
8.4.3 Pointwise Controls and Pointwise Disturbances	p. 246
8.4.4 Boundary Controls and Boundary Observations	p. 247
8.4.5 Data Assimilation Problem and Initial Condition Control	p. 250
8.5 Bilinear-type Robust Control Problems	p. 253
8.5.1 State Problem	p. 254
8.5.2 Differentiability of the Mapping Solution	p. 257
8.5.3 Existence of an Optimal Solution	p. 260
8.5.4 First-order Necessary Conditions	p. 262
8.5.5 Other Situations and Applications	p. 263
8.6 Non-linear Robust Control for Non-linear Evolutive Problems	p. 266
8.6.1 State Equations	p. 267
8.6.2 The Perturbation Problem	p. 268
8.6.3 The Control Framework	p. 268
8.6.4 Initial Condition Control	p. 278
8.6.5 A Remark on the Robust Boundary Control Problem	p. 287
8.6.6 Contraction Mapping and Fixed-point Formulation	p. 290
8.7 Non-linear Time-varying Delay Systems	p. 296
8.7.1 Mathematical Setting	p. 296
8.7.2 Existence and Uniqueness of the Solution	p. 298
8.7.3 The Control Framework	p. 304
8.7.4 Remarks on Time-varying Delays and Control in the Boundary Conditions	p. 314
9 Remarks on Numerical Techniques	p. 319
9.1 Introduction and Studied Problem	p. 319
9.2 Continuous Case	p. 321
9.2.1 Gradient Algorithm	p. 321
9.2.2 Conjugate Gradient Algorithm	p. 322
9.2.3 Lagrange-Newton Method	p. 324
9.3 Discrete Problem	p. 328
9.3.1 Approximation of Robust Control Problems	p. 328
9.3.2 Discrete Gradient Algorithm	p. 329
9.3.3 Multi-grid Gradient Method	p. 331
Part III Applications in the Biological and Physical Sciences: Modeling and Stabilization
10 Vortex Dynamics in Superconductors and Ginzburg-Landau-type Models	p. 339
10.1 Introduction	p. 339
10.1.1 Assumptions and Notation	p. 343
10.1.2 Preliminary Results	p. 345
10.2 Existence and Uniqueness of the Solution of the MTDGL Model	p. 345
10.3 The Perturbation Problem	p. 346
10.3.1 Formulation of the Perturbation Problem	p. 346
10.3.2 Existence and Stability Results	p. 347
10.4 Differentiability of the Operator Solution	p. 348
10.5 Robust Control Problems	p. 350
10.5.1 Control in the External Magnetic Field	p. 350
10.5.2 Control in the Initial Condition of the Vector Potential	p. 360
11 Multi-scale Modeling of Alloy Solidification and Phase-field Model	p. 369
11.1 Introduction	p. 370
11.1.1 Assumptions and Notations	p. 374
11.1.2 Preliminary Results	p. 375
11.2 Existence, Uniqueness and a Maximum Principle	p. 376
11.2.1 Existence and Uniqueness Results	p. 376
11.2.2 A Maximum Principle	p. 376
11.3 The Perturbation Problem	p. 378
11.4 Differentiability of the Operator Solution	p. 380
11.5 Robust Control Problems	p. 382
11.5.1 Disturbance in the Forcing of the Phase-field Parameter	p. 382
11.5.2 Distributed Disturbance in the Initial Condition of the Phase-field Variable	p. 389
12 Large-scale Ocean in the Climate System	p. 395
12.1 Introduction and Formulation of the Problem	p. 395
12.1.1 Motivation	p. 395
12.1.2 Primitive Equations and Study Domain	p. 397
12.2 The Perturbation Problem	p. 400
12.2.1 Preliminary Results and Weak Formulations	p. 400
12.2.2 Existence, Uniqueness and Regularity of the Solution	p. 405
12.2.3 Comments on the Asymptotic Behavior	p. 408
12.3 Robust Control Problems	p. 410
12.3.1 Differentiability of the Operator Solution	p. 411
12.3.2 Existence of an Optimal Solution	p. 413
12.3.3 Optimality Conditions	p. 415
12.4 Primitive Ocean Equations with Vertical Viscosity	p. 418
13 Heat Transfer Laws on Temperature Distribution in Biological Tissues	p. 427
13.1 Introduction	p. 427
13.1.1 Motivation and Statement of the Problem	p. 427
13.1.2 Thermal Damage Calculations	p. 429
13.1.3 Background and Motivation	p. 430
13.1.4 Assumptions and Notations	p. 431
13.2 The State System	p. 432
13.2.1 Existence and Stability Results	p. 432
13.2.2 A Maximum Principle	p. 436
13.3 The Perturbation Problem	p. 437
13.3.1 Formulation of the Perturbation Problem	p. 437
13.3.2 Existence and Stability Results	p. 438
13.4 Robust Control Problems	p. 439
13.4.1 Formulation of the Control Problem and Differentiability	p. 439
13.4.2 Existence of an Optimal Solution	p. 442
13.4.3 Optimality Conditions	p. 443
13.5 Other Situations	p. 445
13.5.1 Data Assimilation	p. 445
13.5.2 Boundary Disturbance	p. 446
13.5.3 Finite Number of Measurements	p. 447
13.5.4 Union of a Finite Number of Subdomains	p. 448
14 Lotka-Volterra-type Systems with Logistic Time-varying Delays	p. 451
14.1 Introduction and Mathematical Setting	p. 451
14.1.1 Motivation	p. 451
14.1.2 Studied Equations	p. 452
14.2 Existence and Uniqueness of the Solution	p. 454
14.3 The Perturbation Problem	p. 459
14.4 Robust Control Problems	p. 460
14.4.1 Formulation of the Control Problem and Differentiability	p. 460
14.4.2 Existence of an Optimal Solution	p. 462
14.4.3 Optimality Conditions	p. 464
14.5 Other Situations	p. 468
14.5.1 Disturbance in the Parameter Function p	p. 468
14.5.2 Remarks on Boundary Control and Habitat Hostility	p. 470
15 Other Systems	p. 473
15.1 Micropolar Fluids and Blood Pressure	p. 473
15.1.1 Introduction and Mathematical Setting	p. 473
15.1.2 Fluctuation and Robust Regulation of the Blood Pressure	p. 475
15.2 Semiconductor Melt Flow in Crystal Growth	p. 478
15.2.1 Introduction and Mathematical Setting	p. 478
15.2.2 Fluctuation and Robust Regulation of the Melt Flow Motion	p. 479
References	p. 483
Index	p. 499

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