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### Summary

### Summary

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.

### Table of Contents

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 |