Preface ix
Chapter 1. Introduction 1
1.1 Parabolic and Hyperbolic PDE Systems 1
1.2 The Roles of PDE Plant Instability, Actuator Location, Uncertainty Structure, Relative Degree, and Functional Parameters 2
1.3 Class of Parabolic PDE Systems 3
1.4 Backstepping 4
1.5 Explicitly Parametrized Controllers 5
1.6 Adaptive Control 5
1.7 Overview of the Literature on Adaptive Control for Parabolic PDEs 6
1.8 Inverse Optimality 7
1.9 Organization of the Book 7
1.10 Notation 9
PART I: NONADAPTIVE CONTROLLERS 11
Chapter 2. State Feedback 13
2.1 Problem Formulation 13
2.2 Backstepping Transformation and PDE for Its Kernel 14
2.3 Converting the PDE into an Integral Equation 17
2.4 Analysis of the Integral Equation by Successive Approximation Series 19
2.5 Stability of the Closed-Loop System 22
2.6 Dirichlet Uncontrolled End 24
2.7 Neumann Actuation 26
2.8 Simulation 27
2.9 Discussion 27
2.10 Notes and References 33
Chapter 3. Closed-Form Controllers 35
3.1 The Reaction-Diffusion Equation 35
3.2 A Family of Plants with Spatially Varying Reactivity 38
3.3 Solid Propellant Rocket Model 40
3.4 Plants with Spatially Varying Diffusivity 42
3.5 The Time-Varying Reaction Equation 45
3.6 More Complex Systems 50
3.7 2D and 3D Systems 52
3.8 Notes and References 54
Chapter 4. Observers 55
4.1 Observer Design for the Anti-Collocated Setup 55
4.2 Plants with Dirichlet Uncontrolled End and Neumann Measurements 58
4.3 Observer Design for the Collocated Setup 59
4.4 Notes and References 61
Chapter 5. Output Feedback 63
5.1 Anti-Collocated Setup 63
5.2 Collocated Setup 65
5.3 Closed-Form Compensators 67
5.4 Frequency Domain Compensator 71
5.5 Notes and References 72
Chapter 6. Control of Complex-Valued PDEs 73
6.1 State-Feedback Design for the Schrödinger Equation 73
6.2 Observer Design for the Schrödinger Equation 76
6.3 Output-Feedback Compensator for the Schrödinger Equation 79
6.4 The Ginzburg-Landau Equation 81
6.5 State Feedback for the Ginzburg-Landau Equation 83
6.6 Observer Design for the Ginzburg-Landau Equation 98
6.7 Output Feedback for the Ginzburg-Landau Equation 101
6.8 Simulations with the Nonlinear Ginzburg-Landau Equation 104
6.9 Notes and References 107
PART II: ADAPTIVE SCHEMES 109
Chapter 7. Systematization of Approaches to Adaptive Boundary Stabilization of PDEs 111
7.1 Categorization of Adaptive Controllers and Identifiers 111
7.2 Benchmark Systems 113
7.3 Controllers 114
7.4 Lyapunov Design 115
7.5 Certainty Equivalence Designs 117
7.6 Trade-offs between the Designs 121
7.7 Stability 122
7.8 Notes and References 124
Chapter 8. Lyapunov-Based Designs 125
8.1 Plant with Unknown Reaction Coefficient 125
8.2 Proof of Theorem 8.1 128
8.3 Well-Posedness of the Closed-Loop System 132
8.4 Parametric Robustness 134
8.5 An Alternative Approach 135
8.6 Other Benchmark Problems 136
8.7 Systems with Unknown Diffusion and Advection Coefficients 142
8.8 Simulation Results 147
8.9 Notes and References 149
Chapter 9. Certainty Equivalence Design with Passive Identifiers 150
9.1 Benchmark Plant 150
9.2 3D Reaction-Advection-Diffusion Plant 154
9.3 Proof of Theorem 9.2 157
9.4 Simulations 163
9.5 Notes and References 164
Chapter 10. Certainty Equivalence Design with Swapping Identifiers 166
10.1 Reaction-Advection-Diffusion Plant 166
10.2 Proof of Theorem 10.1 169
10.3 Simulations 175
10.4 Notes and References 175
Chapter 11. State Feedback for PDEs with Spatially Varying Coefficients 176
11.1 Problem Statement 176
11.2 Nominal Control Design 177
11.3 Robustness to Error in Gain Kernel 179
11.4 Lyapunov Design 185
11.5 Lyapunov Design for Plants with Unknown Advection and
Diffusion Parameters 190
11.6 Passivity-Based Design 191
11.7 Simulations 195
11.8 Notes and References 197
Chapter 12. Closed-Form Adaptive Output-Feedback Contollers 198
12.1 Lyapunov Design--Plant with Unknown Parameter in the Domain 199
12.2 Lyapunov Design--Plant with Unknown Parameter in the 205
Boundary Condition
12.3 Swapping Design--Plant with Unknown Parameter in the Domain 210
12.4 Swapping Design--Plant with Unknown Parameter in the
Boundary Condition 216
12.5 Simulations 223
12.6 Notes and References 225
Chapter 13. Output Feedback for PDEs with Spatially Varying Coefficients 226
13.1 Reaction-Advection-Diffusion Plant 226
13.2 Transformation to Observer Canonical Form 227
13.3 Nominal Controller 228
13.4 Filters 230
13.5 Frequency Domain Compensator with Frozen Parameters 232
13.6 Update Laws 233
13.7 Stability 235
13.8 Trajectory Tracking 242
13.9 The Ginzburg-Landau Equation 244
13.10 Identifier for the Ginzburg-Landau Equation 246
13.11 Stability of Adaptive Scheme for the Ginzburg-Landau Equation 248
13.12 Simulations 255
13.13 Notes and References 255
Chapter 14. Inverse Optimal Control 261
14.1 Nonadaptive Inverse Optimal Control 262
14.2 Reducing Control Effort through Adaptation 265
14.3 Dirichlet Actuation 267
14.4 Design Example 267
14.5 Comparison with the LQR Approach 268
14.6 Inverse Optimal Adaptive Control 271
14.7 Stability and Inverse Optimality of the Adaptive Scheme 273
14.8 Notes and References 275
Appendix A. Adaptive Backstepping for Nonlinear ODEs--The Basics 277
A.1 Nonadaptive Backstepping--The Known Parameter Case 277
A.2 Tuning Functions Design 279
A.3 Modular Design 289
A.4 Output Feedback Designs 297
A.5 Extensions 303
Appendix B. Poincaré and Agmon Inequalities 305
Appendix C. Bessel Functions 307
C.1 Bessel Function Jn 307
C.2 Modified Bessel Function In 307
Appendix D. Barbalat's and Other Lemmas for Proving Adaptive Regulation 310
Appendix E. Basic Parabolic PDEs and Their Exact Solutions 313
E.1 Reaction-Diffusion Equation with Dirichlet Boundary Conditions 313
E.2 Reaction-Diffusion Equation with Neumann Boundary Conditions 315
E.3 Reaction-Diffusion Equation with Mixed Boundary Conditions 315
References 317
Index 327

This book introduces a comprehensive methodology for adaptive control design of parabolic partial differential equations with unknown functional parameters, including reaction-convection-diffusion systems ubiquitous in chemical, thermal, biomedical, aerospace, and energy systems. Andrey Smyshlyaev and Miroslav Krstic develop explicit feedback laws that do not require real-time solution of Riccati or other algebraic operator-valued equations. The book emphasizes stabilization by boundary control and using boundary sensing for unstable PDE systems with an infinite relative degree. The book also presents a rich collection of methods for system identification of PDEs, methods that employ Lyapunov, passivity, observer-based, swapping-based, gradient, and least-squares tools and parameterizations, among others.
Including a wealth of stimulating ideas and providing the mathematical and control-systems background needed to follow the designs and proofs, the book will be of great use to students and researchers in mathematics, engineering, and physics. It also makes a valuable supplemental text for graduate courses on distributed parameter systems and adaptive control.

Andrey Smyshlyaev is assistant project scientist at the University of California, San Diego. Miroslav Krstic is the Sorenson Distinguished Professor and the founding director of the Cymer Center for Control Systems and Dynamics at the University of California, San Diego. Smyshlyaev and Krstic are the authors of Boundary Control of PDEs.