A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering

by Banks; H. T.

• ISBN13: 9781439880838
• ISBN10: 1439880832
• Edition: 1st
• Format: Hardcover
• Copyright: 2012-06-18
• Publisher: Chapman & Hall

Summary

Reflecting research and results on functional analysis over the past 35 years, this text clearly describes various applied problems from engineering, biomathematics, and science. The author reviews the main results, stating that they resemble familiar finite dimensional problems for which there is abundant theory and methods for solution. He then uses analytic techniques to argue well-posedness, or what it is the engineer and scientist wants to do. By using this kind of approach, the solution to the problem takes on mathematical rigor without compromising the fidelity to the original model and application.

Table of Contents

Introduction to Functional Analysis in Applications	p. 1
Example 1: The Heat Equation	p. 1
Some Preliminaries: Hilbert, Banach, and Other Spaces Useful in Operator Theory	p. 4
Return to Example 1: Heat Equation	p. 8
Example 2: General Transport Equation	p. 8
Example 3: Delay Systems-Insect/Insecticide Models	p. 10
Example 4: Probability Measure Dependent Systems - Maxwell's Equations	p. 13
Example 5: Structured Population Models	p. 19
Semigroups and Infinitesimal Generators	p. 21
Basic Principles of Semigroups	p. 21
Infinitesimal Generators	p. 22
Generators	p. 25
Introduction to Generation Theorems	p. 25
Hille-Yosida Theorems	p. 25
Results from the Hille-Yosida Proof	p. 26
Corollaries to Hille-Yosida	p. 27
Lumer-Phillips and Dissipative Operators	p. 28
Examples Using Lumer-Phillips Theorem	p. 34
Return to Example 1: The Heat Equation	p. 34
Return to Example 2: The General Transport Equation	p. 35
Return to Example 3: Delay Systems	p. 36
Return to Example 4: Maxwell's Equations	p. 40
Adjoint Operators and Dual Spaces	p. 47
Adjoint Operators	p. 47
Computation of A*: An Example from the Heat Equation	p. 47
Self-Adjoint Operators and Dissipativeness	p. 48
Dual Spaces and Strong, Weak, and Weak* Topologies	p. 49
Summary of Topologies on X and X*	p. 53
Examples of Spaces and Their Duals	p. 54
Return to Dissipativeness for General Banach Spaces	p. 57
More on Adjoint Operators	p. 58
Adjoint Operator in a Hilbert Space	p. 59
Special Case	p. 59
Examples of Computing Adjoints	p. 59
Gelfand Triple, Sesquilinear Forms, and Lax-Milgram	p. 63
Example 6: The Cantilever Beam	p. 63
The Beam Equation in the Form x = Ax + F	p. 65
A as an Infinitesimal Generator	p. 66
Dissipativeness of A	p. 67
R(¿I - A)=X for some ¿	p. 67
Gelfand Triples	p. 68
Duality Pairing	p. 69
Sesquilinear Forms	p. 69
Representations	p. 70
Lax-Milgram-bounded form	p. 71
Discussion of Ax = f with A bounded	p. 72
Example-The Steady State Heat Equation in H10 (¿)	p. 74
Lax-Milgram-unbounded form	p. 75
The Concept of DA	p. 77
V-elliptic	p. 79
Summary Remarks and Motivation	p. 80
Analytic Semigroups	p. 81
Example 1: The Heat Equation (again)	p. 82
Example 2: The Transport Equation (again)	p. 84
Example 6: The Beam Equation (again)	p. 86
Summary of Results on Analytic Semigroup Generation by Sesquilinear Forms	p. 89
Tanabe Estimates (on "Regular Dissipative Operators")	p. 90
Infinitesimal Generators in a General Banach Space	p. 93
Abstract Cauchy Problems	p. 95
General Second-Order Systems	p. 103
Introduction to Second-Order Systems	p. 103
Results for ¿2 V-elliptic	p. 104
Results for ¿2 H-semielliptic	p. 105
Stronger Assumptions for ¿2	p. 107
Weak Formulations for Second-Order Systems	p. 109
Model Formulation	p. 109
Discussion of the Model	p. 113
Theorems 9.1 and 9.2: Proofs	p. 116
Inverse or Parameter Estimation Problems	p. 123
Approximation and Convergence	p. 126
Some Further Remarks	p. 134
"Weak" or "Variational Form"	p. 137
Finite Element Approximations and the Trotter-Kato Theorems	p. 143
Finite Elements	p. 143
Trotter-Kato Approximation Theorem	p. 146
Delay Systems: Linear and Nonlinear	p. 151
Linear Delay Systems and Approximation	p. 151
Modeling of Viral Delays in HIV Infection Dynamics	p. 155
Nonlinear Delay Systems	p. 161
State Approximation and Convergence for Nonlinear Delay Systems	p. 165
Fixed Delays versus Distributed Delays	p. 170
First Principles Modeling of Distributed Delays	p. 173
Weak* Convergence and the Prohorov Metric in Inverse Problems	p. 177
Populations with Aggregate Data, Uncertainty, and PBM	p. 177
Type I: Individual Dynamics/Aggregate Data Inverse Problems	p. 177
Type II: Aggregate Dynamics/Aggregate Data Inverse Problems	p. 178
A Prohorov Metric Framework for Inverse Problems	p. 180
Metrics on Probability Spaces	p. 182
The Prohorov Metric	p. 183
Robust Statistics	p. 184
The Levy Metric	p. 185
The Bounded Lipschitz Metric	p. 186
Other Metrics	p. 187
Example 5: The Growth Rate Distribution Model and Inverse Problem in Marine Populations	p. 187
The Prohorov Metric in Optimization and Optimal Design Problems	p. 197
Two Player Min-Max Games with Uncertainty	p. 197
Problem Formulation	p. 198
Theoretical Results	p. 202
Optimal Design Techniques	p. 205
Optimal Design Formulations	p. 206
Theoretical Summary	p. 209
Design Strategy Examples	p. 210
Generalized Curves and Relaxed Controls of Variational Theory	p. 211
Preisach Hysteresis in Smart Materials	p. 214
NPML and Mixing Distributions in Statistical Estimation	p. 217
Control Theory for Distributed Parameter Systems	p. 219
Motivation	p. 219
Abstract Formulation	p. 221
Infinite Dimensional LQR Control: Full State Feedback	p. 223
The Finite Horizon Control Problem	p. 225
The Infinite Horizon Control Problem	p. 227
Families of Approximate Control Problems	p. 229
The Finite Horizon Problem: Approximate Control Gains	p. 230
The Infinite Horizon Problem: Approximate Control Gains	p. 232
Example 6 Again!	p. 237
References	p. 241
Index	p. 265
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