A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering  book cover
1st Edition

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

ISBN 9781138374638
Published September 18, 2018 by Chapman and Hall/CRC
280 Pages 9 B/W Illustrations

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Book Description

A Modern Framework Based on Time-Tested Material
A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering presents functional analysis as a tool for understanding and treating distributed parameter systems. Drawing on his extensive research and teaching from the past 20 years, the author explains how functional analysis can be the basis of modern partial differential equation (PDE) and delay differential equation (DDE) techniques.

Recent Examples of Functional Analysis in Biology, Electromagnetics, Materials, and Mechanics
Through numerous application examples, the book illustrates the role that functional analysis—a classical subject—continues to play in the rigorous formulation of modern applied areas. The text covers common examples, such as thermal diffusion, transport in tissue, and beam vibration, as well as less traditional ones, including HIV models, uncertainty in noncooperative games, structured population models, electromagnetics in materials, delay systems, and PDEs in control and inverse problems. For some applications, computational aspects are discussed since many problems necessitate a numerical approach.

Table of Contents

Introduction to Functional Analysis in Applications
Example 1: Heat Equation
Some Preliminaries: Hilbert, Banach, and Other Spaces Useful in Operator Theory
Return to Example 1: Heat Equation
Example 2: General Transport Equation
Example 3: Delay Systems–Insect/Insecticide Models
Example 4: Probability Measure Dependent Systems — Maxwell’s Equations

Example 5: Structured Population Models

Semigroups and Infinitesimal Generators
Basic Principles of Semigroups
Infinitesimal Generators

Introduction to Generation Theorems
Hille-Yosida Theorems
Results from the Hille-Yosida Proof
Corollaries to Hille-Yosida
Lumer-Phillips and Dissipative Operators
Examples Using Lumer-Phillips Theorem

Adjoint Operators and Dual Spaces
Adjoint Operators
Dual Spaces and Strong, Weak, and Weak* Topologies
Examples of Spaces and Their Duals
Return to Dissipativeness for General Banach Spaces
More on Adjoint Operators
Examples of Computing Adjoints

Gelfand Triple, Sesquilinear Forms, and Lax-Milgram
Example 6: The Cantilever Beam
The Beam Equation in the Form x derivative = Ax + F
Gelfand Triples
Sesquilinear Forms
Lax-Milgram (Bounded Form)
Lax-Milgram (Unbounded Form)
Summary Remarks and Motivation

Analytic Semigroups
Example 1: The Heat Equation (again)
Example 2: The Transport Equation (again)
Example 6: The Beam Equation (again)
Summary of Results on Analytic Semigroup Generation by Sesquilinear Forms
Tanabe Estimates (on "Regular Dissipative Operators")
Infinitesimal Generators in a General Banach Space

Abstract Cauchy Problems

General Second-Order Systems
Introduction to Second-Order Systems
Results for σ2 V-elliptic
Results for σ2 H-semielliptic
Stronger Assumptions for σ2

Weak Formulations for Second-Order Systems
Model Formulation
Discussion of the Model
Theorems 9.1 and 9.2: Proofs

Inverse or Parameter Estimation Problems
Approximation and Convergence
Some Further Remarks

"Weak" or "Variational Form"

Finite Element Approximations and the Trotter-Kato Theorems
Finite Elements
Trotter-Kato Approximation Theorem

Delay Systems: Linear and Nonlinear
Linear Delay Systems and Approximation
Modeling of Viral Delays in HIV Infection Dynamics
Nonlinear Delay Systems
State Approximation and Convergence for Nonlinear Delay Systems
Fixed Delays versus Distributed Delays

Weak* Convergence and the Prohorov Metric in Inverse Problems
Populations with Aggregate Data, Uncertainty, and PBM
A Prohorov Metric Framework for Inverse Problems
Metrics on Probability Spaces
Example 5: The Growth Rate Distribution Model and Inverse Problem in Marine Populations

The Prohorov Metric in Optimization and Optimal Design Problems
Two Player Min-Max Games with Uncertainty
Optimal Design Techniques
Generalized Curves and Relaxed Controls of Variational Theory
Preisach Hysteresis in Smart Materials
NPML and Mixing Distributions in Statistical Estimation

Control Theory for Distributed Parameter Systems
Abstract Formulation
Infinite Dimensional LQR Control: Full State Feedback
The Finite Horizon Control Problem
The Infinite Horizon Control Problem

Families of Approximate Control Problems
The Finite Horizon Problem Approximate Control Gains
The Infinite Horizon Problem Approximate Control Gains



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H.T. Banks is a Distinguished University Professor and Drexel Professor of Mathematics at North Carolina State University, where he is also the director of the Center for Research in Scientific Computation and co-director of the Center for Quantitative Sciences in Biomedicine. He currently serves on the editorial boards of 14 journals and has published over 425 papers in applied mathematics and engineering journals. A fellow of the IEEE, IoP, SIAM, and AAAS, Dr. Banks has received numerous honors, including the W.T. and Idalia Reid Prize in Applied Mathematics from SIAM, the Lord Robert May Prize from the Journal of Biological Dynamics, and Best Paper Awards from the ASME and ACS.


"The book under review has the valuable advantage of being of interest to both mathematicians and engineers. … appropriate tools are carefully introduced and discussed in detail, and they are readily applied to practical situations related to the models derived from the generic examples. The main thrust of the book consists of those parts and topics of functional analysis that are fundamental to rigorous discussions of practical differential equations and delay systems as they arise in diverse applications and in particular in control and estimation."
—Larbi Berrahmoune, Mathematical Reviews, May 2013