1st Edition

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

By H.T. Banks Copyright 2012
    280 Pages 9 B/W Illustrations
    by Chapman & Hall

    280 Pages 9 B/W Illustrations
    by Chapman & Hall

    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.

    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




    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