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

1. Auflage 2012
ISBN: 978-1-4398-8084-5
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 280 Seiten

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.

Zielgruppe

Researchers and graduate students in mathematics, engineering, physics, and chemistry.

Autoren/Hrsg.

Banks, H.T.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Interdisziplinäres Wissenschaften Wissenschaften Interdisziplinär Naturwissenschaften, Technik, Medizin

Inhaltsverzeichnis

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

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 s2 V-elliptic

Results for s2 H-semielliptic
Stronger Assumptions for s2

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

Motivation
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

References
Index

Über Autor(innen)

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.