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

A Modern Framework Based on Time-Tested MaterialA 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 MechanicsThrough 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.

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.

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 TheoremsHille-Yosida Theorems Results from the Hille-Yosida ProofCorollaries 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-MilgramExample 6: The Cantilever Beam The Beam Equation in the Form x derivative = Ax + F Gelfand TriplesSesquilinear Forms Lax-Milgram (Bounded Form)Lax-Milgram (Unbounded Form)Summary Remarks and Motivation

Analytic SemigroupsExample 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 FormsTanabe Estimates (on "Regular Dissipative Operators") Infinitesimal Generators in a General Banach Space

Abstract Cauchy Problems

General Second-Order SystemsIntroduction to Second-Order SystemsResults for σ2 V-elliptic Results for σ2 H-semiellipticStronger Assumptions for σ2

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

Inverse or Parameter Estimation ProblemsApproximation and ConvergenceSome Further Remarks

"Weak" or "Variational Form"

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

Delay Systems: Linear and Nonlinear Linear Delay Systems and ApproximationModeling of Viral Delays in HIV Infection DynamicsNonlinear 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 PBMA Prohorov Metric Framework for Inverse Problems Metrics on Probability SpacesExample 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 MotivationAbstract Formulation Infinite Dimensional LQR Control: Full State Feedback The Finite Horizon Control ProblemThe Infinite Horizon Control Problem

Families of Approximate Control ProblemsThe Finite Horizon Problem Approximate Control GainsThe Infinite Horizon Problem Approximate Control Gains

References
Index