1st Edition

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

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*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

**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

*σ*

_{2 }

*V*-elliptic

Results for

*σ*

_{2}

*H*-semielliptic

Stronger Assumptions for

*σ*

_{2}

**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

### Biography

**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