An Introduction to Identification Problems via Functional Analysis

Alfredo Lorenzi

• Publisher: De Gruyter
• Year: 2014
• Print ISBN: 9789067643498
• eBook ISBN: 9783110940923
• Edition: 1
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

this monograph is based on two courses in computational mathematics and operative research, which were given by the author in recent years to doctorate and PhD students. The text focuses on an aspect of the theory of inverse problems, which is usually referred to as identification of parameters (numbers, vectors, matrices, functions) appearing in differential – or integrodifferential – equations. The parameters of such equations are either quite unknown or partially unknown, however knowledge about these is usually essential as they describe the intrinsic properties of the material or substance under consideration.

Table of contents

• Introduction
• Chapter 1. Parameter Identification in Linear Ordinary Differential Equations
• 1.1. Introduction
• 1.2. Examples of inverse problems
• 1.3. Appendix. Exponential of a square matrix
• Chapter 2. Identification Problems in Hilbert Spaces
• 2.1. An identification problem related to a first-order differential equation
• 2.2. Analysis of the dependence on the data of the solution ({yn}, λ) to problem (2.1.5), (2.1.6)
• 2.3. The minimization method
• 2.4. A further identification problem
• 2.5. A generalization to the vector case
• 2.6. An identification problem for a second-order differential equation
• Chapter 3. Proper Riemann Integrals for Banach-valued Functions. Curvilinear Integrals and Banach-valued Holomorphic Functions
• 3.1. Proper Riemann Integrals: basic properties
• 3.2. Curvilinear Integrals and Banach-valued Holomorphic Functions
• Chapter 4. Riemann-Stieltjes Integrals of Banach-valued Functions
• 4.1. Riemann-Stieltjes integrals over compact intervals
• Chapter 5. Improper Riemann Integrals for Banach Space-valued functions
• 5.1. Improper integrals
• Chapter 6. Banach Algebras and Spectral Analysis for Linear Bounded Operators
• 6.1. Introduction
• 6.2. Basic properties of the spectrum of an element
• 6.3. Integration of Banach algebra-valued functions
• 6.4. Banach algebra-valued holomorphic functions and the spectral theorem
• Chapter 7. Identifying Parameters in First-Order Partial Differential Equations
• 7.1. An identification problem relative to a first-order linear partial differential equation
• 7.2. An identification problem relative to a non-linear first-order partial differential equation
• Chapter 8. Identification Problems relative to Linear Bounded Operators I
• 8.1. An identification problem relative to a first-order differential equation
• 8.1.1. The singular case
• 8.1.2. The supersingular case
• 8.1.3. Continuous dependence on the data
• 8.2. An identification problem relative to a second-order differential equation
• 8.3. A particular case
• 8.4. An integro-differential identification problem
• 8.5. A onedimensional integro-differential identification problem
• Chapter 9. Identification Problems Relative to Linear Bounded Operators II
• 9.1. An abstract control problem
• 9.2. A concrete example
• Chapter 10. Analysis of the Continuous Dependence on the Data
• 10.1. Construction of an abstract model for the analysis of the continuous dependence on the data
• 10.2. Continuous dependence on the data of the solution to the identification problem (8.4.1)
• 10.3. Appendix. Gronwall’s generalized inequality
• Chapter 11. Linear Closed operators and Analytic Semigroups of Linear Bounded Operators
• 11.1. Linear closed operators
• 11.2. Resolvent set and spectrum of a linear operator
• 11.3. Sectorial operators
• Chapter 12. Cauchy Problems for Linear Abstract Differential Equations Relative to Sectorial Operators and Applications
• 12.1. Abstract differential equations and analytic semigroups. Applications to Cauchy problems
• 12.2. Applications
• Chapter 13. Identification Problems for Linear Abstract Differential Equations Relative to Sectorial Operators and Applications
• 13.1. An abstract identification problem
• 13.2. An application to a concrete case
• 13.3. An identification problem relative to an abstract non-autonomous first-order differential equation
• 13.4. Analysis of a Cauchy problem with coefficients depending on time only
• 13.5. Analysis of a Cauchy problem with non-homogeneous boundary value conditions on a bounded interval
• 13.6. Solving the identification problem (13.3.1)—(13.3.4)
• Bibliography
• List of Symbols