Turnpike Phenomenon and Symmetric Optimization Problems (Springer Optimization and Its Applications, 190)

✍ Scribed by Alexander J. Zaslavski

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 339
• Category: Library

✦ Synopsis

Written by a leading expert in turnpike phenomenon, this book is devoted to the study of symmetric optimization, variational and optimal control problems in infinite dimensional spaces and turnpike properties of their approximate solutions. The book presents a systematic and comprehensive study of general classes of problems in optimization, calculus of variations, and optimal control with symmetric structures from the viewpoint of the turnpike phenomenon. The author establishes generic existence and well-posedness results for optimization problems and individual (not generic) turnpike results for variational and optimal control problems. Rich in impressive theoretical results, the author presents applications to crystallography and discrete dispersive dynamical systems which have prototypes in economic growth theory.
This book will be useful for researchers interested in optimal control, calculus of variations turnpike theory and their applications, such as mathematicians, mathematical economists, and researchers in crystallography, to name just a few.

✦ Table of Contents

Preface
Contents
1 Introduction
1.1 Generic Existence of Solutions of Minimization Problems
1.2 Optimization Problems Arising in Crystallography
1.3 Symmetric Optimization Problems
1.4 Turnpike Property for Variational Problems
1.5 Variational Problems with Symmetric Integrands
1.6 Notation
1.7 Exercises
2 Symmetric Optimization Problems
2.1 A Generic Approach in Optimization
2.2 A Generic Variational Principle
2.3 The First Generic Result
2.4 The Second and Third Generic Results
2.5 Proofs of Theorems 2.5 and 2.6
2.6 The Fourth Result
2.7 The Fifth Generic Result
2.8 Auxiliary Results for Theorems 2.14 and 2.15
2.9 Proof of Theorems 2.14 and 2.15
2.10 σ-Porous Sets in a Metric Space
2.11 A Variational Principle and Porosity
2.12 Well-Posedness and Porosity for Classes of Minimization Problems
2.13 An Auxiliary Result
2.14 A Well-Posedness Result
2.15 An Extension of Theorem 2.4
2.16 Extensions of Theorems 2.5 and 2.6
2.17 Extension of Theorem 2.10
2.18 An Extension of Theorem 2.14
3 Parametric Optimization
3.1 Generic Variational Principle
3.2 Concretization of the Hypothesis (H)
3.3 The First Generic Existence Results
3.4 The Second Generic Existence Result
4 Infinite Dimensional Control
4.1 Banach Space Valued Functions
4.2 Unbounded Operators
4.3 C0 Semigroup
4.4 Evolution Equations
4.5 C0 Groups
4.6 Admissible Control Operators
4.7 Examples
5 Symmetric Variational Problems
5.1 Preliminaries
5.2 The First Weak Turnpike Result
5.3 Auxiliary Results for the Turnpike Property
5.4 A Turnpike Result
5.5 The Second Weak Turnpike Result
5.6 The Space of Integrands
5.7 Auxiliary Results for Theorem 5.9
5.8 Proof of Theorem 5.9
5.9 Stability of the Weak Turnpike
5.10 Two Extensions of Theorem 5.9
5.11 The Turnpike Phenomenon in the Regions Close to the Right End Points
6 Infinite Dimensional Optimal Control
6.1 Preliminaries
6.2 The First Turnpike Result
6.3 The Space of Integrands
6.4 The Turnpike Property
6.5 The Second Weak Turnpike Result
6.6 A Well-Posedness Result
6.7 Stability of the Turnpike Phenomenon
6.8 An Auxiliary Result for Theorem 6.11
6.9 Proof of Theorem 6.11
6.10 Stability of the Weak Turnpike Phenomenon
6.11 The First Extension of Theorem 6.11
6.12 The Second Extension of Theorem 6.11
6.13 The Turnpike Property in the Regions Close to the Right end Points
7 Optimization Problems Arising in Crystallography
7.1 Preliminaries
7.2 Auxiliary Results
7.3 A Basic Lemma
7.4 Proof of Theorem 7.2
7.5 A Porosity Result
7.6 The Set MMr is Porous
7.7 Auxiliary Results
7.8 Proof of Theorem 7.11
8 Discrete Dispersive Dynamical Systems
8.1 Dynamical Systems with a Lyapunov Function
8.2 Proof of Theorem 8.1
8.3 Proofs of Propositions 8.2, 8.3, and 8.5
8.4 Proof of Theorem 8.6
8.5 Proof of Theorem 8.7
8.6 Proof of Theorem 8.9
8.7 The First Weak Turnpike Result
8.8 An Auxiliary Result
8.9 Proof of Theorem 8.10
8.10 The Second Weak Turnpike Result
8.11 An Auxiliary Result
8.12 Proof of Theorem 8.12
8.13 Stability Results
8.14 An Auxiliary Result
8.15 Proof of Theorem 8.14
8.16 Proof of Theorem 8.15
References
Index