This book is devoted to the study of the turnpike phenomenon arising in optimal control theory. Special focus is placed on Turnpike results, in sufficient and necessary conditions for the turnpike phenomenon and in its stability under small perturbations of objective functions. The most important feature of this book is that it develops a large, general class of optimal control problems in metric space. Additional value is in the provision of solutions to a number of difficult and interesting problems in optimal control theory in metric spaces. Mathematicians working in optimal control, optimization, and experts in applications of optimal control to economics and engineering, will find this book particularly useful.
All main results obtained in the book are new. The monograph contains nine chapters. Chapter 1 is an introduction. Chapter 2 discusses Banach space valued functions, set-valued mappings in infinite dimensional spaces, and related continuous-time dynamical systems. Some convergence results are obtained. In Chapter 3, a discrete-time dynamical system with a Lyapunov function in a metric space induced by a set-valued mapping, is studied. Chapter 4 is devoted to the study of a class of continuous-time dynamical systems, an analog of the class of discrete-time dynamical systems considered in Chapter 3. Chapter 5 develops a turnpike theory for a class of general dynamical systems in a metric space with a Lyapunov function. Chapter 6 contains a study of the turnpike phenomenon for discrete-time nonautonomous problems on subintervals of half-axis in metric spaces, which are not necessarily compact. Chapter 7 contains preliminaries which are needed in order to study turnpike properties of infinite-dimensional optimal control problems. In Chapter 8, sufficient and necessary conditions for the turnpike phenomenon for continuous-time optimal control problems on subintervals of the half-axis in metric spaces, is established. In Chapter 9, the examination continues of the turnpike phenomenon for the continuous-time optimal control problems on subintervals of half-axis in metric spaces discussed in Chapter 8.
Author(s): Alexander J. Zaslavski
Series: Springer Optimization and Its Applications, 201
Publisher: Springer
Year: 2023
Language: English
Pages: 365
City: Cham
Preface
Contents
1 Introduction
1.1 Turnpike Property for Variational Problems
1.2 Notation
1.3 General Dynamical Systems with a Lyapunov Function
1.4 Continuous-Time Nonautonomous Problems on Half-Axis
2 Differential Inclusions
2.1 Banach Space-Valued Functions
2.2 Set-Valued Mappings and a Convergence Result
2.3 Dynamical Systems
2.4 Auxiliary Results
2.5 Proof of Proposition 2.7
2.6 Proof of Proposition 2.8
2.7 Proof of Theorem 2.11
2.8 An Example
3 Discrete-Time Dynamical Systems
3.1 Preliminaries and Main Results
3.2 Proof of Proposition 3.1
3.3 Proofs of Propositions 3.2 and 3.3
3.4 Proof of Theorem 3.5
3.5 Proof of Theorem 3.6
3.6 Proof of Theorem 3.7
4 Continuous-Time Dynamical Systems
4.1 Preliminaries and Main Results
4.2 Proof of Proposition 4.1
4.3 Proof of Proposition 4.2
4.4 Proof of Theorem 4.5
4.5 Proof of Theorem 4.6
4.6 Proof of Theorem 4.7
5 General Dynamical Systems with a Lyapunov Function
5.1 Preliminaries and Two Turnpike Results
5.2 Two Auxiliary Results
5.3 Proof of Proposition 5.2
5.4 Proof of Proposition 5.3
5.5 Proof of Theorem 5.4
5.6 An Auxiliary Result for Theorem 5.5
5.7 Proof of Theorem 5.5
5.8 Proofs of Propositions 5.6 and 5.7
5.9 A Weak Turnpike Result
5.10 Turnpike Results
5.11 Auxiliary Results
5.12 Proof of Theorem 5.15
5.13 Auxiliary Results for Theorem 5.16
5.14 Proofs of Theorems 5.16 and 5.17
5.15 Extensions of Theorems 5.16 and 5.17
5.16 An Auxiliary Result for Theorem 5.24
5.17 Proof of Theorem 5.24
5.18 Generalizations of Theorems 5.16 and 5.17
5.19 Proof of Theorem 5.28
5.20 Continuity of the Function π
5.21 Examples
6 Discrete-Time Nonautonomous Problems on Half-Axis
6.1 Preliminaries and Boundedness Results
6.2 Turnpike Properties
6.3 Perturbed Problems
6.4 Examples
6.5 The Space of Objective Functions and the Stability Result
6.6 Auxiliary Results
6.7 Proof of Theorem 6.22
6.8 Problems with Discount
6.9 Auxiliary Results for Theorem 6.25
6.10 Proof of Theorem 6.25
6.11 Genericity Results for Discrete-Time Problems
6.12 Proof of Theorem 6.29
6.13 Examples of the Space M
6.14 Extensions of the Generic Results
6.15 Smooth Integrands
7 Infinite Dimensional Control
7.1 Unbounded Operators
7.2 C0 Semigroup
7.3 Evolution Equations
7.4 C0 Groups
7.5 Admissible Control Operators
7.6 Examples
8 Continuous-Time Nonautonomous Problems on Half-Axis
8.1 Preliminaries
8.2 The First Class of Problems
8.3 The Second Class of Problems
8.4 A0 and the First and Second Classes of Problems
8.5 The Third Class of Problems
8.6 Boundedness Results
8.7 Turnpike Results
8.8 Lower Semicontinuity Property
8.9 Perturbed Problems
8.10 Auxiliary Results for Theorems 8.5 and Its Proof
8.11 Proof of Theorem 8.6
8.12 Proof of Theorem 8.10
8.13 Proof of Theorem 8.25
8.14 Proof of Theorem 8.26
8.15 An Auxiliary Result
8.16 Proof of Proposition 8.28
8.17 Auxiliary Results for Theorem 8.29
8.18 Proof of Theorem 8.29
8.19 Proof of Theorem 8.33
8.20 The Strong Turnpike Property
8.21 Auxiliary Results for Theorem 8.39
8.22 Proof of Theorem 8.39
8.23 Examples
9 Stability and Genericity Results
9.1 Preliminaries
9.2 Stability of TP
9.3 Auxiliary Results for Theorem 9.2
9.4 Proof of Theorem 9.2
9.5 Problems with Discount
9.6 Auxiliary Results for Theorem 9.5
9.7 Proof of Theorem 9.5
9.8 Genericity
9.9 Proof of Theorem 9.9
9.10 A Turnpike Result
9.11 Spaces od Smooths Integrands
9.12 Assumption (A5) for the First Class of Problems
9.13 Assumption (A5) for the Second Class of Problems
9.14 Assumption (A5) for the Third Class of Problems
References
Index
