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## Über dieses Buch

This book is devoted to the study of the turnpike phenomenon and describes the existence of solutions for a large variety of infinite horizon optimal control classes of problems. Chapter 1 provides introductory material on turnpike properties. Chapter 2 studies the turnpike phenomenon for discrete-time optimal control problems. The turnpike properties of autonomous problems with extended-value integrands are studied in Chapter 3. Chapter 4 focuses on large classes of infinite horizon optimal control problems without convexity (concavity) assumptions. In Chapter 5, the turnpike results for a class of dynamic discrete-time two-player zero-sum game are proven.

This thorough exposition will be very useful for mathematicians working in the fields of optimal control, the calculus of variations, applied functional analysis and infinite horizon optimization. It may also be used as a primary text in a graduate course in optimal control or as supplementary text for a variety of courses in other disciplines. Researchers in other fields such as economics and game theory, where turnpike properties are well known, will also find this Work valuable.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The study of optimal control problems and variational problems defined on infinite intervals and on sufficiently large intervals has been a rapidly growing area of research [3, 4, 8, 11–13, 18, 22, 24–27, 32, 34, 39–42, 44, 50, 51, 56, 70] which has various applications in engineering [1, 29, 76], in models of economic growth [2, 14–17, 21, 23, 28, 33, 38, 43, 46–48, 56], in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals [7, 49], and in the theory of thermodynamical equilibrium for materials [20, 30, 35–37].
Alexander J. Zaslavski

### Chapter 2. Turnpike Properties of Discrete-Time Problems

Abstract
In this chapter we study the structure of approximate solutions of an autonomous discrete-time control system with a compact metric space of states X. This control system is described by a bounded upper semicontinuous function $$v: X \times X \rightarrow R^{1}$$ which determines an optimality criterion and by a nonempty closed set Ω ⊂ X × X which determines a class of admissible trajectories (programs). We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. When X is a compact convex subset of a finite-dimensional Euclidean space, the set Ω is convex, and the function v is strictly concave we obtain a full description of the structure of approximate solutions.
Alexander J. Zaslavski

### Chapter 3. Variational Problems with Extended-Valued Integrands

Abstract
In this chapter we study turnpike properties of approximate solutions of an autonomous variational problem with a lower semicontinuous integrand $$f: R^{n} \times R^{n} \rightarrow R^{1} \cup \{\infty \}$$, where R n is the n-dimensional Euclidean space.
Alexander J. Zaslavski

### Chapter 4. Infinite Horizon Problems

Abstract
In this chapter we establish the existence of solutions for classes of nonconvex (nonconcave) infinite horizon discrete-time optimal control problems. These classes contain optimal control problems arising in economic dynamics which describe general one-sector and two-sector models with nonconcave utility functions representing the preferences of the planner.
Alexander J. Zaslavski

### Chapter 5. Dynamic Discrete-Time Zero-Sum Games

Abstract
In this chapter we study the existence and structure of solutions for dynamic discrete-time two-player zero-sum games and establish a turnpike result. This result describes the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals. We also show that for each initial state there exists a pair of overtaking equilibria strategies over an infinite horizon.
Alexander J. Zaslavski

### Backmatter

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