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## Über dieses Buch

The structure of approximate solutions of autonomous discrete-time optimal control problems and individual turnpike results for optimal control problems without convexity (concavity) assumptions are examined in this book. In particular, the book focuses on the properties of approximate solutions which are independent of the length of the interval, for all sufficiently large intervals; these results apply to the so-called turnpike property of the optimal control problems. By encompassing the so-called turnpike property the approximate solutions of the problems are determined primarily by the objective function and are fundamentally independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. This book also explores the turnpike phenomenon for two large classes of autonomous optimal control problems. It is illustrated that the turnpike phenomenon is stable for an optimal control problem if the corresponding infinite horizon optimal control problem possesses an asymptotic turnpike property. If an optimal control problem belonging to the first class possesses the turnpike property, then the turnpike is a singleton (unit set). The stability of the turnpike property under small perturbations of an objective function and of a constraint map is established. For the second class of problems where the turnpike phenomenon is not necessarily a singleton the stability of the turnpike property under small perturbations of an objective function is established. Containing solutions of difficult problems in optimal control and presenting new approaches, techniques and methods this book is of interest for mathematicians working in optimal control and the calculus of variations. It also can be useful in preparation courses for graduate students.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
The study of the existence and the structure of solutions of optimal control problems defined on infinite intervals and on sufficiently large intervals has recently been a rapidly growing area of research. See, for example, [3, 4, 7–11, 13, 14, 16, 18–22, 25, 27, 32–34, 36, 44, 45, 52, 55] and the references mentioned therein. These problems arise in engineering [1, 23, 57], in models of economic growth [2, 5, 12, 13, 17, 22, 26, 31, 35, 38, 39, 40, 45], in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals [6, 41] and in the theory of thermodynamical equilibrium for materials [15, 24, 28–30]. In this chapter we discuss the structure of solutions of a discrete-time optimal control system describing a general model of economic dynamics.
Alexander J. Zaslavski

### 2. Optimal Control Problems with Singleton Turnpikes

Abstract
In this chapter we study the structure of solutions of a discrete-time control system with a compact metric space of states X which arises in economic dynamics. This control system is described by a nonempty closed set $$\Omega \subset X \times X$$ which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous objective function $$v:X\times X \to R^1$$ which determines an optimality criterion. We show the stability of the turnpike phenomenon under small perturbations of the objective function v and the set Ω.
Alexander J. Zaslavski

### 3. Optimal Control Problems with Discounting

Abstract
In this chapter we continue our study of the structure of approximate solutions of the discrete-time optimal control problems with a compact metric space of states X and with a singleton turnpike. These problems are described by a nonempty closed set $$\Omega \subset X \times X$$ which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous objective function $$v:X\times X \to R^1$$ which determines an optimality criterion. We show the stability of the turnpike phenomenon under small perturbations of the objective function v and the set Ω in the case with discounting. The results of the chapter generalize the results obtained in [54] for the discounting case with a perturbation only on the objective function.
Alexander J. Zaslavski

### 4. Optimal Control Problems with Nonsingleton Turnpikes

Abstract
In this chapter we study stability of the turnpike phenomenon for approximate solutions for a general class of discrete-time optimal control problems with nonsingleton turnpikes and with a compact metric space of states. This class of optimal control problems is identified with a complete metric space of objective functions. We show that the turnpike phenomenon is stable under perturbations of an objective function if the corresponding infinite horizon optimal control problem possesses an asymptotic turnpike property.
Alexander J. Zaslavski

### Backmatter

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