# Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems

• Alexander J. Zaslavski
Book

Part of the SpringerBriefs in Optimization book series (BRIEFSOPTI)

1. Front Matter
Pages i-x
2. Alexander J. Zaslavski
Pages 1-7
3. Alexander J. Zaslavski
Pages 9-45
4. Alexander J. Zaslavski
Pages 47-63
5. Alexander J. Zaslavski
Pages 65-103
6. Back Matter
Pages 105-109

### Introduction

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.

### Keywords

approximate solutions asymptotic turnpike property autonomous problems discrete-time optimal control problems nonconcave problems

#### Authors and affiliations

• Alexander J. Zaslavski
• 1
1. 1.Department of MathematicsTechnion- Israel Institute of TechnHaifaIsrael

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-08034-5
• Copyright Information The Author 2014
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-319-08033-8
• Online ISBN 978-3-319-08034-5
• Series Print ISSN 2190-8354
• Series Online ISSN 2191-575X
• Buy this book on publisher's site
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