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

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

1. 1.Department of MathematicsTechnion- Israel Institute of TechnHaifaIsrael

### Bibliographic information

• Book Title Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems
• Authors Alexander J. Zaslavski
• Series Title SpringerBriefs in Optimization
• Series Abbreviated Title SpringerBriefs in Optimization
• 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 Mathematics and Statistics (R0)
• Softcover ISBN 978-3-319-08033-8
• eBook ISBN 978-3-319-08034-5
• Series ISSN 2190-8354
• Series E-ISSN 2191-575X
• Edition Number 1
• Number of Pages X, 109
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

“The book is intended for graduate students and researchers specialized in the areas of control, optimization and game theories, and their applications. The book is well organized. The presentation is clear and logical. The proofs of the stated results are rigorous. … I believe that the book is an excellent addition to the existing literature on the turnpike theory and its applications and that it has the potential for stimulating further research in the area.” (V. G. Gaĭtsgori, Mathematical Reviews, October, 2015)

“The book is a continuation of several papers and also a previous book by the author devoted to the study of the structure of approximate solutions of nonconvex (nonconcave) discrete-time optimal control problems. … The book could be addressed to postgraduate students as well as to control engineers and researches.” (Bozhidar Cheshankov, zbMATH, Vol. 1305, 2015)