Turnpike Phenomenon and Infinite Horizon Optimal Control

Abstract

Aims and ScopeOptimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences.The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximation techniques and heuristic approaches. This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher's location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) PrefaceThe monograph is devoted to the study of the structure of approximate solutions of n...

Efficient Control Discretization Based on Turnpike Theory for Dynamic Optimization

“…The extension of turnpike theory to nonconvex problems has led to generalized definitions of the turnpike, in which it may no longer be a steady state, but a time-dependent trajectory [31,33]. Nonetheless, the interest in the present work is in a steady-state turnpike for generally nonconvex problems.…”

“…For relatively large t f , the optimal trajectories will traverse close to the optimal steady state for most of the time horizon, and the transient phases will be short in comparison. The extension of turnpike theory to nonconvex problems has led to generalized definitions of the turnpike, in which it may no longer be a steady state, but a time-dependent trajectory [31,33].…”

“…Within the proposed approach, it is seen that both the main and variant formulations yield the same solution. The variant formulation requires slightly fewer iterations to converge, which could be due to the smaller search space resulting from Constraint (33). Nevertheless, its convergence time is slightly higher; this can be attributed to the higher cost per iteration resulting from the auxiliary differential equation representing Constraint (33) and corresponding sensitivities.…”

Using nonlinear model predictive control for dynamic decision problems in economics

“…In mathematical literature, turnpike phenomena are investigated rather for different types of optimal control problems than for specific models, see e.g., Zaslavski (2006) and Zaslavski (2014) for a collection of turnpike theorems or Damm et al (2014) for an exponential turnpike theorem (i.e. σ c from Assumption 3.1 (ii) is exponentially decaying in N ).…”

On the Relation Between Turnpike Properties for Finite and Infinite Horizon Optimal Control Problems

“…Recently, the turnpike property has also attracted interest in areas different from mathematical economics, see, e.g., [5,14,12,8]. This interest stems from the fact that it was realized that this property considerably simplifies the computation of (approximately) optimal trajectories in all areas of optimal control, either directly by constructive synthesis techniques as in [1] or indirectly via a receding horizon approach as in economic model predictive control [6,7].…”