Optimal Control Problems Arising in Mathematical Economics (Monographs in Mathematical Economics, 5) 9811692971, 9789811692970

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Monographs in Mathematical Economics  5

Alexander J. Zaslavski

Optimal Control Problems Arising in Mathematical Economics

Monographs in Mathematical Economics Volume 5

Editor-in-Chief Toru Maruyama, Professor Emeritus, Keio University, Tokyo, Japan Series Editors Shigeo Kusuoka, Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan Jean-Michel Grandmont, CREST-CNRS, Malakoff CX, France R. Tyrrell Rockafellar, Department of Mathematics, University of Washington, Seattle, USA

This series, consisting of research monographs and advanced textbooks for graduate students, is designed to bring together mathematicians interested in obtaining challenging new stimuli from economic theories and economists seeking effective mathematical tools for their research. The scope of the series includes but is not limited to: • Economic theories in various fields based on rigorous mathematical reasoning • Mathematical methods (e.g., analysis, algebra, geometry, probability) motivated by economic theories • Mathematical results of potential relevance to economic theory • Historical study of mathematical economics Comparable, existing monographs series are mainly organized from the viewpoint of “users” of mathematics, and are thus of limited interest to mathematicians. This series in contrast aims at genuine interaction between economists and mathematicians. Most economic phenomena are described by (1) optimizing behaviors of agents (consumers, firms, governments, etc.) and (2) equilibria generated through the interactions of these agents. Consequently, the most basic mathematical subjects for economics fall into two categories: • Optimization theory, static and dynamic. Basic linear / nonlinear programming, calculus of variations, optimal control, dynamic programming etc. provide key mathematical tools. Modern developments in convex analysis, nonlinear functional analysis, set-valued analysis, and non-smooth analysis form indispensable foundations. • Equilibrium theory. Existence in finite- / infinite-dimensional settings, characterization, computational algorithms, dynamic adjustment processes, correspondence with economic structures. Some fixed-point theorems and variational inequalities were created to solve the existence problem of equilibria. The stability theory and the viability theory of differential equations play important roles in the dynamic aspects of the equilibrium theory. Further, differential topology is a basic tool for the geometric analysis of equilibrium manifolds. Economic phenomena significant in the real world often have a dynamic character. For instance, business cycles and movements of asset prices are governed by dynamic laws expressed in terms of differential or difference equations. Nonlinear analysis, harmonic analysis, and stochastic calculus play significant roles in solving dynamic economic problems. Econometric methods, including time series analysis and forecasting, also require sophisticated mathematical tools. Finally, significant developments in game theory and interaction with mathematical logic should be mentioned. All of these topics and phenomena come within the scope of this series, aimed at cooperation between economists and mathematicians and the development of their respective disciplines.

Alexander J. Zaslavski

Optimal Control Problems Arising in Mathematical Economics

Alexander J. Zaslavski Department of Mathematics Technion – Israel Institute of Technology Rishon LeZion, Israel

ISSN 2364-8279 ISSN 2364-8287 (electronic) Monographs in Mathematical Economics ISBN 978-981-16-9297-0 ISBN 978-981-16-9298-7 (eBook) https://doi.org/10.1007/978-981-16-9298-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

For more than 70 years now, there has been a lot of research activity regarding the turnpike theory and infinite horizon optimal control. This activity stems from Samuelson’s discovery of the turnpike property of optimal control problems. The term was first coined by P. Samuelson in 1948 when he showed that an efficient expanding economy would spend most of the time in the vicinity of a balanced equilibrium path (also called a von Neumann path). To have the turnpike property means, roughly speaking, that the approximate solutions of the problems are determined mainly by the objective function (integrand) and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. Since that discovery, many developments have taken place in this field, including, in particular, studies of problems which arise in engineering, in models of economic growth, in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals, in the theory of thermodynamical equilibrium for materials, in game theory, and in optimal control problems with partial differential equations. We present the results on properties of approximate solutions of discrete-time optimal control problems which are independent of the length of the interval, for all sufficiently large intervals. More precisely, the book is devoted to the study of two large classes of discrete-time optimal control problems arising in mathematical economics. This is the first time that the turnpike theory for these classes of problems has been presented in a book. Nonautonomous optimal control problems, the first class, are determined by a sequence of objective functions and a sequence of constraint maps. They correspond to a general model of economic growth. We are interested in turnpike properties of approximate solutions and in the stability of the turnpike phenomenon under small perturbations of objective functions and constraint maps. The second class, autonomous optimal control problems, corresponds to another general class of models of economic dynamics which includes the Robinson–Solow–Srinivasan (RSS) model as a particular case. In the 1960s, the RSS model was introduced by famous economists Robinson, Solow, and Srinivasan and studied by Robinson, Okishio, and Stiglitz. In 2005, Ali Khan and Mitra revisited this model, and their seminal paper led to a number of v

vi

Preface

interesting and important developments. Some of these developments were collected in our book [138]. There exists a rich literature on the study of discrete-time optimal control problems related to various models of economic dynamics. See, for example, [131, 135] and the references mentioned therein. But this variety does not contain the Robinson–Solow–Srinivasan model. This happens because there are qualitative differences between models considered in [131, 135] and the RSS model. In the models of [131, 135], one considers a finite number of goods, and the state of these models is a vector in a finite-dimensional space with nonnegative coordinates, where each coordinate corresponds to a certain type of goods. If a state of the model is good, then all its coordinates should be positive, and if the model has a turnpike (a singleton), then all its coordinates should be positive too. In the RSS model, one considers a finite number of machines, and the state of the model is a vector in a finite-dimensional space with nonnegative coordinates, where each coordinate corresponds to a certain type of machine. It is assumed that among these machines there is the best one. Therefore, on large time intervals, the amounts of other machines should tend to zero. If the RSS model has a turnpike, then all its coordinates should be zero, except for the coordinate corresponding to the best machine. Mathematically, it leads to the fact that if a model of [131, 135] has a turnpike x, then (x, x) is an interior point of the set of admissible pairs. In other words, for this model, a local controllability property holds. This property plays an important role in the analysis of the turnpike phenomenon. Since, for the RSS model, the turnpike has only one positive coordinate, this local controllability property does not hold. Fortunately, the Robinson–Solow–Srinivasan model has the so-called monotonicity property and some weak version of the local controllability property. In this book, we develop the turnpike theory for the large class of autonomous optimal control problems possessing the weak version of the local controllability property and the monotonicity property which includes the RSS model as a particular case. The combination of these two properties allows us to analyze the turnpike phenomenon for this class of models. Note that in our recent book [139], we analyzed the turnpike phenomenon for another class of optimal control problems possessing some strict contraction property and the monotonicity property which also includes the RSS model as a particular case. In Chap. 1, we discuss turnpike properties for a large class of discrete-time optimal control problems studied in [131] and for the Robinson–Solow–Srinivasan model. In Chap. 2, we introduce the first class of optimal control problems and study its turnpike property. This class of problems is also discussed in Chaps. 3– 6. In Chap. 3, we study the stability of the turnpike phenomenon under small perturbations of the objective functions. Analogous results for problems with discounting are considered in Chap. 4. In Chap. 5, we study the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint maps. Analogous results for problems with discounting are established in Chap. 6. The results of Chaps. 2–4 were obtained in [130, 132, 133], while the results

Preface

vii

of Chaps. 5 and 6 are new. The second class of problems is studied in Chaps. 7–9. In Chap. 7, which is based on [118], we study the turnpike properties. The stability of the turnpike phenomenon under small perturbations of the objective functions is established in Chap. 8. In Chap. 9, we establish the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint maps. The results of Chaps. 8 and 9 are new. In Chap. 10, which is based on [98], we study optimal control problems related to a model of knowledge-based endogenous economic growth and show the existence of trajectories of unbounded economic growth and provide estimates for the growth rate. One of the main features of this book is the study of the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint maps, which is presented in Chaps. 5, 6, and 9. This is a highly difficult topic because a small perturbation of constraint maps changes a class of admissible trajectories (programs). In [125], this topic was considered for a subclass of the problems studied in Chaps. 2–6. This subclass consists of problems which are autonomous and their turnpike is a singleton. In Chaps. 5 and 6, optimal control problems are not necessarily autonomous and their turnpikes are not necessarily a singleton. The author believes that this book will be useful for researchers interested in the turnpike theory, infinite horizon optimal control, models of economic growth, and their applications. Rishon LeZion, Israel March 22, 2021

Alexander J. Zaslavski

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 The Turnpike Phenomenon . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Nonconcave (Nonconvex) Problems . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Stability of the Turnpike Phenomenon . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Nonautonomous Control Systems Without Constraints . . . . . . . . . . . . 1.6 Nonautonomous Constrained Control Systems . . . . . . . . . . . . . . . . . . . . . 1.7 The Robinson–Solow–Srinivasan Model . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8 Overtaking Optimal Programs for the RSS Model . . . . . . . . . . . . . . . . . 1.9 Turnpike Properties of the RSS Model .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.10 Autonomous Optimal Control Problems .. . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 5 7 8 10 12 17 24 27 35

2

Turnpike Conditions for Optimal Control Systems . . . . . . . . . . . . . . . . . . . . 2.1 Preliminaries.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 TP Implies ATP and Property (P). . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7 Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

37 37 40 45 48 55 58 61 63 67

3

Nonautonomous Problems with Perturbed Objective Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 69 3.1 Preliminaries.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 69 3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 73 3.3 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 76 3.4 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81 3.5 Proof of Proposition 3.4.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93 3.6 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 96 3.7 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101 ix

x

Contents

4

Nonautonomous Problems with Discounting . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Preliminaries.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Proofs of Theorems 4.1 and 4.4 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

105 105 109 113 131

5

Stability of the Turnpike Phenomenon for Nonautonomous Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Preliminaries and Stability Results . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Proofs of Theorems 5.1 and 5.2 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Proofs of Theorems 5.3 and 5.4 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Proof of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Proofs of Theorems 5.6 and 5.7 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

135 135 143 159 163 167 171 175

Stability of the Turnpike for Nonautonomous Problems with Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Preliminaries and the Main Results . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Proofs of Theorems 6.2 and 6.3 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Proof of Theorem 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

177 177 180 189 194

7

Turnpike Properties for Autonomous Problems . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Preliminaries and the Main Results . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Proof of Theorem 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5 Proofs of Theorems 7.4 and 7.5 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 Proofs of Theorems 7.6, 7.7 and 7.8.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.7 The Robinson–Solow–Srinivasan Model . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.8 A Model of Economic Dynamics.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.9 Equivalence of Optimality Criteria .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.10 Proof of Theorem 7.22 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.11 Weak Turnpike Theorems .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.12 Proof of Theorem 7.23 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

199 199 204 214 215 218 220 223 224 230 231 239 240

8

Autonomous Problems with Perturbed Objective Functions . . . . . . . . . . 8.1 Preliminaries and the Main Results . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Proofs of Theorems 8.1 and 8.2 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Proofs of Theorems 8.3 and 8.4 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5 Optimal Control Systems with Discounting . . . .. . . . . . . . . . . . . . . . . . . . 8.6 An Auxiliary Result for Theorems 8.9 and 8.10 .. . . . . . . . . . . . . . . . . . . 8.7 Proof of Theorem 8.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.8 Proof of Theorem 8.10 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.9 Existence of Overtaking Optimal Programs . . . .. . . . . . . . . . . . . . . . . . . .

245 245 250 256 265 268 269 278 283 288

6

Contents

9

xi

Stability Results for Autonomous Problems . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Preliminaries and the Main Results . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Proof of Theorem 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5 Proof of Theorem 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Optimal Control Problems with Discounting . . .. . . . . . . . . . . . . . . . . . . . 9.7 An Auxiliary Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8 Proof of Theorem 9.15 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

293 293 300 302 322 331 334 335 343

10 Models with Unbounded Endogenous Economic Growth . . . . . . . . . . . . . 10.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Existence of Unbounded Growth and Balanced Growth Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Proof of Theorem 10.5 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.5 Proof of Theorem 10.6 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

351 351 353 357 367 368

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 371 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 377

Chapter 1

Introduction

The study of optimal control problems defined on infinite intervals and on sufficiently large intervals has been a rapidly growing area of research [4–6, 9–15, 19, 21–27, 29, 33, 37, 40, 57, 58, 63, 65, 66, 71, 72, 74, 77, 88, 101, 108, 109, 114, 116, 120, 122, 124, 125, 135, 136] which has various applications in engineering [3, 18, 60, 105], in models of economic growth [7, 16–18, 28, 34–36, 38, 39, 57, 59, 64, 70, 75, 85–87, 89, 90, 100, 105, 113, 117, 123, 127–129, 131, 134, 138], in infinite discrete models of solid-state physics related to dislocations in onedimensional crystals [8, 79, 102], in control with partial differential equations [31, 32, 78, 97, 137] and in the theory of thermodynamical equilibrium for materials [20, 61, 67–69]. In this chapter we discuss turnpike properties and optimality criteria over an infinite horizon for two classes of dynamic optimization problems. Problems of the first class determine a general model of economic growth, while problems from the second class are related to the Robinson–Solow–Srinivasan model.

1.1 The Turnpike Phenomenon In this book our goal is to study the structure of approximate solutions over large intervals for discrete-time constrained optimal control problems. In this chapter we discuss the structure of approximate solutions of a discrete-time control system with a compact metric space of states X. First we consider an autonomous case. The control system is described by a bounded upper semicontinuous function v : X × X → 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

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_1

1

2

1 Introduction

study the problems T −1

−1 v(xi , xi+1 ) → max, {(xi , xi+1 )}Ti=0 ⊂ Ω, x0 = z, xT = y,

(P1)

i=0 T −1

−1 v(xi , xi+1 ) → max, {(xi , xi+1 )}Ti=0 ⊂ Ω, x0 = z

(P2)

i=0

and T −1

−1 v(xi , xi+1 ) → max, {(xi , xi+1 )}Ti=0 ⊂ Ω,

(P3)

i=0

where T ≥ 1 is an integer and the points y, z ∈ X. In the classical turnpike theory the objective function v possesses the turnpike property (TP) if there exists a point x¯ ∈ X (a turnpike) such that the following condition holds: For each positive number  there exists an integer L ≥ 1 such that for each integer T ≥ 2L and each solution {xi }Ti=0 ⊂ X of the problem (P1) the inequality ρ(xi , x) ¯ ≤  is true for all i = L, . . . , T − L. It should be mentioned that the constant L depends neither on T nor on y, z. The turnpike phenomenon has the following interpretation. If one wishes to reach a point A from a point B by a car in an optimal way, then one should turn to a turnpike, spend most of the time on it and then leave the turnpike to reach the required point. In the classical turnpike theory [28, 70, 87, 100] the space X is a compact convex subset of a finite-dimensional Euclidean space, the set Ω is convex and the function v is strictly concave. Under these assumptions the turnpike property can be established and the turnpike x¯ is a unique solution of the maximization problem v(x, x) → max, (x, x) ∈ Ω. In this situation it is shown that for each program {xt }∞ t =0 either the sequence −1  T t =0

v(xt , xt +1 ) − T v(x, ¯ x) ¯

∞ T =1

is bounded (in this case the program {xt }∞ t =0 is called (v)-good) or it diverges to −∞. Moreover, it is also established that any (v)-good program converges to the turnpike x. ¯ In the sequel this property is called the asymptotic turnpike property. Recently it was shown that the turnpike property is a general phenomenon which holds for large classes of variational and optimal control problems without convexity

1.1 The Turnpike Phenomenon

3

assumptions. (See, for example, [105] and the references mentioned therein.) For these classes of problems a turnpike is not necessarily a singleton but may instead be an nonstationary trajectory (in the discrete time nonautonomous case) or an absolutely continuous function on the interval [0, ∞) (in the continuous time nonautonomous case) or a compact subset of the space X (in the autonomous case). For classes of problems considered in [105], using the Baire category approach, it was shown that the turnpike property holds for a generic (typical) problem. We are also interested in individual (non-generic) results describing the structure of approximate solutions. We study the problems (P1)–(P3) with the constraint −1 {(xi , xi+1 )}Ti=0 ⊂ Ω where Ω is an arbitrary nonempty closed subset of X × X. Clearly, these constrained problems are more difficult and less understood than their unconstrained prototypes in [105]. They are also more realistic from the point of view of mathematical economics. As we have mentioned before, in general a turnpike is not necessarily a singleton. Nevertheless, problems of the type (P1)–(P3) for which the turnpike is a singleton are of great importance because of the following reasons: there are many models of economic growth for which a turnpike is a singleton; if a turnpike is a singleton, then approximate solutions have very simple structure and this is very important for applications; if a turnpike is a singleton, then it can be easily calculated as a solution of the problem v(x, x) → max, (x, x) ∈ Ω. Such problems were studied in [107, 118, 131, 134]. The turnpike property is very important for applications. Suppose that our objective function v has the turnpike property and we know a finite number of “approximate” solutions of the problem (P1). Then we know the turnpike x, ¯ or at least its approximation, and the constant L (see the definition of (TP)) which is an estimate for the time period required to reach the turnpike. This information can be useful if we need to find an “approximate” solution of the problem (P1) with a new time interval [m1 , m2 ] and the new values z, y ∈ X at the end points m1 and m2 . Namely, instead of solving this new problem on the “large” interval [m1 , m2 ] we can find an “approximate” solution of the problem (P1) on the “small” interval [m1 , m1 + L] with the values z, x¯ at the end points and an “approximate” solution of the problem (P1) on the “small” interval [m2 − L, m2 ] with the values x, ¯ y at the end points. Then the concatenation of the first solution, the constant sequence xi = x, ¯ i = m1 + L, . . . , m2 − L and the second solution is an “approximate” solution of the problem (P1) on the interval [m1 , m2 ] with the values z, y at the end points. Sometimes as an “approximate” solution of the problem (P1) we can choose 2 any admissible sequence {xi }m i=m1 satisfying xm1 = z, xm2 = y and xi = x¯ for all i = m1 + L, . . . , m2 − L. As we have already mentioned above, in the classical turnpike theory the space X is a compact convex subset of a finite-dimensional Euclidean space, the set Ω is convex, the function v is strictly concave and the turnpike x¯ is a unique solution of the maximization problem v(x, x) → max, (x, x) ∈ Ω.

4

1 Introduction

See [28, 70, 87, 100] and the literature mentioned therein, where our optimal control problems correspond to models of economic dynamics. In these models X is the space of states and Ω = {(x, y) ∈ X × X : y ∈ a(x)}, where a : X → 2X \ {∅} is a technological map (technology) which is, of course, set-valued and has a convex graph. If at some instant of time t ≥ 0, the economy state is xt ∈ X, then at the instant of time t +1, the state of the economy can be any point xt +1 ∈ a(xt ) obtained by using the technology a. Then v(xt , xt +1 ) is some economical characteristic of the admissible pair (xt , xt +1 ) (for example, consumption at moment t) which we would like to maximize. For a given initial point x and a large natural number T we consider all the programs {xt }Tt=0 such that x0 = x and the goal is to maximize the total economic characteristic (consumption). Then it turns out that the turnpike phenomenon takes place when T is sufficiently large. Since the set of states X is convex, the cost function v is concave, the technology map a is set-valued and its graph is convex too, the tools of convex analysis and set-valued analysis play an important role. So this was the first class of optimal control problems studied in the turnpike theory related to models of economic dynamics. The second class of optimal control problems considered in the economic literature is problems with technological changes. At some moment it became clear that because of technological changes a model of economic growth should be defined by a sequence of set-valued maps at : X → 2X \ {∅}, t = 0, 1, . . . and a program is a sequence {xt }Tt=0 , where T is a natural number, satisfying xt +1 ∈ a(xt ) for all t = 0, . . . , T − 1. Of course, the cost functions now also depend on t. For this second class of problems the turnpike phenomenon was studied also under assumptions that the space of states X is convex, the cost functions vt are concave, the technology maps at are set-valued and their graphs are convex. For this class of problems the turnpike is not a singleton but a nonstationary program. The turnpike theory for the third class of optimal control problems related to mathematical economics was developed in our research summarized in [131]. The research leads us to understanding that the turnpike phenomenon holds for large classes of autonomous (time-independent) problems without any convexity– concavity assumption. This class of problems contains the first one and a turnpike for its problems is a singleton. It was shown that the turnpike phenomenon holds if all good programs converge to the same point, which will be the turnpike. The stability of the turnpike phenomenon under small perturbations were established too. For details see Sects. 1.2–1.4. Thus the study of these three classes of problems is a well-established area of research. But as we have already mentioned before, nonautonomous constrained problems with technological changes and without convexity–concavity assumptions are more realistic from the point of view of real world applications. This is the fourth new class of problems and its study will be one of the main topics of this book. For details see Sect. 1.6.

1.2 Nonconcave (Nonconvex) Problems

5

1.2 Nonconcave (Nonconvex) Problems Let (X, ρ) be a compact metric space, Ω be a nonempty closed subset of X × X and let v : X × X → R 1 be a bounded upper semicontinuous function. A sequence {xt }∞ t =0 ⊂ X is called an (Ω)-program (or just a program if the set Ω is understood) if (xt , xt +1 ) ∈ Ω for all nonnegative integers t. A sequence {xt }Tt=0 where T ≥ 1 is an integer is called an (Ω)-program (or just a program if the set Ω is understood) if (xt , xt +1 ) ∈ Ω for all integers t ∈ [0, T − 1]. We consider the problems T −1

−1 v(xi , xi+1 ) → max, {(xi , xi+1 )}Ti=0 ⊂ Ω, x0 = y, xT = z,

i=0 T −1

−1 v(xi , xi+1 ) → max, {(xi , xi+1 )}Ti=0 ⊂ Ω, x0 = z

i=0

and T −1

−1 v(xi , xi+1 ) → max, {(xi , xi+1 )}Ti=0 ⊂ Ω,

i=0

where T ≥ 1 is an integer and the points y, z ∈ X. We suppose that there exist a point x¯ ∈ X and a positive number c¯ such that the following assumptions hold: (A1) (x, ¯ x) ¯ is an interior point of Ω and v : X × X → R 1 is continuous at the point ( x, ¯ x); ¯ T −1 (A2) ¯ x) ¯ + c¯ for any natural number T and any program t =0 v(xt , xt +1 ) ≤ T v(x, {xt }Tt=0 . Property (A2) implies that for each program {xt }∞ t =0 either the sequence −1  T

v(xt , xt +1 ) − T v(x, ¯ x) ¯

t =0

∞ T =1

 −1 is bounded or limT →∞ [ Tt =0 v(xt , xt +1 ) − T v(x, ¯ x)] ¯ = −∞. A program {xt }∞ is called (v)-good if the sequence t =0 −1  T t =0

is bounded.

v(xt , xt +1 ) − T v(x, ¯ x) ¯

∞ T =1

6

1 Introduction

We suppose that the following assumption holds: (A3) (The asymptotic turnpike property) For every (v)-good program {xt }∞ t =0 , limt →∞ ρ(xt , x) ¯ = 0. Note that properties (A1)–(A3) hold for models of economic dynamics considered in the classical turnpike theory. For each positive number M denote by XM the set of all points x ∈ X for which there exists a program {xt }∞ t =0 such that x0 = x and that for all natural numbers T the following inequality holds: T −1

v(xt , xt +1 ) − T v(x, ¯ x) ¯ ≥ −M.

t =0

It is not difficult to see that ∪{XM : M ∈ (0, ∞)} is the set of all points x ∈ X for which there exists a (v)-good program {xt }∞ t =0 satisfying x0 = x. Let T ≥ 1 be an integer and Δ ≥ 0. A program {xi }Ti=0 ⊂ X is called (Δ)optimal if for any program {xi }Ti=0 satisfying x0 = x0 , the inequality T −1

v(xi , xi+1 ) ≥

i=0

T −1

v(xi , xi+1 )−Δ

i=0

holds. The turnpike theory for problems (P1) and (P2) is presented in [131]. In particular, in Chapter 2 of [131] we prove the following turnpike result for approximate solutions of our second optimization problem stated above. Theorem 1.1 Let , M be positive numbers. Then there exist a natural number L and a positive number δ such that for each integer T > 2L and each (δ)-optimal program {xt }Tt=0 which satisfies x0 ∈ XM there exist nonnegative integers τ1 , τ2 ≤ L such that ρ(xt , x) ¯ ≤  for all t = τ1 , . . . , T −τ2 and if ρ(x0 , x) ¯ ≤ δ, then τ1 = 0. An analogous turnpike result for approximate solutions of our first optimization problem is also proved in Chapter 2 of [131]. ∞ A program {xt }∞ t =0 is called (v)-overtaking optimal if for each program {yt }t =0 satisfying y0 = x0 the inequality lim sup

T −1

T →∞ t =0

[v(yt , yt +1 ) − v(xt , xt +1 )] ≤ 0

holds. In Chapter 2 of [131] we prove the following result which establishes the existence of an overtaking optimal program.

1.3 Examples

7

Theorem 1.2 Assume that x ∈ X and that there exists a (v)-good program {xt }∞ t =0 such that x0 = x. Then there exists a (v)-overtaking optimal program {xt∗ }∞ t =0 such that x0∗ = x.

1.3 Examples Example 1.3 Let (X, ρ) be a compact metric space, Ω be a nonempty closed subset of X × X, x¯ ∈ X, (x, ¯ x) ¯ be an interior point of Ω, π : X → R 1 be a continuous function, α be a real number and L : X × X → [0, ∞) be a continuous function such that for each (x, y) ∈ X × X the equality L(x, y) = 0 holds if and only if (x, y) = (x, ¯ x). ¯ Set v(x, y) = α − L(x, y) + π(x) − π(y) for all x, y ∈ X. It is not difficult to see that assumptions (A1), (A2) and (A3) hold. Example 1.4 Let X be a compact convex subset of the Euclidean space R n with the norm | · | induced by the scalar product ·, · , let ρ(x, y) = |x − y|, x, y ∈ R n , Ω be a nonempty closed subset of X × X, a point x¯ ∈ X, (x, ¯ x) ¯ be an interior point of Ω and let v : X × X → R 1 be a strictly concave continuous function such that v(x, ¯ x) ¯ = sup{v(z, z) : z ∈ X and (z, z) ∈ Ω}. We assume that there exists a positive constant r¯ such that ¯ |y − x| ¯ ≤ r¯ } ⊂ Ω. {(x, y) ∈ R n × R n : |x − x|, It is a well-known fact of convex analysis [73, 82] that there exists a point l ∈ R n such that v(x, y) ≤ v(x, ¯ x) ¯ + l, x − y for any point (x, y) ∈ X × X. Set L(x, y) = v(x, ¯ x) ¯ + l, x − y − v(x, y) for all (x, y) ∈ X × X. It is not difficult to see that this example is a particular case of Example 1.3. Therefore assumptions (A1), (A2) and (A3) hold.

8

1 Introduction

1.4 Stability of the Turnpike Phenomenon Let (X, ρ) be a compact metric space and Ω be a nonempty closed subset of X × X. We denote by M(Ω) the set of all bounded functions u : Ω → R 1 . For every function w ∈ M(Ω) define w = sup{|w(x, y)| : (x, y) ∈ Ω}. For each x, y ∈ X, each integer T ≥ 1 and each u ∈ M(Ω) set σ (u, T , x) = sup

−1  T

 u(xi , xi+1 ) : {xi }Ti=0 is an (Ω)-program and x0 = x ,

i=0

σ (u, T , x, y) = sup

−1  T

 u(xi , xi+1 ) : {xi }Ti=0 is an (Ω)-program and x0 = x, xT = y ,

i=0

σ (u, T ) = sup

−1  T

 u(xi , xi+1 ) : {xi }Ti=0 is an (Ω)-program .

i=0

(Here we use the convention that the supremum of an empty set is −∞). For every pair of points x, y ∈ X, every pair of nonnegative integers T1 , T2 which 2 −1 ⊂ M(Ω) define satisfies T1 < T2 and every finite sequence of functions {ut }Tt =T 1 T2 −1 σ ({ut }t=T , T1 , T2 , x) = sup 1

2 −1  T

ut (xt , xt+1 ) :

t=T1

 2 is an (Ω) -program and x = x , {xt }Tt=T T 1 1 2 −1 , T1 , T2 , x, y) = sup σ ({ut }Tt=T 1

2 −1  T

ut (xt , xt+1 ) :

t=T1

 2 is an (Ω) -program and x = x, x = y , {xt }Tt=T T T 1 2 1 T2 −1 , T1 , T2 ) = sup σ ({ut }t=T 1

2 −1  T

 2 ut (xt , xt+1 ) : {xt }Tt=T is an (Ω)-program . 1

t=T1

Let v ∈ M(Ω) be an upper semicontinuous function. Set v(x, y) = −v − 1 for all (x, y) ∈ (X × X) \ Ω.

1.4 Stability of the Turnpike Phenomenon

9

Suppose that there exist a point x¯ ∈ X and real positive constants c¯ such that assumptions (A1)–(A3) from Sect. 1.2 hold. Let T ≥ 1 be an integer. Denote by YT the set of all x ∈ X for which there exists a program {xt }Tt=0 such that x0 = x, ¯ xT = x. Denote by Y¯T the set of all x ∈ X for T which there exists a program {xt }t =0 such that x0 = x, xT = x. ¯ The following two theorems stated below are proved in [131]. They show that the turnpike phenomenon holds for approximate solutions of the optimal control problems of the types (P1) and (P2) with objective functions ut , t = 0, . . . , T − 1 belonging to a small neighborhood of v. Theorem 1.5 Let  > 0, L0 be a positive integer and M0 be a positive number. Then there exist a natural number L and a positive number δ <  such that for −1 every natural number T > 2L, every finite sequence of functions {ut }Tt =0 ⊂ M(Ω) for which ut − v ≤ δ, t = 0 . . . T − 1, and every (Ω)-program {xt }Tt=0 for which x0 ∈ Y¯L0 , xT ∈ YL0 , T −1 t =0

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , 0, T , x0 , xT ) − M0

and τ +L−1  t =τ

+L−1 ut (xt , xt +1 ) ≥ σ ({ut }τt =τ , τ, τ + L, xτ , xτ +L ) − δ

for every nonnegative integer τ ≤ T − L there exist integers τ1 ∈ [0, L], τ2 ∈ [T − L, T ] such that ρ(xt , x) ¯ ≤ , t = τ1 , . . . , τ2 . Moreover, if ρ(x0 , x) ¯ ≤ δ, then τ1 = 0 and if ρ(xT , x) ¯ ≤ δ, then τ2 = T . Theorem 1.6 Let  > 0, L0 be a positive integer and M0 be a positive number. Then there exist a positive integer L and a positive number δ <  such that for −1 every natural number T > 2L, every finite sequence {ut }Tt =0 ⊂ M(Ω) for which ut − v ≤ δ, t = 0 . . . , T − 1 and every (Ω)-program {xt }Tt=0 for which x0 ∈ Y¯L0 ,

T −1 t =0

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , 0, T , x0 ) − M0

10

1 Introduction

and τ +L−1  t =τ

+L−1 ut (xt , xt +1 ) ≥ σ ({ut }τt =τ , τ, τ + L, xτ , xτ +L ) − δ

for every integer τ ∈ [0, T − L] there exist a pair of integers τ1 ∈ [0, L], τ2 ∈ [T − L, T ] such that ρ(xt , x) ¯ ≤ , t = τ1 , . . . , τ2 . ¯ ≤ δ, then τ1 = 0. Moreover, if ρ(x0 , x)

1.5 Nonautonomous Control Systems Without Constraints In this section, which is based on [104], we discuss the structure of solutions of the optimization problems m 2 −1

2 vi (zi , zi+1 ) → min, {zi }m i=m1 ⊂ X and zm1 = x, zm2 = y,

(P )

i=m1

where vi : X × X → R 1 , i = 0, ±1, ±2, . . . is a continuous function defined on a metric space X and x, y ∈ X. Let Z = {0, ±1, ±2, . . . } be the set of all integers, (X, ρ) be a compact metric space and let vi : X × X → R 1 , i = 0, ±1, ±2, . . . be a sequence of continuous functions such that sup{|vi (x, y)| : x, y ∈ X, i ∈ Z} < ∞ and which satisfy the following assumption: (A) For each positive number  there exists a positive number δ such that if i ∈ Z and if points x1 , x2 , y1 , y2 ∈ X satisfy ρ(xj , yj ) ≤ δ, j = 1, 2, then |vi (x1 , x2 ) − vi (y1 , y2 )| ≤ . For each pair of points y, z ∈ X and each pair of integers n1 , n2 > n1 put σ (n1 , n2 , y, z) = inf

σ (n1 , n2 ) = inf

⎧ 2 −1 ⎨n ⎩

i=n1

⎧ 2 −1 ⎨n ⎩

i=n1

⎫ ⎬

2 vi (xi , xi+1 ) : {xi }ni=n ⊂ X, xn1 = y, xn2 = z , 1 ⎭

⎫ ⎬

2 vi (xi , xi+1 ) : {xi }ni=n ⊂X . 1 ⎭

1.5 Nonautonomous Control Systems Without Constraints

11

Choose a real number d0 > 0 such that |vi (x, y)| ≤ d0 , x, y ∈ X, i ∈ Z. A sequence {yi }∞ i=−∞ ⊂ X is called good if there exists a positive number c such that for each pair of integers m1 , m2 > m1 , m 2 −1

vi (yi , yi+1 ) ≤ σ (m1 , m2 , ym1 , ym2 ) + c.

i=m1

We say that the sequence {vi }∞ i=−∞ has the turnpike property (TP) if there exists a sequence { x i }∞ ⊂ X which satisfies the following condition: i=−∞ For each  > 0 there are δ > 0 and a natural number N such that for each pair 2 of integers T1 , T2 ≥ T1 + 2N and each sequence {yi }Ti=T ⊂ X which satisfies 1 T 2 −1

vi (yi , yi+1 ) ≤ σ (T1 , T2 , yT1 , yT2 ) + δ

i=T1

there are integers τ1 ∈ {T1 , . . . , T1 + N}, τ2 ∈ {T2 − N, . . . , T2 } such that: xi ) ≤ , i = τ1 , . . . , τ2 ; (i) ρ(yi , (ii) if ρ(yT1 , xT1 ) ≤ δ, then τ1 = T1 and if ρ(yT2 , xT2 ) ≤ δ, then τ2 = T2 . ∞ The sequence { x i }∞ i=−∞ ⊂ X is called the turnpike of {vi }i=−∞ . ∞ Assume that { xi }i=−∞ ⊂ X. How do we verify if the sequence of cost functions {vi }∞ has (TP) and { x i }∞ i=−∞ i=−∞ is its turnpike? In [104] we introduced three ∞ properties (a), (b) and (c) and showed that {vi }∞ i=−∞ has (TP) if and only if {vi }i=−∞ possesses properties (a), (b) and (c). Property (a) means that all good sequences have the same asymptotic behavior. Property (b) means that for each pair of integers 2 m1 , m2 > m1 the sequence { x i }m i=m1 is a unique solution of problem (P) with x = xm1 , y = xm2 and that if a sequence {yi }∞ i=−∞ ⊂ X is a solution of problem (P) for each pair of integers m1 , m2 > m1 with x = ym1 , y = ym2 , then yi = xi for 2 all integers i. Property (c) means that if a sequence {yi }m ⊂ X is an approximate i=m1 solution of problem (P) and m2 − m1 is large enough, then there is j ∈ [m1 , m2 ] such that yj is close to xj . The next theorem was obtained in [104]. ∞ Theorem 1.7 Let { x i }∞ i=−∞ ⊂ X. Then the sequence {vi }i=−∞ possesses the ∞ turnpike property and { xi }i=−∞ is its turnpike if and only if the following properties hold:

(a) If {yi }∞ i=−∞ ⊂ X is good, then lim ρ(yi , xi ) = 0 , lim ρ(yi , xi ) = 0.

i→∞

i→−∞

12

1 Introduction

(b) For each pair of integers m1 , m2 > m1 , m 2 −1

vi ( xi , xi+1 ) = σ (m1 , m2 , xm1 , xm2 )

i=m1

and if a sequence {yi }∞ i=−∞ ⊂ X satisfies m 2 −1

vi (yi , yi+1 ) = σ (m1 , m2 , ym1 , ym2 )

i=m1

for each pair of integers m1 , m2 > m1 , then yi = xi , i ∈ Z; (c) For each positive number  there exist a positive number δ and an integer L ≥ 1 such that for each integer m and each sequence {yi }m+L i=m ⊂ X which satisfies m+L−1 

vi (yi , yi+1 ) ≤ σ (m, m + L, ym , ym+L ) + δ

i=m

there exists an integer j ∈ {m, . . . , m + L} for which ρ(yj , xj ) ≤ . It should be mentioned that properties (a)–(c) easily follow from the turnpike property. However, it is very nontrivial to show that properties (a)–(c) are sufficient for this property.

1.6 Nonautonomous Constrained Control Systems In this book our first goal is to study the turnpike properties of nonautonomous discrete-time optimal control systems arising in economic dynamics which are determined by sequences of lower semicontinuous objective functions and by a sequence of constraints maps. To have these properties means that the approximate solutions of the problems are determined mainly by the objective functions, and are essentially independent of the choice of intervals and endpoint conditions, except in regions close to the endpoints. These nonautonomous optimal control problems are natural extensions of autonomous constrained problems and of nonautonomous problems without constraints discussed before. For each nonempty set Y denote by B(Y ) the set of all bounded functions f : Y → R 1 and for each f ∈ B(Y ) set f  = sup{|f (y)| : y ∈ Y }. For each nonempty compact metric space Y denote by C(Y ) the set of all continuous functions f : Y → R 1 .

1.6 Nonautonomous Constrained Control Systems

13

Let (X, ρ) be a compact metric space with the metric ρ. The set X × X is equipped with the metric ρ1 defined by ρ1 ((x1 , x2 ), (y1 , y2 )) = ρ(x1 , y1 ) + ρ(x2 , y2 ), (x1 , x2 ), (y1 , y2 ) ∈ X × X. For each integer t ≥ 0 let Ωt be a nonempty closed subset of the metric space X × X. Let T ≥ 0 be an integer. A sequence {xt }∞ t =T ⊂ X is called a program if (xt , xt +1 ) ∈ Ωt for all integers t ≥ T . 2 ⊂ X is called Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A sequence {xt }Tt =T 1 a program if (xt , xt +1 ) ∈ Ωt for all integers t satisfying T1 ≤ t < T2 . We assume that there exists a program {xt }∞ t =0 . Denote by M the set of all sequences of functions {ft }∞ such that for each integer t ≥ 0 t =0 ft ∈ B(Ωt ) and that sup{ft  : t = 0, 1, . . . } < ∞. ∞ For each pair of sequences {ft }∞ t =0 , {gt }t =0 ∈ M set ∞ d({ft }∞ t =0 , {gt }t =0 ) = sup{ft − gt  : t = 0, 1, . . . }.

It is easy to see that d : M × M → [0, ∞) is a metric on M and that the metric space (M, d) is complete. Let {ft }∞ t =0 ∈ M. We consider the following optimization problems: T 2 −1 t =T1 T 2 −1 t =T1 T 2 −1 t =T1

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program, 1

(PT1 ,T2 )

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program and xT1 = y, 1

(PT1 ,T2 )

(y)

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program and xT1 = y, xT2 = z, 1

(y,z)

(PT1 ,T2 ) where y, z ∈ X and integers T1 , T2 satisfy 0 ≤ T1 < T2 .

14

1 Introduction

For each y, z ∈ X and each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 set U ({ft }∞ t=0 , T1 , T2 ) = inf

2 −1  T

 2 ft (xt , xt+1 ) : {xt }Tt=T is a program , 1

t=T1

U ({ft }∞ t=0 , T1 , T2 , y) = inf

2 −1  T

 2 ft (xt , xt+1 ) : {xt }Tt=T is a program and xT1 = y , 1

t=T1

U ({ft }∞ t=0 , T1 , T2 , y, z) = inf

2 −1  T

ft (xt , xt+1 ) :

t=T1

 2 {xt }Tt=T is a program and x = y, x = z . T T 1 2 1

Here we assume that the infimum over an empty set is ∞. We study the structure of (y) (y,z) approximate solutions of problems (PT1 ,T2 ), PT1 ,T2 and PT1 ,T2 which are defined as follows. Let M ≥ 0, y, z ∈ X and let integers T1 , T2 satisfy 0 ≤ T1 < T2 . A program (y,z) 2 is called an (M)-approximate solution of problem PT1 ,T2 if {xt }Tt =T 1 xT1 = y, xT2 = z and

T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 , y, z) + M. (y)

It is called an (M)-approximate solution of problem PT1 ,T2 if xT1 = y and

T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 , y) + M.

2 The program {xt }Tt =T is called an (M)-approximate solution of problem PT1 ,T2 if 1

T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 ) + M.

A program {xt }∞ t =0 is called an (M)-approximate solution of the corresponding infinite horizon problem if for each pair of integers T1 ≥ 0, T2 > T1 , T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 ) + M.

Denote by Mreg the set of all sequences of functions {fi }∞ i=0 ∈ M for which f and constants c > 0, γ > 0 such that the following there exist a program {xt }∞ f f t =0 conditions hold:

1.6 Nonautonomous Constrained Control Systems

15

(C1) the function ft is lower semicontinuous for all integers t ≥ 0; (C2) for each pair of integers T1 ≥ 0, T2 > T1 , T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 ) + cf ; f

f

(C3) for each  > 0 there exists δ > 0 such that for each integer t ≥ 0 and each f f (x, y) ∈ Ωt satisfying ρ(x, xt ) ≤ δ, ρ(y, xt +1) ≤ δ we have f

f

|ft (xt , xt +1 ) − ft (x, y)| ≤ ; f

(C4) for each integer t ≥ 0, each (xt , xt +1 ) ∈ Ωt satisfying ρ(xt , xt ) ≤ γf and f each (xt +1 , xt +2 ) ∈ Ωt +1 satisfying ρ(xt +2, xt +2 ) ≤ γf there is x ∈ X such that (xt , x) ∈ Ωt , (x, xt +2 ) ∈ Ωt +1 ; moreover, for each  > 0 there exists δ ∈ (0, γf ) such that for each integer t ≥ f 0, each (xt , xt +1 ) ∈ Ωt and each (xt +1, xt +2 ) ∈ Ωt +1 satisfying ρ(xt , xt ) ≤ δ and ρ(xt +2 , xt +2 ) ≤ δ there is x ∈ X such that f

(xt , x) ∈ Ωt , (x, xt +2 ) ∈ Ωt +1 , ρ(x, xt +1) ≤ . f

¯ reg the closure of Mreg in (M, d). Denote by Mc,reg the set of Denote by M all sequences {fi }∞ i=0 ∈ Mreg such that fi ∈ C(Ωi ) for all integers i ≥ 0 and by ¯ c,reg the closure of Mc,reg in (M, d). M We study the optimization problems stated above with the sequence of objective functions {fi }∞ i=0 ∈ Mreg . Our study is based on the relation between these finite horizon problems and the corresponding infinite horizon optimization problem f ∞ determined by {fi }∞ i=0 . Note that the condition (C2) means that the program {xt }t =0 is an approximate solution of this infinite horizon problem. f ∞ Let {fi }∞ i=0 ∈ Mreg , a program {xi }i=0 , cf > 0 and γf > 0 be such that (C1)–(C4) hold. ∞ A program {xt }∞ t =S , where S ≥ 0 is an integer, is called ({fi }i=0 )-good if the sequence −1  T i=S

is bounded.

fi (xi , xi+1 ) −

T −1 i=S

f

f

fi (xi , xi+1 )

∞ T =S+1

16

1 Introduction

We say that the sequence {fi }∞ i=0 possesses an asymptotic turnpike property f ∞ (ATP) with {xi }i=0 being the turnpike if for each integer S ≥ 0 and each ({fi }∞ i=0 )good program {xi }∞ i=S , f

lim ρ(xi , xi ) = 0.

i→∞

We say that the sequence {fi }∞ i=0 possesses a turnpike property (TP) if for each  > 0 and each M > 0 there exist δ > 0 and a natural number L such that for each 2 pair of integers T1 ≥ 0, T2 ≥ T1 + 2L and each program {xt }Tt =T which satisfies 1 T 2 −1

∞ fi (xi , xi+1 ) ≤ min{U ({fi }∞ i=0 , T1 , T2 , xT1 , xT2 ) + δ, U ({fi }i=0 , T1 , T2 ) + M},

i=T1 f

the inequality ρ(xi , xi ) ≤  holds for all integers i = T1 + L, . . . , T2 − L. f ∞ The sequence {xi }∞ i=0 is called the turnpike of {fi }i=0 . In [130] we obtained the following two turnpike results. Theorem 1.8 The sequence {fi }∞ i=0 possesses the turnpike property if and only if {fi }∞ possesses (ATP) and the following property: i=0 (P) For each  > 0 and each M > 0 there exist δ > 0 and a natural number L +L such that for each integer T ≥ 0 and each program {xt }Tt =T which satisfies T +L−1 

fi (xi , xi+1 )

i=T ∞ ≤ min{U ({fi }∞ i=0 , T , T + L, xT , xT +L ) + δ, U ({fi }i=0 , T , T + L) + M} f

there is an integer j ∈ {T , . . . , T + L} for which ρ(xj , xj ) ≤ . Property (P) means that if a natural number L is large enough and a program +L is an approximate solution of the corresponding finite horizon problem, {xt }Tt =T f then there is j ∈ {T , . . . , T + L} such that xj is close to xj . We denote by Card(A) the cardinality of the set A. Theorem 1.9 Assume that the sequence {fi }∞ i=0 possesses (ATP) and property (P),  > 0 and M > 0. Then there exists a natural number L such that for each pair of 2 which satisfies integers T1 ≥ 0, T2 > T1 + L and each program {xt }Tt =T 1 T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({fi }∞ i=0 , T1 , T2 ) + M

1.7 The Robinson–Solow–Srinivasan Model

17

the following inequality holds: f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > }) ≤ L. The class of nonautonomous optimal control problems, discussed in this section, is studied in Chaps. 2–6. Theorems 1.8 and 1.9 are proved in Chap. 2. In Chap. 3 we study the stability of the turnpike phenomenon under small perturbations of objective functions. Chapter 4 contains stability results for problems with discounting. In Chap. 5 we analyze the stability of the turnpike phenomenon under small perturbations of objective functions and constraint maps. Stability results for problems with discounting, under small perturbations of objective functions and constraint maps, are established in Chap. 6.

1.7 The Robinson–Solow–Srinivasan Model In this section we discuss the Robinson–Solow–Srinivasan model (or the RSS model for short) which was introduced in the sixties by British economist Joan Robinson who was a central figure in what became known as post-Keynesian economics, Robert Solow who was awarded the John Bates Clark Medal in 1961 and the Nobel Memorial Prize in Economic Sciences in 1987 and T. N. Srinivasan, a foreign associate of the National Academy of Sciences of the USA [80, 92, 93] and was studied by Joan Robinson, Nobuo Okishio, a former President of the Japan Association of Economics and Econometric, and Joseph Stiglitz, a recipient of the Nobel Memorial Prize in Economic Sciences [76, 81, 94–96]. Recently, the Robinson–Solow–Srinivasan model was studied by M. Ali Khan and T. Mitra [41– 49], M. Ali Khan and Piazza [50–52], M. Ali Khan and A. J. Zaslavski [53–56] and A. J. Zaslavski [103, 106, 110–113, 115–118, 120–123, 126–128]. Many results on the RSS model are collected in [138]. Thus the study of the RSS model is a well-established area of research. We use the following notation. 1 ) be the set of real (non-negative) numbers and let R n be the nLet R 1 (R+ dimensional Euclidean space with the non-negative orthant n = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n}. R+

For every pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , define their inner product by xy =

n  i=1

xi yi

18

1 Introduction

and let x  y, x > y, x ≥ y have their usual meaning. Namely, for a given pair of vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ R n , we say that x ≥ y, if xi ≥ yi for all i = 1, . . . , n, x > y if x ≥ y and x = y, and x  y if xi > yi for all i = 1, . . . , n. For every z ∈ R 1 set z = max{i : i ≤ z is an integer}. n, Let e(i), i = 1, . . . , n, be the ith unit vector in R n , and e be an element of R+ n all of whose coordinates are unity. For every x ∈ R , denote by x its Euclidean norm in R n . Let a = (a1 , . . . , an )  0, b = (b1 , . . . , bn )  0, d ∈ (0, 1), ci = bi /(1 + dai ), i = 1, . . . , n. These parameters define an economy capable of producing a finite number n of alternative types of machines. For every i = 1, . . . , n, one unit of machine of type i requires ai > 0 units of labor to construct it, and together with one unit of labor, each unit of it can produce bi > 0 units of a single consumption good. Thus, the production possibilities of the economy are represented by an (labor) inputcoefficients vector, a = (a1 , . . . , an )  0 and an output-coefficients vector, b = (b1 , . . . , bn )  0. Without loss of generality we assume that the types of machines are numbered such that b1 ≥ b2 · · · ≥ bn . We assume that all machines depreciate at a rate d ∈ (0, 1). Thus the effective labor cost of producing a unit of output on a machine of type i is given by (1 + dai )/bi : the direct labor cost of producing unit output, and the indirect cost of replacing the depreciation of the machine in this production. We consider the reciprocal of the effective labor cost, the effective output that takes the depreciation into account, and denote it by ci for the machine of type i. In this section we assume that there is a unique machine type σ at which effective labor cost (1 + dai )/bi is minimal, or at which the effective output per man bi /(1 + dai ) is maximal. Thus assume the following:

there exists σ ∈ {1, . . . , n} such that for all i ∈ {1, . . . , n}\{σ }, cσ > ci .

(1.1)

For each nonnegative integer t let x(t) = (x1 (t), . . . , xn (t)) ≥ 0 denote the amounts of the n types of machines that are available in time-period t, and let z(t + 1) = (z1 (t + 1), . . . , zn (t + 1)) ≥ 0 be the gross investments in the n types of machines during period t + 1. Hence, z(t + 1) = (x(t + 1) − x(t)) + dx(t), the sum of net investment and of depreciation. Let y(t) = (y1 (t), . . . , yn (t)) be the amounts of the n types of machines used for production of the consumption good, by(t), during period t + 1. Let the total labor force of the economy be stationary and positive. We normalize it to be unity. It is clear that gross investment z(t + 1), representing the production of new machines of the various types, requires az(t +1) units of labor in period t. Also y(t), representing the use of available machines for manufacture of the consumption good, requires ey(t) units of labor in period t. Thus, the availability of labor constrains employment in the consumption and investment

1.7 The Robinson–Solow–Srinivasan Model

19

sectors is described by az(t + 1) + ey(t) ≤ 1. Note that the flow of consumption and of investment (new machines) are in gestation during the period and available at the end of it. We now give a formal description of this technological structure. A sequence {x(t), y(t)}∞ t =0 is called a program if for each integer t ≥ 0 n n (x(t), y(t)) ∈ R+ × R+ , x(t + 1) ≥ (1 − d)x(t), 0 ≤ y(t) ≤ x(t),

a(x(t + 1) − (1 − d)x(t)) + ey(t) ≤ 1.

(1.2)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A pair of sequences 2 2 −1 ({x(t)}Tt =T , {y(t)}Tt =T ) 1 1

n is called a program if x(T2 ) ∈ R+ and for each integer t satisfying T1 ≤ t < T2 relations (1.2) hold. We associate with every program {x(t), y(t)}∞ t =0 its gross investment sequence {z(t + 1)}∞ such that t =0

z(t + 1) = x(t + 1) − (1 − d)x(t), t = 0, 1, . . . and a consumption sequence {by(t)}∞ t =0 . In Sect. 1.1 we discussed four classes of optimal control problems related to mathematical economics. One feels that the RSS model belongs to the first class. Indeed for the first class of problems a technological map is a general set-valued map with a convex graph. In the case of the RSS model the technological map is given by n n a(x) = (1 − d)x + {y ∈ R+ : ay ≤ 1}, x ∈ R+ ,

a very particular case. Note that when the RSS model was introduced, convex and set-valued analysis were not so much developed and their tools did not allow the study of models with more general technological mappings. There are also qualitative differences between the general model of Sect. 1.1 and the RSS model. In the general model one considers n goods and the state of the model is a vector n where for each integer i ∈ {1, . . . , n}, x is the amount of x = (x1 , . . . , xn ) ∈ R+ i the ith type good. If the state of the model is good, then all its coordinates should be positive, and if the model has a turnpike (a singleton) all its coordinates should be positive too. In the RSS model one considers n types of machines and the state of the model n where for each integer i ∈ {1, . . . , n}, x is the is a vector x = (x1 , . . . , xn ) ∈ R+ i amount of the ith type machine. Moreover, by (1.1) there exists the best machine of type σ . Therefore on large time intervals the amounts of machines of type i = σ should tend to zero. If the RSS model has a turnpike x = (x1 , . . . , xn ) then xi = 0 for all i = σ and the situation is completely different. If a singleton x is a turnpike for the general model, then (x, x) is a interior point of the set of all

20

1 Introduction

admissible pairs. The RSS model does not possesses this property, which is called a local controllability property. The local controllability property plays an important role in the analysis of the turnpike phenomenon. In this analysis we often need for a given finite number of programs to build their 2 concatenation, which is a program too. Assume that we have programs {xt }m t =m1 m3 and {xt }t =m2 +1 . In the case of unconstrained problems their concatenation is a program too. In the case of constrained problems the situation is more difficult and less understood. Nevertheless, if xm2 and xm2 +1 are close to the turnpike, then the concatenation is still a program because of the local controllability property. It turns out that this fact is sufficient for the analysis of the turnpike phenomenon. But it does not hold for the RSS model. However, it was shown in [138] that the turnpike theory presented in [131] is extended for the Robinson–Solow–Srinivasan model. This is possible because the model possesses some interesting and important features. Namely, the RSS model has the so-called monotonicity property and some weak version of the local controllability property. The RSS model possesses the following monotonicity property: n For every admissible (x, y) and every x˜ ∈ R+ which satisfies x˜ ≥ x there exists y˜ ≥ y such that the pair (x, ˜ y) ˜ is admissible and the value of the cost function at (x, y) does not exceed the value of the cost function at the point (x, ˜ y). ˜ The RSS model also has the following weak controllability property in a neighborhood of the turnpike x: ˜ For every pair of points x, y which are close to the turnpike there exists a point y ≥ y which is also close to the turnpike such that the pair (x, y ) is admissible. The combination of these two properties allows us to construct an analog of the m3 2 concatenation of the programs {xt }m t =m1 and {xt }t =m2 +1 when xm2 and xm2 +1 are m3 close to the turnpike. Namely, they guarantee the existence of the program {zt }t =m 1 such that zt = xt , t = m1 , . . . , m2 , zt ≥ xt , t = m2 + 1, . . . , m3 . This is very useful in our analysis of the turnpike phenomenon for the RSS model. The results presented in this section were obtained in [41]. Proposition 1.10 For every program {x(t), y(t)}∞ t =0 there exists a constant m(x(0)) > 0, depending only on x(0), such that x(t) ≤ m(x(0))e for all nonnegative integers t. This result shows that our study of the model can be actually reduced to the case when all its states belong to a bounded set. Let w : [0, ∞) → R 1 be a continuous strictly increasing concave and differentiable function which represents the preferences of the planner. We use the following optimality criterion.

1.7 The Robinson–Solow–Srinivasan Model

21

A program {x ∗ (t), y ∗ (t)}∞ t =0 is weakly optimal if lim inf T →∞

T  [w(by(t)) − w(by ∗ (t))] ≤ 0 t =0

∗ for every program {x(t), y(t)}∞ t =0 satisfying x(0) = x (0). Set n n Ω = {(x, x ) ∈ R+ × R+ : x − (1 − d)x ≥ 0 and a(x − (1 − d)x) ≤ 1}. n We have a correspondence Λ : Ω → R+ given by n : 0 ≤ y ≤ x and ey ≤ 1 − a(x − (1 − d)x)}, (x, x ) ∈ Ω. Λ(x, x ) = {y ∈ R+

For any (x, x ) ∈ Ω define u(x, x ) = max{w(by) : y ∈ Λ(x, x )}. n such that ( x, x ) is a solution to the problem: A golden-rule stock is x ∈ R+

maximize u(x, x ) subject to (i) x ≥ x; (ii) (x, x ) ∈ Ω. For i = 1, . . . , n set

qi = ai bi /(1 + dai ), p i = w (bσ (1 + daσ )−1 ) qi , y = (1 + daσ )−1 e(σ ). The following useful lemma plays an important role in the study of the RSS model. Lemma 1.11 w(b y ) ≥ w(by) + p x − p x for any (x, x ) ∈ Ω and for any y ∈ Λ(x, x ). Theorem 1.12 There exists a unique golden-rule stock x = (1 + daσ )−1 e(σ ). We use the following notion of good programs introduced by Gale [28] and used in optimal control [18, 105, 131]. 1 A program {x(t), y(t)}∞ t =0 is called good if there exists M ∈ R such that T 

(w(by(t)) − w(b y )) ≥ M for all integers T ≥ 0.

t =0

A program is called bad if lim

T →∞

T  (w(by(t)) − w(b y )) = −∞. t =0

22

1 Introduction

n . Then there exists a good program Proposition 1.13 Let x0 ∈ R+

{x(t), y(t)}∞ t =0 which satisfies x(0) = x0 . Proposition 1.14 Let {x(t), y(t)}∞ t =0 be a program. Then there exists a constant M(x(0)) ≥ 0 such that for every pair of nonnegative integers t1 ≤ t2 , t2 

(w(by(t)) − w(b y )) ≤ M(x(0)).

t =t1

Proposition 1.14 easily implies the following result. Proposition 1.15 Every program which is not good is bad. For any (x, x ) ∈ Ω and any y ∈ Λ(x, x ) set δ(x, y, x ) = p (x − x ) − (w(by) − w(b y )).

(1.3)

We say that a program {x(t), y(t)}∞ t =0 has the average turnpike property if lim T −1

T →∞

T −1

(x(t), y(t)) = ( x, y ).

t =0

Proposition 1.16 Assume that a program {x(t), y(t)}∞ t =0 is good. Then it has the average turnpike property. The next result easily follows from Lemma 1.11 and (1.3). Proposition 1.17 Assume that {x(t), y(t)}∞ t =0 is a program. Then for every integer t ≥ 0, δ(x(t), y(t), x(t + 1)) ≥ 0 and for every natural number T , T 

(w(by(t)) − w(b y ))

t =0

=p (x(0) − x(T + 1)) −

T 

δ(x(t), y(t), x(t + 1)).

t =0

Proposition 1.17 implies the following result.

1.7 The Robinson–Solow–Srinivasan Model

23

Proposition 1.18 A program {x(t), y(t)}∞ t =0 is good if and only if ∞ 

δ(x(t), y(t), x(t + 1)) < ∞.

t =0 n For every x0 ∈ R+ define

Δ(x0 ) = inf

∞ 

δ(x(t), y(t), x(t + 1)) :

t =0

 . {x(t), y(t)}∞ is a program such that x(0) = x 0 t =0 n Proposition 1.19 Let x0 ∈ R+ . Then

0 ≤ Δ(x0) < ∞ and there exists a program {x (t), y (t)}∞ t =0 such that x (0) = x0 , Δ(x0 ) =

∞ 

δ(x (t), y (t), x (t + 1)).

t =0

Proposition 1.20 Assume that a program {x(t), y(t)}∞ t =0 satisfies Δ(x(0)) =

∞ 

δ(x(t), y(t), x(t + 1)).

t =0

Then it is weakly optimal. n . Then there exists a weakly optimal program Theorem 1.21 Let x0 ∈ R+ ∞ {x(t), y(t)}t =0 satisfying x(0) = x0 . If x0 = x , then the program {x(t), y(t)}∞ t =0 satisfying

x(t) = y(t) = x , t = 0, 1, . . . is weakly optimal. The following auxiliary result plays an important role in the study of the RSS model. Lemma 1.22 Let ξσ = 1 − d − aσ−1 .

24

1 Introduction

The von Neumann facet {(x, x ) ∈ Ω : there exists y ∈ Λ(x, x ) such that δ(x, y, x ) = 0} is a subset of the set {(x, x ) ∈ Ω : xi = xi = 0, i ∈ {1, . . . , n} \ {σ }, xσ = aσ−1 + ξσ xσ } with the equality if the function w is linear. If the function w is strictly concave, then the face is the singleton {( x, x )}. It should be mentioned that any weakly optimal program is good.

1.8 Overtaking Optimal Programs for the RSS Model In this section we continue to use the assumptions introduced in Sect. 1.7. The following three theorems were obtained in [103]. Theorem 1.23 Assume that the function w is strictly concave. Then for every good program {x(t), y(t)}∞ t =0 , x, x ). lim (x(t), y(t)) = (

t →∞

Set ξσ = 1 − d − (1/aσ ). Theorem 1.24 Assume that ξσ = −1. Then x, x) lim (x(t), y(t)) = (

t →∞

for every good program {x(t), y(t)}∞ t =0 . A program {x ∗ (t), y ∗ (t)}∞ t =0 is overtaking optimal if lim sup

T 

T →∞ t =0

[w(by(t)) − w(by ∗ (t))] ≤ 0

∗ for every program {x(t), y(t)}∞ t =0 which satisfies x(0) = x (0).

Theorem 1.25 Assume that for every good program {x(t), y(t)}∞ t =0 , lim (x(t), y(t)) = ( x, x ).

t →∞

1.8 Overtaking Optimal Programs for the RSS Model

25

n there is an overtaking optimal program {x(t), y(t)}∞ Then for every point x0 ∈ R+ t =0 such that x(0) = x0 .

Corollary 1.26 Assume that the function w is strictly concave. Then for every point n x0 ∈ R+ there exists an overtaking optimal program {x(t), y(t)}∞ t =0 satisfying x(0) = x0 . n there is an Corollary 1.27 Assume that ξσ = −1. Then for every point x0 ∈ R+ ∞ overtaking optimal program {x(t), y(t)}t =0 such that x(0) = x0 .

The following three theorems were obtained in [53]. Theorem 1.28 Assume that for each good program {u(t), v(t)}∞ t =0 , lim (u(t), v(t)) = ( x, x ).

t →∞

Then for each program {x(t), y(t)}∞ t =0 the following conditions are equivalent: ∞ (i) t =0 δ(x(t), y(t), x(t + 1)) = Δ(x(0)). (ii) {x(t), y(t)}∞ t =0 is overtaking optimal. (iii) {x(t), y(t)}∞ t =0 is weakly optimal. Theorem 1.29 Assume that at least one of the following conditions holds: (a) w is strictly concave.  −1. (b) ξσ = Let M0 ,  > 0. Then there exists a natural number T0 such that for each overtaking optimal program {x(t), y(t)}∞ t =0 satisfying x(0) ≤ M0 e and each integer t ≥ T0 , x(t) − x , y(t) − x  ≤ . Theorem 1.30 Assume that at least one of the following conditions holds: (a) w is strictly concave. (b) ξσ =  −1. Let  > 0. Then there is δ > 0 such that for each overtaking optimal program {x(t), y(t)}∞ x  ≤ δ the following inequality holds: t =0 satisfying x(0) − x(t) − x , y(t) − x ≤  for all integers t ≥ 0. In [110] we studied the structure of good programs of the RSS model and proved the following three results.

26

1 Introduction

Theorem 1.31 Let a program {x(t), y(t)}∞ t =0 be good. Then for each i {1, . . . , n} \ {σ }, ∞ 

∈

xi (t) < ∞,

t =0 ∞ 

(xσ (t) − yσ (t)) < ∞

t =0

and the sequence {

T −1 t =0

xσ (t) − T (1 + daσ )−1 }∞ T =1 is bounded.

Theorem 1.32 Let the function w be linear. Then a program {x(t), y(t)}∞ t =0 is good if and only if for each i ∈ {1, . . . , n} \ {σ }, ∞ 

xi (t) < ∞,

t =0 ∞ 

(xσ (t) − yσ (t)) < ∞

t =0

and the sequence {

T −1 t =0

xσ (t) − T (1 + daσ )−1 }∞ T =1 is bounded.

Theorem 1.33 Let w ∈ C 2 , w

(b x ) = 0 and let for every good program {u(t), v(t)}∞ , t =0 lim (u(t), v(t)) = ( x, x ).

t →∞

Then a program {x(t), y(t)}∞ t =0 is good if and only if for each i ∈ {1, . . . , n} \ {σ }, ∞ 

xi (t) < ∞,

t =0 ∞ 

(xσ (t) − yσ (t)) < ∞,

t =0 ∞ 

(yσ (t) − xσ )2 < ∞,

t =0

and the sequence {

T −1 t =0

xσ (t) − T (1 + daσ )−1 }∞ T =1 is bounded.

1.9 Turnpike Properties of the RSS Model

27

1.9 Turnpike Properties of the RSS Model In this section we discuss the turnpike properties for the Robinson–Solow– Srinivasan model. To have these properties means that the approximate solutions of the problems are essentially independent of the choice of an interval and endpoint conditions. We show that these turnpike properties hold and that they are stable under perturbations of an objective function. The turnpike for the RSS model is the golden-rule stock x . Clearly, ( x, x ) is not an interior point of the set Ω. Therefore assumption (A1) from Sect. 1.2 does not hold for the RSS model. We continue to use the assumptions introduced in Sect. 1.7. n Let z ∈ R+ and T ≥ 1 be a natural number. Set U (z, T ) = sup

−1  T

w(by(t)) :

t =0

 −1 ({x(t)}Tt=0, {y(t)}Tt =0 ) is a program such that x(0) = z . n , T1 , T2 be integers, 0 ≤ T1 < T2 . Clearly U (z, T ) is a finite number. Let x0 , x1 ∈ R+ Define

U (x0 , x1 , T1 , T2 ) = sup

2 −1  T

t =T1

2 2 −1 w(by(t)) : ({x(t)}Tt =T , {y(t)}Tt =T ) 1 1

 is a program such that x(T1 ) = x0 , x(T2 ) ≥ x1 . (Here we suppose that a supremum over an empty set is −∞.) Clearly U (x0 , x1 , T1 , T2 ) < ∞. n and any integer T ≥ 1, U (z, T ) = U (z, 0, 0, T ). It is also clear that for any z ∈ R+ In this section we assume that the following asymptotic turnpike property holds: (ATP) Each good program {x(t), y(t)}∞ t =0 converges to the golden-rule stock ( x, x ):

x, x ). lim (x(t), y(t)) = (

t →∞

With Card(A) we denote in the sequel the cardinality of a finite set A. The following two turnpike results were obtained in [55]. Theorem 1.34 Let M,  be positive numbers and Γ ∈ (0, 1). Then there exists a n satisfying natural number L such that for each integer T > L, each z0 , z1 ∈ R+ T −1 T −1 z0 ≤ Me and az1 ≤ Γ d and each program ({x(t)}t =0, {y(t)}t =0 ) which satisfies x(0) = z0 , x(T ) ≥ z1 ,

T −1 t =0

w(by(t)) ≥ U (z0 , z1 , 0, T ) − M,

28

1 Introduction

the following inequality holds: Card{i ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > } ≤ L. Theorem 1.35 Let M,  be positive numbers and Γ ∈ (0, 1). Then there exist a natural number L and a positive number γ such that for each integer T > 2L, n satisfying z ≤ Me and az ≤ Γ d −1 and each program each z0 , z1 ∈ R+ 0 1 T −1 T ({x(t)}t =0, {y(t)}t =0 ) which satisfies x(0) = z0 , x(T ) ≥ z1 ,

T −1

w(by(t)) ≥ U (z0 , z1 , 0, T ) − γ

t =0

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. Moreover, if x(0) − x  ≤ γ then τ1 = 0 and if x(T ) − x  ≤ γ then τ2 = T . In [121] we continued to study the turnpike phenomenon for the RSS model and proved the following three turnpike results which are extensions of Theorem 1.35. Theorem 1.36 Suppose that for each good program {u(t), v(t)}∞ t =0 , lim (u(t), v(t)) = ( x, x ).

t →∞

Let M,  be positive numbers and Γ ∈ (0, 1). Then there exist a natural number n L and a positive number γ such that for each integer T > 2L, each z0 , z1 ∈ R+ T −1 −1 T satisfying z0 ≤ Me and az1 ≤ Γ d and each program ({x(t)}t =0 , {y(t)}t =0 ) which satisfies x(0) = z0 , x(T ) ≥ z1 , τ +L−1 

w(by(t)) ≥ U (x(τ ), x(τ + L), 0, L) − γ for all τ ∈ {0, . . . , T − L}

t =τ

and T −1 t =T −L

w(by(t)) ≥ U (x(T − L), z1 , 0, L) − γ

1.9 Turnpike Properties of the RSS Model

29

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. Moreover, if x(0) − x  ≤ γ then τ1 = 0 and if x(T ) − x  ≤ γ then τ2 = T . Theorem 1.37 Suppose that for each good program {u(t), v(t)}∞ t =0 , x, x ). lim (u(t), v(t)) = (

t →∞

Let M,  be positive numbers. Then there exist a natural number L and a positive n satisfying z0 ≤ Me number γ such that for each integer T > 2L, each z0 ∈ R+ T −1 T and each program ({x(t)}t =0, {y(t)}t =0 ) which satisfies x(0) = z0 , τ +L−1 

w(by(t)) ≥ U (x(τ ), x(τ + L), 0, L) − γ for all τ ∈ {0, . . . , T − L}

t =τ

and T −1

w(by(t)) ≥ U (x(T − L), L) − γ

t =T −L

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ then τ2 = T . Moreover, if x(0) − x  ≤ γ then τ1 = 0 and if x(T ) − Theorem 1.38 Suppose that for each good program {u(t), v(t)}∞ t =0 , lim (u(t), v(t)) = ( x, x ).

t →∞

Let M,  be positive numbers. Then there exist a natural number L and a positive n satisfying z ≤ Me number γ such that for each integer T > 2L, each z0 ∈ R+ 0 T −1 T and each program ({x(t)}t =0, {y(t)}t =0 ) which satisfies x(0) = z0 , T −1 t =0

w(by(t)) ≥ U (z0 , T ) − γ

30

1 Introduction

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. Moreover, if x(0) − x  ≤ γ then τ1 = 0 and x(T ) − x  ≤ γ then τ2 = T . n → R 1 define For every positive number M and every function φ : R+

φM = sup{|φ(z)| : z ∈ R n and 0 ≤ z ≤ Me}. n → R 1 , i = T1 , . . . , T2 − 1 be Let integers T1 , T2 satisfy 0 ≤ T1 < T2 , wi : R+ n n bounded on bounded subsets of R+ functions. For every pair of points z0 , z1 ∈ R+ define T2 −1 U ({wt }t=T , z0 , z1 ) = sup 1

2 −1  T

wt (y(t)) :

t=T1 2 2 −1 , {y(t)}Tt=T ) ({x(t)}Tt=T 1 1

T −1

2 , z0 ) = sup U ({wt }t=T 1

2 −1  T

is a program such that x(T1 ) = z0 ,

 x(T2 ) ≥ z1 ,

wt (y(t)) :

t=T1 T −1

T

2 2 , {y(t)}t=T ) ({x(t)}t=T 1 1

is a program such that x(T1 ) = z0

 .

(Here we assume that the supremum over an empty set is −∞.) It is not difficult to see that the following result holds. n → R1 , Lemma 1.39 Let integers T1 , T2 satisfy 0 ≤ T1 < T2 and wi : R+ n i = T1 , . . . , T2 − 1 be bounded on bounded subsets of R+ upper semicontinuous functions. Then the following assertions hold. n 2 2 −1 there exists a program ({x(t)}Tt =T , {y(t)}Tt =T ) such 1. For every point z0 ∈ R+ 1 1 that

x(T1 ) = z0 ,

T 2 −1 t =T1

2 −1 wt (y(t)) = U ({wt }Tt =T , z0 ). 1

n 2 −1 2. For every pair of points z0 , z1 ∈ R+ such that U ({wt }Tt =T , z0 , z1 ) is finite there 1 2 2 −1 exists a program ({x(t)}Tt =T , {y(t)}Tt =T ) such that x(0) = z0 , x(T2 ) ≥ z1 and 1 1

T 2 −1 t =T1

2 −1 wt (y(t)) = U ({wt }Tt =T , z0 , z1 ). 1

The following stability results were obtained in [121]. They show that the turnpike phenomenon is stable under small perturbations of the utility functions.

1.9 Turnpike Properties of the RSS Model

31

Theorem 1.40 Suppose that for each good program {u(t), v(t)}∞ t =0 , lim (u(t), v(t)) = ( x, x ).

t →∞

Let M > max{(ai d)−1 : i = 1, . . . , n},  > 0 and Γ ∈ (0, 1). Then there exist a natural number L and a positive number γ˜ such that for each integer T > 2L, each n satisfying z ≤ Me and az ≤ Γ d −1 , each finite sequence of functions z0 , z1 ∈ R+ 0 1 n n wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ and such that wi − w(b(·))M ≤ γ˜ −1 ) such for every integer i ∈ {0, . . . , T − 1} and every program ({x(t)}Tt=0, {y(t)}Tt =0 that

x(0) = z0 , x(T ) ≥ z1 , τ +L−1  t =τ

+L−1 wt (y(t)) ≥ U ({wt }τt =τ , x(τ ), x(τ + L)) − γ˜

for every τ ∈ {0, . . . , T − L} and T −1 t =T −L

−1 wt (y(t)) ≥ U ({wt }Tt =T −L , x(T − L), z1 ) − γ˜ ,

there exist integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. x  ≤ γ˜ , then τ2 = T . Moreover, if |x(0) − x  ≤ γ˜ , then τ1 = 0 and if x(T ) − Theorem 1.41 Suppose that for each good program {u(t), v(t)}∞ t =0 , lim (u(t), v(t)) = ( x, x ).

t →∞

Let M > max{(ai d)−1 : i = 1, . . . , n} and  > 0. Then there exist a natural number L and a positive number γ˜ such that for each integer T > 2L, each z0 ∈ n satisfying z ≤ Me, each finite sequence of functions w : R n → R 1 , i = R+ 0 i + n and such that 0, . . . , T − 1 which are bounded on bounded subsets of R+ wi − w(b(·))M ≤ γ˜

32

1 Introduction

−1 for each i ∈ {0, . . . , T −1} and each program ({x(t)}Tt=0 , {y(t)}Tt =0 ) which satisfies

x(0) = z0 , τ +L−1  t =τ

+L−1 wt (y(t)) ≥ U ({wt }τt =τ , x(τ ), x(τ + L)) − γ˜ ,

for each integer τ ∈ {0, . . . , T − L} and T −1 t =T −L

−1 wt (y(t)) ≥ U ({wt }Tt =T −L , x(T − L)) − γ˜

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. Moreover, if x(0) − x  ≤ γ then τ1 = 0 and if x(T ) − x  ≤ γ then τ2 = T . Theorem 1.42 Suppose that for each good program {u(t), v(t)}∞ t =0 , lim (u(t), v(t)) = ( x, x ).

t →∞

Let M > max{(ai d)−1 : i = 1, . . . , n},  > 0 and Γ ∈ (0, 1). Then there exist a natural number L, a positive number γ and λ > 1 such that for each integer T > n satisfying z ≤ Me and az ≤ Γ d −1 , each finite sequence of 2L, each z0 , z1 ∈ R+ 0 1 n functions wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on bounded subsets n and such that w − w(b(·)) of R+ i M ≤ γ for each i ∈ {0, . . . , T − 1}, each −1 sequence {αi }Ti=0 ⊂ (0, 1] such that for each i, j ∈ {0, . . . , T − 1} satisfying |j − −1 i| ≤ L the inequality αi αj−1 ≤ λ holds and each program ({x(t)}Tt=0 , {y(t)}Tt =0 ) such that x(0) = z0 , x(T ) ≥ z1 , τ +L−1  t =τ

+L−1 αt wt (y(t)) ≥ U ({αt wt }τt =τ , x(τ ), x(τ + L)) − γ ατ

for each integer τ ∈ {0, . . . , T − L} and T −1 t =T −L

−1 αt wt (y(t)) ≥ U ({αt wt }Tt =T −L , z1 ) − γ αT −L

1.9 Turnpike Properties of the RSS Model

33

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. Moreover, if x(0) − x  ≤ γ then τ1 = 0 and if x(T ) − x  ≤ γ then τ2 = T . Theorem 1.43 Suppose that for each good program {u(t), v(t)}∞ t =0 , lim (u(t), v(t)) = ( x, x ).

t →∞

Let M > max{(ai d)−1 : i = 1, . . . , n} and  > 0. Then there exist a natural number L, a positive number γ and λ > 1 such that for each integer T > 2L, each n satisfying z ≤ Me, each finite sequence of functions w : R n → R 1 , z0 ∈ R+ 0 i + n and such that i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ wi − w(b(·))M ≤ γ −1 ⊂ (0, 1] such that for each for each i ∈ {0, . . . , T − 1}, each sequence {αi }Ti=0

i, j ∈ {0, . . . , T − 1} satisfying |j − i| ≤ L the inequality αi αj−1 ≤ λ holds and each program

−1 ) such that ({x(t)}Tt=0, {y(t)}Tt =0

x(0) = z0 , τ +L−1  t =τ

+L−1 αt wt (y(t)) ≥ U ({αt wt }τt =τ , x(τ ), x(τ + L)) − γ ατ

for each integer τ ∈ {0, . . . , T − L} and T −1 t =T −L

−1 αt wt (y(t)) ≥ U ({αt wt }Tt =T −L , x(T − L)) − γ αT −L

there are integers τ1 , τ2 such that τ1 ∈ [0, L], τ2 ∈ [T − L, T ], x(t) − x , y(t) − x  ≤  for all t = τ1 , . . . , τ2 − 1. Moreover, if x(0) − x  ≤ γ then τ1 = 0 and if x(T ) − x  ≤ γ then τ2 = T . The following results were obtained in [126].

34

1 Introduction

Theorem 1.44 Suppose that for each good program {u(t), v(t)}∞ t =0 , lim (u(t), v(t)) = ( x, x ).

t →∞

Let M > max{(ai d)−1 : i = 1, . . . , n}, M0 > 0,  > 0 and Γ ∈ (0, 1). Then there exist a natural number L and a positive number γ˜ such that for each n satisfying z ≤ Me and az ≤ Γ d −1 , each finite integer T > L, each z0 , z1 ∈ R+ 0 1 n sequence of functions wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on n and such that bounded subsets of R+ wi − w(b(·))M ≤ γ˜ −1 ) such that for each i ∈ {0, . . . , T − 1} and each program ({x(t)}Tt=0, {y(t)}Tt =0

x(0) = z0 , x(T ) ≥ z1 , T −1 t =0

−1 wt (y(t)) ≥ U ({wt }Tt =0 , z0 , z1 ) − M0

the following inequality holds: Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ L. Theorem 1.45 Suppose that for each good program {u(t), v(t)}∞ t =0 , lim (u(t), v(t)) = ( x, x ).

t →∞

Let M > max{(ai d)−1 : i = 1, . . . , n}, M0 > 0 and  > 0. Then there exist a natural number L and a positive number γ˜ such that for each integer T > L, each n n z0 ∈ R+ satisfying z0 ≤ Me, each finite sequence of functions wi : R+ → R1 , n i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ and such that wi − w(b(·))M ≤ γ˜ −1 for each i ∈ {0, . . . , T −1} and each program ({x(t)}Tt=0 , {y(t)}Tt =0 ) which satisfies

x(0) = z0 , T −1 t =0

−1 wt (y(t)) ≥ U ({wt }Tt =0 , z0 ) − M0

1.10 Autonomous Optimal Control Problems

35

the following inequality holds: Card({t ∈ {0, . . . , T − 1} : max{x(t) − x , y(t) − x } > }) ≤ L.

1.10 Autonomous Optimal Control Problems In this chapter we discussed turnpike properties and optimality criteria over an infinite horizon for two classes of dynamic optimization problems. Problems of the first class determine a general model of economic growth, while problems from the second class are related to the Robinson–Solow–Srinivasan model. The turnpike for the RSS model is the golden-rule stock x and ( x, x ) is not an interior point of the set of all admissible pairs Ω. Therefore assumption (A1) from Sect. 1.2 does not hold for the RSS model and it cannot be treated as a particular case of the general model. Nevertheless, the turnpike theory was developed for the RSS model. This was possible because the model possesses some interesting and important features. n define More precisely, for every x ∈ R+ n a(x) = {x ∈ R+ : (x, x ) ∈ Ω}. n we have It is not difficult to see that for every pair of points x, y ∈ R+

H (a(x), a(y)) ≤ (1 − d)x − y, where H (·, ·) is the Hausdorff metric. In other words, the set-valued mapping a is a strict contraction. The RSS model possesses the following monotonicity property: n For every (x, y) ∈ Ω and every x˜ ∈ R+ which satisfies x˜ ≥ x there exists y˜ ∈ a(x) ˜ such that y˜ ≥ y, u(x, ˜ y) ˜ ≥ u(x, y). The RSS model also has the following weak controllability property in a neighborhood of x: ˜ There exists a positive number r¯ such that for every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ there exists a point y ∈ X satisfying the inequality

y ≥ y and the inclusion (x, y ) ∈ Ω. Moreover, for every positive number  there exists a positive number δ such that for every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ δ, there exists a point y ∈ X for which y ≥ y, (x, y ) ∈ Ω and y − x ¯ ≤ . These properties play a crucial role in the analysis of the turnpike phenomenon for the RSS model. This understanding led us to the idea to develop a turnpike theory

36

1 Introduction

for large classes of optimal control problems possessing the weak controllability property and the monotonicity property described above, which includes the RSS model as a particular case. This theory is developed in Chaps. 7–9 of this book. Turnpike properties for this class of problems are proved in Chap. 7. In Chap. 8 we study the stability of the turnpike phenomenon under small perturbations of objective functions. In Chap. 9 we analyze the stability of the turnpike phenomenon under small perturbations of objective functions and constraint maps. The results of Chaps. 8 and 9 are new. In Chap. 10 we analyze models with unbounded endogenous economic growth.

Chapter 2

Turnpike Conditions for Optimal Control Systems

We study necessary and sufficient conditions for turnpike properties of approximate solutions of nonautonomous discrete-time optimal control systems arising in economic dynamics which are determined by sequences of lower semicontinuous objective functions. To have these properties means that the approximate solutions of the problems are determined mainly by the objective functions, and are essentially independent of the choice of intervals and endpoint conditions, except in regions close to the endpoints.

2.1 Preliminaries For each nonempty set Y denote by B(Y ) the set of all bounded functions f : Y → R 1 and for each f ∈ B(Y ) set f  = sup{|f (y)| : y ∈ Y }. For each nonempty compact metric space Y denote by C(Y ) the set of all continuous functions f : Y → R 1 . Let (X, ρ) be a compact metric space with the metric ρ. The set X × X is equipped with the metric ρ1 defined by ρ1 ((x1 , x2 ), (y1 , y2 )) = ρ(x1 , y1 ) + ρ(x2 , y2 ), (x1 , x2 ), (y1 , y2 ) ∈ X × X. For each integer t ≥ 0 let Ωt be a nonempty closed subset of the metric space X × X.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_2

37

38

2 Turnpike Conditions for Optimal Control Systems

Let T ≥ 0 be an integer. A sequence {xt }∞ t =T ⊂ X is called a program if (xt , xt +1 ) ∈ Ωt for all integers t ≥ T . 2 Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A sequence {xt }Tt =T ⊂ X is called 1 a program if (xt , xt +1 ) ∈ Ωt for all integers t satisfying T1 ≤ t < T2 . We assume that there exists a program {xt }∞ t =0 . Denote by M the set of all sequences of functions {ft }∞ such that for each integer t ≥ 0 t =0 ft ∈ B(Ωt )

(2.1)

sup{ft  : t = 0, 1, . . . } < ∞.

(2.2)

and that

∞ For each pair of sequences {ft }∞ t =0 , {gt }t =0 ∈ M set ∞ d({ft }∞ t =0 , {gt }t =0 ) = sup{ft − gt  : t = 0, 1, . . . }.

(2.3)

It is easy to see that d : M × M → [0, ∞) is a metric on M and that the metric space (M, d) is complete. Let {ft }∞ t =0 ∈ M. We consider the following optimization problems: T 2 −1 t =T1 T 2 −1 t =T1 T 2 −1 t =T1

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program, 1

(PT1 ,T2 )

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program and xT1 = y, 1

(PT1 ,T2 )

(y)

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program and xT1 = y, xT2 = z, 1

(y,z)

(PT1 ,T2 ) where y, z ∈ X and integers T1 , T2 satisfy 0 ≤ T1 < T2 . For each y, z ∈ X and each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 set U ({ft }∞ t =0 , T1 , T2 ) = inf

2 −1  T

t =T1

 2 ft (xt , xt +1 ) : {xt }Tt =T is a program , 1 (2.4)

U ({ft }∞ t =0 , T1 , T2 , y)

= inf

2 −1  T

ft (xt , xt +1 ) :

t =T1

 2 is a program and x = y , {xt }Tt =T T 1 1

(2.5)

2.1 Preliminaries

39

U ({ft }∞ t =0 , T1 , T2 , y, z) = inf

2 −1  T

ft (xt , xt +1 ) :

t =T1

 2 {xt }Tt =T is a program and x = y, x = z . T T 1 2 1 (2.6) Here we assume that the infimum over an empty set is ∞. In this chapter we study (y) (y,z) the structure of approximate solutions of problems (PT1 ,T2 ), PT1 ,T2 and PT1 ,T2 which are defined as follows. Let M ≥ 0, y, z ∈ X and let integers T1 , T2 satisfy 0 ≤ T1 < T2 . A program (y,z) 2 is called an (M)-approximate solution of problem PT1 ,T2 if {xt }Tt =T 1 xT1 = y, xT2 = z and

T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 , y, z) + M. (y)

It is called an (M)-approximate solution of problem PT1 ,T2 if xT1 = y and

T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 , y) + M.

2 The program {xt }Tt =T is called an (M)-approximate solution of problem PT1 ,T2 if 1

T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 ) + M.

A program {xt }∞ t =0 is called an (M)-approximate solution of the corresponding infinite horizon problem if for each pair of integers T1 ≥ 0, T2 > T1 , T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 ) + M.

Denote by Mreg the set of all sequences of functions {fi }∞ i=0 ∈ M for which f ∞ there exist a program {xt }t =0 and constants cf > 0, γf > 0 such that the following conditions hold: (C1) the function ft is lower semicontinuous for all integers t ≥ 0; (C2) for each pair of integers T1 ≥ 0, T2 > T1 , T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 ) + cf ; f

f

40

2 Turnpike Conditions for Optimal Control Systems

(C3) for each  > 0 there exists δ > 0 such that for each integer t ≥ 0 and each f f (x, y) ∈ Ωt satisfying ρ(x, xt ) ≤ δ, ρ(y, xt +1) ≤ δ we have f

f

|ft (xt , xt +1 ) − ft (x, y)| ≤ ; f

(C4) for each integer t ≥ 0, each (xt , xt +1 ) ∈ Ωt satisfying ρ(xt , xt ) ≤ γf and f each (xt +1 , xt +2 ) ∈ Ωt +1 satisfying ρ(xt +2, xt +2 ) ≤ γf there is x ∈ X such that (xt , x) ∈ Ωt , (x, xt +2 ) ∈ Ωt +1 ; moreover, for each  > 0 there exists δ ∈ (0, γf ) such that for each integer t ≥ f 0, each (xt , xt +1 ) ∈ Ωt and each (xt +1, xt +2 ) ∈ Ωt +1 satisfying ρ(xt , xt ) ≤ δ and ρ(xt +2 , xt +2 ) ≤ δ there is x ∈ X such that f

(xt , x) ∈ Ωt , (x, xt +2 ) ∈ Ωt +1 , ρ(x, xt +1) ≤ . f

¯ reg the closure of Mreg in (M, d). Denote by Mc,reg the set of Denote by M all sequences {fi }∞ i=0 ∈ Mreg such that fi ∈ C(Ωi ) for all integers i ≥ 0 and by ¯ c,reg the closure of Mc,reg in (M, d). M We study the optimization problems stated above with the sequence of objective functions {fi }∞ i=0 ∈ Mreg . Our study is based on the relation between these finite horizon problems and the corresponding infinite horizon optimization problem f ∞ determined by {fi }∞ i=0 . Note that the condition (C2) means that the program {xt }t =0 is an approximate solution of this infinite horizon problem. We are interested in turnpike properties of approximate solutions of our optimization problems, which are independent of the length of the interval T2 − T1 , for all sufficiently large intervals. To have these properties means that the approximate solutions of the problems are determined mainly by the objective functions, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. In this chapter we establish the turnpike property for the optimal control system is determined by a nonstationary sequence of objective functions {ft }∞ t =0 and by a nonstationary sequence of sets of admissible pairs {Ωt }∞ t =0 . The results of this chapter were obtained in [130].

2.2 Main Results ∞ Let {fi }∞ i=0 ∈ Mreg , a program {xi }i=0 , cf > 0 and γf > 0 be such that (C1)–(C4) hold. We begin with the following useful result. f

2.2 Main Results

41

Proposition 2.1 Let S ≥ 0 be an integer and {xi }∞ i=S be a program. Then either the T −1 T −1 f f sequence { i=S fi (xi , xi+1 ) − i=S fi (xi , xi+1 )}∞ T =S+1 is bounded or −1

T

lim

T →∞

fi (xi , xi+1 ) −

i=S

T −1

 f f fi (xi , xi+1 ) = ∞.

(2.7)

i=S

Proof By (C2), for each natural number T > S, T −1

fi (xi , xi+1 ) −

i=S

T −1

f

f

fi (xi , xi+1 )

i=S

≥ U ({fi }∞ i=0 , S, T ) −

T −1

f

f

fi (xi , xi+1 ) ≥ −cf .

i=S

Then in order to prove the proposition it is sufficient to show that if the sequence −1  T

fi (xi , xi+1 ) −

i=S

T −1

f

f

fi (xi , xi+1 )

i=S

∞

(2.8)

T =S+1

is not bounded from above, then (2.7) holds. Assume that the sequence (2.8) is not bounded from above. Let Q be any positive number. Then there is an integer T0 > S such that T 0 −1

f

f

[fi (xi , xi+1 ) − fi (xi , xi+1 )] > Q.

(2.9)

i=S

It follows from (2.9) and (C2) that for each integer T > T0 , T −1

fi (xi , xi+1 ) −

i=S

=

T −1

f

f

fi (xi , xi+1 )

i=S

T 0 −1 i=S

fi (xi , xi+1 ) −

T 0 −1

f

f

fi (xi , xi+1 ) +

i=S

> Q + U ({fi }∞ i=0 , T0 , T ) −

T −1

fi (xi , xi+1 ) −

i=T0 T −1

f

T −1

f

f

fi (xi , xi+1 )

i=T0

f

fi (xi , xi+1 ) ≥ Q − cf .

i=T0

Since Q is any positive number we conclude that (2.7) holds. Proposition 2.1 is proved.

42

2 Turnpike Conditions for Optimal Control Systems

∞ A program {xt }∞ t =S , where S ≥ 0 is an integer, is called ({fi }i=0 )-good if the sequence −1  T

fi (xi , xi+1 ) −

i=S

T −1

f

f

fi (xi , xi+1 )

i=S

∞ T =S+1

is bounded. We say that the sequence {fi }∞ i=0 possesses an asymptotic turnpike property (or f being the turnpike if for each integer S ≥ 0 and each briefly ATP) with {xi }∞ i=0 ∞ , ({fi }∞ )-good program {x } i i=S i=0 f

lim ρ(xi , xi ) = 0.

i→∞

We say that the sequence {fi }∞ i=0 possesses a turnpike property (or briefly TP) if for each  > 0 and each M > 0 there exist δ > 0 and a natural number L such that 2 for each pair of integers T1 ≥ 0, T2 ≥ T1 + 2L and each program {xt }Tt =T which 1 satisfies T 2 −1

fi (xi , xi+1 ) ≤ min{U ({fi }∞ i=0 , T1 , T2 , xT1 , xT2 ) + δ,

i=T1

U ({fi }∞ i=0 , T1 , T2 ) + M}, f

the inequality ρ(xi , xi ) ≤  holds for all integers i = T1 + L, . . . , T2 − L. f ∞ The sequence {xi }∞ i=0 is called the turnpike of {fi }i=0 . In this chapter we prove the following results. Theorem 2.2 The sequence {fi }∞ i=0 possesses the turnpike property if and only if {fi }∞ possesses ATP and the following property: i=0 (P) For each  > 0 and each M > 0 there exist δ > 0 and a natural number L +L such that for each integer T ≥ 0 and each program {xt }Tt =T which satisfies T +L−1 

fi (xi , xi+1 )

i=T ∞ ≤ min{U ({fi }∞ i=0 , T , T + L, xT , xT +L ) + δ, U ({fi }i=0 , T , T + L) + M} f

there is an integer j ∈ {T , . . . , T + L} for which ρ(xj , xj ) ≤ . Property (P) means that if a natural number L is large enough and a program +L is an approximate solution of the corresponding finite horizon problem, {xt }Tt =T f then there is j ∈ {T , . . . , T + L} such that xj is close to xj . We denote by Card(A) the cardinality of the set A.

2.2 Main Results

43

Theorem 2.3 Assume that the sequence {fi }∞ i=0 possesses ATP and property (P),  > 0 and M > 0. Then there exists a natural number L such that for each pair of 2 which satisfies integers T1 ≥ 0, T2 > T1 + L and each program {xt }Tt =T 1 T 2 −1

ft (xt , xt +1 ) ≤ U ({fi }∞ i=0 , T1 , T2 ) + M

t =T1

the following inequality holds: f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > }) ≤ L. ∞ Let S ≥ 0 be an integer. A program {xt }∞ t =S is called ({fi }i=0 )-minimal if for each integer T > S, T −1

ft (xt , xt +1 ) = U ({fi }∞ i=0 , S, T , xS , xT ).

t =S ∞ A program {xt }∞ t =S is called ({fi }i=0 )-overtaking optimal if for each program

satisfying xS = xS ,

{xt }∞ t =S

lim sup T →∞

−1  T

ft (xt , xt +1 ) −

t =S

T −1 t =S

 ft (xt , xt +1 ) ≤ 0.

Theorem 2.4 Assume that the sequence {fi }∞ i=0 possesses ATP, z ∈ X, S ≥ 0 is an integer and that there exists an ({fi }∞ )-good program {xt }∞ i=0 t =S satisfying xS = z. ∞ ∗ Then there exists an ({fi }i=0 )-overtaking optimal program {xt∗ }∞ t =S satisfying xS = z. Theorem 2.5 Assume that the sequence {fi }∞ i=0 possesses ATP, z ∈ X, S ≥ 0 ∞ is an integer and that there exists an ({fi }∞ i=0 )-good program {x¯ t }t =S satisfying ∞ x¯S = z. Let a program {xt }t =S satisfy xS = z. Then the following properties are equivalent: ∞ (i) {xt }∞ t =S is an ({fi }i=0 )-overtaking optimal program; ∞ ∞ (ii) the program {xt }t =S is ({fi }∞ i=0 )-minimal and ({fi }i=0 )-good; ∞ ∞ (iii) the program {xt }t =S is ({fi }i=0 )-minimal and satisfies f

lim ρ(xt , xt ) = 0.

t →∞

Now we show that {fi }∞ i=0 is approximated by elements of Mreg possessing TP.

44

2 Turnpike Conditions for Optimal Control Systems

For each r ∈ (0, 1) and all integers i ≥ 0 set f

fi(r)(x, y) = fi (x, y) + rρ(x, xi ), (x, y) ∈ Ωi .

(2.10)

Clearly, {fi(r)}∞ i=0 ∈ Mreg for all r ∈ (0, 1) and ∞ lim d({fi(r) }∞ i=0 , {fi }i=0 ) = 0.

r→0+

Proposition 2.6 Let r ∈ (0, 1). Then the sequence {fi }∞ i=0 possesses TP with f ∞ {xi }i=0 being the turnpike. (r)

Proof By Theorem 2.2 it is sufficient to show that {fi(r)}∞ i=0 possesses ATP and property (P). Assume that S ≥ 0 is an integer and that a program {xi }∞ i=S is (r) ∞ ({fi }i=0 )-good. Then there is c1 > 0 such that −1 T −1  T    f f (r) ft (xt , xt +1 ) − ft(r) (xt , xt +1 ) ≤ c1 for all integers T > S.  t =S

(2.11)

t =S

 f f By Proposition 2.1 and (2.16), ∞ t =S ρ(xt , xt ) < ∞, limt →∞ ρ(xt , xt ) = 0 and ATP holds. Let  > 0 and M > 0. Choose a natural number L >  −1 r −1 (M + cf ) + 1.

(2.12)

+L Assume that an integer T ≥ 0 and that a program {xt }Tt =T satisfies T +L−1 

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , T , T + L) + M. (r)

(r)

(2.13)

i=T

Relations (2.10) and (2.13) and (C2) imply that M+

T +L−1 

f

f

fi (xi , xi+1 ) ≥

i=T

T +L−1 

fi(r) (xi , xi+1 )

i=T

=

T +L−1 

fi (xi , xi+1 ) + r

i=T

≥

T +L−1 

f

ρ(xi , xi )

i=T f

f

fi (xi , xi+1 ) − cf + r

i=T

L min{ρ(xi , xi ) : i = T , . . . , T + L − 1} ≤ r −1 (M + cf ). f

T +L−1 

T +L−1  i=T

f

ρ(xi , xi ),

2.3 TP Implies ATP and Property (P)

45

Together with (2.12) this implies that f

min{ρ(xi , xi ) : i = T , . . . , T + L − 1} ≤ . Thus property (P) holds and Proposition 2.6 is proved. In Sect. 2.3 we show that TP implies ATP and property (P). Section 2.4 contains auxiliary results. Proof of Theorem 2.2 is given in Sect. 2.5, while Theorem 2.3 is proved in Sect. 2.6. Proof of Theorem 2.5 is given in Sect. 2.7. Theorem 2.4 is proved in Sect. 2.8.

2.3 TP Implies ATP and Property (P) ∞ Proposition 2.7 Let S ≥ 0 be an integer and a let program {xi }∞ i=S be ({fi }i=0 )good. Then there is a number c > 0 such that for each pair of integers T1 ≥ S and T2 > T1 , T 2 −1

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , T1 , T2 ) + c

(2.14)

i=T1

and the following property holds: (iv) for each  > 0 there exists a natural number L such that for each integer T1 ≥ L and each integer T2 > T1 , T 2 −1

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , T1 , T2 , xT1 , xT2 ) + .

i=T1

Proof By definition, there is a number c0 > 0 such that for each integer T > S, −1 T −1  T    f f fi (xi , xi+1 ) − fi (xi , xi+1 ) ≤ c0 .  i=S

(2.15)

i=S

Set c = 2c0 + cf .

(2.16)

By (2.15) and (C2), for each integer T > S, T −1 i=S

fi (xi , xi+1 ) ≤

T −1 i=S

fi (xi , xi+1 ) + c0 ≤ U ({fi }∞ i=0 , S, T ) + cf + c0 . f

f

(2.17)

46

2 Turnpike Conditions for Optimal Control Systems

Let integers T1 > S and T2 > T1 . By (2.15) and (C2), T 2 −1

fi (xi , xi+1 ) =

i=T1

T 2 −1

fi (xi , xi+1 ) −

T 1 −1

i=S

≤

T 2 −1

fi (xi , xi+1 )

i=S f f fi (xi , xi+1 ) + c0

−

i=S

=

T 2 −1

T 1 −1

f

f

fi (xi , xi+1 ) + c0

i=S

fi (xi , xi+1 ) + 2c0 ≤ U ({fi }∞ i=0 , T1 , T2 ) + cf + 2c0 . f

f

i=T1

(2.18) By (2.16), (2.17) and (2.18) inequality (2.14) holds for each integer T1 ≥ S and each integer T2 > T1 . Let us show that property (iv) holds. Assume the contrary. Then there exist  > 0

∞ and sequences of natural numbers {Tk }∞ k=1 , {Tk }k=1 such that for each integer k ≥ 1, S < Tk < Tk < Tk+1 , Tk −1



(2.19)

fi (xi , xi+1 ) > U ({fi }∞ i=0 , Tk , Tk , xTk , xTk ) + .

(2.20)

i=Tk

By (2.19) and (2.20) there exists a program {yi }∞ i=S such that yi = xi , i = S, . . . , T1 and that for all natural numbers k, yTk = xTk , yTk = xTk , Tk −1



Tk −1

fi (xi , xi+1 ) >

i=Tk



fi (yi , yi+1 ) + /2,

i=Tk

yt = xt , t = Tk , . . . , Tk+1 .

(2.21)

By (2.14) and (2.21), for each integer p ≥ 1, Tp −1

0≤



fi (yi , yi+1 ) − U ({fi }∞ i=0 , S, Tp )

i=S Tp −1

≤

 i=S

Tp −1

fi (yi , yi+1 ) −

 i=S

fi (xi , xi+1 ) + c ≤ p(−/2) + c → −∞ as p → ∞.

2.3 TP Implies ATP and Property (P)

47

The contradiction we have reached proves that property (iv) holds. Proposition 2.7 is proved. Proposition 2.8 Assume that TP holds. Then ATP and property (P) holds. Proof Let us show that ATP holds. Assume that S ≥ 0 is an integer and the program ∞ {xi }∞ i=S is ({fi }i=0 )-good. By Proposition 2.7, there is c > 0 such that inequality (2.14) holds for all integers T1 ≥ S, T2 > T1 . Let  > 0. By TP there exist δ > 0 and a natural number L0 such that for each 2 pair of integers T1 ≥ 0, T2 ≥ T1 + 2L0 and each program {zi }Ti=T satisfying 1 T 2 −1

fi (zi , zi+1 ) ≤ min{U ({fi }∞ i=0 , T1 , T2 , zT1 , zT2 ) + δ,

i=T1

U ({fi }∞ i=0 , T1 , T2 ) + c},

(2.22)

we have f

ρ(xi , xi ) ≤  for all integers i = T1 + L0 , . . . , T2 − L0 .

(2.23)

By Proposition 2.7 (see property (i)) there exists a natural number L1 > S such that for each integer T1 ≥ L1 and each integer T2 > T1 , T 2 −1

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , T1 , T2 , xT1 , xT2 ) + δ.

(2.24)

i=T1

Assume that integers T1 ≥ L1 , T2 ≥ T1 + 2L0 .

(2.25)

Then (2.14) and (2.24) hold. In view of (2.14) and (2.24), inequality (2.22) holds. By (2.22) and (2.25) relation (2.13) is true. Since (2.23) holds for any pair of integers T1 , T2 satisfying (2.25) we conclude that f

ρ(xi , xi ) ≤  for all integers i ≥ L1 + L0 . f

Since  is an arbitrary positive number we conclude that limt →∞ ρ(xt , xt ) = 0 and ATP holds. Clearly (P) holds. Proposition 2.8 is proved.

48

2 Turnpike Conditions for Optimal Control Systems

2.4 Auxiliary Results Lemma 2.9 Let  > 0. Then there exists δ > 0 such that for each pair of integers 2 satisfying T1 ≥ 0, T2 > T1 and each program {xi }Ti=T 1 f

T 2 −1

ρ(xTj , xTj ) ≤ δ, j = 1, 2,

(2.26)

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , T1 , T2 , xT1 , xT2 ) + δ

(2.27)

i=T1

the following inequality holds: T 2 −1

fi (xi , xi+1 ) ≤

i=T1

T 2 −1

f

f

fi (xi , xi+1 ) + .

(2.28)

i=T1

Proof By (C3) there exists δ0 ∈ (0, ) such that for each integer t ≥ 0 and each f f (x, y) ∈ Ωt satisfying ρ(x, xt ) ≤ δ0 , ρ(y, xt +1 ) ≤ δ0 , f

f

|ft (xt , xt +1 ) − ft (x, y)| ≤ /8.

(2.29)

By (C4) there is a positive number δ1 < min{δ0 , γf }

(2.30)

such that for each integer t ≥ 0, each (ut , ut +1 ) ∈ Ωt and each (u t +1 , u t +2 ) ∈ Ωt +1 satisfying ρ(ut , xt ) ≤ δ1 , ρ(u t +2 , xt +2 ) ≤ δ1 f

f

(2.31)

there is x ∈ X such that (ut , x) ∈ Ωt , (x, u t +2 ) ∈ Ωt +1 , ρ(x, xt +1 ) ≤ δ0 . f

(2.32)

By (C4) there is a positive number (2.33)

δ < min{δ1 , /8}

such that for each integer t ≥ 0, each (ut , ut +1 ) ∈ Ωt and each (u t +1 , u t +2 ) ∈ Ωt +1 satisfying ρ(ut , xt ) ≤ δ, ρ(u t +2 , xt +2 ) ≤ δ f

f

(2.34)

2.4 Auxiliary Results

49

there is x ∈ X such that (ut , x) ∈ Ωt , (x, u t +2 ) ∈ Ωt +1 , ρ(x, xt +1 ) ≤ δ1 . f

(2.35)

2 Assume that integers T1 ≥ 0, T2 > T1 and that a program {xi }Ti=T satisfies 1 (2.26) and (2.27). In order to prove the lemma it is sufficient to show that (2.28) holds. There are cases:

T2 = T1 + 1;

(2.36)

T2 = T1 + 2;

(2.37)

T2 = T1 + 3;

(2.38)

T2 > T1 + 3.

(2.39)

Assume that (2.36) holds. Then by the choice of δ0 , (2.26), (2.29), (2.33) and (2.39), f

f

fT1 (xT1 , xT1 +1 ) ≤ fT1 (xT1 , xT1 +1 ) + /8 and (2.28) holds. Assume that (2.37) holds. Then by (2.26), (2.33), (2.37) and the choice of δ1 (see (2.31) and (2.32)) there exists x˜T1 +1 ∈ X such that f

(xT1 , x˜T1 +1 ) ∈ ΩT1 , (x˜T1 +1 , xT1 +2 ) ∈ ΩT1 +1 , ρ(x˜T1 +1 , xT1 +1 ) ≤ δ0 .

(2.40)

By (2.26), (2.30), (2.33), (2.40) and the choice of δ0 (see (2.29)), f

f

|fT1 (xT1 , xT1 +1 ) − fT1 (xT1 , x˜T1 +1 )| ≤ /8, f

f

|fT1 +1 (xT1 +1 , xT1 +2 ) − fT1 +1 (x˜T1 +1 , xT1 +2 )| ≤ /8. Together with (2.27) and (2.40) this implies that T 1 +1

fi (xi , xi+1 ) ≤ fT1 (xT1 , x˜T1 +1 ) + fT1 +1 (x˜T1 +1 , xT1 +2 ) + δ

i=T1 f

f

f

f

≤ fT1 (xT1 , xT1 +1 ) + fT1 +1 (xT1 +1 , xT1 +2 ) + /4 + δ. Together with (2.33) this implies (2.28). Assume that (2.38) holds. By (2.26), (2.38) and the choice of δ (see (2.34), (2.35)) there exists x˜T1 +1 ∈ X such that f

f

(xT1 , x˜T1 +1 ) ∈ ΩT1 , (x˜T1 +1 , xT1 +2 ) ∈ ΩT1 +1 , ρ(x˜T1 +1 , xT1 +1 ) ≤ δ1 .

(2.41)

50

2 Turnpike Conditions for Optimal Control Systems

By (2.26), (2.30), (2.33), (2.41) and the choice of δ1 (see (2.31) and (2.32)) there exists x˜T1 +2 ∈ X such that f

(x˜T1 +1 , x˜T1 +2 ) ∈ ΩT1 +1 , (x˜T1 +2 , xT1 +3 ) ∈ ΩT1 +2 , ρ(x˜T1 +2 , xT1 +2 ) ≤ δ0 . (2.42) By (2.26), (2.30), (2.33), (2.41), (2.42) and the choice of δ0 (see (2.29)), f

f

max{|fT1 (xT1 , x˜T1 +1 ) − fT1 (xT1 , xT1 +1 )|, f

f

f

f

|fT1 +1 (x˜T1 +1 , x˜T1 +2 ) − fT1 +1 (xT1 +1 , xT1 +2 )|, |fT1 +2 (x˜T1 +2 , xT1 +3 ) − fT1 +2 (xT1 +2 , xT1 +3 )|} ≤ /8. Together with (2.27), (2.41) and (2.42) this implies that T 1 +2

fi (xi , xi+1 ) ≤ δ + fT1 (xT1 , x˜T1 +1 ) + fT1 +1 (x˜T1 +1 , x˜T1 +2 )

i=T1

+ fT1 +2 (x˜T1 +2 , xT1 +3 ) ≤ δ + (3/8) +

T 1 +2

f

f

fi (xi , xi+1 ).

i=T1

Together with (2.33) this implies (2.28). Assume that (2.39) holds. By (2.26), (2.39) and the choice of δ (see (2.33)– (2.35)), there exist x˜T1 +1 ∈ X and x˜T2 −1 ∈ X such that f

f

(xT1 , x˜T1 +1 ) ∈ ΩT1 , (x˜T1 +1 , xT1 +2 ) ∈ ΩT1 +1 , ρ(x˜T1 +1 , xT1 +1 ) ≤ δ1 , f

f

(xT2 −2 , x˜T2 −1 ) ∈ ΩT2 −2 , (x˜T2 −1 , xT2 ) ∈ ΩT2 −1 , ρ(x˜T2 −1 , xT2 −1 ) ≤ δ1 .

(2.43) (2.44)

Put x˜T1 = xT1 , x˜T2 = xT2 , f

x˜i = xi for all i ∈ {T1 , . . . , T2 } \ {T1 , T1 + 1, T2 − 1, T2 }.

(2.45)

2 By (2.39) and (2.43)–(2.45), {x˜i }Ti=T is a program. By (2.26), (2.30), (2.33), (2.43)– 1 (2.45) and the choice of δ0 (see (2.29)) for all i ∈ {T1 , T1 + 1, T2 − 2, T2 − 1},

f

f

|fi (x˜i , x˜i+1 ) − fi (xi , xi+1 )| ≤ /8.

(2.46)

2.4 Auxiliary Results

51

By (2.27), (2.45) and (2.46), T 2 −1

fi (xi , xi+1 ) ≤

i=T1

T 2 −1

fi (x˜i , x˜i+1 ) + δ ≤

i=T1

T 2 −1

f

f

fi (xi , xi+1 ) + 4(/8) + δ.

i=T1

Together with (2.33) this implies (2.28). Thus (2.22) holds in all the cases and Lemma 2.9 is proved. Lemma 2.10 Let  > 0. Then there exists δ > 0 such that for each pair of integers 2 satisfying T1 > 0, T2 > T1 + 2 and each program {xi }Ti=T 1 f

f

ρ(xT1 +1 , xT1 +1 ) ≤ δ, ρ(xT2 −1 , xT2 −1 ) ≤ δ, T 2 −2

(2.47)

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , T1 + 1, T2 − 1, xT1 +1 , xT2 −1 ) + δ

(2.48)

i=T1 +1 2 +1 such that there exists a program {x˜i }Ti=T 1 −1

f

f

x˜ T1 −1 = xT1 −1 , x˜T2 +1 = xT2 +1 , x˜i = xi , i = T1 + 1, . . . .T2 − 1

(2.49)

and that the following inequality holds: T2 

fi (x˜i , x˜i+1 ) ≤

i=T1 −1

T2 

f

f

fi (xi , xi+1 ) + .

(2.50)

i=T1 −1

Proof By Lemma 2.9 there exists δ0 ∈ (0, γf ) such that for each pair of integers 2 T1 ≥ 0, T2 > T1 and each program {yt }Tt =T satisfying 1 f

ρ(yTj , xTj ) ≤ δ0 , j = 1, 2,

T 2 −1

fi (yi , yi+1 ) ≤ U ({fi }∞ i=0 , T1 , T2 , yT1 , yT2 ) + δ0

i=T1

(2.51) we have T 2 −1 i=T1

fi (yi , yi+1 ) ≤

T 2 −1

f

f

fi (xi , xi+1 ) + /8.

(2.52)

i=T1

By (C3) there exists δ1 ∈ (0, γf )

(2.53)

52

2 Turnpike Conditions for Optimal Control Systems f

such that for each integer t ≥ 0 and each (x, y) ∈ Ωt satisfying ρ(x, xt ) ≤ δ1 , f ρ(y, xt +1) ≤ δ1 we have f

f

|ft (xt , xt +1 ) − ft (x, y)| ≤ /8.

(2.54)

δ2 ∈ (0, γf )

(2.55)

By (C4) there exists

such that for each integer t ≥ 0, each (yt , yt +1 ) ∈ Ωt and each (yt +1 , yt +2) ∈ Ωt +1 satisfying ρ(yt , xt ) ≤ δ2 , ρ(yt +2 , xt +2 ) ≤ δ2 f

f

(2.56)

there is y ∈ X such that (yt , y) ∈ Ωt , (y, yt +2) ∈ Ωt +1 , ρ(y, xt +1) ≤ δ1 . f

(2.57)

Set δ = min{δ0 , δ1 , δ2 }.

(2.58)

Assume that an integer T1 > 0, an integer T2 > T1 + 2 and that a program 2 satisfies (2.47) and (2.48). By (2.47), (2.48), the choice of δ0 (see (2.51), {xi }Ti=T 1 (2.52)) and (2.58), T 2 −2

fi (xi , xi+1 ) ≤

i=T1 +1

T 2 −2

f

f

fi (xi , xi+1 ) + /8.

(2.59)

i=T1 +1

By (2.47) and the choice of δ2 (see (2.55)–(2.57)) and (2.58) there exist x˜T1 , x˜T2 ∈ X such that f

f

(2.60)

f

(2.61)

(xT1 −1 , x˜T1 ) ∈ ΩT1 −1 , (x˜T1 , xT1 +1 ) ∈ ΩT1 , ρ(x˜T1 , xT1 ) ≤ δ1 , f

(x˜T2 , xT2 +1 ) ∈ ΩT2 , (xT2 −1 , x˜T2 ) ∈ ΩT2 −1 , ρ(x˜T2 , xT2 ) ≤ δ1 . Put f

f

x˜T1 −1 = xT1 −1 , x˜T2 +1 = xT2 +1 , x˜i = xi , i ∈ {T1 , . . . , T2 } \ {T1 , T2 }.

(2.62)

2 +1 By (2.60)–(2.62), {x˜i }Ti=T is a program. In order to complete the proof of the 1 −1 lemma it is sufficient to show that inequality (2.50) is valid. By (2.60)–(2.62), the

2.4 Auxiliary Results

53

choice of δ1 (see (2.54)), (2.47) and (2.58), f

f

|fi (x˜i , x˜i+1 ) − fi (xi , xi+1 )| ≤ /8 for all i = T1 − 1, T1 , T2 − 1, T2 .

(2.63)

By (2.59), (2.62) and (2.63), T2 

fi (x˜i , x˜i+1 ) =

i=T1 −1

T 2 −2

fi (xi , xi+1 ) + fT1 −1 (x˜T1 −1 , x˜T1 ) + fT1 (x˜T1 , x˜T1 +1 )

i=T1 +1

+ fT2 −1 (x˜T2 −1 , x˜T2 ) + fT2 (x˜T2 , x˜T2 +1 ) ≤

T 2 −2 i=T1 +1

f

f

f

f

fi (xi , xi+1 ) + /8 + fT1 −1 (xT1 −1 , xT1 ) + /8 f

f

+ fT1 (xT1 , xT1 +1 ) + /8 f

f

f

f

+ fT2 −1 (xT2 −1 , xT2 ) + /8 + fT2 (xT2 , xT2 +1 ) + /8 =

T2 

f

f

fi (xi , xi+1 ) + 5/8.

i=T1 −1

Lemma 2.10 is proved. Lemma 2.11 Assume that {fi }∞ i=0 possesses ATP and let  > 0. Then there exist δ > 0 and a natural number L such that for each pair of integers T2 > T1 ≥ L and 2 satisfying each program {xi }Ti=T 1 f

f

xT1 = xT1 , xT2 = xT2 ,

T 2 −1

fi (xi , xi+1 ) ≤

i=T1

T 2 −1

f

f

fi (xi , xi+1 ) + δ

i=T1

f

the inequality ρ(xt , xt ) ≤  holds for all t = T1 , . . . , T2 . Proof Assume that the lemma is not true. Then there exist sequences of natural ∞ numbers {Tk }∞ k=1 , {Sk }k=1 such that for each natural number k, Tk < Sk < Tk+1 k and there exists a program {xi(k) }Si=T such that k

(k)

f

(k)

f

xTk = xTk , xSk = xSk , S k −1 i=Tk

(k) fi (xi(k) , xi+1 )≤

S k −1 i=Tk

(2.64) fi (xi , xi+1 ) + 2−k , f

f

(2.65)

54

2 Turnpike Conditions for Optimal Control Systems f

max{ρ(xi(k), xi ) : i = Tk , . . . , Sk } > .

(2.66)

Define a sequence {xi }∞ i=0 ⊂ X as follows: for each integer k ≥ 1, xi = xi(k) , i = Tk , . . . , Sk ,

(2.67)

xi = xi for all integers i ≥ 0 such that i ∈ ∪∞ k=1 {Tk , . . . , Sk }. f

(2.68)

By (2.64), (2.67) and (2.68) {xi }∞ i=0 is a well-defined program. By (2.64), (2.65), (2.67) and (2.68) for each integer p ≥ 1, Sp 

fi (xi , xi+1 ) ≤

i=0

Sp 

f f fi (xi , xi+1 )

+

p 

i=0

2−i .

i=1

∞ Combined with Proposition 2.1 this implies that the program {xi }∞ i=0 is ({fi }i=0 )good. In view of ATP there is a natural number τ such that f

ρ(xt , xt ) ≤ /2 for all integers t ≥ τ.

(2.69)

By (2.67) and (2.69) for each integer k ≥ 1 satisfying Tk > τ f

ρ(xt(k), xt ) ≤ /2, t = Tk , . . . , Sk . This contradicts (2.66). The contradiction we have reached completes the proof of Lemma 2.11. Lemma 2.12 Assume that {fi }∞ i=0 possesses ATP and let  > 0. Then there exist δ > 0 and a natural number L such that for each pair of integers T1 , T2 satisfying 2 T1 > L, T2 > T1 + 2 and each program {xi }Ti=T satisfying 1 f

f

ρ(xT1 +1 , xT1 +1 ) ≤ δ, ρ(xT2 −1 , xT2 −1 ) ≤ δ, T 2 −2

(2.70)

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , T1 + 1, T2 − 1, xT1 +1 , xT2 −1 ) + δ

(2.71)

i=T1 +1

the following inequality holds: f

ρ(xi , xi ) ≤ , i = T1 + 1, . . . , T2 − 1. Proof By Lemma 2.11, there exist δ1 > 0 and a natural number L such that for 2 each pair of integers S2 > S1 ≥ L and each program {xi }Si=S satisfying 1 f

xSi = xSi , i = 1, 2,

S 2 −1 i=S1

fi (xi , xi+1 ) ≤

S 2 −1 i=S1

f

f

fi (xi , xi+1 ) + δ1

(2.72)

2.5 Proof of Theorem 2.2

55

we have f

ρ(xi , xi ) ≤ , i = S1 , . . . , S2 .

(2.73)

By Lemma 2.10, there exist δ > 0 such that for each pair of integers T1 > 0, 2 satisfying T2 > T1 + 2 and each program {xi }Ti=T 1 f

f

ρ(xT1 +1 , xT1 +1 ) ≤ δ, ρ(xT2 −1 , xT2 −1 ) ≤ δ, T 2 −2

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , T1 + 1, T2 − 1, xT1 +1 , xT2 − 1) + δ

(2.74) (2.75)

i=T1 +1 2 +1 there exists a program {x˜i }Ti=T such that 1 −1

f

f

x˜T1 −1 = xT1 −1 , x˜T2 +1 = xT2 +1 , x˜ i = xi , i = T1 + 1, . . . .T2 − 1, T2 

fi (x˜i , x˜i+1 ) ≤

i=T1 −1

T2 

f

f

fi (xi , xi+1 ) + δ1 .

(2.76) (2.77)

i=T1 −1

2 Assume that an integer T1 > L, an integer T2 > T1 + 2 and a program {xi }Ti=T 1 satisfies (2.70) and (2.71). By (2.70), (2.71) and the choice of δ (see (2.74), (2.75)) 2 +1 there exists a program {x˜i }Ti=T which satisfies (2.76), (2.77). By (2.76), (2.77), 1 −1 the choice of δ1 (see (2.72), (2.73)),

f

ρ(x˜i , xi ) ≤ , i = T1 − 1, . . . , T2 + 1. Together with (2.76) this implies that f

ρ(xi , xi ) ≤ , i = T1 + 1, . . . , T2 − 1. Lemma 2.12 is proved.

2.5 Proof of Theorem 2.2 By Proposition 2.8, TP implies ATP and property (P). Assume that ATP and property (P) hold. Let  > 0 and M > 0. By Lemma 2.12 there exist δ0 > 0 and a natural number L0 such that for each pair of integers S1 , S2 satisfying S1 > L0 , S2 > S1 + 2 and

56

2 Turnpike Conditions for Optimal Control Systems

2 each program {xi }Si=S satisfying 1

f

f

ρ(xS1 +1 , xS1 +1 ) ≤ δ0 , ρ(xS2 −1 , xS2 −1 ) ≤ δ0 , S 2 −2

(2.78)

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , S1 + 1, S2 − 1, xS1 +1 , xS2 −1 ) + δ0

(2.79)

i=S1 +1

the following inequality holds: f

ρ(xi , xi ) ≤ , i = S1 + 1, . . . , S2 − 1.

(2.80)

By property (P) there exist δ ∈ (0, δ0 )

(2.81)

and a natural number L1 such that for each integer T ≥ 0 and each program +L1 {xt }Tt =T which satisfies T +L 1 −1 

(2.82)

fi (xi , xi+1 )

i=T ∞ ≤ min{U ({fi }∞ i=0 , T , T + L1 , xT , xT +L1 ) + δ, U ({fi }i=0 , T , T + L1 ) + 3cf + M},

there is an integer j such that f

j ∈ {T , . . . , T + L1 }, ρ(xj , xj ) ≤ δ0 .

(2.83)

Choose a natural number L ≥ 4L0 + 4L1 .

(2.84)

2 Assume that a pair of integers T1 ≥ 0, T2 ≥ T1 + 2L and that a program {xt }Tt =T 1 satisfies

T 2 −1

∞ fi (xi , xi+1 ) ≤ min{U ({fi }∞ i=0 , T1 , T2 , xT1 , xT2 ) + δ, U ({fi }i=0 , T1 , T2 ) + M}.

i=T1

(2.85) In order to complete the proof of the theorem it is sufficient to show that f

ρ(xi , xi ) ≤  for all i = T1 + L, . . . , T2 − L.

(2.86)

2.5 Proof of Theorem 2.2

57

Let integers S1 , S2 satisfy T1 < S1 < S2 < T2 . By (2.85) and (C2) S 2 −1

fi (xi , xi+1 ) =

i=S1

T 2 −1

fi (xi , xi+1 ) −

i=T1

S 1 −1

fi (xi , xi+1 ) −

i=T1

T 2 −1

fi (xi , xi+1 )

i=S2

∞ ∞ ≤ U ({fi }∞ i=0 , T1 , T2 ) + M − U ({fi }i=0 , T1 , S1 ) − U ({fi }i=0 , S2 , T2 )

≤

T 2 −1

f

f

fi (xi , xi+1 ) + M −

i=T1

=

S 2 −1

S 1 −1

f

f

fi (xi , xi+1 ) + cf −

i=T1

T 2 −1

f

f

fi (xi , xi+1 ) + cf

i=S2

fi (xi , xi+1 ) + 2cf + M ≤ U ({fi }∞ i=0 , S1 , S2 ) + 3cf + M. f

f

i=S1

Thus S 2 −1

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , S1 , S2 ) + 3cf + M

(2.87)

i=S1

for all pairs of integers S1 , S2 satisfying T1 < S1 < S2 < T2 . By (2.84), (2.85), (2.87) and the choice of δ (see (2.81)–(2.83)) there exist integers τ1 ∈ {L1 +T1 +2L0 , . . . , T1 +2L0 +2L1 }, τ2 ∈ {T2 −2L1 , . . . , T2 −L1 }

(2.88)

such that ρ(xτi , xτfi ) ≤ δ0 , i = 1, 2.

(2.89)

τ2 − τ1 ≥ 2L0 + L.

(2.90)

By (2.84) and (2.88),

By (2.85) and (2.88), τ 2 −1

fi (xi , xi+1 ) ≤ U ({fi }∞ i=0 , τ1 , τ2 , xτ1 , xτ2 ) + δ.

(2.91)

i=τ1

By (2.81), (2.88), (2.89), (2.90), (2.91) and the choice of L0 and δ0 (see (2.78)– (2.80)), f

ρ(xi , xi ) ≤  for all i = τ1 , . . . , τ2 . Together with (2.84) and (2.88) this implies (2.86). Theorem 2.2 is proved.

58

2 Turnpike Conditions for Optimal Control Systems

2.6 Proof of Theorem 2.3 We suppose that the sum over an empty set is zero. By ATP, property (P) and Theorem 2.2, {fi }∞ i=0 possesses TP. By TP there exist δ ∈ (0, 1) and a natural number L0 such that the following property holds: 2 (v) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + 2L0 and each program {xt }τt =τ 1 which satisfies τ 2 −1 t =τ1

ft (xt , xt +1 ) ≤ min{U ({fi }∞ i=0 , τ1 , τ2 , xτ1 , xτ2 ) + δ, U ({fi }∞ i=0 , τ1 , τ2 ) + 2M + 2cf } f

the inequality ρ(xi , xi ) ≤  holds for all integers i = τ1 + L0 , . . . , τ2 − L0 . Choose a natural number L > (4L0 + 3)(δ −1 M + 1).

(2.92)

2 satisfies Assume that integers T1 ≥ 0, T2 > T1 + L and that a program {xt }Tt =T 1

T 2 −1

ft (xt , xt +1 ) ≤ U ({fi }∞ i=0 , T1 , T2 ) + M.

(2.93)

t =T1

Set t0 = T1 .

(2.94) q

By induction we define a finite strictly increasing sequence of integers {ti }i=0 ⊂ [T1 , T2 ] where q is a natural number such that: tq = T2 ;

(2.95)

(vi) for each integer i satisfying 0 ≤ i < q − 1, ti+1 −1

 t =ti

ft (xt , xt +1 ) > U ({fi }∞ i=0 , ti , ti+1 , xti , xti+1 ) + δ;

(2.96)

(vii) if an integer i satisfies 0 ≤ i ≤ q − 1 and (2.96), then ti+1 −2

ti+1 > ti + 1 and

 t =ti

ft (xt , xt +1 ) ≤ U ({fi }∞ i=0 , ti , ti+1 − 1, xti , xti+1 −1 ) + δ. (2.97)

2.6 Proof of Theorem 2.3

59

Assume that an integer p ≥ 0 and we have already defined a strictly increasing p sequence of integers {ti }i=0 ⊂ [T1 , T2 ] such that tp < T2 and that for each integer i satisfying 0 ≤ i < p, (2.96) and (2.97) hold. (Note that for p = 0 our assumption holds.) We define tp+1 . There are two cases: T 2 −1 t =tp T 2 −1 t =tp

ft (xt , xt +1 ) ≤ U ({fi }∞ i=0 , tp , T2 , xtp , xT2 ) + δ;

(2.98)

ft (xt , xt +1 ) > U ({fi }∞ i=0 , tp , T2 , xtp , xT2 ) + δ.

(2.99)

Assume that (2.98) holds. Then we set q = p + 1, tq = T2 , the construction of the sequence is completed and properties (vi) and (vii) hold. Assume that (2.99) holds. Set tp+1 = min{S ∈ {tp + 1, . . . , T2 } : S−1  t =tp

ft (xt , xt +1 ) > U ({fi }∞ i=0 , tp , S, xtp , xS ) + δ}.

(2.100)

Clearly, tp+1 is well-defined. If tp+1 = T2 , then we set q = p + 1, the construction is completed and it is not difficult to see that (vi) and (vii) hold. Assume that tp+1 < T2 . Then it is easy to see that the assumption made for p is also true for p + 1. Clearly, our construction is completed after a final number of steps and let tq = T2 be its last element, where q is a natural number. It follows from the construction that properties (vi) and (vii) hold. By (2.93) and property (vi), M≥

T 2 −1

ft (xt , xt +1 ) − U ({fi }∞ i=0 , T1 , T2 )

t =T1

≥

T 2 −1

ft (xt , xt +1 ) − U ({fi }∞ i=0 , T1 , T2 , xT1 , xT2 )

t =T1

≥

−1   ti+1  t =ti

ft (xt , xt +1 ) − U ({fj }∞ j =0 , ti , ti+1 , xti , xti+1 ) :

 i is an integer, 0 ≤ i < q − 1 ≥ δ(q − 1), q ≤ δ −1 M + 1.

(2.101)

60

2 Turnpike Conditions for Optimal Control Systems

Set A = {i ∈ {0, . . . , q − 1} : ti+1 − ti > 2L0 }.

(2.102)

Let j ∈ A. By (vi), (vii) and (2.102), tj+1 −2



t =tj

ft (xt , xt +1 ) ≤ U ({fi }∞ i=0 , tj , tj +1 − 1, xtj , xtj+1 −1 ) + δ.

(2.103)

By (2.93) and (C2), tj+1 −2



ft (xt , xt +1 ) =

t =tj

T 2 −1

ft (xt , xt +1 )

t =T1

−



{ft (xt , xt +1 ) : t is an integer, T1 ≤ t < tj }

T 2 −1

−

ft (xt , xt +1 )

t =tj+1 −1

≤

T 2 −1 t =T1

− −

f

f

ft (xt , xt +1 ) + M + cf



f

f

{ft (xt , xt +1 ) : t is an integer, T1 ≤ t < tj } + cf

T 2 −1 t =tj+1 −1

f

f

ft (xt , xt +1 )

tj+1 −2

=



t =tj

f

f

ft (xt , xt +1 ) + M + 2cf

≤ U ({fi }∞ i=0 , tj , tj +1 − 1) + 2M + 2cf .

(2.104)

By (2.102)–(2.104) and property (v), f

ρ(xt , xt ) ≤  for all integers t = tj + L0 , . . . , tj +1 − 1 − L0

(2.105)

2.7 Proof of Theorem 2.5

61

for all j ∈ A. By (2.102) and (2.105), f

{t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > } ⊂

 {{ti , . . . , ti+1 } : i ∈ {0, . . . , q − 1} \ A}

∪{{ti , . . . , ti + L0 − 1} ∪ {ti+1 − 1 − L0 , . . . , ti+1 } : i ∈ A} and in view of (2.92) and (2.101), f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > }) ≤ q(2L0 + 1) + qL0 + q(L0 + 2) = q(4L0 + 3) ≤ (4L0 + 3)(δ −1 M + 1) < L. Theorem 2.3 is proved.

2.7 Proof of Theorem 2.5 It is clear that property (i) implies property (ii) and property (ii) implies property (iii). ∞ Assume that property (iii) holds and show that {xt }∞ t =S is an ({fi }i=0 ) -overtaking optimal program. Let a program {xt }∞ t =S satisfy xS = xS .

(2.106)

In order to complete the proof of the theorem it is sufficient to show that lim sup T →∞

−1  T

ft (xt , xt +1 ) −

t =S

T −1 t =S

 ft (xt , xt +1) ≤ 0.

(2.107)

In view of Proposition 2.1, we may assume without loss of generality that the ∞ program {xt }∞ t =S is ({fi }i=0 )-good. Assume that (2.107) does not hold. Then there exist a strictly increasing sequence of natural numbers {Tk }∞ k=1 and  > 0 such that for any integer k ≥ 1, T k −1

T k −1

ft (xt , xt +1 ) + 2.

(2.108)

lim ρ(xt , xt ) = 0, lim ρ(xt , xt ) = 0.

(2.109)

ft (xt , xt +1 ) ≥

t =S

t =S

By ATP, f

t →∞

f

t →∞

62

2 Turnpike Conditions for Optimal Control Systems

By (C3) there is a number 0 such that 0 ∈ (0, /4), 0 < γf /4

(2.110) f

and that for each integer t ≥ 0 and each (x, y) ∈ Ωt satisfying ρ(x, xt ) ≤ 0 , f ρ(y, xt +1) ≤ 0 , f

f

|ft (xt , xt +1 ) − ft (x, y)| ≤ /8.

(2.111)

By (C4) there is a positive number (2.112)

δ < 0 /2

such that the following property holds: (viii) For each integer t ≥ 0, each (ut , ut +1 ) ∈ Ωt and each (u t +1 , u t +2 ) ∈ Ωt +1 satisfying ρ(ut , xt ) ≤ δ, ρ(u t +2 , xt +2 ) ≤ δ f

f

there is x ∈ X such that (ut , x) ∈ Ωt , (x, u t +2 ) ∈ Ωt +1, ρ(x, xt +1) ≤ 0 . f

By (2.109), there is a natural number τ0 such that for each integer t ≥ τ0 , ρ(xt , xt ) ≤ δ, ρ(xt , xt ) ≤ δ. f

f

(2.113)

Choose a natural number k such that Tk > τ0 + 4.

(2.114)

By (2.113) and (2.114), ρ(xt , xt ) ≤ δ, ρ(xt , xt ) ≤ δ forall t = Tk , Tk + 1, Tk + 2, Tk + 3. f

f

(2.115)

By (2.115) and property (viii), there is z ∈ X such that (xT k , z) ∈ ΩTk , (z, xTk +2 ) ∈ ΩTk +1 , ρ(z, xTk +1 ) ≤ 0 .

(2.116)

x˜t = xt , t = S, . . . , Tk , x˜ Tk +1 = z, x˜Tk +2 = xTk +2 .

(2.117)

f

Define

2.8 Proof of Theorem 2.4

63 T +2

k By (2.116) and (2.117), {x˜t }t =S is a program. By (2.106) and (2.117),

x˜ S = xS .

(2.118)

By (2.117), T k +1

ft (xt , xt +1 ) −

t =S

=

T k −1

T k +1

ft (x˜t , x˜t +1 )

t =S

ft (xt , xt +1 ) −

t =S

T k −1 t =S

ft (xt , xt +1 )

+ fTk (xTk , xTk +1 ) + fTk +1 (xTk +1 , xTk +2 ) − fTk (xT k , z) − fTk +1 (z, xTk +2 ). (2.119) By (2.112), (2.115), (2.116) and the choice of 0 (see (2.111)), f

f

|fTk (xTk , xTk +1 ) − fTk (xTk , xTk +1 )| ≤ /8, f

f

|fTk +1 (xTk +1 , xTk +2 ) − fTk +1 (xTk +1 , xTk +2 )| ≤ /8, |fTk (xT k , z) − fTk (xTk , xTk +1 )| ≤ /8, f

f

f

f

|fTk +1 (z, xTk +2 ) − fTk +1 (xTk +1 , xTk +2 )| ≤ /8.

(2.120)

By (2.108), (2.119) and (2.120), T k +1

ft (xt , xt +1 ) −

t =S

T k +1

ft (x˜t , x˜t +1 ) ≥ 2 − 4(/4) = .

t =S

This contradicts property (iii). The contradiction we have reached proves (2.107). Theorem 2.5 is proved.

2.8 Proof of Theorem 2.4 By definition there is a number c0 > 0 such that −1 T −1  T    f f f (x , x ) − ft (xt , xt +1 ) ≤ c0 for all integers t ≥ S + 1.  t t t +1 t =S

t =S

(2.121)

64

2 Turnpike Conditions for Optimal Control Systems (T )

For each integer T > S there is a program {xi }Ti=S such that (T )

xS

T −1 t =S

= z,

(2.122)

ft (xt(T ) , xt(T+1) ) = U ({fi }∞ i=0 , S, T , z).

(2.123)

Let an integer T > S. By (2.121)–(2.123), T −1 t =S

ft (xt(T ) , xt(T+1) ) ≤

T −1

ft (xt , xt +1 ) ≤

T −1

t =S

t =S

f

f

ft (xt , xt +1 ) + c0 .

(2.124)

By (2.122) and (2.124) for each pair of integers τ1 , τ2 satisfying S ≤ τ1 < τ2 ≤ T , τ 2 −1 t =τ1

ft (xt(T ) , xt(T+1) ) ≤

T −1 t =S

ft (xt(T ) , xt(T+1) )

 (T ) (T ) {ft (xt , xt +1 ) : t is an integer, S ≤ t < τ1 }  (T ) (T ) − {ft (xt , xt +1 ) : t is an integer, τ2 ≤ t ≤ T − 1}

−

≤

T −1 t =S

f

f

ft (xt , xt +1 ) + c0

 f f {ft (xt , xt +1 ) : t is an integer, S ≤ t < τ1 } + cf  f f − {ft (xt , xt +1 ) : t is an integer, τ2 ≤ t < T2 } + cf −

=

τ 2 −1 t =τ1

f

f

ft (xt , xt +1 ) + 2cf + c0 .

(2.125)

Clearly, there exists a strictly increasing sequence of natural numbers {Tk }∞ k=1 such that T1 > S and for each integer t ≥ S there exists xt∗ = lim xt

(Tk )

k→∞

.

(2.126)

By (2.125) and (2.126), for each pair of integers τ1 , τ2 satisfying S ≤ τ1 < τ2 , τ 2 −1 t =τ1

ft (xt∗ , xt∗+1 ) ≤

τ 2 −1 t =τ1

f

f

ft (xt , xt +1 ) + 2cf + c0 .

(2.127)

2.8 Proof of Theorem 2.4

65

∞ By (2.127) and Proposition 2.1 the program {xt∗ }∞ t =S is ({fi }i=0 )-good and by ATP

lim ρ(xt∗ , xt ) = 0. f

(2.128)

t →∞

In view of Theorem 2.4, in order to complete the proof of the theorem it is sufficient ∞ to show that the program {xt∗ }∞ t =S is ({fi }i=0 )-minimal. S0 Assume the contrary. Then there exist an integer S0 > S, a program {xt }t =S and a number Δ > 0 such that xS = xS∗ = z, xS 0 = xS∗0 , S 0 −1 t =S

ft (xt∗ , xt∗+1 ) >

S 0 −1 t =S

(2.129) ft (xt , xt +1 ) + Δ.

(2.130)

By (C3) there is a number 0 such that 0 ∈ (0, Δ/4), 0 < γf /4

(2.131) f

such that for each integer t ≥ 0 and each (x, y) ∈ Ωt satisfying ρ(x, xt ) ≤ 0 , f ρ(y, xt +1) ≤ 0 , f

f

|ft (xt , xt +1 ) − ft (x, y)| ≤ Δ/8.

(2.132)

By (C4) there is a number δ ∈ (0, 0 /4)

(2.133)

such that the following property holds: (ix) For each integer t ≥ 0, each (ut , ut +1 ) ∈ Ωt and each (u t +1 , u t +2 ) ∈ Ωt +1 satisfying ρ(ut , xt ) ≤ 2δ, ρ(u t +2 , xt +2 ) ≤ 2δ f

f

there is x ∈ X such that (ut , x) ∈ Ωt , (x, u t +2 ) ∈ Ωt +1, ρ(x, xt +1) ≤ 0 . f

By (2.128), there is a natural number S1 > S0 such that ρ(xt∗ , xt ) ≤ δ for all integers t ≥ S1 . f

(2.134)

66

2 Turnpike Conditions for Optimal Control Systems

By (2.126), there is a natural number k such that Tk > S1 + 4,

(2.135)

ρ(xt∗ , xt(Tk ) ) ≤ δ for all t = S, . . . , S1 + 4, .

(2.136)

k) ft (xt∗ , xt∗+1 ) ≤ ft (xt(Tk ) , xt(T+1 )+(8−1 Δ)(S1 +2)−1 , t = S, . . . , S1 +1.

(2.137)

By (2.134) and (2.136), for all integers t = S1 , . . . , S1 + 4, f

ρ(xt , xt(Tk ) ) ≤ 2δ.

(2.138)

By property (ix), (2.134), (2.135) and (2.138), there is y ∈ X such that k (xS∗1 , y) ∈ ΩS1 , (y, xS1 +2 ) ∈ ΩS1 +1 , ρ(y, xS1 +1 ) ≤ 0 .

f

(T )

(2.139)

Define x˜t = xt , t = S, . . . , S0 , x˜t = xt∗ , t = S0 + 1, . . . , S1 , (Tk )

x˜S1 +1 = y, x˜t = xt

, t = S1 + 2, . . . , Tk .

(2.140)

k By (2.129), (2.139) and (2.140), {x˜t }Tt =S is a program. By (2.122), (2.129), (2.130), (2.133)–(2.135), (2.137)–(2.140) and the choice of 0 (see (2.132)),

T k −1

(Tk )

ft (xt

t =S

=

S 1 +1

(Tk )

ft (xt

t =S

=

S 1 +1 t =S

−

S 1 −1

S 1 +1 t =S

T k −1

ft (x˜t , x˜t +1 )

t =S

(T )

k , xt +1 )−

k) ft (xt(Tk ) , xt(T+1 )−

t =S0

≥

(T )

k , xt +1 )−

S 1 +1

ft (x˜t , x˜t +1 )

t =S S 0 −1 t =S

ft (xt , xt +1 )

k) ft (xt∗ , xt∗+1 ) − fS1 (xS∗1 , y) − fS1 +1 (y, xS(T1 +2 )

ft (xt∗ , xt∗+1 ) − 8−1 Δ −

S 0 −1 t =S

ft (xt , xt +1 ) − fS1 (xS1 , xS1 +1 ) − Δ/8 f

f

2.9 An Example

67

f

f

− fS1 +1 (xS1 +1 , xS1 +2 ) − Δ/8 −

=

S 0 −1 t =S

ft (xt∗ , xt∗+1 ) −

S 0 −1 t =S

S 1 −1 t =S0

ft (xt∗ , xt∗+1 )

ft (xt , xt +1 ) + fS1 (xS∗1 , xS∗1 +1 )

+ fS1 +1 (xS∗1 +1 , xS∗1 +2 ) − fS1 (xS1 , xS1 +1 ) − fS1 +1 (xS1 +1 , xS1 +2 ) f

f

f

f

− (3/8)Δ > Δ − (3/8)Δ − (2/8)Δ > Δ/8. Combined with (2.129) and (2.140) this contradicts (2.122) and (2.123). The ∞ contradiction we have reached proves that the program {xt∗ }∞ t =0 is ({fi }i=0 )minimal. Theorem 2.4 is proved.

2.9 An Example We use the notation, definitions and assumptions of Sect. 2.1. For each integer t ≥ 0 let Ωt be a nonempty closed subset of the metric space f X × X. Assume that {xt }∞ t =0 is a program, r∗ > 0, for every integer t ≥ 0, f

f

{(z1 , z2 ) ∈ X × X : ρ1 ((z1 , z2 ), (xt , xt +1 )) ≤ r∗ } ⊂ Ωt , Lt : X × X → [0, ∞), πt : X → R 1 are continuous functions and f

f

Lt (xt , xt +1 ) = 0, ft (z1 , z2 ) = Lt (z1 , z2 ) + πt (z1 ) − πt (z2 ), z1 , z2 ∈ X, sup{Lt (z1 , z2 ) : z1 , z2 ∈ X, t = 0, 1, . . . } < ∞, sup{|πt (z)| : z ∈ X, t = 0, 1, . . . } < ∞. It is not difficult to see that assumptions (C1)–(C4) hold and ATP and property (P) hold if for every integer t ≥ 0, f

{(z1 , z2 ) ∈ X × X : Lt (z1 , z2 ) = 0} ⊂ {xt } × X and if each  > 0 there is δ > 0 such that for each integer t ≥ 0, each (z1 , z2 ) ∈ f X × X satisfying L(z1 , z2 ) ≤ δ, we have ρ(z1 , xt ) ≤ .

Chapter 3

Nonautonomous Problems with Perturbed Objective Functions

We study turnpike properties of approximate solutions of nonautonomous discretetime optimal control systems which are determined by sequences of lower semicontinuous objective functions. To have these properties means that the approximate solutions of the problems are determined mainly by the objective functions, and are essentially independent of the choice of intervals and endpoint conditions, except in regions close to the endpoints. We show that these turnpike properties are stable under small perturbations of the objective functions.

3.1 Preliminaries We use the notation, definitions and assumptions introduced in Chap. 2. Recall that for every nonempty set Y we denote by B(Y ) the collection of all bounded functions f : Y → R 1 and that for every function f ∈ B(Y ) we set f  = sup{|f (y)| : y ∈ Y }. For every nonempty compact metric space Y denote by C(Y ) the set of all continuous functions f : Y → R 1 . Let (X, ρ) be a compact metric space with the metric ρ. The set X × X is equipped with the metric ρ1 defined by ρ1 ((x1 , x2 ), (y1 , y2 )) = ρ(x1 , y1 ) + ρ(x2 , y2 ), (x1 , x2 ), (y1 , y2 ) ∈ X × X. For every nonnegative integer t let Ωt be a nonempty closed subset of the metric space X × X. Let T be a nonnegative integer. A sequence {xt }∞ t =T ⊂ X is called a program if (xt , xt +1 ) ∈ Ωt for all integers t ≥ T . © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_3

69

70

3 Nonautonomous Problems with Perturbed Objective Functions

2 Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A sequence {xt }Tt =T ⊂ X is called 1 a program if (xt , xt +1 ) ∈ Ωt for all integers t satisfying T1 ≤ t < T2 . We suppose that there exists a program {xt }∞ t =0 . Denote by M the set of all sequences of functions {ft }∞ t =0 such that for every nonnegative integer t ≥ 0

ft ∈ B(Ωt )

(3.1)

sup{ft  : t = 0, 1, . . . } < ∞.

(3.2)

and that

∞ For every pair of sequences of functions {ft }∞ t =0 , {gt }t =0 ∈ M define ∞ d({ft }∞ t =0 , {gt }t =0 ) = sup{ft − gt  : t = 0, 1, . . . }.

(3.3)

It is not difficult to see that d : M × M → [0, ∞) is a metric on M and that the metric space (M, d) is complete. Let {ft }∞ t =0 ∈ M. We consider the following optimization problems: T 2 −1 t =T1 T 2 −1 t =T1 T 2 −1 t =T1

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program, 1

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program and xT1 = y, 1

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program and xT1 = y, xT2 = z, 1

where y, z ∈ X and integers T1 , T2 satisfy 0 ≤ T1 < T2 . For every pair of points y, z ∈ X and every pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 define U ({ft }∞ t=0 , T1 , T2 ) = inf

2 −1  T

 T2 ft (xt , xt+1 ) : {xt }t=T is a program , 1

(3.4)

t=T1

U ({ft }∞ t=0 , T1 , T2 , y) = inf

2 −1  T

 2 ft (xt , xt+1 ) : {xt }Tt=T is a program and x = y , T 1 1

t=T1

(3.5) U ({ft }∞ t=0 , T1 , T2 , y, z) = inf

2 −1  T

ft (xt , xt+1 ) :

t=T1

 2 is a program and x = y, x = z . {xt }Tt=T T T 1 2 1

(3.6)

3.1 Preliminaries

71

Here we assume that the infimum over an empty set is ∞. Denote by Mreg the set of all sequences of functions {fi }∞ i=0 ∈ M for which f and positive numbers c and γf such that the there exist a program {xt }∞ f t =0 following assumptions hold: (C1) the function ft is lower semicontinuous for every nonnegative integer t; (C2) for every pair of nonnegative integers T2 > T1 , T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 ) + cf ; f

f

(C3) for every positive number  there exists a positive number δ such that for every f nonnegative integer t and every point (x, y) ∈ Ωt for which ρ(x, xt ) ≤ δ and f ρ(y, xt +1) ≤ δ the inequality f

f

|ft (xt , xt +1 ) − ft (x, y)| ≤  is true; (C4) for every nonnegative integer t, every point (xt , xt +1 ) ∈ Ωt for which f ρ(xt , xt ) ≤ γf and every point (xt +1 , xt +2 ) ∈ Ωt +1 for which ρ(xt +2, xt +2 ) ≤ γf there exists a point x ∈ X satisfying that f

(xt , x) ∈ Ωt , (x, xt +2 ) ∈ Ωt +1 ; moreover, for every positive number  there exists a number δ ∈ (0, γf ) such that for every nonnegative integer t, every point (xt , xt +1 ) ∈ Ωt and every f f point (xt +1 , xt +2 ) ∈ Ωt +1 for which ρ(xt , xt ) ≤ δ and ρ(xt +2, xt +2 ) ≤ δ there exists a point x ∈ X such that (xt , x) ∈ Ωt , (x, xt +2 ) ∈ Ωt +1 , ρ(x, xt +1) ≤ . f

¯ reg the closure of Mreg in (M, d). Denote by Mc,reg the set of Denote by M all sequences {fi }∞ i=0 ∈ Mreg such that fi ∈ C(Ωi ) for every nonnegative integer i ¯ and by Mc,reg the closure of Mc,reg in (M, d). We study the optimization problems stated above with the sequence of objective functions {fi }∞ i=0 ∈ Mreg . Our study is based on the relation between these finite horizon problems and the corresponding infinite horizon optimization problem f ∞ determined by {fi }∞ i=0 . Note that the condition (C2) means that the program {xt }t =0 is an approximate solution of this infinite horizon problem. f ∞ Let {fi }∞ i=0 ∈ Mreg and let a program {xi }i=0 and positive numbers cf and γf be such that assumptions (C1)–(C4) hold.

72

3 Nonautonomous Problems with Perturbed Objective Functions

∞ Recall that a program {xt }∞ t =S , where S ≥ 0 is an integer, is called ({fi }i=0 )-good if the sequence −1  T

fi (xi , xi+1 ) −

i=S

T −1

f

f

fi (xi , xi+1 )

i=S

∞ T =S+1

is bounded. Recall that the sequence {fi }∞ i=0 possesses an asymptotic turnpike property (ATP) f being the turnpike if for each integer S ≥ 0 and each ({fi }∞ with {xi }∞ i=0 i=0 )-good ∞ program {xi }i=S , f

lim ρ(xi , xi ) = 0.

i→∞

Recall that the sequence {fi }∞ i=0 has a turnpike property (TP) if for every positive number  and every positive number M there exist a positive number δ and an integer L ≥ 1 such that for every pair of integers T1 ≥ 0, T2 ≥ T1 + 2L and every 2 program {xt }Tt =T satisfying 1 T 2 −1

fi (xi , xi+1 )

i=T1 ∞ ≤ min{U ({fi }∞ i=0 , T1 , T2 , xT1 , xT2 ) + δ, U ({fi }i=0 , T1 , T2 ) + M}, f

the inequality ρ(xi , xi ) ≤  is true for all natural numbers i = T1 + L, . . . , T2 − L. f ∞ The sequence {xi }∞ i=0 is called the turnpike of {fi }i=0 . ∞ By Theorem 2.2, the sequence {fi }i=0 possesses the turnpike property if and only if {fi }∞ i=0 has ATP and the following property: (P) For every positive number  and every positive number M there exist a positive number δ and an integer L ≥ 1 such that for every nonnegative integer +L T and every program {xt }Tt =T satisfying T +L−1 

fi (xi , xi+1 )

i=T ∞ ≤ min{U ({fi }∞ i=0 , T , T + L, xT , xT +L ) + δ, U ({fi }i=0 , T , T + L) + M} f

there exists an integer j ∈ {T , . . . , T + L} such that ρ(xj , xj ) ≤ . Property (P) means that if a natural number L is large enough and a program +L {xt }Tt =T is an approximate solution of the corresponding finite horizon problem, f then there is j ∈ {T , . . . , T + L} such that xj is close to xj .

3.2 Main Results

73

In this chapter we assume that {fi }∞ i=0 possesses TP and show that the turnpike property is stable under small perturbations of the objective functions. The results of this chapter were obtained in [132].

3.2 Main Results We suppose that the sum over an empty set is zero. f ∞ Let {fi }∞ i=0 ∈ Mreg and let a program {xi }i=0 , cf > 0 and γf ∈ (0, 1) be such that assumptions (C1)–(C4) hold. We suppose that {fi }∞ i=0 has ATP and property (P). Then in view of Theorem 2.2, the sequence of functions {fi }∞ i=0 has TP. In this chapter we prove the following three results which show that the turnpike property is stable under small perturbations of the objective functions. Theorem 3.1 Let  ∈ (0, 1) and M > 0. Then there exist a natural number L0 and a real number δ0 ∈ (0, min{, γf }) such that for each integer L1 ≥ L0 the following assertion holds with δ = (8L1 )−1 δ0 . Assume that integers T1 ≥ 0, T2 > T1 + 2L1 , {gi }∞ i=0 ∈ M satisfies ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ q

2 and that a program {xt }Tt =T and a finite sequence of integers {Si }i=0 (where q is a 1 natural number) satisfy

S0 = T1 , T2 ≥ Sq > T2 − L1 , Si+1 − Si ∈ [L0 , L1 ] for all integers i ∈ [0, q − 1], Si+1 −1



gt (xt , xt+1 ) ≤ U ({gj }∞ j =0 , Si , Si+1 ) + M for all i = 0, . . . , q − 1,

t=Si T 2 −1

gt (xt , xt+1 ) ≤ U ({gj }∞ j =0 , Sq−1 , T2 ) + M,

t=Sq−1 Si+2 −1



gt (xt , xt+1 ) ≤ U ({gj }∞ j =0 , Si , Si+2 , xSi , xSi+2 ) + δ0

t=Si T 2 −1 t=Sq−2

gt (xt , xt+1 ) ≤ U ({gj }∞ j =0 , Sq−2 , T2 , xSq−2 , xT2 ) + δ0 .

for all i = 0, . . . , q − 2,

74

3 Nonautonomous Problems with Perturbed Objective Functions

Then f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L1 , T2 − L1 ]. Theorem 3.2 Let  ∈ (0, 1) and M > 0. Then there exist a positive number δ < min{, γf } and a natural number L such that for each pair of integers T1 ≥ 0, T2 > T1 + 2L, each {gi }∞ i=0 ∈ M satisfying ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ, 2 and each program {xt }Tt =T which satisfies 1

T 2 −1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , T1 , T2 ) + M

t =T1

and τ +L−1  t =τ

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , τ, τ + L, xτ , xτ +L ) + δ

for each integer τ ∈ [T1 , T2 − L], the following inequality holds: f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L, T2 − L]. We denote by Card(A) the cardinality of the set A. Theorem 3.3 Let  ∈ (0, 1) and M > 0. Then there exist a positive number δ and a natural number L such that for each pair of integers T1 ≥ 0, T2 > T1 + L, each {gi }∞ i=0 ∈ M satisfying ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ

and such that the function gt is lower semicontinuous for all integers t ≥ 0, and 2 which satisfies each program {xt }Tt =T 1 T 2 −1 t =T1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , T1 , T2 ) + M

3.2 Main Results

75

the following inequality holds: f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > }) ≤ L. In the proof of Theorem 3.3 we use the following result. Proposition 3.4 There exist a positive number δ, a natural number L and a real number c0 > 0 such that for each {gi }∞ i=0 ∈ M satisfying ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ

and such that the function gt is lower semicontinuous for all integers t ≥ 0, there g exists a program {xt }∞ t =0 such that g

f

ρ(xt , xt ) ≤ γf /4 for all integers t ≥ L and that for each pair of integers S1 ≥ 0, S2 > S1 , S 2 −1 t =S1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , S1 , S2 ) + c0 . g

g

We prove the following generic result which shows that most of the elements ¯ c,reg possess the ¯ reg and M (in the sense of the Baire category) of the spaces M turnpike property. Many results of this kind for classes of variational problems are collected in [105]. Note that the generic approach of [105] is not limited to the turnpike property, but is also applicable to other problems in the optimization theory [119]. ¯ reg or M ¯ c,reg . Then there exists an everywhere Theorem 3.5 Let A be either M dense subset F ⊂ A which is a countable intersection of open subsets of A such ∗ ∞ that for each {gi }∞ i=0 ∈ F there exist a program {xt }t =0 and numbers c∗ > 0, γ∗ > 0 such that the following properties hold: (a) for each sufficiently large integer t ≥ 0, each (xt , xt +1 ) ∈ Ωt and each (xt +1 , xt +2 ) ∈ Ωt +1 satisfying ρ(xt , xt∗ ) ≤ γ∗ , ρ(xt +2, xt∗+2 ) ≤ γ∗ there is x ∈ X such that (xt , x) ∈ Ωt , (x, xt +2) ∈ Ωt +1 ;

76

3 Nonautonomous Problems with Perturbed Objective Functions

(b) for each pair of integers S1 ≥ 0, S2 > S1 , S 2 −1 t =S1

gt (xt∗ , xt∗+1 ) ≤ U ({gi }∞ i=0 , S1 , S2 ) + c∗ ;

(c) for each  > 0 and each M > 0 there exist a positive number δ and a natural number L such that for each pair of integers T1 ≥ 0, T2 > T1 + 2L, each {fi }∞ i=0 ∈ M satisfying ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ, 2 each program {xt }Tt =T which satisfies 1

T 2 −1

ft (xt , xt +1 ) ≤ U ({fi }∞ i=0 , T1 , T2 ) + M

t =T1

and τ +L−1  t =τ

ft (xt , xt +1 ) ≤ U ({fi }∞ i=0 , τ, τ + L, xτ , xτ +L ) + δ

for each integer τ ∈ [T1 , T2 − L], the following inequality holds: ρ(xt , xt∗ ) ≤  for all integers t ∈ [T1 + L, T2 − L]. Theorem 3.1 is proved in Sect. 3.3, Theorem 3.2 is proved in Sect. 3.4, the proof of Proposition 3.4 is given in Sects. 3.5 and 3.6 contains the proof of Theorem 3.3. Theorem 3.5 is proved in Sect. 3.7.

3.3 Proof of Theorem 3.1 In view of TP there are an integer L ≥ 1 and a positive number δ0 < min{, γf } for which the following property is true:

(3.7)

3.3 Proof of Theorem 3.1

77

2 (P1) For every pair of integers τ1 ≥ 0, τ2 ≥ τ1 + 2L and every program {xt }τt =τ 1 satisfying

τ 2 −1

fi (xi , xi+1 ) ≤ min{U ({fi }∞ i=0 , τ1 , τ2 , xτ1 , xτ2 ) + 2δ0 ,

i=τ1

U ({fi }∞ i=0 , τ1 , τ2 ) + 2M + cf + 2} we have f

ρ(xi , xi ) ≤  for every integer i = τ1 + L, . . . , τ2 − L. Fix an integer L0 > 4L.

(3.8)

δ = (8L1 )−1 δ0 .

(3.9)

Let L1 ≥ L0 be a integer and let

Assume that integers T1 , T2 satisfy T1 ≥ 0, T2 > T1 + 2L1 ,

(3.10)

a sequence of functions {gi }∞ i=0 ∈ M satisfies ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ,

(3.11) q

2 a program {xt }Tt =T and a finite sequence of integers {Si }i=0 (where q is a natural 1 number) satisfy

S0 = T1 , T2 ≥ Sq > T2 − L1 , Si+1 − Si ∈ [L0 , L1 ] for each integer i satisfying 0 ≤ i ≤ q − 1

(3.12) (3.13)

(note that (3.12), (3.13) imply that q ≥ 2), for each integer i ∈ [0, q − 1], Si+1 −1



t =Si T 2 −1 t =Sq−1

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Si , Si+1 ) + M,

(3.14)

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Sq−1 , T2 ) + M,

(3.15)

78

3 Nonautonomous Problems with Perturbed Objective Functions

and for every integer i ∈ [0, q − 2], Si+2 −1



t =Si T 2 −1 t =Sq−2

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Si , Si+2 , xSi , xSi+2 ) + δ0 ,

(3.16)

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Sq−2 , T2 , xSq−2 , xT2 ) + δ0 .

(3.17)

In order to complete the proof of the theorem it is sufficient to show that f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L1 , T2 − L1 ]. Assume that i ∈ [0, q − 1] is an integer. In view of (3.7), (3.9), (3.11), (3.13) and (3.14), Si+1 −1



Si+1 −1

ft (xt , xt+1 ) ≤

t=Si



gt (xt , xt+1 ) + δ(Si+1 − Si )

t=Si

≤ U ({gj }∞ j=0 , Si , Si+1 ) + M + δ(Si+1 − Si ) ≤ U ({fj }∞ j=0 , Si , Si+1 ) + M + 2δ(Si+1 − Si ) ∞ ≤ U ({fj }∞ j=0 , Si , Si+1 ) + M + 2δL1 ≤ U ({fj }j=0 , Si , Si+1 ) + M + 1.

(3.18) By (3.7), (3.9), (3.11)–(3.13) and (3.15), T 2 −1 t =Sq−1

ft (xt , xt +1 ) ≤

T 2 −1

gt (xt , xt +1 ) + δ(T2 − Sq−1 )

t =Sq−1

≤ U ({gj }∞ j =0 , Sq−1 , T2 ) + M + 2L1 δ ≤ U ({fj }∞ j =0 , Sq−1 , T2 ) + δ(T2 − Sq−1 ) + M + 2L1 δ ≤ U ({fj }∞ j =0 , Sq−1 , T2 ) + M + 4L1 δ ≤ U ({fj }∞ j =0 , Sq−1 , T2 ) + M + 1.

(3.19)

3.3 Proof of Theorem 3.1

79

Equations (3.9), (3.11), (3.13), and (3.16) imply that for every integer i ∈ [0, q − 2], we have Si+2 −1



Si+2 −1



ft (xt , xt +1 ) ≤

t =Si

gt (xt , xt +1 ) + δ(Si+2 − Si )

t =Si

≤ U ({gj }∞ j =0 , Si , Si+2 , xSi , xSi+2 ) + δ0 + 2L1 δ ≤ U ({fj }∞ j =0 , Si , Si+2 , xSi , xSi+2 ) + δ(Si+2 − Si ) + δ0 + 2L1 δ ≤ U ({fj }∞ j =0 , Si , Si+2 , xSi , xSi+2 ) + δ0 + 4L1 δ ≤ U ({fj }∞ j =0 , Si , Si+2 , xSi , xSi+2 ) + 2δ0 .

(3.20)

In view of (3.9), (3.11)–(3.13) and (3.17), T 2 −1

T 2 −1

ft (xt , xt +1 ) ≤

t =Sq−2

gt (xt , xt +1 ) + δ(T2 − Sq−2 )

t =Sq−2

≤ U ({gj }∞ j =0 , Sq−2 , T2 , xSq−2 , xT2 ) + δ0 + 3L1 δ ≤ U ({fj }∞ j =0 , Sq−2 , T2 , xSq−2 , xT2 ) + 3L1 δ + δ0 + 3L1 δ ≤ U ({fj }∞ j =0 , Sq−2 , T2 , xSq−2 , xT2 ) + 2δ0 .

(3.21)

Assumption (C2) and (3.18) imply that for every integer i ∈ [0, q − 2], we have Si+2 −1



Si+1 −1

ft (xt , xt +1 ) ≤

t =Si



Si+2 −1

ft (xt , xt +1 ) +

t =Si



ft (xt , xt +1 )

t =Si+1

∞ ≤ U ({fj }∞ j =0 , Si , Si+1 ) + M + 1 + U ({fj }j =0 , Si+1 , Si+2 ) + M + 1 Si+2 −1

≤



t =Si

ft (xt , xt +1 ) + 2M + 2 ≤ U ({fj }∞ j =0 , Si , Si+2 ) + cf + 2M + 2. f

f

(3.22) It follows from (3.18), (3.19) and assumption (C2) that T 2 −1

Sq−1 −1

ft (xt , xt +1 ) =

t =Sq−2



t =Sq−2

ft (xt , xt +1 ) +

T 2 −1

ft (xt , xt +1 )

t =Sq−1

∞ ≤ U ({fj }∞ j =0 , Sq−2 , Sq−1 ) + M + 1 + U ({fj }j =0 , Sq−1 , T2 ) + M + 1

≤

T 2 −1 t =Sq−2

ft (xt , xt +1 ) + 2M + 2 ≤ U ({fj }∞ j =0 , Sq−2 , T2 ) + cf + 2M + 2. f

f

(3.23)

80

3 Nonautonomous Problems with Perturbed Objective Functions

Property (P1), (3.8), (3.13), (3.20), (3.22) imply that for every integer i ∈ [0, q − 2], we have f

ρ(xt , xt ) ≤  for all integers t = Si + L, . . . , Si+2 − L.

(3.24)

In view of (3.8), (3.12), (3.13), (3.21), (3.23) and property (P1), we have f

ρ(xt , xt ) ≤  for all integers t = Sq−2 + L, . . . , T2 − L.

(3.25)

Assume that an integer τ ∈ [T1 + L1 , T2 − L1 ].

(3.26) f

In order to complete the proof of the theorem it is sufficient to show that ρ(xτ , xτ ) ≤ . Equations (3.12) and (3.26) imply that there exists a nonnegative integer i such that 0 ≤ i < q, τ ∈ [Si , Si+1 ].

(3.27)

In view of (3.12), (3.13), (3.26) and (3.27), S1 ≤ T1 + L1 and we may assume without loss of generality that i ≥ 1. By (3.24), f

ρ(xt , xt ) ≤  for all integers t = Si−1 + L, . . . , Si+1 − L.

(3.28)

There are two cases: τ ≤ Si+1 − L;

(3.29)

τ > Si+1 − L.

(3.30)

If Eq. (3.29) is true, then by (3.8), (3.13), (3.27) and (3.28), ρ(xτ , xτf ) ≤ . Assume that (3.30) is true. There are two cases: i + 1 < q;

(3.31)

i + 1 = q.

(3.32)

3.4 Proof of Theorem 3.2

81

Assume that (3.31) is valid. Then (3.24) is true and by (3.8), (3.13) and (3.27), ρ(xτ , xτf ) ≤ . Assume that (3.32) holds. Then it follows from (3.8), (3.13), (3.25)–(3.27), (3.32) f that ρ(xτ , xτ ) ≤ . Theorem 3.1 is proved.

3.4 Proof of Theorem 3.2 Fix a number d0 > sup{fi  : i = 0, 1, . . . }.

(3.33)

Property (P) implies that there exist δ0 ∈ (0, γf /4)

(3.34)

and an integer L0 ≥ 1 such that the following property holds: +L0 satisfying (P2) For every nonnegative integer T and every program {xt }Tt =T T +L 0 −1 

fi (xi , xi+1 )

i=T ∞ ≤ min{U ({fi }∞ i=0 , T , T + L0 , xT , xT +L0 ) + δ0 , U ({fi }i=0 , T , T + L0 ) + cf + 4} f

there exists an integer j ∈ {T , . . . , T + L0 } such that ρ(xj , xj ) ≤ γf /4. In view of Theorem 3.1, there exist an integer L1 ≥ 4 and δ1 ∈ (0, min{, γf })

(3.35)

such that for every natural number L2 ≥ L1 , every pair of integers T1 ≥ 0, T2 > T1 + 2L2 and every sequence {gi }∞ i=0 ∈ M which satisfies ∞ −1 d({ft }∞ t =0 , {gt }t =0 ) ≤ (8L2 ) δ1

(3.36)

the following property holds: q 2 (P3) Assume that {xt }Tt =T is a program, q ≥ 1 is an integer, {Si }i=1 is a finite 1 sequence of integers such that: S1 = T1 , T2 ≥ Sq > T2 − L2 ,

82

3 Nonautonomous Problems with Perturbed Objective Functions

for every natural number i satisfying 1 ≤ i ≤ q − 1, Si+1 − Si ∈ [L1 , L2 ], Si+1 −1



t =Si T 2 −1 t =Sq−1

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Si , Si+1 ) + M + cf + 8d0 + 1,

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Sq−1 , T2 ) + M + cf + 8d0 + 1,

and for every integer i ∈ [1, q − 2], Si+2 −1



t =Si T 2 −1 t =Sq−2

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Si , Si+2 , xSi , xSi+2 ) + δ1 ,

gi (xi , xi+1 ) ≤ U ({gj }∞ j =0 , Sq−2 , T2 , xSq−2 , xT2 ) + δ1 .

Then f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L2 , T2 − L2 ]. Fix an integer p0 ≥ 6 + M + 8(L0 + 1)(d0 + 1) + L1

(3.37)

L2 = (p0 + 4)L0 + 2L1 .

(3.38)

L ≥ 4L0 p0 + L2

(3.39)

δ < min{(4L0 )−1 δ0 , (8L2 )−1 δ1 }.

(3.40)

T1 ≥ 0, T2 > T1 + 2L,

(3.41)

and set

Choose an integer

and a positive number

Assume that integers

3.4 Proof of Theorem 3.2

83

{gi }∞ i=0 ∈ M satisfies ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ,

(3.42)

2 a program {xt }Tt =T satisfies 1

T 2 −1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , T1 , T2 ) + M

(3.43)

t =T1

and τ +L−1  t =τ

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , τ, τ + L, xτ , xτ +L ) + δ

(3.44)

for each integer τ ∈ [T1 , T2 − L]. In order to complete the proof of the theorem it is sufficient to show that f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L, T2 − L].

(3.45)

We show that the following property holds: (P4) If an integer S satisfies f

S ∈ [T1 , T2 − L0 ], min{ρ(xt , xt ) : t = S, . . . , S + L0 } > γf /4,

(3.46)

then S+L 0 −1 

gt (xt , xt +1 ) ≥

t =S

S+L 0 −1  t =S

f

f

gt (xt , xt +1 ) + 3.

(3.47)

Assume that an integer S satisfies (3.46). By (3.39) and (3.44), S+L 0 −1 

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , S, S + L0 , xS , xS+L0 ) + δ.

(3.48)

t =S

In view of (3.42) and (3.48), S+L 0 −1  t =S

ft (xt , xt +1 ) ≤

S+L 0 −1 

gt (xt , xt +1 ) + δL0

t =S

≤ U ({gi }∞ i=0 , S, S + L0 , xS , xS+L0 ) + δ + δL0 ≤ U ({fi }∞ i=0 , S, S + L0 , xS , xS+L0 ) + δ(2L0 + 1) ≤ U ({fi }∞ i=0 , S, S + L0 , xS , xS+L0 ) + δ0 .

(3.49)

84

3 Nonautonomous Problems with Perturbed Objective Functions

Property (P2), (3.46) and (3.49) imply that S+L 0 −1 

ft (xt , xt +1 ) ≥ U ({fi }∞ i=0 , S, S + L0 ) + 4 + cf .

(3.50)

t =S

Assumption (C2) and (3.50) imply that S+L 0 −1 

ft (xt , xt +1 ) ≥

t =S

S+L 0 −1  t =S

f

f

ft (xt , xt +1 ) + 4.

(3.51)

It follows from (3.34), (3.40), (3.42) and (3.51) that S+L 0 −1 

gt (xt , xt+1 ) ≥

S+L 0 −1 

t=S

≥

S+L 0 −1 

ft (xt , xt+1 ) − δL0 ≥

S+L 0 −1 

t=S f

f

f

ft (xt , xt+1 ) + 4 − δL0

t=S

f

gt (xt , xt+1 ) + 4 − 2δL0 ≥

S+L 0 −1 

t=S

f

f

gt (xt , xt+1 ) + 3

t=S

and (3.47) holds. Thus property (P4) holds. We show that there exists an integer j ∈ {T1 , . . . , T2 } f such that ρ(xj , xj ) ≤ γf /4. Assume the contrary. Then f

ρ(xi , xi ) > γf /4 for all integers i = T1 , . . . , T2 .

(3.52)

There exists an integer p ≥ 1 for which pL0 ≤ T2 − T1 < (p + 1)L0 .

(3.53)

In view of (3.41) and (3.49), we have p ≥ 2p0 .

(3.54)

j ∈ [0, p − 1].

(3.55)

Assume that an integer

Property (P4), (3.52) and (3.55) imply that T1 +(j +1)L0 −1 t =T1 +j L0

gt (xt , xt +1 ) ≥

T1 +(j +1)L0 −1 t =T1 +j L0

f

f

gt (xt , xt +1 ) + 3.

3.4 Proof of Theorem 3.2

85

It follows from the inequality above, (3.33), (3.40), (3.42), (3.43), (3.53), (3.54) and (3.56) that M≥

T 2 −1

gt (xt , xt+1 ) −

t=T1

=

T 2 −1

f

f

gt (xt , xt+1 )

t=T1

p−1   T1 +(j+1)L  0 −1

gt (xt , xt+1 ) −

T1 +(j+1)L  0 −1

t=T1 +jL0

j=0

f

f



gt (xt , xt+1 )

t=T1 +jL0

 f f + {gt (xt , xt+1 ) − gt (xt , xt+1 ) : t is an integer such that T1 + pL0 ≤ t < T2 } ≥ 3p − 2L0 sup{gt  : t = 0, 1, . . . } ≥ 3p − 2L0 (d0 + 1) ≥ p0 − 2L0 (d0 + 1), p0 ≤ M + 2L0 (d0 + 1).

This contradicts (3.37). The contradiction we have reached proves that f

min{ρ(xi , xi ) : i = T1 , . . . , T2 } ≤ γf /4.

(3.56)

Assume that an integer τ satisfies T1 ≤ τ, ρ(xτ , xτf ) ≤ γf /4, L0 (1 + p0 ) + τ ≤ T2 .

(3.57)

We show that there is an integer S such that f

T2 ≥ S ≥ τ + L0 , ρ(xS , xS ) ≤ γf /4.

(3.58)

Assume the contrary. Then f

ρ(xS , xS ) > γf /4 for all integers S = τ + L0 , , . . . , T2 .

(3.59)

In view of (3.57) there exists an integer p ≥ 1 such that pL0 ≤ T2 − (τ + L0 ) < (p + 1)L0 .

(3.60)

Equations (3.57) and (3.60) imply that p0 ≤ p.

(3.61)

Property (P4), (3.59) and (3.60) imply that τ +(j +2)L  0 −1 t=τ +(j +1)L0

gt (xt , xt+1 ) ≥

τ +(j +2)L  0 −1

f

f

gt (xt , xt+1 ) + 3 for all integers j ∈ [0, p − 1].

t=τ +(j +1)L0

(3.62)

86

3 Nonautonomous Problems with Perturbed Objective Functions

Assumption (C4) and Eq. (3.57) imply that there exists a point z ∈ X such that f

(xτ , z) ∈ Ωτ , (z, xτ +2 ) ∈ Ωτ +1 .

(3.63)

2 for which In view of (3.63), there exists a program {x˜t }Tt =T 1

f

x˜t = xt , t = T1 , . . . , τ, x˜τ +1 = z, x˜t = xt , t = τ + 2, . . . , T2 .

(3.64)

It follows from (3.33), (3.42), (3.43), (3.60)–(3.62) and (3.64) that M≥

T 2 −1

gt (xt , xt +1 ) −

t =T1

=

gt (xt , xt +1 ) −

t =τ

≥

gt (x˜t , x˜t +1 )

t =T1

τ +L 0 −1 

−

T 2 −1

τ +L 0 −1 

T 2 −1

gt (x˜t , x˜t +1 ) +

t =τ

gt (xt , xt +1 )

t =τ +L0

f f gτ +L0 (x˜τ +L0 , x˜τ +L0 +1 ) + gτ +L0 (xτ +L0 , xτ +L0 +1 ) −

p−1 +2)L0 −1   τ +(j  j =0

gt (xt , xt +1 ) −

t =τ +(j +1)L0

τ +(j  +2)L0 −1 t =τ +(j +1)L0

f

f

T 2 −1 t =τ +L0

f

f

gt (xt , xt +1 )



gt (xt , xt +1 )

− 2(L0 + 1) sup{gt  : t = 0, 1, . . . }  f f + {gt (xt , xt +1 ) − gt (xt , xt +1 ) : t is an integer such that τ + (p + 1)L0 ≤ t < T2 } ≥ 3p − 4(1 + L0 ) sup{gt  : t = 0, 1, . . . } ≥ 3p − 4(1 + L0 )(d0 + 1), p0 ≤ p ≤ M + 4(1 + L0 )(d0 + 1). This contradicts (3.37). The contradiction we have reached proves that there is an integer S satisfying (3.58). Thus we have shown that the following property holds: (P5) For every integer τ satisfying (3.57) there exists an integer S satisfying (3.58). We show that the following property holds: (P6) For every integer τ such that T2 ≥ τ, τ − L0 (1 + p0 ) ≥ T1 , ρ(xτ , xτf ) ≤ γf /4

(3.65)

there exists an integer S for which f

T1 ≤ S ≤ τ − L0 , ρ(xS , xS ) ≤ γf /4.

(3.66)

3.4 Proof of Theorem 3.2

87

Let an integer τ satisfy (3.65). We show that there exists an integer S for which (3.66) is true. Assume the contrary. Then f

ρ(xt , xt ) > γf /4 for all integers t = T1 , . . . , τ − L0 .

(3.67)

In view of (3.65), there exists an integer p ≥ 1 such that pL0 ≤ τ − L0 − T1 < (p + 1)L0 .

(3.68)

Equations (3.65) and (3.68) imply that p0 ≤ p.

(3.69)

Property (P4), (3.67) and (3.68) imply that τ −(j +1)L  0 −1

gt (xt , xt+1 ) ≥

t=τ −(j +2)L0

τ −(j +1)L  0 −1

f

f

gt (xt , xt+1 ) + 3 for all integers j ∈ [0, p − 1].

t=τ −(j +2)L0

(3.70) In view of (3.65) and assumption (C4) there exists a point z ∈ X such that f

(xτ −2 , z) ∈ Ωτ −2 , (z, xτ ) ∈ Ωτ −1 .

(3.71)

2 such that Equation (3.71) implies that there exists a program {x˜t }Tt =T 1

f

x˜t = xt , t = T1 , . . . , τ − 2, x˜ τ −1 = z, x˜t = xt , t = τ, . . . , T2 .

(3.72)

It follows from (3.33), (3.42), (3.43), (3.68), (3.70) and (3.72) that M≥

=

T 2 −1

gt (xt , xt +1) −

t =T1

t =T1

τ −1

τ −1 

gt (xt , xt +1 ) −

t =T1

=

T 2 −1

+ +

gt (x˜t , x˜t +1 )

t =T1

τ −L 0 −1  t =T1

gt (x˜t , x˜t +1 )

gt (xt , xt +1 ) −

τ −1−L 0 t =T1

f

f

gt (xt , xt +1 )

f f gτ −L0 −1 (xτ −L0 −1 , xτ −L0 ) − gτ −L0 −1 (x˜τ −L0 −1 , x˜τ −L0 )



{gt (xt , xt +1 ) − gt (x˜t , x˜t +1 ) : t is an integer such that τ − L0 ≤ t < τ }

88

3 Nonautonomous Problems with Perturbed Objective Functions

≥

p−1 +1)L0 −1   τ −(j  j =0

t =τ −(j +2)L0

f

f

(gt (xt , xt +1 ) − gt (xt , xt +1 )



− 2(2L0 + 1) sup{gt  : t = 0, 1, . . . } ≥ 3p − 2(1 + 2L0 )(d0 + 1), p0 ≤ p ≤ M + 2(1 + 2L0 )(d0 + 1). This contradicts (3.37). The contradiction we have reached proves that there exists an integer S satisfying equation (3.66) and that property (P6) holds. Properties (P5) and (P6) and Eq. (3.56) imply that there exist an integer q ≥ 1 q and a finite strictly increasing sequence of integers {Si }i=1 ⊂ [T1 , T2 ] such that for every integer t ∈ {T1 , . . . , T2 }, f

ρ(xt , xt ) ≤ γf /4 if and only if t ∈ {S1 , . . . , Sq },

(3.73)

q ≥ 2,

(3.74)

S1 < T1 + L0 (1 + p0 ), Sq > T2 − L0 (1 + p0 ).

(3.75)

Let an integer k ∈ [1, q − 1].

(3.76)

Sk+1 − Sk ≤ (p0 + 2)L0 .

(3.77)

Sk+1 − Sk > (p0 + 2)L0 .

(3.78)

We show that

Assume the contrary. Then

In view of (3.78), there exists an integer p ≥ 1 such that p > p0 , (p + 2)L0 ≤ Sk+1 − Sk < (p + 3)L0 .

(3.79)

In view of (3.73), f

ρ(xt , xt ) > γf /4, t = Sk + 1, . . . , Sk+1 − 1.

(3.80)

j ∈ [0, p − 1].

(3.81)

Let an integer

3.4 Proof of Theorem 3.2

89

Property (P4) and Eqs. (3.79)–(3.81) imply that Sk +(j +2)L0 −1

gt (xt , xt +1 ) ≥

t =Sk +(j +1)L0

Sk +(j +2)L0 −1 t =Sk +(j +1)L0

f

f

gt (xt , xt +1 ) + 3.

(3.82)

It follows from (3.73) and assumption (C4) that there exist points z1 , z2 ∈ X such that f

(xSk , z1 ) ∈ ΩSk , (z, xSk +2 ) ∈ ΩSk +1 ,

(3.83)

f

(xSk+1 −2 , z2 ) ∈ ΩSk+1 −2 , (z2 , xSk +1 ) ∈ ΩSk+1 −1 .

(3.84)

2 such that In view of (3.83) and (3.84), there exists a program {x˜t }Tt =T 1

x˜t = xt , t = T1 , . . . , Sk , x˜Sk +1 = z1 , f

x˜t = xt , t = Sk + 2, . . . , Sk+1 − 2, x˜Sk+1 −1 = z2 , x˜t = xt , t = Sk+1 , . . . , T2 .

(3.85)

It follows from (3.33), (3.42), (3.43), (3.79), (3.82) and (3.85) that M≥

T 2 −1

gt (xt , xt +1 ) −

t =T1

T 2 −1

Sk+1 −1

=



Sk+1 −1

gt (xt , xt +1 ) −

t =Sk

=



gt (x˜t , x˜t +1 )

t =Sk

Sk +L 0 −1 t =Sk

+

gt (x˜t , x˜t +1 )

t =T1

f

p−1 +2)L0 −1   Sk +(j j =0

f

(gt (xt , xt +1 ) − gt (xt , xt +1 ))

t =Sk +(j +1)L0

gt (xt , xt +1 ) −

Sk +(j +2)L0 −1 t =Sk +(j +1)L0

f

f



gt (xt , xt +1 )

 f f + {gi (xi , xi+1 ) − gi (x˜i , x˜i+1 ) : i = Sk , Sk + 1, Sk+1 − 2, Sk+1 − 1} Sk+1 −1

+



(gt (xt , xt +1 ) − gt (x˜t , x˜t +1 ))

t =Sk +(p+1)L0

≥ −2L0 (sup{gt  : t = 0, 1, . . . }) + 3p − 8(sup{gt  : t = 0, 1, . . . }) − 4L0 sup{gt  : t = 0, 1, . . . }

90

3 Nonautonomous Problems with Perturbed Objective Functions

≥ 3p − (d0 + 1)(6L0 + 8), p0 ≤ p ≤ M + (8 + 6L0 )(d0 + 1). This contradicts (3.37). The contradiction we have reached proves (3.77) for all integers k ∈ [1, q − 1]. Now we show that property (P3) implies Eq. (3.45). We set S˜1 = S1 . Assume that j ≥ 1 is an integer and that we have already defined a finite strictly increasing sequence of integers {S˜i }i=1 ⊂ {S1 , . . . , Sq } j

(3.86)

such that for every integer i satisfying 1 ≤ i < j , we have S˜i+1 − S˜i ∈ [L1 , L2 ].

(3.87)

(Evidently, for j = 1 our assumption is true.) If S˜j > Sq − L2 , then the construction is completed. Assume that S˜j ≤ Sq − L2 .

(3.88)

Then in view of Eqs. (3.38) and (3.88), we have S˜j + L1 ≤ Sq − L2 + L1 ≤ Sq − (p0 + 4)L0 .

(3.89)

It follows from (3.86) and (3.89) that there exists an integer k ≥ 2 such that S˜j + L1 ∈ [Sk−1 , Sk ].

(3.90)

Equations (3.38), (3.77), (3.86), (3.89), and (3.90) imply that S˜j ≤ Sk−1 , k < q. We set S˜j +1 = Sk and in view of Eq. (3.90), we have L1 ≤ S˜j +1 − S˜j ≤ L1 + Sk − Sk−1 ≤ L1 + (p0 + 2)L0 ≤ L2 and the assertion made for j is also true for j + 1. Clearly, our construction is completed after a finite number of steps and we obtain a finite strictly increasing sequence of integers {S˜i }ki=1 ⊂ {S1 , . . . , Sq }, where k ≥ 1 is an integer, such that S˜1 = S1 , S˜k > Sq − L2 , for all integers i satisfying 1 ≤ i < k, (3.87) holds. (3.91)

3.4 Proof of Theorem 3.2

91

It follows from (3.39), (3.41), (3.75) and (3.91) that S˜k − S˜1 > Sq − L2 − S1 > T2 − T1 − 2L0 (p0 + 1) − L2 > 2L − 2L0 (p0 + 1) − L2 > 2L2 .

(3.92)

In view of (3.40) and (3.42), Eq. (3.36) is valid. We apply property (P3) to the program {xt }

S˜ k . t =S˜ 1

In view of Eq. (3.44), for every pair of integers τ1 , τ2 satisfying S˜1 ≤ τ1 < τ2 ≤ S˜k , τ2 − τ1 ≤ L,

(3.93)

we have τ 2 −1 t =τ1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , τ1 , τ2 , xτ1 , xτ2 ) + δ.

(3.94)

Let an integer j satisfy j ∈ [1, k − 1].

(3.95)

In view of (3.73), (3.87) and (3.91), we have 4 ≤ L1 ≤ S˜j +1 − S˜j ≤ L2 , f Sj

(3.96) f ) Sj+1

ρ(xS˜j , x ˜ ) ≤ γf /4, ρ(xS˜j+1 , x ˜

≤ γf /4.

(3.97)

Assumption (C4), (3.96) and (3.97) imply that there exist points z1 , z2 ∈ X such that f ) Sj +2

(xS˜j , z1 ) ∈ ΩS˜j , (z1 , x ˜ f , z2 ) Sj+1 −2

(x ˜

∈ ΩS˜j +1 ,

∈ ΩS˜j+1 −2 , (z2 , xS˜j+1 ) ∈ ΩS˜j+1 −1 .

In view of (3.98), there is a program {x˜t }

S˜ j+1 t =S˜ j

(3.98)

such that

x˜S˜j = xS˜j , x˜S˜j +1 = z1 , x˜t = xt , t = S˜j + 2, . . . , S˜j +1 − 2, f

x˜S˜j+1 −1 = z2 , x˜S˜j+1 = xS˜j+1 .

(3.99)

92

3 Nonautonomous Problems with Perturbed Objective Functions

It follows from (3.39), (3.42), (3.93), (3.94), (3.96) and (3.99) that S˜j+1 −1



S˜j+1 −1

gt (xt , xt +1 ) ≤

t =S˜ j



gt (x˜t , x˜t +1 ) + δ

t =S˜ j S˜j+1 −1

≤



ft (x˜t , x˜t +1 ) + δ(S˜j +1 − S˜j ) + δ

t =S˜ j S˜j+1 −1

≤



ft (x˜t , x˜t +1 ) + δ(L2 + 1).

(3.100)

t =S˜ j

By (3.33), (3.99), (3.100) and (C2), S˜ j+1 −1



S˜ j+1 −1

gt (xt , xt +1 ) ≤

t =S˜ j



ft (x˜t , x˜t +1 ) + δ(L2 + 1)

t =S˜ j S˜ j+1 −1

≤



t =S˜ j

f

f

ft (xt , xt +1 ) + 8d0 + δ(L2 + 1)

˜ ˜ ≤ U ({fi }∞ i=0 , Sj , Sj +1 ) + cf + 8d0 + δ(L2 + 1).

(3.101)

It follows from (3.40), (3.42), (3.96) and (3.101) that S˜ j+1 −1



˜ ˜ gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , Sj , Sj +1 ) + cf + 8d0 + δ(2L2 + 1)

t =S˜ j

˜ ˜ ≤ U ({gi }∞ i=0 , Sj , Sj +1 ) + cf + 8d0 + 1.

(3.102)

It follows from (3.36), (3.92)–(3.94), (3.96), (3.102) and property (P3) applied to the program {x˜t }

S˜ k , t =S˜ 1

that

f ρ(xt , xt ) ≤  for all integers t ∈ [S˜1 + L2 , S˜k − L2 ].

(3.103)

In view of (3.39), (3.75) and (3.91), we have S˜1 + L2 = S1 + L2 < T1 + L2 + L0 (1 + p0 ) < T1 + L,

(3.104)

S˜k − L2 ≥ Sq − 2L2 > T2 − L0 (1 + p0 ) − 2L2 > T2 − L.

(3.105)

3.5 Proof of Proposition 3.4

93

It follows from (3.103)–(3.105) that f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L, T2 − L]. This completes the proof of Theorem 3.2.

3.5 Proof of Proposition 3.4 In view of Theorem 3.2, there is a positive number δ < γf /4 and an integer L ≥ 1 such that the following property holds: (P7) For every sequence of functions {gi }∞ i=0 ∈ M satisfying ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ, 2 every pair of integers T1 ≥ 0, T2 > T1 + 2L, every program {xt }Tt =T satisfying 1

T 2 −1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , T1 , T2 ) + δ

t =T1

we have f

ρ(xt , xt ) ≤ γf /4 for all integers t ∈ [T1 + L, T2 − L]. Fix an integer c0 > (4L + 8)(sup{fi } : i = 0, 1, . . . , } + 1).

(3.106)

Assume that a sequence of functions {gi }∞ i=0 ∈ M satisfies ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ

(3.107)

and that the function gt is lower semicontinuous for all nonnegative integers t. For (T ) every natural number T there exists a program {xt }Tt=0 such that T −1 t =0

gt (xt(T ) , xt(T+1) ) = U ({gi }∞ i=0 , 0, T ).

We show that the following property holds:

(3.108)

94

3 Nonautonomous Problems with Perturbed Objective Functions

(P8) For every natural number T and every pair of integers S1 , S2 satisfying 0 ≤ S1 < S2 ≤ T , we have S 2 −1 t =S1

gt (xt(T ) , xt(T+1) ) ≤ U ({gi }∞ i=0 , S1 , S2 ) + c0 .

(3.109)

Assume that integers S1 , S2 , T satisfy T ≥ 1, 0 ≤ S1 < S2 ≤ T . There are two cases: S2 − S1 ≤ 2L + 4;

(3.110)

S2 − S1 > 2L + 4.

(3.111)

Assume that Eq. (3.110) is true. In view of (3.106), (3.107) and (3.110), we have 2 −1  S    (T ) (T ) gt (xt , xt +1 ) − U ({gi }∞ , S , S )  i=0 1 2 

t =S1

≤ 2(S2 − S1 ) sup{gi } : i = 0, 1, . . . , } ≤ 2(2L + 4)(sup{fi } : i = 0, 1, . . . , } + 1) < c0 .

(3.112)

2 Assume that (3.111) is valid. Evidently, there is a program {xt }St =S such that 1

S 2 −1

gt (xt , xt +1 ) = U ({gi }∞ i=0 , S1 , S2 ).

(3.113)

t =S1

Property (P7), (3.107), (3.108), (3.111) and (3.113) imply that f

ρ(xt(T ) , xt ) ≤ γf /4, t = L, . . . , T − L, f

ρ(xt , xt ) ≤ γf /4, t = S1 + L, . . . , S2 − L.

(3.114) (3.115)

Assumption (C4), (3.114) and (3.115) imply that there exist points z1 , z2 ∈ X such that ) , z1 ) ∈ ΩS1 +L , (z1 , xS1 +L+2 ) ∈ ΩS1 +L+1 , (xS(T1 +L (T )

(xS2 −L−2 , z2 ) ∈ ΩS2 −L−2 , (z2 , xS2 −L ) ∈ ΩS2 −L−1 .

(3.116)

3.5 Proof of Proposition 3.4

95

Define (T )

x˜t = xt

, t = 0, . . . , S1 + L, x˜S1 +L+1 = z1 ,

x˜t = xt , t = S1 + L + 2, . . . , S2 − L − 2, x˜S2 −L−1 = z2 , x˜t = xt(T ) , t = S2 − L, . . . , T .

(3.117)

In view of (3.116) and (3.117), {x˜t }Tt=0 is a program. It follows from (3.106), (3.107), (3.111) and (3.117) that S 2 −1 2 −1  S    gt (xt , xt +1 ) − gt (x˜t , x˜t +1 )  t =S1

t =S1

+L+1  S 1  = gt (xt , xt +1 ) + t =S1

−

S 2 −1

gt (xt , xt +1 )

t =S2 −L−2

S 1 +L+1

gt (x˜t , x˜t +1 ) −

t =S1

  gt (x˜t , x˜t +1 )

S 2 −1 t =S2 −L−2

≤ sup{gi  : i = 0, 1, . . . }(4L + 8) ≤ (4L + 8)(sup{fi  : i = 0, 1, . . . } + 1) < c0 .

(3.118)

In view of (3.108), (3.113), (3.117) and (3.118), we have 0≤

T 

gt (x˜t , x˜t +1 ) −

t =0

=

S 2 −1

S 2 −1

gt (x˜t , x˜t +1 ) −

t =S1

(T )

gt (xt

(T )

, xt +1 ) ≤ c0 +

S 2 −1

gt (xt , xt +1 ) + c0 −

S 2 −1 t =S1

S 2 −1

(T )

, xt +1 )

(T )

gt (xt

t =S1

t =S1 S 2 −1

(T )

gt (xt

t =0

t =S1

≤

T 

(T )

, xt +1 )

gt (xt(T ) , xt(T+1) ),

gt (xt , xt +1 ) ≤ c0 + U ({gi }∞ i=0 , S1 , S2 ).

t =S1

Thus in both cases (3.109) is valid. Thus property (P8) holds. Property (P7), (3.107) and (3.108) imply that for every natural number T > 2L, we have (T )

ρ(xt

f

, xt ) ≤ γf /4, t = L, . . . , T − L.

(3.119)

96

3 Nonautonomous Problems with Perturbed Objective Functions

There exists a strictly increasing sequence of natural numbers {Tk }∞ k=1 such that for every nonnegative integer t there exists g

(Tk )

xt = lim xt k→∞

.

(3.120)

In view (3.120), for every integer t ≥ L, g

f

ρ(xt , xt ) ≤ γf /4. Property (P8) and Eq. (3.120) imply that for every pair of integers S1 ≥ 0, S2 > S1 , we have S 2 −1 t =S1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , S1 , S2 ) + c0 . g

g

Proposition 3.4 is proved.

3.6 Proof of Theorem 3.3 We suppose that the sum over an empty set is zero. In view of Proposition 3.4, there are δ0 > 0 and integers L0 ≥ 1, c0 ≥ 1 such that the following property holds: (P9) For every sequence of functions {gi }∞ i=0 ∈ M which satisfies ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ0

and such that the function gt is lower semicontinuous for every nonnegative integer g t there exists a program {xt }∞ t =0 such that g

f

ρ(xt , xt ) ≤ γf /4 for all integers t ≥ L0

(3.121)

and that for every pair of integers S1 ≥ 0, S2 > S1 , we have S 2 −1 t =S1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , S1 , S2 ) + c0 . g

g

(3.122)

Theorem 3.2 implies that there exist a positive number δ1 < min{, γf } and an integer L1 ≥ 1 such that the following property holds:

(3.123)

3.6 Proof of Theorem 3.3

97

(P10) For every sequence of functions {gi }∞ i=0 ∈ M which satisfies ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ1 , 2 satisfying every pair of integers T1 ≥ 0, T2 > T1 + 2L1 and every program {xt }Tt =T 1

T 2 −1

gt (xt , xt +1 ) ≤ min{U ({gi }∞ i=0 , T1 , T2 , xT1 , xT2 ) + δ1 ,

t =T1

U ({gi }∞ i=0 , T1 , T2 ) + M + c0 } we have f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L1 , T2 − L1 ]. Let δ = min{δ1 , δ2 }

(3.124)

L > (4L1 + 6)(δ −1 M + 1).

(3.125)

and an integer

Assume that a pair of integers T1 ≥ 0, T2 > T1 + L,

(3.126)

a sequence of functions {gi }∞ i=0 ∈ M satisfies ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ,

(3.127)

the function gt is lower semicontinuous for every nonnegative integer t and that a 2 program {xt }Tt =T satisfies 1 T 2 −1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , T1 , T2 ) + M.

(3.128)

t =T1

In order to complete the proof of the theorem it is sufficient to show that f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > }) ≤ L. Property (P9), (3.124) and (3.127) imply that there exists a program {xt }∞ t =0 such that Eq. (3.121) is valid and the following property hold: g

98

3 Nonautonomous Problems with Perturbed Objective Functions

(P11) Eq. (3.122) is true for every pair of integers S1 ≥ 0, S2 > S1 . We show that the following property holds: (P12) For every pair of integers S1 , S2 satisfying T1 ≤ S1 < S2 ≤ T2 , we have S 2 −1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , S1 , S2 ) + M + 2c0 .

t =S1

Assume that integers S1 , S2 satisfy T1 ≤ S1 < S2 ≤ T2 . In view of (3.122), (3.128) and property (P11), we have S 2 −1

gt (xt , xt +1 ) =

t =S1

T 2 −1

gt (xt , xt +1 )

t =T1

 {gt (xt , xt +1 ) : t is an integer, T1 ≤ t < S1 }  − {gt (xt , xt +1 ) : t is an integer, S2 ≤ t < T2 }

−

≤

T 2 −1 t =T1

g

g

gt (xt , xt +1 ) + M

 g g {gt (xt , xt +1 ) : t is an integer, T1 ≤ t < S1 } + c0  g g − {gt (xt , xt +1 ) : t is an integer, S2 ≤ t < T2 } + c0 −

=

S 2 −1 t =S1

g

g

gt (xt , xt +1 ) + M + 2c0 .

Thus property (P12) holds. Put t0 = T1 .

(3.129) q

By induction we define a finite strictly increasing sequence of integers {ti }i=0 ⊂ [T1 , T2 ] where q ≥ 1 is an integer such that: tq = T2 ;

(3.130)

(P13) for every integer i satisfying 0 ≤ i < q − 1, we have ti+1 −1

 t =ti

gt (xt , xt +1 ) > U ({gj }∞ j =0 , ti , ti+1 , xti , xti+1 ) + δ;

(3.131)

3.6 Proof of Theorem 3.3

99

(P14) if an integer i satisfies 0 ≤ i ≤ q − 1 and Eq. (3.131), then we have ti+1 −2



ti+1 > ti + 1 and

t =ti

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , ti , ti+1 − 1, xti , xti+1 −1 ) + δ. (3.132)

Assume that p is a nonnegative integer and that we have already defined a strictly p increasing sequence of integers {ti }i=0 ⊂ [T1 , T2 ] such that tp < T2 and that for every nonnegative integer i < p, Eqs. (3.131) and (3.132) are valid. (Note that for p = 0 our assumption holds.) We define tp+1 . There are two cases: T 2 −1 t =tp T 2 −1 t =tp

gt (xt , xt +1) ≤ U ({gi }∞ i=0 , tp , T2 , xtp , xT2 ) + δ;

(3.133)

gt (xt , xt +1 ) > U ({gi }∞ i=0 , tp , T2 , xtp , xT2 ) + δ.

(3.134)

Assume that Eq. (3.133) is valid. Then we set q = p+1, tq = T2 , the construction of the sequence is completed and properties (P13), (P14) hold. Assume that Eq. (3.134) is true. Define tp+1 = min{S ∈ {tp + 1, . . . , T2 } : S−1  t =tp

gt (xt , xt +1 ) > U ({gi }∞ i=0 , tp , S, xtp , xS ) + δ}.

(3.135)

Evidently, tp+1 is well-defined. If tp+1 = T2 , then we set q = p+1, the construction is completed and it is not difficult to see that properties (P13) and (P14) hold. Assume that tp+1 < T2 . Then it is not difficult to see that the assumption made for p is also true for p + 1. Clearly, our construction is completed after a final number of steps and let tq = T2 be its last element, where q ≥ 1 is an integer. It follows from the construction that properties (P13) and (P14) hold. In view of Eq. (3.128) and property (P13), we have M≥

T 2 −1

gt (xt , xt +1 ) − U ({gi }∞ i=0 , T1 , T2 )

t =T1

≥

−1   ti+1  t =ti

gt (xt , xt +1 ) − U ({gj }∞ j =0 , ti , ti+1 , xti , xti+1 ) :

100

3 Nonautonomous Problems with Perturbed Objective Functions

i is an integer, 0 ≤ i < q − 1



≥ δ(q − 1), q ≤ δ −1 M + 1.

(3.136)

Define A = {i ∈ {0, . . . , q − 1} : ti+1 − ti > 2L1 + 2}.

(3.137)

Let j ∈ A. Property (P14) implies that tj+1 −2



t =tj

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , tj , tj +1 − 1, xtj , xtj+1 −1 ) + δ.

(3.138)

In view of (P12), we have tj+1 −2



t =tj

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , tj , tj +1 − 1) + M + 2c0 .

(3.139)

It follows from (3.124), (3.127), (3.137)–(3.139) and property (P10) that f

ρ(xt , xt ) ≤  for all integers t = tj + L1 , . . . , tj +1 − 1 − L1 .

(3.140)

Equations (3.125), (3.130), and (3.140) imply that f

{t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > } ⊂



{{ti , . . . , ti+1 } : i ∈ {0, . . . , q − 1} \ A}

  {{ti , . . . , ti + L1 − 1} {ti+1 − 1 − L1 , . . . , ti+1 } : i ∈ A}

and by (3.137), we have f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > }) ≤ q(2L1 + 3) + qL1 + q(L1 + 2) ≤ q(4L1 + 5) ≤ (4L1 + 5)(δ −1 M + 1) < L. This completes the proof of Theorem 3.3.

3.7 Proof of Theorem 3.5

101

3.7 Proof of Theorem 3.5 ∞ For every sequence of functions {fi }∞ i=0 ∈ Mreg let a program {xt }t =0 , γf ∈ (0, 1), cf > 0 be such that assumptions (C1)–(C4) hold. Let Areg be the set of all sequences of functions {fi }∞ i=0 ∈ Mreg ∩ A which has the turnpike property. Proposition 2.6 implies that Areg is an everywhere subset of A. In view of Proposition 3.4, for every sequence of functions {fi }∞ i=0 ∈ Areg ∞ ) ≥ 1 and an open there exist a positive constant c({fi }∞ ), an integer L({f } i i=0 i=0 ∞ neighborhood U({fi }∞ i=0 ) of {fi }i=0 in A such that the following property holds: ∞ (P15) For every sequence of functions {gi }∞ i=0 ∈ U({fi }i=0 ) there exists a g ∞ program {xt }t =0 such that f

ρ(xt , xt ) ≤ γf /4 for all integers t ≥ L({fi }∞ i=0 ), g

f

for each pair of integers S1 ≥ 0, S2 > S1 , S 2 −1 t =S1

∞ gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , S1 , S2 ) + c({fi }i=0 ). g

g

Let {fi }∞ i=0 ∈ Areg , n be a natural number. Theorem 3.2 implies that there ∞ exist an open neighborhood U({fi }∞ i=0 , n) of {fi }i=0 in A, a positive number ∞ ∞ δ({fi }i=0 , n) and an integer L({fi }i=0 , n) ≥ 1 such that the following property holds: ∞ (P16) For every sequence of functions {gi }∞ i=0 ∈ U({fi }i=0 , n), every pair of T2 integers T1 ≥ 0, T2 > T1 + 2L({fi }∞ i=0 , n), every program {xt }t =T1 satisfying T 2 −1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , T1 , T2 ) + n

t =T1

and τ +L({fi }∞ i=0 ,n)−1



gt (xt , xt +1 )

t =τ ∞ ≤ U ({gi }∞ ) + δ({fi }∞ i=0 , τ, τ + L({fi }i=0 , n), xτ , xτ +L({fi }∞ i=0 , n) i=0 ,n)

for every integer τ ∈ [T1 , T2 − L({fi }∞ i=0 , n)] we have ∞ ρ(xt , xt ) ≤ (4n)−1 for all integers t ∈ [T1 + L({fi }∞ i=0 , n), T2 − L({fi }i=0 , n)]. f

102

3 Nonautonomous Problems with Perturbed Objective Functions

Set F=



∞ {U({fi }∞ i=0 ) : {fi }i=0 ∈ Areg }

∞   



 ∞ {U({fi }∞ i=0 , n) : {fi }i=0 ∈ Areg } .

(3.141)

n=1

Evidently, F is a countable intersection of open everywhere dense subsets of the complete metric space A. ∞ Let {gi }∞ i=0 ∈ F . In view of Eq. (3.141) there exist {fi }i=0 ∈ Areg such that ∞ {gi }∞ i=0 ∈ U({fi }i=0 ).

(3.142)

Let {xt }∞ t =0 be as guaranteed by property (P15). Put g

∞ c∗ = c({fi }∞ i=0 ), γ∗ = γf /4, L∗ = L({fi }i=0 ).

(3.143)

It follows from Eqs. (3.142), (3.143), property (P15) and assumption (C4) that the following properties hold: (P17) for every pair of integers S1 ≥ 0, S2 > S1 , we have S 2 −1 t =S1

gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , S1 , S2 ) + c∗ ; g

g

(3.144)

(P18) for every integer t ≥ L∗ , every (xt , xt +1 ) ∈ Ωt and every (xt +1, xt +2 ) ∈ Ωt +1 which satisfy ρ(xt , xt ) ≤ γ∗ , ρ(xt +2 , xt +2 ) ≤ γ∗ there exists a point x ∈ X such that g

g

(xt , x) ∈ Ωt , (x, xt +2) ∈ Ωt +1 . Assume that , M are positive numbers. Fix an integer n ≥ 1 such that n > M + c∗ , (4n)−1 < .

(3.145)

In view of (3.141), there exist {fi(n) }∞ i=0 ∈ Areg

(3.146)

3.7 Proof of Theorem 3.5

103

such that ∞ {gi }∞ i=0 ∈ U({fi }i=0 , n). (n)

(3.147)

Property (P17) implies that there exists an integer L¯ ≥ 1 such that for every pair of ¯ S2 > S1 , we have integers S1 ≥ L, S 2 −1 t =S1

(n) ∞ gt (xt , xt +1 ) ≤ U ({gi }∞ i=0 , S1 , S2 , xS1 , xS2 ) + δ({fi }i=0 , n). g

g

(3.148)

It follows from Eqs. (3.144)–(3.148) and property (P16) that g f ρ(xt , xt ) ≤ (4n)−1 for all integers t ≥ L¯ + L({fi(n) }∞ i=0 , n).

(3.149)

Choose a natural number ¯ L > L({fi(n) }∞ i=0 , n) + L.

(3.150)

∞ {hi }∞ i=0 ∈ U({fi }i=0 , n),

(3.151)

T1 ≥ 0, T2 > T1 + 2L,

(3.152)

Assume that (n)

a pair of integers

2 is a program such that {xt }Tt =T 1

T 2 −1

ht (xt , xt +1 ) ≤ U ({hi }∞ i=0 , T1 , T2 ) + M

(3.153)

t =T1

and for every integer τ ∈ [T1 , T2 − L], we have τ +L−1  t =τ

∞ ht (xt , xt +1 ) ≤ U ({hi }∞ i=0 , τ, τ +L, xτ , xτ +L )+δ({fi }i=0 , n). (n)

(3.154)

Property (P14) and Eqs. (3.145), (3.146), (3.150), (3.151), (3.153), and (3.154) imply that (fn )

ρ(xt , xt

) ≤ (4n)−1

(n) ∞ for all integers t = T1 + L({fi(n) }∞ i=0 , n), . . . , T2 − L({fi }i=0 , n).

(3.155)

104

3 Nonautonomous Problems with Perturbed Objective Functions

In view of (3.145), (3.149), (3.150), (3.152), (3.155), for all integers t = T1 + L, . . . , T2 − L, we have ρ(xt , xt ) ≤ 2n−1 < . g

This completes the proof of Theorem 3.5.

Chapter 4

Nonautonomous Problems with Discounting

We study existence and the turnpike property of solutions of discrete-time control systems with discounting and with a compact metric space of states. To have the turnpike property means that the approximate solutions of the problems are determined mainly by the objective functions, and are essentially independent of the choice of intervals and endpoint conditions, except in regions close to the endpoints. We show that this turnpike property is stable under small perturbations of the objective functions.

4.1 Preliminaries We use the notation, definitions and assumptions introduced in Chap. 2. Recall that for each nonempty set Y we denote by B(Y ) the set of all bounded functions f : Y → R 1 and for each f ∈ B(Y ) set f  = sup{|f (y)| : y ∈ Y }. For each nonempty compact metric space Y denote by C(Y ) the set of all continuous functions f : Y → R 1 . Let (X, ρ) be a compact metric space with the metric ρ. The set X × X is equipped with the metric ρ1 defined by ρ1 ((x1 , x2 ), (y1 , y2 )) = ρ(x1 , y1 ) + ρ(x2 , y2 ), (x1 , x2 ), (y1 , y2 ) ∈ X × X. For each integer t ≥ 0 let Ωt be a nonempty closed subset of the metric space X × X. Let T ≥ 0 be an integer. A sequence {xt }∞ t =T ⊂ X is called a program if (xt , xt +1 ) ∈ Ωt for all integers t ≥ T . © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_4

105

106

4 Nonautonomous Problems with Discounting

2 Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A sequence {xt }Tt =T ⊂ X is called 1 a program if (xt , xt +1 ) ∈ Ωt for all integers t satisfying T1 ≤ t < T2 . We assume that there exists a program {xt }∞ t =0 . Denote by M the set of all sequences of functions {ft }∞ t =0 such that for each integer t ≥ 0

ft ∈ B(Ωt )

(4.1)

sup{ft  : t = 0, 1, . . . } < ∞.

(4.2)

and that

∞ For each pair of sequences {ft }∞ t =0 , {gt }t =0 ∈ M set ∞ d({ft }∞ t =0 , {gt }t =0 ) = sup{ft − gt  : t = 0, 1, . . . }.

(4.3)

It is easy to see that d : M × M → [0, ∞) is a metric on M and that the metric space (M, d) is complete. Let {ft }∞ t =0 ∈ M. We consider the following optimization problems: T 2 −1 t =T1 T 2 −1 t =T1 T 2 −1 t =T1

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program, 1

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program and xT1 = y, 1

2 ft (xt , xt +1 ) → min s. t. {xt }Tt =T is a program and xT1 = y, xT2 = z, 1

where y, z ∈ X and integers T1 , T2 satisfy 0 ≤ T1 < T2 . For each y, z ∈ X and each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 set U ({ft }∞ t=0 , T1 , T2 ) = inf

2 −1  T

 2 ft (xt , xt+1 ) : {xt }Tt=T is a program , 1

(4.4)

t=T1

U ({ft }∞ t=0 , T1 , T2 , y) = inf

2 −1  T

 2 ft (xt , xt+1 ) : {xt }Tt=T is a program and x = y , T 1 1

t=T1

(4.5) U ({ft }∞ t=0 , T1 , T2 , y, z) = inf

2 −1  T

ft (xt , xt+1 ) :

t=T1

 2 {xt }Tt=T is a program and x = y, x = z . T T 1 2 1

(4.6)

4.1 Preliminaries

107

Here we assume that the infimum over an empty set is ∞. Denote by Mreg the set of all sequences of functions {fi }∞ i=0 ∈ M for which f and constants c > 0, γ > 0 such that the following there exist a program {xt }∞ f f t =0 assumptions hold: (C1) the function ft is lower semicontinuous for all integers t ≥ 0; (C2) for each pair of integers T1 ≥ 0, T2 > T1 , T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 ) + cf ; f

f

(C3) for each  > 0 there exists δ > 0 such that for each integer t ≥ 0 and each f f (x, y) ∈ Ωt satisfying ρ(x, xt ) ≤ δ, ρ(y, xt +1) ≤ δ we have f

f

|ft (xt , xt +1 ) − ft (x, y)| ≤ ; f

(C4) for each integer t ≥ 0, each (xt , xt +1 ) ∈ Ωt satisfying ρ(xt , xt ) ≤ γf and f each (xt +1 , xt +2 ) ∈ Ωt +1 satisfying ρ(xt +2, xt +2 ) ≤ γf there is x ∈ X such that (xt , x) ∈ Ωt , (x, xt +2 ) ∈ Ωt +1 ; moreover, for each  > 0 there exists δ ∈ (0, γf ) such that for each integer t ≥ f 0, each (xt , xt +1 ) ∈ Ωt and each (xt +1, xt +2 ) ∈ Ωt +1 satisfying ρ(xt , xt ) ≤ δ and ρ(xt +2 , xt +2 ) ≤ δ there is x ∈ X such that f

(xt , x) ∈ Ωt , (x, xt +2 ) ∈ Ωt +1 , ρ(x, xt +1) ≤ . f

¯ reg the closure of Mreg in (M, d). Denote by Mc,reg the set of Denote by M all sequences {fi }∞ i=0 ∈ Mreg such that fi ∈ C(Ωi ) for all integers i ≥ 0 and by ¯ c,reg the closure of Mc,reg in (M, d). M f ∞ Let {fi }∞ i=0 ∈ Mreg and let a program {xi }i=0 , cf > 0 and γf > 0 be such that (C1)–(C4) hold. ∞ A program {xt }∞ t =S , where S ≥ 0 is an integer, is called ({fi }i=0 )-good if the sequence −1  T i=S

is bounded.

fi (xi , xi+1 ) −

T −1 i=S

f

f

fi (xi , xi+1 )

∞ T =S+1

108

4 Nonautonomous Problems with Discounting

Recall that the sequence {fi }∞ i=0 possesses an asymptotic turnpike property (ATP) f ∞ with {xi }i=0 being the turnpike if for each integer S ≥ 0 and each ({fi }∞ i=0 )-good program {xi }∞ i=S , f

lim ρ(xi , xi ) = 0.

i→∞

Recall that the sequence {fi }∞ i=0 possesses a turnpike property (TP) if for each  > 0 and each M > 0 there exist δ > 0 and a natural number L such that for each 2 pair of integers T1 ≥ 0, T2 ≥ T1 + 2L and each program {xt }Tt =T which satisfies 1 T 2 −1

fi (xi , xi+1 )

i=T1 ∞ ≤ min{U ({fi }∞ i=0 , T1 , T2 , xT1 , xT2 ) + δ, U ({fi }i=0 , T1 , T2 ) + M}, f

the inequality ρ(xi , xi ) ≤  holds for all integers i = T1 + L, . . . , T2 − L. f ∞ The sequence {xi }∞ i=0 is called the turnpike of {fi }i=0 . ∞ By Theorem 2.2, the sequence {fi }i=0 possesses the turnpike property if and only if {fi }∞ i=0 possesses ATP and the following property: (P) For each  > 0 and each M > 0 there exist δ > 0 and a natural number L such +L that for each integer T ≥ 0 and each program {xt }Tt =T which satisfies T +L−1 

fi (xi , xi+1 )

i=T ∞ ≤ min{U ({fi }∞ i=0 , T , T + L, xT , xT +L ) + δ, U ({fi }i=0 , T , T + L) + M} f

there is an integer j ∈ {T , . . . , T + L} for which ρ(xj , xj ) ≤ . In the sequel we use the following notation. For each y, z ∈ X, each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 and each finite sequence of functions gi ∈ B(Ωi ), i = T1 , . . . , T2 − 1 set T2 −1 U ({gi }i=T , T1 , T2 ) = inf 1

2 −1  T

 2 gt (xt , xt+1 ) : {xt }Tt=T is a program , 1

(4.7)

t=T1 T2 −1 U ({gi }i=T , T1 , T2 , y) = inf 1

2 −1  T

 2 gt (xt , xt+1 ) : {xt }Tt=T is a program and x = y , T 1 1

t=T1

(4.8)

4.2 Main Results

109

T2 −1 U ({gi }i=T , T1 , T2 , y, z) = inf 1

2 −1  T

gt (xt , xt+1 ) :

t=T1

 2 {xt }Tt=T is a program and x = y, x = z . T T 1 2 1

(4.9)

Here again we assume that the infimum over an empty set is ∞.

4.2 Main Results We suppose that the sum over an empty set is zero. f ∞ Let {fi }∞ i=0 ∈ Mreg and let a program {xi }i=0 , cf > 0 and γf ∈ (0, 1) be such that (C1)–(C4) hold. We suppose that {fi }∞ i=0 possesses ATP and property (P). Then by Theorem 2.2, {fi }∞ possesses TP. i=0 Let S ≥ 0 be an integer. A point x ∈ X is called ({fi }∞ i=0 , S)-good if there exists ∞ such that x = x. an ({fi }∞ )-good program {x } t t =S S i=0 A point x ∈ X is called ({fi }∞ i=0 , S, M)-good, where M is a positive number, if there exists a program {xt }∞ t =S such that xS = x and for all integers T > S, T −1 t =S

ft (xt , xt +1 ) −

T −1 t =S

f

f

ft (xt , xt +1 ) ≤ M.

The following theorem is the main result of this chapter. Theorem 4.1 Let M > 0 and  ∈ (0, γf ). Then there exist a number δ ∈ (0, ), a natural number L and λ ∈ (0, 1) such that for each pair of integers T1 ≥ 0, T2 > T1 + 2L, each {gi }∞ i=0 ∈ M such that gi is a lower semicontinuous function for all integers i ≥ 0 and that ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ, 2 −1 each finite sequence {αi }Ti=T ⊂ (0, 1] such that 1

αi αj−1 ≥ λ for all i, j ∈ {T1 , . . . , T2 − 1} satisfying |i − j | ≤ L, αi αj−1 ≥ λ for all i, j ∈ {T1 , . . . , T2 − 1} satisfying j ≥ i 2 and each program {xt }Tt =T such that the point xT1 is ({fi }∞ i=0 , T1 , M)-good and 1

T 2 −1 t =T1

2 −1 αt gt (xt , xt +1 ) = U ({αi gi }Ti=T , T1 , T2 , xT1 ) 1

110

4 Nonautonomous Problems with Discounting

the following inequality holds: f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L, T2 − L]. Theorem 4.1 establishes the turnpike property of solutions of optimal finite horizon problems associated with the objective functions αi gi , i = T1 , . . . , T2 − 1, where the sequence of functions {gi }∞ i=0 ∈ M and the sequence of discount T2 −1 coefficients {αt }t =T1 ⊂ (0, 1] satisfy the assumptions of the theorem. Roughly speaking, the turnpike property holds if the sequence of functions {gi }∞ i=0 belongs T2 −1 to a small neighborhood {fi }∞ and the discount coefficients {α } ⊂ (0, 1] are t t =T1 i=0 changed rather slowly. Let S ≥ 0 be an integer and gi ∈ B(Ωi ) for all integers i ≥ S. A program ∞

∞ {xt }∞ t =S is called ({gi }i=S )-overtaking optimal if for each program {xt }t =S satisfying

xS = xS , lim sup T →∞

−1

T

gt (xt , xt +1 ) −

t =S

T −1 t =S

 gt (xt , xt +1 ) ≤ 0.

Note that the existence of a ({gi }∞ i=S )-overtaking optimal program when the functions {gi }∞ i=S tend to zero rapidly is a well-known fact. Here we present a version of this result. Theorem 4.2 Let {gi }∞ i=0 ⊂ M be such that for each integer t ≥ 0 the function gt is lower semicontinuous and ∞ 

gi  < ∞,

i=0

S ≥ 0 be an integer and let z ∈ X be such that there exists a program {xt }∞ t =S satisfying xS = z. ∗ ∞ Then there exists a ({gi }∞ i=S )-overtaking optimal program {xt }t =S satisfying ∗ xS = z. Proof Clearly, for any program {zt }∞ t =S , ∞  t =S

|gt (zt , zt +1 )| ≤

∞ 

gi  < ∞.

i=0

Set Δ = inf

∞  t =S

 gt (yt , yt +1 ) : {yt }∞ t =S is a program and yS = z .

(4.10)

4.2 Main Results

111

Clearly, Δ is well defined and finite. In order to prove Theorem 4.2 it is sufficient to show that there is a program {xi∗ }∞ i=S such that ∞ 

xS∗ = z,

t =S

gt (xt∗ , xt∗+1 ) = Δ.

By (4.10), for each integer k ≥ 1, there is a program {xt }∞ t =S such that (k)

xS(k) = z,

∞  t =S

gt (xt(k) , xt(k) +1 ) ≤ Δ + 1/k.

(4.11)

Extracting a subsequence and re-indexing if necessary we may assume without loss of generality that for any integer t ≥ S there exists xt∗ = lim xt . (k)

(4.12)

k→∞

∗ Clearly, {xi∗ }∞ i=S is a program satisfying xS = z. Let  > 0. Then there is a natural number S0 > S such that ∞ 

gt  < .

(4.13)

t =S0

By (4.11)–(4.13) for all integers T > S0 , T −1 t =S

gt (xt∗ , xt∗+1 ) ≤ lim inf k→∞

≤ lim inf k→∞

∞  t =S

T −1 t =S

(k)

(k)

gt (xt , xt +1 )

 (k) (k) gt (xt , xt +1 ) +  ≤ lim (Δ + k −1 + ) = Δ +  k→∞

and ∞  t =S

gt (xt∗ , xt∗+1 ) ≤ Δ + .

Since  is any positive number we conclude that Theorem 4.2 is proved.

∞

t =S

gt (xt∗ , xt∗+1 ) ≤ Δ.

Theorem 4.3 Let M > 0,  = γf /4 and let δ ∈ (0, ), a natural number L and λ ∈ (0, 1) be as guaranteed by Theorem 4.1.

112

4 Nonautonomous Problems with Discounting

Let {gi }∞ i=0 ∈ M be such that for each integer t ≥ 0, the function gt is lower semicontinuous and that ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ

(4.14)

{αi }∞ i=0 ⊂ (0, 1], lim αi = 0,

(4.15)

and let i→∞

αi αj−1 ≥ λ for all nonnegative integers i, j satisfying |i − j | ≤ L

(4.16)

αi αj−1 ≥ λ for all nonnegative integers i, j satisfying j ≥ i.

(4.17)

and

Then for each integer S ≥ 0 and each ({fi }∞ i=0 , S, M)-good z ∈ X there exists a (S,z) ∞ (S,z) }t =S such that xS = z and that the following property holds: program {xt For each γ > 0 there is a natural number n0 such that for each integer S ≥ 0, each integer T ≥ S + n0 and each ({fi }∞ i=0 , S, M)-good point z ∈ X, −1 |U ({αt gt }Tt =S , S, T , z) −

T −1 t =S

αt gt (xt(S,z), xt(S,z) +1 )| ≤ γ .

It is clear that Theorem 4.3 establishes the existence of ({αt gt }∞ t =S )-overtaking optimal program when (4.14)–(4.17) hold. Roughly speaking, an ({αt gt }∞ t =S )overtaking optimal program exists if {gi }∞ belongs to a small neighborhood of i=0 ∞ tends to zero slowly. {fi }∞ and the sequence of the discount coefficients {α } i i=0 i=0 The following result establishes the turnpike property for overtaking optimal programs. Theorem 4.4 Let M > 0 and  ∈ (0, γf ). Then there exist a number δ ∈ (0, ), a natural number L and λ ∈ (0, 1) such that for each {gi }∞ i=0 ∈ M, where gi is a lower semicontinuous function for all integers i ≥ 0, which satisfies ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ,

each integer T1 ≥ 0, each sequence {αi }∞ i=T1 ⊂ (0, 1] such that αi αj−1 ≥ λ for all integers i, j ≥ T1 satisfying |i − j | ≤ L, αi αj−1 ≥ λ for all integers i, j satisfying j ≥ i ≥ T1

4.3 Proofs of Theorems 4.1 and 4.4

113

∞ and each ({αi gi }∞ t =T1 )-overtaking optimal program {xt }t =T1 for which the point xT1 ∞ is ({fi }i=0 , T1 , M)-good, the following inequality holds: f

ρ(xt , xt ) ≤  for all integers t ≥ T1 + L. Theorems 4.1 and 4.4 are proved in Sect. 4.3. Theorem 4.3 is proved in Sect. 4.4. These results were obtained in [133].

4.3 Proofs of Theorems 4.1 and 4.4 Choose d0 > sup{fi  : i = 0, 1, . . . }.

(4.18)

We prove Theorems 4.1 and 4.4 simultaneously. Choose a number M1 > M + 4.

(4.19)

By property (P) there exist δ0 ∈ (0, γf /4) and a natural number L0 such that the following property holds: +L0 (P1) For each integer T ≥ 0 and each program {xt }Tt =T which satisfies T +L 0 −1 

fi (xi , xi+1 )

i=T

≤ min{U ({fi }∞ i=0 , T , T + L0 , xT , xT +L0 ) + δ0 , U ({fi }∞ i=0 , T , T + L0 ) + cf + 4 + M1 } f

there is an integer j ∈ {T , . . . , T + L0 } for which ρ(xj , xj ) ≤ γf /4. By Theorem 3.1 there exist a natural number L1 ≥ 4 and a real number δ1 ∈ (0, min{, γf })

(4.20)

such that for each integer L2 ≥ L1 , each pair of integers T1 ≥ 0, T2 > T1 + 2L2 and each {gi }∞ i=0 ∈ M satisfying ∞ −1 d({ft }∞ t =0 , {gt }t =0 ) ≤ (8L2 ) δ1

(4.21)

114

4 Nonautonomous Problems with Discounting

the following property holds: q

2 is a program, q is a natural number, {Si }i=1 is a finite (P2) Assume that {xt }Tt =T 1 sequence of integers such that:

S1 = T1 , T2 ≥ Sq > T2 − L2 , for each integer i satisfying 1 ≤ i ≤ q − 1, Si+1 − Si ∈ [L1 , L2 ], Si+1 −1



t =Si T 2 −1 t =Sq−1

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Si , Si+1 ) + M1 + cf + 8(d0 + 1) + 4, gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Sq−1 , T2 ) + M1 + cf + 8(d0 + 1) + 4,

for each integer i ∈ [1, q − 2], Si+2 −1



t =Si T 2 −1 t =Sq−2

gt (xt , xt +1 ) ≤ U ({gj }∞ j =0 , Si , Si+2 , xSi , xSi+2 ) + δ1 ,

gi (xi , xi+1 ) ≤ U ({gj }∞ j =0 , Sq−2 , T2 , xSq−2 , xT2 ) + δ1 .

Then f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L2 , T2 − L2 ]. Choose a natural number p0 ≥ 6 + M1 + 16(L0 + 4)(d0 + 4) + L1 .

(4.22)

L2 = (p0 + 4)L0 + 2L1 .

(4.23)

L ≥ 4L0 p0 + 4L2 ,

(4.24)

Set

Choose a natural number

4.3 Proofs of Theorems 4.1 and 4.4

115

a positive number δ < min{(16L0 )−1 δ0 , (48L2 )−1 δ1 }

(4.25)

and a number λ ∈ (0, 1) such that 18L0 (1 + d0 )(1 − λ)λ−1 < δ0 , λp0 > 2−1 , 96L2 (1 + d0 )(1 − λ) < δ1 .

(4.26)

Assume that {gi }∞ i=0 ∈ M, for each integer i ≥ 0 the function gi is lower semicontinuous, ∞ d({fi }∞ i=0 , {gi }i=0 ) ≤ δ,

(4.27)

T1 ≥ 0 is an integer and x˜ ∈ X is an ({fi }∞ i=0 , T1 , M)-good point. In the case of Theorem 4.1 we assume that an integer 2 −1 T2 > T1 + 2L, {αi }Ti=T ⊂ (0, 1], 1

(4.28)

αi αj−1 ≥ λ for all i, j ∈ {T1 , . . . , T2 − 1} satisfying |i − j | ≤ L,

(4.29)

αi αj−1 ≥ λ for all i, j ∈ {T1 , . . . , T2 − 1} satisfying j ≥ i

(4.30)

2 and that a program {xt }Tt =T satisfies 1

˜ xT1 = x,

T 2 −1 t =T1

2 −1 αt gt (xt , xt +1 ) = U ({αi gi }Ti=T , T1 , T2 , xT1 ). 1

(4.31)

In the case of Theorem 4.4 we assume that {αi }∞ i=T1 ⊂ (0, 1],

(4.32)

αi αj−1 ≥ λ for all integers i, j ≥ T1 satisfying |i − j | ≤ L,

(4.33)

αi αj−1 ≥ λ for all integers i, j satisfying j ≥ i ≥ T1

(4.34)

∞ and that an ({αi gi }∞ t =T1 )-overtaking optimal program {xt }t =T1 satisfies

xT1 = x. ˜

(4.35)

In order to complete the proof in the case of Theorem 4.1 it is sufficient to show that f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L, T2 − L]

116

4 Nonautonomous Problems with Discounting

and in the case of Theorem 4.4 it is sufficient to show that f

ρ(xt , xt ) ≤  for all integers t ≥ T1 + L. Since the point x˜ is ({fi }∞ i=0 , T1 , M)-good it follows from (4.31) and (4.35) that there is a program {˜zt }∞ such that z˜ T1 = x˜ and that for each integer T > T1 , t =T1 T −1

T −1

ft (˜zt , z˜ t +1 ) ≤

t =T1

t =T1

f

f

ft (xt , xt +1 ) + M.

(4.36)

In the case of Theorem 4.1 set I = [T1 , T2 ] and in the case of Theorem 4.4 set I = [T1 , ∞). We show that the following property holds: (P3) If an integer S satisfies f

[S, S + L0 ] ⊂ I, min{ρ(xt , xt ) : t = S, . . . , S + L0 } > γf /4,

(4.37)

then S+L 0 −1 

αt gt (xt , xt +1 ) ≥

t =S

S+L 0 −1  t =S

f

f

αt gt (xt , xt +1 ) + 3αS .

Assume that an integer S satisfies (4.37). By (4.31) in the case of Theorem 4.1 ∞ and the ({αi gi }∞ t =T1 )-overtaking optimality of the program {xt }t =T1 in the case of Theorem 4.4, S+L 0 −1 

0 −1 αS−1 αt gt (xt , xt +1 ) = U ({αS−1 αi gi }S+L , S, S + L0 , xS , xS+L0 ). i=S

t =S

(4.38) By (4.24)–(4.27), (4.30) and (4.33), for each integer q ≥ 0 and each program q+L {yt }t =q 0 , S+L 0 −1 0 −1  S+L      ft (yt , yt +1 ) − αS−1 αt gt (yt , yt +1 )  t =S

≤ L0 max{ft −

t =S

αS−1 αt gt 

: t = S, . . . , S + L0 − 1}

≤ L0 max{ft − gt  + gt − αS−1 αt gt  : t = S, . . . , S + L0 − 1} ≤ L0 δ + L0 max{|1 − αS−1 αt |gt  : t = S, . . . , S + L0 − 1} ≤ L0 δ + L0 (1 + d0 )(1 − λ)λ−1 < δ0 /8.

(4.39)

4.3 Proofs of Theorems 4.1 and 4.4

117

By (4.38), (4.39) and (C2), S+L 0 −1 

ft (xt , xt +1 )
γf /4 for all integers i ∈ I.

(4.50)

Assume the contrary. Then f

4.3 Proofs of Theorems 4.1 and 4.4

119

In the case of Theorem 4.4 set p = ∞.

(4.51)

In the case of Theorem 4.1 choose a natural number p such that pL0 ≤ T2 − T1 < (p + 1)L0 .

(4.52)

By (4.18), (4.28), (4.51), and (4.52), p ≥ 2p0 .

(4.53)

0 ≤ j, j + 1 ≤ p.

(4.54)

Assume that an integer j satisfies

By (4.50), (4.52) and property (P3), T1 +(j +1)L0 −1

αt gt (xt , xt +1 ) ≥

t =T1 +j L0

T1 +(j +1)L0 −1 t =T1 +j L0

f

f

αt gt (xt , xt +1 ) + 3αT1 +j L0 .

(4.55)

Consider the case of Theorem 4.1. By (4.22), (4.24), (4.26), (4.27), (4.29)– (4.31), (4.48), (4.52), (4.53) and (4.55), 0≥

T 2 −1

αt gt (xt , xt+1 ) −

t=T1

=

T 2 −1

αt gt (xt , xt+1 ) −

j =0

T 2 −1

f

f

αt gt (xt , xt+1 ) +

t=T1

p−1 +1)L0 −1   T1 +(j

+

αt gt (x˜t , x˜t+1 )

t=T1

t=T1

≥

T 2 −1



t=T1 +j L0

αt gt (xt , xt+1 ) −

T 2 −1

f

f

αt gt (xt , xt+1 ) −

t=T1 T1 +(j +1)L0 −1

T 2 −1

αt gt (x˜t , x˜t+1 )

t=T1

 f f αt gt (xt , xt+1 )

t=T1 +j L0

f f {αt gt (xt , xt+1 ) − αt gt (xt , xt+1 )

: t is an integer such that T1 + pL0 ≤ t < T2 }

− 2(L0 + 2)αT1 λ−1 sup{||gi || : i is an integer and i ≥ 0} ≥3

p−1 

αT1 +j L0 − 4(L0 + 2)αT1 λ−1 sup{gi  : i is an integer and i ≥ 0}

j =0

≥3

p−1  j =0

αT1 +j L0 − 4(L0 + 2)αT1 λ−1 (d0 + 1)

120

4 Nonautonomous Problems with Discounting

≥3

p 0 −1

αT1 +j L0 − 4(L0 + 2)αT1 λ−1 (d0 + 1)

j =0

≥ 3αT1

p 0 −1

λj − 4(L0 + 2)αT1 λ−1 (d0 + 1)

j =0

≥ 3αT1 (p0 /2) − 8(L0 + 2)αT1 (d0 + 1) = αT1 (p0 − 8(L0 + 2)(d0 + 1)) > 4αT1 ,

a contradiction. The contradiction we have reached proves that in the case of Theorem 4.1 there is an integer j ∈ I such that (4.49) holds. ∞ Consider the case of Theorem 4.4. Since {xt }∞ t =T1 is an ({αi gi }t =T1 )-overtaking optimal program, it follows from (4.22), (4.26), (4.27), (4.33)–(4.35), (4.48) and (4.55) that 0 ≥ lim sup

T



T →∞

t=T1

≥ lim sup

T 

 αt gt (x˜t , x˜t+1 )

t=T1

T1 +kL 0 −1

k→∞

+

αt gt (xt , xt+1 ) −

αt gt (xt , xt+1 ) −

T1 +kL 0 −1

t=T1

T1 +kL 0 −1

f

t=T1 f

f

αt gt (xt , xt+1 ) −

t=T1

f

αt gt (xt , xt+1 )

T1 +kL 0 −1

 αt gt (x˜t , x˜t+1 )

t=T1

k−1 

 ≥ lim sup 3 αT1 +j L0 − 2(L0 + 2)αT1 λ−1 sup{gi  : i is an integer and i ≥ 0} k→∞

≥3

p 0 −1

j =0

αT1 +j L0 − 2(L0 + 2)αT1 λ−1 (d0 + 1)

j =0

≥ 3αT1

p 0 −1

λj − 2(L0 + 2)αT1 λ−1 (d0 + 1)

j =0

≥ 3αT1 (p0 /2) − 4(L0 + 2)αT1 (d0 + 1) > αT1 ,

a contradiction. The contradiction we have reached proves that in the case of Theorem 4.4 there is an integer j ∈ I such that (4.49) holds. Assume that an integer τ satisfies T1 ≤ τ, L0 (1 + p0 ) + τ ∈ I, ρ(xτ , xτf ) ≤ γf /4.

(4.56)

We show that there is an integer S such that f

S ∈ I, S ≥ τ + L0 , ρ(xS , xS ) ≤ γf /4.

(4.57)

4.3 Proofs of Theorems 4.1 and 4.4

121

Assume the contrary. Then f

ρ(xS , xS ) > γf /4 for all integers S ∈ I satisfying S ≥ τ + L0 .

(4.58)

In the case of Theorem 4.4 set p = ∞.

(4.59)

In the case of Theorem 4.1 there is an integer p such that pL0 ≤ T2 − (τ + L0 ) < (p + 1)L0 .

(4.60)

p0 ≤ p.

(4.61)

1 ≤ j ≤ p.

(4.62)

By (4.56) and (4.60),

Assume that an integer j satisfies

By (4.56), (4.60), (4.62) and property (P3), τ +(j  +1)L0 −1

αt gt (xt , xt +1 ) ≥

t =τ +j L0

τ +(j  +1)L0 −1 t =τ +j L0

f

f

αt gt (xt , xt +1 ) + 3ατ +j L0 .

(4.63)

f

(4.64)

By (4.56) and (C4) there is ξ ∈ X such that (xτ , ξ ) ∈ Ωτ , (ξ, xτ +2 ) ∈ Ωτ +1 . By (4.64) there is a program {ξt }t ∈I such that f

ξt = xt , t = T1 , . . . , τ, ξτ +1 = ξ, ξt = xt

for all integers t ∈ I satisfying t ≥ τ + 2.

(4.65)

Consider the case of Theorem 4.1. By (4.26), (4.28)–(4.31), (4.60), (4.63) and (4.65), 0≥

T 2 −1

αt gt (xt , xt+1 ) −

t=T1

=

T 2 −1 t=τ

T 2 −1

αt gt (ξt , ξt+1 )

t=T1

αt gt (xt , xt+1 ) −

T 2 −1 t=τ

αt gt (ξt , ξt+1 )

122

=

4 Nonautonomous Problems with Discounting τ +L 0 −1 

αt gt (xt , xt+1 ) −

τ +L 0 −1 

t=τ T 2 −1

+

αt gt (xt , xt+1 ) −

t=τ +L0

≥

T 2 −1

αt gt (xt , xt+1 ) −

− ατ λ

T 2 −1

f

f

αt gt (xt , xt+1 )

2(L0 + 4) sup{gt  : t = 0, 1, . . . }

p  τ +(j  +1)L0 −1  j =1

+

αt gt (ξt , ξt+1 )

t=τ +L0 −1



T 2 −1 t=τ +L0

t=τ +L0

≥

αt gt (ξt , ξt+1 )

t=τ

τ +(j  +1)L0 −1

αt gt (xt , xt+1 ) −

t=τ +j L0

 f f αt gt (xt , xt+1 )

t=τ +j L0 f

f

{αt gt (xt , xt+1 ) − αt gt (xt , xt+1 ) :

t is an integer such that τ + (p + 1)L0 ≤ t < T2 } − 2ατ λ−1 (L0 + 4)(d0 + 1) ≥

p 

3ατ +j L0 − 4ατ λ−1 (L0 + 4)(d0 + 1)

j =1

≥ 3ατ

p0 

λj − 8ατ (L0 + 4)(d0 + 1) ≥ 3ατ (p0 /2) − 8ατ (L0 + 4)(d0 + 1)

j =1

≥ ατ (p0 − 8(L0 + 4)(d0 + 2)) ≥ ατ ,

a contradiction. The contradiction we have reached proves that in the case of Theorem 4.1 there is an integer S which satisfies (4.57). ∞ Consider the case of Theorem 4.4. Since {xt }∞ t =T1 is an ({αi gi }t =T1 )-overtaking optimal program, it follows from (4.65) that 0 ≥ lim sup T →∞

−1

T

αt gt (xt , xt +1 ) −

t =T1

T −1

 αt gt (ξt , ξt +1 ) .

t =T1

By (4.22), (4.24), (4.26), (4.27), (4.33), (4.34), (4.63), (4.65) and (4.66), 0 ≥ lim sup T →∞

= lim sup T →∞

−1

T

αt gt (xt , xt+1 ) −

t=T1 −1

T t=τ

T −1

 αt gt (ξt , ξt+1 )

t=T1

αt gt (xt , xt+1 ) −

T −1 t=τ

 αt gt (ξt , ξt+1 )

(4.66)

4.3 Proofs of Theorems 4.1 and 4.4

= lim sup

0 −1

τ +L 

T →∞

+

123

αt gt (xt , xt+1 ) −

τ +L 0 −1 

t=τ

T −1

αt gt (ξt , ξt+1 )

t=τ T −1

αt gt (xt , xt+1 ) −

t=τ +L0

 αt gt (ξt , ξt+1 )

t=τ +L0

≥ lim sup − 2L0 ατ λ−1 sup{gi  : i is an integer and i ≥ 0} k→∞

+

τ +(k+1)L  0 −1

τ +(k+1)L  0 −1

αt gt (xt , xt+1 ) −

t=τ +L0

f

f

αt gt (xt , xt+1 )

t=τ +L0

 − 4ατ λ−1 sup{gi  : i is an integer and i ≥ 0}

≥ lim sup − 2(d0 + 1)(L0 + 2)ατ λ−1 k→∞

+

+1)L0 −1 k  τ +(j   j =1

αt gt (xt , xt+1 ) −

τ +(j  +1)L0 −1

t=τ +j L0

 f f αt gt (xt , xt+1 )

t=τ +j L0

≥ −2(d0 + 1)(L0 + 2)ατ λ−1 + lim sup

k 

k→∞ j =1 −1

≥ −2(d0 + 1)(L0 + 2)ατ λ

+

p0 

3ατ +j L0

3ατ +j L0

j =1

≥ −2(d0 + 1)(L0 + 2)ατ λ−1 + 3ατ

p0 

λj ≥ −4(d0 + 1)(L0 + 2)ατ + 3ατ (p0 /2)

j =1

≥ ατ (p0 − 4(d0 + 1)(L0 + 2)) > ατ ,

a contradiction. The contradiction we have reached proves that in the case of Theorem 4.4 there is an integer S which satisfies (4.57). Thus we have shown that the following property holds: (P4) For each integer τ satisfying (4.56) there is an integer S satisfying (4.57). We show that the following property holds: (P5) For each integer τ satisfying τ ∈ I, τ − L0 (1 + p0 ) ≥ T1 , ρ(xτ , xτf ) ≤ γf /4

(4.67)

there is an integer S such that f

T1 ≤ S ≤ τ − L0 , ρ(xS , xS ) ≤ γf /4.

(4.68)

124

4 Nonautonomous Problems with Discounting

Let an integer τ satisfy (4.67). We show that there is an integer S satisfying (4.68). Assume the contrary. Then f

ρ(xt , xt ) > γf /4 for all integers t = T1 , . . . , τ − L0 .

(4.69)

By (4.67), there is a natural number p such that p0 ≤ p, pL0 ≤ τ − L0 − T1 < (p + 1)L0 .

(4.70)

Assume that an integer j ∈ [1, p].

(4.71)

By (4.69)–(4.71) and property (P3), τ −j L0 −1 

αt gt (xt , xt +1 ) ≥

t =τ −(j +1)L0

τ −j L0 −1  t =τ −(j +1)L0

f

f

αt gt (xt , xt +1 )+3ατ −(j +1)L0 .

(4.72)

We continue to consider the program {x˜t }∞ t =T1 satisfying (4.48). By (4.67) and (C4) there is η ∈ X such that f

(η, xτ ) ∈ Ωτ −1 , (xτ −2 , η) ∈ Ωτ −2 .

(4.73)

By (4.48), (4.67), (4.73) and (4.22) there is a program {ηt }τt=T1 such that ηt = x˜t , t = T1 , . . . , τ − 2, ητ −1 = η, ητ = xτ .

(4.74)

By (4.22), (4.31), (4.35), (4.48), (4.67) and (4.74), f

ηT1 = x˜ T1 = x˜ = xT1 , ηt = xt for all integers t = T1 + L0 + 2, . . . , τ − 2. (4.75) By (4.31) and (4.75) (in the case of Theorem 4.1) and ({αi gi }∞ t =T1 )-overtaking optimality of the program {xt }∞ (in the case of Theorem 4.4), (4.22), (4.26)– t =T1 (4.30), (4.33), (4.34), (4.67), (4.69), (4.72) and (4.74), 0≥

τ −1 

αt gt (xt , xt+1 ) −

t=T1

=

p 

τ −1 

αt gt (ηt , ηt+1 )

t=T1 τ −j L0 −1



(αt gt (xt , xt+1 ) − αt gt (ηt , ηt+1 ))

j =1 t=τ −(j +1)L0

+



{αt gt (xt , xt+1 ) − αt gt (ηt , ηt+1 ) :

4.3 Proofs of Theorems 4.1 and 4.4

125

t is an integer such that T1 ≤ t < τ − (p + 1)L0 }  + {αt gt (xt , xt+1 ) − αt gt (ηt , ηt+1 ) : t is an integer such that τ − L0 ≤ t < τ } =

p   j =1

τ −j L0 −1  t=τ −(j +1)L0

τ −L 0 −1 

+

αt gt (xt , xt+1 ) −

t=τ −(j +1)L0 f

f

αt gt (xt , xt+1 ) −

t=τ −(p+1)L0

 f f αt gt (xt , xt+1 )

τ −j L0 −1 

τ −L 0 −1 

αt gt (ηt , ηt+1 )

t=τ −(p+1)L0

− 4L0 sup{gt  : t = 0, 1, . . . }αT1 λ−1 ≥

p 

3ατ −(j +1)L0 − 2 sup{gi  : i = 0, 1, . . . }αT1 λ−1 (L0 + 4)

j =1

− 4L0 αT1 λ−1 sup{gi  : i is a nonnegative integer} ≥ 3ατ −(p+1)L0

p−1 

λj − αT1 λ−1 (d0 + 1)8(L0 + 2)

j =0

≥ 3αT1 λ

p−1 

λj − 2αT1 (d0 + 1)8(L0 + 2) ≥ 3αT1 (p0 /2) − 16αT1 (d0 + 1)(L0 + 2)

j =0

≥ αT1 ((p0 − 16(d0 + 1)(L0 + 2)) > αT1 ,

a contradiction. The contradiction we have reached proves that there is an integer S satisfying (4.68) and that property (P5) holds. In the case of Theorem 4.1 it follows from (4.28), (4.49), (4.54) and properties (P4) and (P5) that there exist a natural number q and a finite strictly increasing q sequence of integers {Si }i=1 ⊂ [T1 , T2 ] such that for each t ∈ {T1 , . . . , T2 }, f

ρ(xt , xt ) ≤ γf /4 if and only if t ∈ {S1 , . . . , Sq },

(4.76)

q ≥ 2, S1 < T1 + L0 (1 + p0 ), Sq > T2 − L0 (1 + p0 ).

(4.77)

In the case of Theorem 4.4 it follows from (4.49) and properties (P4) and (P5) that there exists a strictly increasing sequence of integers {Si }∞ i=1 ⊂ [T1 , ∞) such that for each integer t ∈ [T1 , ∞), f

ρ(xt , xt ) ≤ γf /4 if and only if t ∈ {Si : i is a natural number}, S1 < T1 + L0 (1 + p0 ).

(4.78)

126

4 Nonautonomous Problems with Discounting

In the case of Theorem 4.4 set q = ∞.

(4.79)

1 ≤ k and k + 1 ≤ q.

(4.80)

Sk+1 − Sk ≤ (p0 + 2)L0 .

(4.81)

Sk+1 − Sk > (p0 + 2)L0 .

(4.82)

Let an integer k satisfy

We show that

Assume the contrary. Then

By (4.82) there is a natural number p such that p > p0 , (p + 2)L0 ≤ Sk+1 − Sk < (p + 3)L0 .

(4.83)

By (4.76), f

ρ(xt , xt ) > γf /4 for all integers t = Sk + 1, . . . , Sk+1 − 1.

(4.84)

Let an integer j ∈ [0, p − 1].

(4.85)

By (4.83)–(4.85) and property (P3), Sk +(j +2)L0 −1

αt gt (xt , xt +1 ) ≥

t =Sk +(j +1)L0

Sk +(j +2)L0 −1 t =Sk +(j +1)L0

f

f

αt gt (xt , xt +1 ) + 3αSk +(j +1)L0 . (4.86)

By (4.22), (4.76), (4.82) and (C4) there exist η1 , η2 ∈ X such that f

(xSk , η1 ) ∈ ΩSk , (η1 , xSk +2 ) ∈ ΩSk +1 , f

(xSk+1 −2 , η2 ) ∈ ΩSk+1 −2 , (η2 , xSk +1 ) ∈ ΩSk+1 −1 .

(4.87)

4.3 Proofs of Theorems 4.1 and 4.4

127

By (4.22), (4.82) and (4.87) there is a program { xt }t ∈I such that xt = xt , t = T1 , . . . , Sk , xSk +1 = η1 , f

xSk+1 −1 = η2 , xt = xt , t = Sk + 2, . . . , Sk+1 − 2, xt = xt for all integers t ∈ I satisfying t ≥ Sk+1 .

(4.88)

By (4.31), (4.88) (in the case of Theorem 4.1) and ({αi gi }∞ i=T1 )-overtaking optimality of the program {xt }∞ t =T1 (in the case of Theorem 4.4), (4.26), (4.27), (4.29), (4.30), (4.33), (4.34), (4.83) and (4.86), Sk+1 −1

0≥



Sk+1 −1

αt gt (xt , xt+1 ) −

t=T1

≥

αt gt ( xt , xt+1 )

t=T1

Sk+1 −1

=





Sk+1 −1

αt gt (xt , xt+1 ) −



t=Sk

t=Sk

Sk+1 −1

Sk+1 −1



αt gt (xt , xt+1 ) −

t=Sk



αt gt ( xt , xt+1 )

f

f

αt gt (xt , xt+1 )

t=Sk

− 8(sup{gt  : t = 0, 1, . . . }αSk λ−1 ≥

Sk +L 0 −1

f

f

(αt gt (xt , xt+1 ) − αt gt (xt , xt+1 ))

t=Sk

+

p−1 +2)L0 −1   Sk +(j j =0

t=Sk +(j +1)L0

Sk+1 −1



+

 f f αt gt (xt , xt+1 ) − αt gt (xt , xt+1 )

(αt gt (xt , xt+1 ) − αt gt (xt , xt+1 )) − 8αSk λ−1 (d0 + 1) f

f

t=Sk +(p+1)L0

≥ −2αSk λ−1 L0 sup{gt  : t = 0, 1, . . . } +

p−1 

3αSk +(j +1)L0

j =0

− 4αSk λ−1 L0 sup{gt  : t = 0, 1, . . . } − 8αSk λ−1 (d0 + 1) ≥ 3αSk

p 0 −1

λj +1 − 2αSk (6L0 + 8)(d0 + 1) ≥ 3αSk (p0 /2) − 2αSk (6L0 + 8)(d0 + 1)

j =0

≥ αSk (p0 − (12L0 + 16)(d0 + 1)) > αSk ,

a contradiction. The contradiction we have reached proves that the following property holds:

128

4 Nonautonomous Problems with Discounting

(P6) Sk+1 − Sk ≤ (p0 + 2)L0 for all integers k satisfying 1 ≤ k and k + 1 ≤ q. We set S˜1 = S1 .

(4.89)

S∞ = Sq = ∞.

(4.90)

In the case of Theorem 4.4 set

Assume that j is a natural number and that we have already defined a finite strictly increasing sequence of integers {S˜i }i=1 ⊂ {Si : i is a natural number for which i ≤ q} j

(4.91)

such that for each i satisfying 1 ≤ i < j , S˜i+1 − S˜i ∈ [L1 , L2 ].

(4.92)

(Clearly, for j = 1 our assumption holds.) If S˜j + L2 > Sq , then the construction is completed. Assume that S˜j + L2 ≤ Sq .

(4.93)

S˜j + L1 + (p0 + 4)L0 ≤ S˜j + L2 ≤ Sq .

(4.94)

Then by (4.23) and (4.93),

By (4.73), (4.78) and (4.89) there is a natural number k such that S˜j + L1 ∈ [Sk−1 , Sk ].

(4.95)

By (4.91), (4.92), (4.94) and (4.95), S˜j ≤ Sk−1 , k ≤ q.

(4.96)

S˜j +1 = Sk .

(4.97)

We set

By (4.23), (4.95), (4.97) and (P6), L1 ≤ S˜j +1 − S˜j ≤ L1 + Sk − Sk−1 ≤ L1 + (p0 + 2)L0 ≤ L2 and the assertion made for j also holds for j + 1.

4.3 Proofs of Theorems 4.1 and 4.4

129

In the case of Theorem 4.4 we obtain a sequence of integers {S˜i }∞ i=1 such that ∞ ˜ {S˜i }∞ i=1 ⊂ {Si }i=1 , S1 = S1 < T1 + L0 (1 + p0 ),

S˜i+1 − S˜i ∈ [L1 , L2 ] for all integers i ≥ 1.

(4.98)

In the case of Theorem 4.1 our construction is completed after a finite number of steps and we obtain a finite strictly increasing sequence of integers {S˜i }ki=1 , where k is a natural number, such that {S˜i }ki=1 ⊂ {S1 , . . . , Sq }, S˜1 = S1 < T1 + L0 (1 + p0 ), S˜k > Sq − L2 > T2 − L0 (1 + p0 ) − L2 , S˜i+1 − S˜i ∈ [L1 , L2 ] for all integers i satisfying 1 ≤ i < k.

(4.99) (4.100)

In the case of Theorem 4.1, by (4.24), (4.28) and (4.99), S˜k − S˜1 > T2 − T1 − 2L0 (p0 + 1) − L2 > 2L − 2L0 (p0 + 1) − L2 > 2L2 .

(4.101)

In the case of Theorem 4.4 let k be any natural number such that S˜k − S˜1 > 2L2 .

(4.102)

˜

We apply property (P2) to the program {xt }Sk ˜ . By (4.31) (in the case of t =S1 ∞ Theorem 4.1) and ({αi gi }∞ i=T1 )-overtaking optimality of the program {xt }t =T1 (in the case of Theorem 4.4), for each pair of integers τ1 , τ2 satisfying S˜1 ≤ τ1 < τ2 ≤ S˜k , we have τ 2 −1 t =τ1

2 −1 αt gt (xt , xt +1 ) = U ({αi gi }τi=τ , τ1 , τ2 , xτ1 , xτ2 ). 1

(4.103)

By (4.18), (4.24)–(4.27), (4.29) and (4.33) for each pair of integers τ1 , τ2 satisfying S˜1 ≤ τ1 < τ2 ≤ S˜k , τ2 ≤ τ1 + 3L2 ,

(4.104)

130

4 Nonautonomous Problems with Discounting

2 for each program {yt }τt =τ we have 1

τ 2 −1 2 −1  τ    ft (yt , yt +1 ) − ατ−1 α g (y , y )   t t t t +1 1 t =τ1

≤

t =τ1

τ 2 −1

|ft (yt , yt +1 ) − gt (yt , yt +1 )| +

t =τ1

τ 2 −1 t =τ1

|gt (yt , yt +1 )||1 − ατ−1 αt | 1

≤ δ(τ2 − τ1 ) + (τ2 − τ1 )(1 + d0 )|λ − 1|λ−1 ≤ (τ2 − τ1 )δ + 2(τ2 − τ1 )(1 + d0 )|1 − λ| ≤ 3L2 δ + 6L2 (1 + d0 )(1 − λ) < δ1 /8.

(4.105)

By (4.98), (4.100), (4.103) and (4.105), for each integer i satisfying 1 ≤ i ≤ k − 2, S˜ i+2 −1



S˜ i+2 −1

ft (xt , xt +1 ) ≤

t =S˜ i



t =S˜ i

α −1 ˜ αt gt (xt , xt +1 ) + δ1 /8 Si

= U ({α −1 ˜ αt gt } Si

S˜ i+2 −1 ˜ ˜ , Si , Si+2 , xS˜i , xS˜i+2 ) + δ1 /8 t =S˜ i

˜ ˜ ≤ U ({ft }∞ t =0 , Si , Si+2 , xS˜ i , xS˜ i+2 ) + δ1 /4.

(4.106)

Let an integer j satisfy 1 ≤ j < k. By (4.98) and (4.100), 4 ≤ L1 ≤ S˜j +1 − S˜j ≤ L2 , f Sj

(4.107) f ) Sj+1

ρ(xS˜j , x ˜ ) ≤ γf /4, ρ(xS˜j+1 , x ˜

≤ γf /4.

(4.108)

By (4.107), (4.108) and (C4) there is η1 , η2 ∈ X such that f ) Sj +2

(xS˜j , η1 ) ∈ ΩS˜j , (η1 , x ˜ f , η2 ) Sj+1 −2

(x ˜

∈ ΩS˜j +1 ,

∈ ΩS˜j+1 −2 , (η2 , xS˜j+1 ) ∈ ΩS˜j+1 −1 .

(4.109)

Set x¯ S˜j = xS˜j , x˜S˜j +1 = η1 , x¯S˜j+1 −1 = η2 , x¯t = xt , t = S˜j + 2, . . . , S˜j +1 − 2, f

(4.110) x¯ S˜j+1 = xS˜j+1 .

4.4 Proof of Theorem 4.3

131

By (4.109) and (4.110), {x¯t }

S˜ j+1 t =S˜ j

is a program. By (4.98), (4.100), (4.103)–

(4.105), (4.110) and (C2), S˜j+1 −1



S˜j+1 −1

ft (xt , xt+1 ) ≤

t=S˜j



α −1 ˜ αt gt (xt , xt+1 ) + δ1 /8 Sj

t=S˜ j S˜j+1 −1

≤



α −1 ˜ αt gt (x¯t , x¯t+1 ) + δ1 /8 Sj

t=S˜ j S˜j+1 −1

≤



S˜ j+1 −1



ft (x¯t , x¯t+1 ) + δ1 /4 ≤

t=S˜ j

f

f

ft (xt , xt+1 ) + 8(d0 + 1) + 1

t=S˜j

˜ ˜ ≤ U ({fi }∞ i=0 , Sj , Sj +1 ) + cf + 8(d0 + 1) + 1.

(4.111)

By (4.106) which holds for all integers i satisfying 1 ≤ i ≤ k − 2, (4.111) which holds for all integers j satisfying 1 ≤ j < k, (4.25), (4.27), (4.98), (4.100)–(4.102) ˜

and property (P2) applied with the program {xt }Sk ˜ , t =S1

ρ(xt , xt ) ≤  for all integers t ∈ [S˜1 + L2 , S˜k − L2 ]. f

(4.112)

In the case of Theorem 4.1 it follows from (4.24), (4.99) and (4.112) that f

ρ(xt , xt ) ≤  for all integers t ∈ [T1 + L, T2 − L]. In the case of Theorem 4.4, since k is any sufficiently large natural number, it follows from (4.24), (4.98) and (4.112) that f

ρ(xt , xt ) ≤  for all integers t ≥ T1 + L. Theorems 4.1 and 4.4 are proved.

4.4 Proof of Theorem 4.3 Fix d0 > sup{fi  : i = 0, 1, . . . }. In the proof we use the following auxiliary result.

(4.113)

132

4 Nonautonomous Problems with Discounting

Lemma 4.5 Let γ > 0. Then there is a natural number n0 such that for each pair of integers T1 ≥ 0, T2 > T1 + n0 , each integer S ∈ [T1 + n0 , T2 − 1] and each 2 program {xt }Tt =T such that 1 xT1 is ({fi }∞ i=0 , T1 , M)-good, T 2 −1 t =T1

αt gt (xt , xt +1 ) = U ({αt gt }∞ t =0 , T1 , T2 , xT1 )

(4.114) (4.115)

the following inequality holds: S−1  t =T1

αt gt (xt , xt +1 ) ≤ U ({αt gt }∞ t =0 , T1 , S, xT1 ) + γ .

Proof By (4.15) there is a natural number n0 > 4L + 4

(4.116)

such that for all integers t > n0 − L − 4 αt ≤ γ (8L + 8)−1 (d0 + 1)−1 .

(4.117)

Assume that integers T1 ≥ 0, T2 > T1 +n0 , an integer S ∈ [T1 +n0 , T2 −1] and that 2 a program {xt }Tt =T satisfies (4.114) and (4.115). Clearly, there is a program {x˜t }St=T1 1 such that x˜T1 = xT1 ,

S−1 

αt gt (x˜t , x˜t +1 ) = U ({αi gi }∞ i=0 , T1 , S, xT1 ).

(4.118)

t =T1

By the choice of δ and L, (4.10), (4.14), (4.17), (4.114)–(4.116), (4.118) and Theorem 4.1, f

(4.119)

f

(4.120)

ρ(xt , xt ) ≤ γf /4, t = T1 + L, . . . , T2 − L, ρ(x˜t , xt ) ≤ γf /4, t = T1 + L, . . . , S − L. By (4.119), (4.120) and (C4) there is ξ ∈ X such that (x˜S−L−1 , ξ ) ∈ ΩS−L−1 , (ξ, xS−L+1 ) ∈ ΩS−L .

(4.121)

Define yt = x˜t , t = T1 , . . . , S − L − 1, yS−L = ξ, yt = xt , t = S − L + 1, . . . , T2 . (4.122)

4.4 Proof of Theorem 4.3

133

2 By (4.121) and (4.122), {yt }Tt =T is a program. In view of (4.14), (4.113), (4.115), 1 (4.117), (4.118) and (4.122),

0≥

=

T 2 −1

αt gt (xt , xt +1 ) −

t =T1

t =T1

S−L 

S−L 

αt gt (xt , xt +1 ) −

t =T1

≥

S−L−2 

S−1 

αt gt (xt , xt +1 ) −

S−1 

αt gt (yt , yt +1 )

S−L−2 

αt gt (x˜t , x˜t +1 ) − (2αS−L−1 + 2αS−L )(d0 + 1)

t =T1

αt gt (xt , xt +1 ) −

t =T1

≥

αt gt (yt , yt +1 )

t =T1

t =T1

≥

T 2 −1

S−1 

αt gt (x˜t , x˜t +1 ) − 4(d0 + 1)

t =T1

αt gt (xt , xt +1 ) −

t =T1

S−1 



S−1 

 αt

t =S−L−1

αt gt (x˜t , x˜t +1 )

t =T1

− 4(d0 + 1)(L + 1)γ (8L + 8)−1 (d0 + 1)−1 and S−1  t =T1

αt gt (xt , xt +1 ) ≤ U ({αt gt }∞ t =0 , T1 , S, xT1 ) + γ .

Lemma 4.5 is proved. Completion of the proof of Theorem 4.3 Let an integer S ≥ 0 and z ∈ X be an ({fi }∞ i=0 , S, M)-good point. For each (z,S,T ) T }t =S such that integer T > S there is a program {xt xS(z,S,T ) = z,

T −1 t =S

) αt gt (xt(z,S,T ) , xt(z,S,T ) = U ({αt gt }∞ t =0 , S, T , z). +1

(4.123)

Clearly, there exists a strictly increasing sequence of natural numbers {Tj }∞ j =1 such that T1 > S and that for any integer t ≥ S there exists (z,S)

xt

(z,S,Tj )

= lim xt j →∞

.

(4.124)

134

4 Nonautonomous Problems with Discounting (z,S) ∞ }t =S

Clearly, {xt

is a program and (z,S)

xS

= z.

(4.125)

Let γ > 0. By Lemma 4.5 there is a natural number n0 such that the following property holds: (P7) For each pair integer S˜ ≥ 0, each integer T ≥ S˜ + n0 , each integer Q ∈ ˜ [S˜ + n0 , T − 1] and each program {xt }T ˜ such that xS˜ is ({fi }∞ i=0 , S, M)t =S good and that T −1

˜ αt gt (xt , xt +1 ) = U ({αt gt }∞ t =0 , S, T , xS˜ )

t =S˜

the following inequality holds: Q−1 

˜ αt gt (xt , xt +1 ) ≤ U ({αt gt }∞ t =0 , S, Q, xS˜ ) + γ .

t =S˜

Let T ≥ S + n0 be an integer and j be a natural number such that Tj > T . By (P7) (with S˜ = S, T = Tj , Q = T ) and (4.123), T −1

(z,S,Tj )

αt gt (xt

t =S

(z,S,Tj )

, xt +1

) ≤ U ({αt gt }∞ t =0 , S, T , z) + γ .

Together with (4.124) this implies that T −1

(z,S)

αt gt (xt

t =S

Theorem 4.3 is proved.

, xt +1 ) ≤ U ({αt gt }∞ t =0 , S, T , z) + γ . (z,S)

Chapter 5

Stability of the Turnpike Phenomenon for Nonautonomous Problems

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 sequence of nonempty closed sets Ωt ⊂ X × X, t = 0, 1, . . . which determines a class of admissible trajectories (programs) and by a bounded sequence of lower semicontinuous objective functions ft : X ×X → R 1 , t = 0, 1, . . . which determines an optimality criterion. We show the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint sets.

5.1 Preliminaries and Stability Results We use the notation, definitions and assumptions introduced in Chap. 2. Recall that for each nonempty set Y we denote by B(Y ) the set of all bounded functions f : Y → R 1 and for each f ∈ B(Y ) set f  = sup{|f (y)| : y ∈ Y }. For each nonempty compact metric space Y denote by C(Y ) the set of all continuous functions f : Y → R 1 . Denote by Card(B) the cardinality of a set B. Let (X, ρ) be a compact metric space with the metric ρ. The set X × X is equipped with the metric ρ1 defined by ρ1 ((x1 , x2 ), (y1 , y2 )) = ρ(x1 , y1 ) + ρ(x2 , y2 ), (x1 , x2 ), (y1 , y2 ) ∈ X × X. For each x ∈ X and each nonempty set C ⊂ X set ρ(x, C) = inf{ρ(x, y) : y ∈ C}. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_5

135

136

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

For each x ∈ X and each r > 0 set B(x, r) = {y ∈ X : ρ(x, y) ≤ r}. For each (x1 , x2 ) ∈ X × X and each nonempty set C ⊂ X × X set ρ1 ((x1 , x2 ), C) = inf{ρ1 ((x1 , x2 ), (y1 , y2 )) : (y1 , y2 ) ∈ C}. For each pair of nonempty sets A, B ⊂ X set H (A, B) = max{sup{ρ(x, B) : x ∈ A}, sup{ρ(y, A) : y ∈ B}}. For each integer t ≥ 0 let Ωt be a nonempty closed subset of the metric space X × X. We suppose that for each integer t ≥ 0 there exists a set-valued mapping at : X → 2X \ {∅} such that Ωt = {(x, y) : x ∈ X, y ∈ at (x)}.

(5.1)

∞ Let T ≥ 0 be an integer. A sequence {xt }∞ t =T ⊂ X is called an ({at }t =T )-program ∞ (or ({Ωt }t =T )-program) if xt +1 ∈ at (xt ) for all integers t ≥ T . 2 ⊂ X is called Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A sequence {xt }Tt =T 1

2 −1 2 −1 )-program (or ({Ωt }Tt =T )-program) if xt +1 ∈ at (xt ) for all integers t an ({at }Tt =T 1 1 satisfying T1 ≤ t < T2 . ∞ We assume that there exists an ({at }∞ t =0 )-program {xt }t =0 . Denote by M the set of all sequences of functions ft ∈ B(X × X), t = 0, 1, . . . such that

sup{ft  : t = 0, 1, . . . } < ∞.

(5.2)

Let {ft }∞ t =0 ∈ M. For each y, z ∈ X and each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 set U ({ft }∞ t =0 , T1 , T2 )

= inf

2 −1  T

t =T1

 T2 −1 2 ft (xt , xt +1 ) : {xt }Tt =T is an ({a } )-program , t t =T 1 1 (5.3)

U ({ft }∞ t =0 , T1 , T2 , y) = inf

2 −1  T

ft (xt , xt +1 ) :

t =T1

 T2 −1 2 {xt }Tt =T is an ({a } )-program and x = y , t T 1 t =T 1 1

(5.4)

5.1 Preliminaries and Stability Results

U ({ft }∞ t =0 , T1 , T2 , y, z) = inf

137

2 −1  T

ft (xt , xt +1 ) :

t =T1

 T2 −1 2 {xt }Tt =T is an ({a } )-program and x = y, x = z . t T T 1 2 t =T1 1 (5.5) Here we assume that the infimum over an empty set is ∞. f ∞ We suppose that there exist an ({at }∞ t =0 )-program {xt }t =0 and constants cf > 0, γf > 0 such that the following assumptions hold: (C1) The function ft is continuous for all integers t ≥ 0, and moreover, for each  > 0 there exists δ > 0 such that for each integer t ≥ 0 and each pair of points (x1 , x2 ), (y1 , y2 ) ∈ X × X satisfying ρ1 ((x1 , x2 ), (y1 , y2 )) ≤ δ the inequality |ft (x1 , x2 ) − ft (y1 , y2 )| ≤  holds; (C2) for each pair of integers T1 ≥ 0, T2 > T1 , T 2 −1 t =T1

ft (xt , xt +1 ) ≤ U ({ft }∞ t =0 , T1 , T2 ) + cf ; f

f

(C3) for each  > 0 there exists δ > 0 such that for each integer t ≥ 0 and each (x, y) ∈ X satisfying ρ(x, y) ≤ δ the inequality H (at (x), at (y)) ≤  holds; (C4) for each integer t ≥ 0, f

f

B(xt +1 , γf ) ⊂ at (xt ) and f

f

xt +1 ∈ at +1 (x) for every x ∈ B(xt , γf ). Evidently, assumption (C2) is assumption (C2) from Chap. 2, (C1) implies assumption (C1) from Chap. 2, (C1) implies assumption (C3) of Chap. 2 and assumption (C4) from Chap. 2 follows from assumption (C4) above. Therefore all the results of Chap. 2 are valid in our case considered in this chapter.

138

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

∞ ∞ An ({at }∞ t =S )-program {xt }t =S , where S ≥ 0 is an integer, is called ({fi }i=0 )good if the sequence −1  T

T −1

fi (xi , xi+1 ) −

i=S

f

f

fi (xi , xi+1 )

∞

i=S

T =S+1

is bounded. In this chapter we use the asymptotic turnpike property ATP, the turnpike property TP and property (P) introduced in Sect. 2.2. We suppose that {fi }∞ i=0 possesses TP. Fix λ¯ ∈ (0, γf ).

(5.6)

Let t ≥ 0 be an integer and let λ > 0. Denote by E(t, λ) the set of all mappings a : X → 2X \ {∅} such that the following conditions hold: for each x ∈ X and each y ∈ a(x), ρ(y, at (x)) ≤ λ;

(5.7)

for each x ∈ B(xt , λ¯ ), f

¯ ⊂ a(xt ). xt +1 ∈ a(x) and B(xt +1 , λ) f

f

f

(5.8)

Let integers T1 , T2 satisfy 0 ≤ T1 < T2 and let bt : X → 2X \ {∅}, t = T1 , . . . , T2 − 1. 2 2 −1 A sequence {xt }Tt =T ⊂ X is called a ({bt }Tt =T )-program if xt +1 ∈ bt (xt ) for all 1 1 integers t ∈ [T1 , T2 − 1]. 2 −1 ⊂ B(X × X) set For each x, y ∈ X and each finite sequence {ut }Tt =T 1 2 −1 2 −1 , {bt }Tt =T , T1 , T2 , x) U ({ut }Tt =T 1 1

= inf

2 −1  T

ut (xt , xt +1 ) :

t =T1

 T2 −1 2 {xt }Tt =T is an ({b } )-program and x = x , t T 1 t =T1 1 2 −1 2 −1 U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , x, y) = inf 1 1

2 −1  T

ut (xt , xt +1 ) :

t =T1

 2 2 −1 {xt }Tt =T is an ({bt }Tt =T )-program, xT1 = x and xT2 = y , 1 1 2 −1 2 −1 U ({ut }Tt =T , {bt }Tt =T , T1 , T2 ) 1 1

= inf

2 −1  T

(5.9)

(5.10)

ut (xt , xt +1 ) :

t =T1

 T2 −1 2 {xt }Tt =T is an ({b } )-program . t t =T1 1

(5.11)

5.1 Preliminaries and Stability Results

139

(Here we use the convention that the infimum of an empty set is ∞.) 2 −1 , T1 , T2 ) the set of all x ∈ X for which there exists a Denote by Y ({bt }Tt =T 1 2 −1 2 )-program {xt }Tt =T such that xT1 = xT1 and xT2 = x. ({bt }Tt =T 1 1 T2 −1 ¯ Denote by Y ({bt }t =T1 , T1 , T2 ) the set of all x ∈ X for which there exists a

f

2 −1 2 ({bt }Tt =T )-program {xt }Tt =T such that xT1 = x and xT2 = xT2 . 1 1 In this chapter we prove the following stability results. They show the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint sets. In the first two theorems (Theorems 5.1 and 5.2) we establish the stability of TP, while in Theorems 5.3 and 5.4 we deal with the weak turnpike property.

f

Theorem 5.1 Let  be a positive number and let l1 , l2 be natural numbers. Then there exist δ > 0 and a natural number L˜ > l1 + l2 such that for each pair of ˜ each integers T1 ≥ 0, T2 > T1 + 2L, bt ∈ E(t, δ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ, t = T1 , . . . , T2 − 1 2 −1 2 and each ({bt }Tt =T )-program {xt }Tt =T which satisfies 1 1 1 +l1 −1 2 −1 , T1 , T1 + l1 ), xT2 ∈ Y ({bt }Tt =T , T2 − l2 , T2 ), xT1 ∈ Y¯ ({bt }Tt =T 1 2 −l2

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 , xT2 ) + δ 1 1

the following inequality holds: f ˜ ρ(xt , xt ) ≤  for all t = T1 + L˜ 1 , . . . , T2 − L.

Theorem 5.2 Let  be a positive number and let l1 be a natural number. Then there exist δ > 0 and a natural number L˜ > l1 such that for each pair of integers T1 ≥ 0, ˜ each T2 > T1 + 2L, bt ∈ E(t, δ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ, t = T1 , . . . , T2 − 1

140

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

2 −1 2 and each ({bt }Tt =T )-program {xt }Tt =T which satisfies 1 1 1 +l1 −1 , T1 , T1 + l1 ), xT1 ∈ Y¯ ({bt }Tt =T 1

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) + δ 1 1

the following inequality holds: ˜ ρ(xt , xt ) ≤  for all t = T1 + L˜ 1 , . . . , T2 − L. f

Theorem 5.3 Let , M be positive numbers and let l1 , l2 be natural numbers. Then there exist δ > 0 and a natural number L > l1 + l2 such that for each pair of integers T1 ≥ 0, T2 > T1 + L, each bt ∈ E(t, δ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ, t = T1 , . . . , T2 − 1 2 −1 2 )-program {xt }Tt =T which satisfies and each ({bt }Tt =T 1 1 1 +l1 −1 2 −1 xT1 ∈ Y¯ ({bt }Tt =T , T1 , T1 + l1 ), xT2 ∈ Y ({bt }Tt =T , T2 − l2 , T2 ), 1 2 −l2

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 , xT2 ) + M 1 1

the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > }) ≤ L holds. Theorem 5.4 Let , M be positive numbers and let l1 be a natural number. Then there exist δ > 0 and a natural number L > l1 such that for each pair of integers T1 ≥ 0, T2 > T1 + L, each bt ∈ E(t, δ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ, t = T1 , . . . , T2 − 1

5.1 Preliminaries and Stability Results

141

2 −1 2 and each ({bt }Tt =T )-program {xt }Tt =T which satisfies 1 1 1 +l1 −1 , T1 , T1 + l1 ), xT1 ∈ Y¯ ({bt }tT=T 1

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) + M 1 1

the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > }) ≤ L holds. In the following three results the stability of TP is established for programs which are approximate optimal on subintervals of a fixed and sufficiently large length. Theorem 5.5 Let  ∈ (0, λ¯ ) and M be a positive number. Then there exist γ ∈ (0, ) and a natural number L1 such that for each integer L2 ≥ L1 there exists a positive number δ < γ such that the following assertion holds. Assume that T1 ≥ 0, T2 > T1 + 6L2 is a pair of integers, bt ∈ E(t, δ), t = T1 , . . . , T2 − 1, ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfy ut − ft  ≤ δ, t = T1 , . . . , T2 − 1 2 −1 2 and that a ({bt }Tt =T )-program {xt }Tt =T and a finite sequence of integers {Si }i=0 1 1 satisfy

q

S0 = T1 , Si+1 − Si ∈ [L1 , L2 ], i = 0, . . . , q − 1, Sq ∈ (T2 − L2 , T2 ], Si+1 −1



Si+1 −1

ut (xt , xt +1 ) ≤

t =Si



t =Si

f

f

ut (xt , xt +1 ) + M

for each integer i ∈ [0, q − 1], Si+2 −1



t =Si

S

−1

S

−1

i+2 i+2 ut (xt , xt +1 ) ≤ U ({ut }t =S , {bt }t =S , Si , Si+2 , xSi , xSi+2 ) + γ i i

for each integer i ∈ [0, q − 2] and T 2 −1 t =Sq−2

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =S , {bt }Tt =S , Sq−2 , T2 , xSq−2 , xT2 ) + γ . q−2 q−2

142

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

Then f

ρ(xt , xt ) ≤  for all t = T1 + 3L2 , . . . , T2 − 2L2 . Theorem 5.6 Let , M be positive numbers and let l1 , l2 be natural numbers. Then there exist δ > 0 and a natural number L > l1 + l2 such that for each pair of integers T1 ≥ 0, T2 > T1 + 2L, each bt ∈ E(t, δ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ, t = T1 , . . . , T2 − 1 2 −1 2 )-program {xt }Tt =T which satisfies and each ({bt }Tt =T 1 1 1 +l1 −1 2 −1 xT1 ∈ Y¯ ({bt }Tt =T , T1 , T1 + l1 ), xT2 ∈ Y ({bt }Tt =T , T2 − l2 , T2 ), 1 2 −l2

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 , xT2 ) + M 1 1

and τ +L−1  t =τ

+L−1 +L−1 ut (xt , xt +1 ) ≤ U ({ut }τt =τ , {bt }τt =τ , τ, τ + L, xτ , xτ +L ) + δ

for each integer τ ∈ [T1 , T2 − L], the following inequality holds: f

ρ(xt , xt ) ≤  for all t = T1 + L, . . . , T2 − L. Theorem 5.7 Let , M be positive numbers and let l1 be a natural number. Then there exist δ > 0 and a natural number L > l1 such that for each pair of integers T1 ≥ 0, T2 > T1 + 2L, each bt ∈ E(t, δ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ, t = T1 , . . . , T2 − 1 2 −1 2 )-program {xt }Tt =T which satisfies and each ({bt }Tt =T 1 1 1 +l1 −1 xT1 ∈ Y¯ ({bt }tT=T , T1 , T1 + l1 ), 1

5.2 Auxiliary Results T 2 −1 t =T1

143

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) + M 1 1

and τ +L−1  t =τ

+L−1 +L−1 ut (xt , xt +1 ) ≤ U ({ut }τt =τ , {bt }τt =τ , τ, τ + L, xτ , xτ +L ) + δ

for each integer τ ∈ [T1 , T2 − L], the following inequality holds: f

ρ(xt , xt ) ≤  for all t = T1 + L, . . . , T2 − L. Theorems 5.1–5.7 are new.

5.2 Auxiliary Results In order to prove our stability results we need the following useful lemmas. Lemma 5.8 Let  be a positive number and let L be a natural number. Then there exists δ > 0 such that for each integer S ≥ 0, each bt ∈ E(t, δ), t = S, . . . , S + L − 1, each ut ∈ B(X × X), t = S, . . . , S + L − 1 satisfying ut − ft  ≤ δ, t = S, . . . , S + L − 1 S+L−1 )-program {xt }S+L )-program and each ({bt }S+L−1 t =S t =S there exists an ({at }t =S S+L {yt }t =S such that

yS = xS , ρ(xt , yt ) ≤  for all t = S, . . . , S + L and S+L−1 

ft (yt , yt +1 ) ≤

t =S

S+L−1 

ut (xt , xt +1 ) + .

t =S

Proof By (C1), there exists 0 ∈ (0, /2)

144

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

such that the following property holds: (i) For each integer t ≥ 0 and each pair of points (ξ1 , ξ2 ), (η1 , η2 ) ∈ X × X satisfying ρ1 ((ξ1 , ξ2 ), (η1 , η2 )) ≤ 0 we have |ft (ξ1 , ξ2 ) − ft (η1 , η2 )| ≤ (4L)−1 . By (C3), there exists δL ∈ (0, (4L)−1 0 )

(5.12)

H (at (ξ1 ), at (ξ2 )) ≤ 0 /4

(5.13)

such that

for each integer t ≥ 0 and each ξ1 , ξ2 ∈ X satisfying ρ(ξ1 , ξ2 ) ≤ δL . By induction, using (C3), we obtain a sequence of positive numbers δi , i = 0, . . . , L such that for each integer i ∈ {1, . . . , L}, δi−1 < δi /2

(5.14)

H (at (ξ1 ), at (ξ2 )) ≤ δi /2

(5.15)

and

for each integer t ≥ 0 and each ξ1 , ξ2 ∈ X satisfying ρ(ξ1 , ξ2 ) ≤ δi−1 . Set δ = δ0 /2.

(5.16)

Assume that S ≥ 0 is an integer, bt ∈ E(t, δ), t = S, . . . , S + L − 1,

(5.17)

ut ∈ B(X × X), t = S, . . . , S + L − 1,

(5.18)

ut − ft  ≤ δ, t = S, . . . , S + L − 1

(5.19)

S+L−1 )-program. {xt }S+L t =S is an ({bt }t =S

(5.20)

and that

5.2 Auxiliary Results

145

By (5.12), (5.14), (5.16) and (5.19), S+L−1   S+L−1     ut (xt , xt +1 ) − ft (xt , xt +1 ) ≤ Lδ ≤ LδL ≤ 0 /4 < /8.  t =S

(5.21)

t =S

Set yS = xS .

(5.22)

xS+1 ∈ bS (xS ).

(5.23)

In view of (5.20),

Equations (5.17) and (5.23) imply that ρ(xS+1 , aS (xS )) ≤ δ.

(5.24)

By (5.16) and (5.24), there exists yS+1 ∈ aS (yS )

(5.25)

ρ(xS+1 , yS+1 ) ≤ δ ≤ δ0 .

(5.26)

such that

Assume that an integer k ∈ {1, . . . , L} \ {L} and that we defined yt ∈ X, t = S, . . . , S + k such that (5.22) and (5.25) are true and that for all i = 1, . . . , k we have ρ(xS+i , yS+i ) ≤ δi−1 ,

(5.27)

yS+i ∈ aS+i−1 (yS+i−1 ).

(5.28)

H (aS+k (xS+k ), aS+k (yS+k )) ≤ δk /2.

(5.29)

In view of (5.15) and (5.27),

Equations (5.7), (5.17), and (5.20) imply that ρ(xS+k+1 , aS+k (xS+k )) ≤ δ.

(5.30)

It follows from (5.14), (5.16), (5.29), and (5.30) that ρ(xS+k+1 , aS+k (yS+k )) ≤ δ + δk /2 ≤ δk .

(5.31)

146

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

By (5.31), there exists yS+k+1 ∈ aS+k (yS+k ) such that ρ(xS+k+1 , yS+k+1 ) ≤ δk . Thus the assumption made for k also holds for k + 1. There by induction we constructed the sequence {yt }S+L t =S ⊂ X such that yS = xS , yt +1 ∈ at (yt ), t = S, . . . , S + L − 1, ρ(xt , yt ) ≤ δL ≤ 0 /4, t = S, . . . , S + L.

(5.32)

Property (i) and (5.32) imply that for all integers t = S, . . . , S + L − 1, |ft (xt , xt +1 ) − ft (yt , yt +1 )| ≤ (4L)−1 .

(5.33)

By (5.33), S+L−1 

ft (yt , yt +1 ) ≤

t =S

S+L−1 

ft (xt , xt +1 ) + /4

t =S

≤

S+L−1 

ut (xt , xt +1 ) + .

t =S

Lemma 5.8 is proved. Lemma 5.9 Let , M be positive numbers. Then there exists a natural number L such that for each integer L¯ ≥ L there exists δ > 0 such that the following assertion holds. ¯ each For each pair of integers S1 ≥ 0, S2 ∈ [S1 + L, S1 + L], bt ∈ E(t, δ), t = S1 , . . . , S2 − 1,

(5.34)

each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ, t = S1 , . . . , S2 − 1

(5.35)

5.2 Auxiliary Results

147

2 −1 2 and each ({bt }St =S )-program {xt }St =S satisfying 1 1

S 2 −1

ut (xt , xt +1 ) ≤

t =S1

S 2 −1

f

f

ut (xt , xt ) + M

(5.36)

t =S1

the inequality f

min{ρ(xt , xt ) : t = S1 + 1, . . . , S2 } ≤  holds. Proof We may assume without loss of generality that  < 1. By Theorem 2.3 and (C2), there exists a natural number L such that the following property holds: 2 −1 (i) For each pair of integers T1 ≥ 0, T2 ≥ T1 + L, and each ({at }Tt =T )-program 1 2 {xt }Tt =T which satisfies 1

T 2 −1 t =T1

ft (xt , xt +1) ≤

T 2 −1 t =T1

f

f

ft (xt , xt +1 ) + M + 4

we have f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > /4}) < L. Let an integer L¯ ≥ L. By Lemma 5.8, there exists δ0 > 0 such that the following property holds: ¯ each (ii) For each pair of integers T1 ≥ 0, T2 ∈ [T1 + L, T1 + L], bt ∈ E(t, δ0 ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ0 , t = T1 , . . . , T2 − 1 2 −1 2 2 −1 )-program {xt }Tt =T there exists an ({at }Tt =T )-program and each ({bt }Tt =T 1 1 1 2 {yt }Tt =T such that 1

yT1 = xT1 , ρ(xt , yt ) ≤ /4 for all t = T1 , . . . , T2

148

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

and T 2 −1

ft (yt , yt +1 ) ≤

t =T1

T 2 −1

ut (xt , xt +1 ) + /4.

t =T1

Set δ = min{δ0 ,  L¯ −1 }.

(5.37)

¯ are integers, (5.34) holds, ut ∈ Assume that S1 ≥ 0, S2 ∈ [S1 + L, S1 + L] 2 −1 2 )-program {xt }St =S B(X × X), t = S1 , . . . , S2 − 1 satisfy (5.35) and that a ({bt }St =S 1 1 satisfies (5.36). S2 −1 By (5.34), (5.35), (5.37) and property (ii), there exists an ({at }t =S1 )-program 2 {yt }St =S such that 1

y S1 = x S1 , ρ(xt , yt ) ≤ /4 for all t = S1 , . . . , S2 , S 2 −1

ft (yt , yt +1 ) ≤

t =S1

S 2 −1

ut (xt , xt +1 ) + /4.

t =S1

In view of (5.35)–(5.37) and the equation above, S 2 −1

ft (yt , yt +1 ) ≤

t =S1

S 2 −1

ut (xt , xt +1 ) + /4

t =S1

≤

S 2 −1

f

f

ut (xt , xt ) + M + 1/4

t =S1

≤

S 2 −1

¯ + M + 1/4 ft (xt , xt ) + Lδ f

f

f

f

t =S1

≤

S 2 −1

ft (xt , xt ) + M + 2.

t =S1

Property (i) and the equation above imply that f

min{ρ(xt , xt ) : t = S1 + 1, . . . , S2 } ≤ /4. Lemma 5.9 is proved.

(5.38)

5.2 Auxiliary Results

149

Lemma 5.10 Let  be a positive number. Then there exists δ > 0 and a natural number L such that for each natural number L¯ ≥ 2 there exists δ¯ ∈ (0, δ) such that the following assertion holds. ˜ each For each pair of integers S1 > L, S2 ∈ [S1 + 1, S1 + L], ¯ t = S1 − 1, . . . , S2 − 1, bt ∈ E(t, δ),

(5.39)

each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ¯ t = S1 , . . . , S2 − 1 ut − ft  ≤ δ,

(5.40)

2 −1 2 )-program {xt }St =S which satisfies and each ({bt }St =S 1 1

f

f

ρ(xS1 , xS1 ) ≤ δ, ρ(xS2 , xS2 ) ≤ δ, S 2 −1 t =S1

(5.41)

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + δ 1 1

(5.42)

the following inequality holds: f

ρ(xt , xt ) ≤  for all t = S1 , . . . , S2 . Proof By Lemma 2.11, there exist a natural number L and a positive number γ < /4 such that the following property holds: 2 −1 )-program (i) For each pair of integers T1 ≥ L, T2 > T1 , and each ({at }Tt =T 1 2 {xt }Tt =T which satisfies 1

f

x j = x j , j = T1 , T2 , T 2 −1

ft (xt , xt +1 ) ≤

t =T1

T 2 −1 t =T1

f

f

ft (xt , xt +1 ) + 2γ

we have f

ρ(xt , xt ) ≤ /4 for all t = T1 , . . . , T2 . By (C1), there exists δ ∈ (0, min{γ /8, λ¯ /4})

150

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

such that the following property holds: (ii) For each integer t ≥ 0 and each pair of points (ξ1 , ξ2 ), (η1 , η2 ) ∈ X × X satisfying ρ(ξi , ηi ) ≤ 2δ, i = 1, 2 we have |ft (ξ1 , ξ2 ) − ft (η1 , η2 )| ≤ γ /4. Let L¯ be a natural number. By Lemma 5.8, there exists a positive number δ1 < δ/2 such that the following property holds: ¯ each (iii) For each pair of integers S1 ≥ 0, S2 ∈ [S1 + 1, S1 + L], bt ∈ E(t, δ1 ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ1 , t = S1 , . . . , S2 − 1 2 −1 2 2 −1 )-program {xt }St =S there exists an ({at }St =S )-program and each ({bt }St =S 1 1 1 2 {yt }St =S such that 1

y S1 = x S1 , ρ(xt , yt ) ≤ δ/4, t = S1 , . . . , S2 and S 2 −1 t =S1

ft (yt , yt +1 ) ≤

S 2 −1

ut (xt , xt +1 ) + δ/4.

t =S1

Set ¯ −1 δ1 . δ¯ = (2L) Assume that ¯ S1 > L, S2 ∈ [S1 + 1, S1 + L],

(5.43)

5.2 Auxiliary Results

151

are integers, (5.39) holds, ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfy (5.40) and 2 −1 2 )-program {xt }St =S satisfies (5.41) and (5.42). that a ({bt }St =S 1 1

2 −1 By (5.39), (5.40), (5.43) and property (iii), there exists an ({at }St =S )-program 1

2 {yt }St =S such that 1

y S1 = x S1 , ρ(xt , yt ) ≤ δ/4 for all t = S1 , . . . , S2 ,

(5.44)

and S 2 −1

ft (yt , yt +1 ) ≤

t =S1

S 2 −1

ut (xt , xt +1 ) + δ/4.

(5.45)

t =S1

In view of (5.42) and (5.45), S 2 −1 t =S1

2 −1 2 −1 ft (yt , yt +1) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + δ + δ/4. 1 1

(5.46)

Set f

x˜S1 = xS1 , x˜t = xt for all integers t = S1 + 1, . . . , S2 − 1, x˜S2 = xS2 .

(5.47)

2 2 −1 ¯ is a ({bt }St =S )In view of (5.8), (5.41), (5.47) and the inequality δ < λ/4, {x˜t }St =S 1 1 program. It follows from (5.41), (5.47) and property (ii) that

f

f

|ft (x˜t , x˜t +1 ) − ft (xt , xt +1 )| ≤ γ /4, i = S1 , . . . , S2 − 1. By (5.40), (5.43), (5.46)–(5.48), S 2 −1

ft (yt , yt +1 ) ≤

t =S1

S 2 −1

ut (x˜t , x˜t +1 ) + δ + δ/4

t =S1

≤ δ + δ/4 +

S 2 −1

ft (x˜t , x˜t +1 ) + (S2 − S1 )δ¯

t =S1

≤

S 2 −1 t =S1

ft (xt , xt +1 ) + γ /2 + δ + δ/4 + L¯ δ¯ f

f

(5.48)

152

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

≤

S 2 −1 t =S1

≤

S 2 −1 t =S1

f

f

f

f

(5.49)

f

(5.50)

ft (xt , xt +1 ) + γ /2 + (5/4)δ + δ1 /2

ft (xt , xt +1 ) + γ .

In view of (5.41) and (5.44), for t = S1 , S2 , f

ρ(yt , xt ) ≤ ρ(yt , xt ) + ρ(xt , xt ) ≤ δ/4 + δ ≤ 2δ. Set f

f

yS1 −1 = xS1 −1 , yS2 +1 = xS2 +1 .

(5.51)

2 +1 2 It follows from (5.41), (5.50) and (5.51) that {yt }St =S is an ({at }St =S )-program. 1 −1 1 −1 Property (ii), (5.50) and (5.51) imply that for t = S1 − 1, S2 ,

|ft (yt , yt +1 ) − ft (xt , xt +1 )| ≤ 4−1 γ . f

f

(5.52)

By (5.49) and (5.52), S2 

ft (yt , yt +1 ) ≤

S2 

f

f

ft (xt , xt +1 ) + 2γ .

(5.53)

ρ(yt , xt ) ≤ /4 for all t = S1 − 1, . . . , S2 + 1.

(5.54)

t =S1 −1

t =S1 −1

By (5.51) and (5.53), f

By (5.44) and (5.54), for all t = S1 , . . . , S2 , f

f

ρ(xt , xt ) ≤ ρ(xt , yt ) + ρ(yt , xt ) ≤ . This completes the proof of Lemma 5.10. In order to prove our stability results we need the following two lemmas. Lemma 5.11 Let , M0 be positive numbers and l1 , l2 be natural numbers. Then there exists δ > 0 and a natural number L > l1 + l2 such that for each pair of integers S1 ≥ 0, S2 ≥ S1 + L, each bt ∈ E(t, δ), t = S1 , . . . , S2 − 1,

5.2 Auxiliary Results

153

each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ, t = S1 , . . . , S2 − 1, 2 −1 2 )-program {xt }St =S satisfying each ({bt }St =S 1 1 2 −1 xS2 ∈ Y ({bt }St =S , S2 − l2 , S2 ) 2 −l2

and S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + M0 1 1

and each integer S ∈ [S1 , S2 − L] satisfying 1 −1 xS ∈ Y¯ ({bt }S+l , S, S + l1 ) t =S

the inequality f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L} ≤  holds. Lemma 5.12 Let , M0 be positive numbers and l1 be a natural number. Then there exists δ > 0 and a natural number L > l1 such that for each pair of integers S1 ≥ 0, S2 ≥ S1 + L, each bt ∈ E(t, δ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ, t = S1 , . . . , S2 − 1, 2 −1 2 )-program {xt }St =S satisfying each ({bt }St =S 1 1

S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 ) + M0 1 1

and each integer S ∈ [S1 , S2 − L] satisfying 1 −1 , S, S + l1 ) xS ∈ Y¯ ({bt }S+l t =S

154

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

the inequality f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L} ≤  holds. Proof We prove Lemmas 5.11 and 5.12 simultaneously. We may assume without loss of generality that ¯  < λ.

(5.55)

By Lemma 5.9, there exist a natural number L0 and δ ∈ (0, 1) such that the following property holds: (i) For each integer S ≥ 0, each bt ∈ E(t, δ), t = S, . . . , S + L0 − 1, each ut ∈ B(X × X), t = S, . . . , S + L0 − 1 satisfying ut − ft  ≤ δ, t = S, . . . , S + L0 − 1 S+L −1

and each ({bt }t =S 0

S+L −1

)-program {xt }t =S 0

S+L 0 −1 

ut (xt , xt +1 ) ≤

t =S

satisfying

S+L 0 −1 

f

f

ut (xt , xt ) + 1

t =S

the inequality f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L0 } ≤  holds. In the case of Lemma 5.12 set l2 = 1. Choose a natural number k0 such that k0 > 2(sup{ft  : t = 0, 1, . . . } + 1)(L0 + l1 + l2 + 1) + M0 + 1.

(5.56)

Set L = L0 k0 .

(5.57)

Let S1 ≥ 0, S2 ≥ S1 + L be integers, bt ∈ E(t, δ), t = S1 , . . . , S2 − 1,

(5.58)

5.2 Auxiliary Results

155

ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfy ut − ft  ≤ δ, t = S1 , . . . , S2 − 1,

(5.59)

2 −1 2 )-program {xt }St =S satisfy and a ({bt }St =S 1 1 2 −1 xS2 ∈ Y ({bt }St =S , S2 − l2 , S2 ) 2 −l2

(5.60)

and S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + M0 1 1

(5.61)

in the case of Lemma 5.11 and satisfy S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 ) + M0 1 1

(5.62)

in the case of Lemma 5.12. Assume that an integer S ∈ [S1 , S2 − L], 1 −1 xS ∈ Y¯ ({bt }S+l , S, S + l1 ). t =S

(5.63)

In order to complete the proof of Lemmas 5.11 and 5.12 it is sufficient to show that f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L} ≤ . Assume the contrary. Then f

ρ(xt , xt ) >  for all t = S + 1, . . . , S + L.

(5.64)

There are two cases: (1) There is an integer S0 ∈ (S, S2 ] such that f

ρ(xS0 , xS0 ) ≤ ;

(5.65)

ρ(xt , xt ) >  for all t = S + 1, . . . , S2 .

(5.66)

(2) f

156

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

Assume that case (1) holds. In view of (5.64) and (5.65), S0 > S + L. We may assume without loss of generality that f

ρ(xt , xt ) >  for all t = S + 1, . . . , S0 − 1.

(5.67)

1 −1 1 In view of (5.63), there exists a ({bt }S+l )-program {yt }S+l t =S t =S such that

f

yS = xS , yS+l1 = xS+ll .

(5.68)

Set f

yt = xt for all integers t ∈ {S + l1 + 1, . . . , S0 − 1}, y S0 = x S0 .

(5.69) (5.70)

S −1

S

0 0 We claim that {yt }t =S is a ({bt }t =S )-program. In view of (5.8), (5.10), (5.68) and (5.69) it is sufficient to show that

f

xS0 ∈ bS0 (xS0 −1 ).

(5.71)

It is easy to see that (5.71) follows from (5.55) and (5.65). It follows from (5.59), (5.61), (5.64) and (5.68)–(5.70) that S 0 −1

ut (xt , xt +1 ) − M0 ≤

t =S

≤

S 0 −1

S 0 −1

ut (yt , yt +1 )

t =S f

f

ut (xt , xt ) + 2

t =S

≤

S 0 −1

S+l 1 −1 

ut  + 2uS0 −1 

t =S f

f

ut (xt , xt ) + 2(l1 + 1)(sup{ft  : t = 0, 1, . . . } + 1).

(5.72)

t =S

There exists a natural number k such that S0 − S ∈ (kL0 , (k + 1)L0 ].

(5.73)

By (5.57), (5.73) and the inequality S0 > S + L, k ≥ k0 .

(5.74)

5.2 Auxiliary Results

157

It follows from (5.58), (5.59), (5.67), (5.73) and property (i) that for each integer j ∈ [0, k − 1], S+(j  +1)L0 −1

ut (xt , xt +1 ) >

S+(j  +1)L0 −1

t =S+j L0

t =S+j L0

f

f

ut (xt , xt +1 ) + 1.

(5.75)

By (5.59) and (5.72)–(5.75), M0 + 2(l1 + 1)(sup{ft  : t = 0, 1, . . . } + 1) ≥

S 0 −1

ut (xt , xt +1 ) −

t =S

≥

t =S

+1)L0 −1 k−1 S+(j   j =0

S 0 −1

t =S+j L0

f

f

ut (xt , xt +1 ) f

f

(ut (xt , xt +1 ) − ut (xt , xt +1 ))



− 2L0 sup{ut  : t = 0, 1, . . . } ≥ k − 2L0 (sup{ft  : t = 0, 1, . . . } + 1) and k0 ≤ k ≤ M0 + 2(sup{ft  : t = 0, 1, . . . } + 1)(L0 + l1 + 1). This contradicts (5.56). The contradiction we have reached proves that case (1) does not hold. Therefore case (2) holds and (5.66) is true. 1 −1 1 In view of (5.63), there exists a ({bt }S+l )-program {yt }S+l t =S t =S such that (5.68) holds. In the case of Lemma 5.12 set f

yt = xt for all integers t ∈ [S + l1 + 1, S2 ]

(5.76)

2 2 −1 is a ({bt }St =S )-program. and then in view of (5.8), {yt }St =S In the case of Lemma 5.11, in view of (5.60), there exists yt ∈ X, t = S2 − l2 , . . . , S2 such that 2 2 −1 is a ({bt }St =S )-program {yt }St =S 2 −l2 2 −l2

(5.77)

satisfying f

yS2 −l2 = xS2 −l2 , yS2 = xS2 .

(5.78)

In the case of Lemma 5.11 set f

yt = xt for all integers t satisfying S + l1 < t < S2 − l2

(5.79)

158

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

2 2 −1 and then in view of (5.8), (5.68) and (5.77)–(5.79), {yt }St =S is also a ({bt }St =S )program. It follows from (5.61), (5.62), (5.68) and (5.78) that

S 2 −1

ut (xt , xt +1 ) ≤

t =S

S 2 −1

ut (yt , yt +1 ) + M0 .

(5.80)

t =S

By (5.68), (5.76), (5.78) and (5.79), f

yt = xt for all integers t = S + l1 , . . . , S2 − l2 .

(5.81)

By (5.59), (5.80) and (5.81), S 2 −1

ut (xt , xt +1 ) ≤

t =S

≤

S 2 −1

S 2 −1 t =S

ut (yt , yt +1 ) + M0

t =S

t =S

≤

S 2 −1

f

f

f

f

ut (xt , xt +1 ) + 2(l1 + l2 ) sup{ut  : t = 0, 2, . . . } + M0 ut (xt , xt +1 ) + M0 + 2(l1 + l2 )(sup{ft  : t = 0, 2, . . . } + 1).

(5.82)

There exists a natural number k such that S2 − S ∈ [kL0 , (k + 1)L0 ).

(5.83)

By (5.57), (5.63) and (5.83), k ≥ k0 .

(5.84)

It follows from (5.58), (5.59), (5.66), (5.83) and property (i) that for each integer j ∈ [0, k − 1], S+(j  +1)L0 −1 t =S+j L0

ut (xt , xt +1 ) >

S+(j  +1)L0 −1 t =S+j L0

f

f

ut (xt , xt +1 ) + 1.

(5.85)

5.3 Proofs of Theorems 5.1 and 5.2

159

By (5.59) and (5.82)–(5.85), M0 + 2(l1 + l2 )(sup{ft  : t = 0, 2, . . . } + 1) ≥

S 2 −1

ut (xt , xt +1 ) −

t =S

≥

t =S

+1)L0 −1 k−1 S+(j   j =0

S 2 −1

t =S+j L0

f

f

ut (xt , xt +1 ) f

f

(ut (xt , xt +1 ) − ut (xt , xt +1 ))



− 2 L0 (sup{ft  : t = 0, 2, . . . } + 1) ≥ k0 − 2L0 (sup{ft  : t = 0, 2, . . . } + 1) and k0 ≤ 2(sup{ft  : t = 0, 2, . . . } + 1)(L0 + l1 + l2 + 1) + M0 . This contradicts (5.56). The contradiction we have reached completes the proof of Lemmas 5.11 and 5.12.

5.3 Proofs of Theorems 5.1 and 5.2 We prove Theorems 5.1 and 5.2 simultaneously. By Lemma 5.10, there exist ¯ , 1}) γ ∈ (0, min{λ, and a natural number L0 such that the following property holds: (i) For each integer L ≥ 2 there exists γL ∈ (0, γ ) such that for each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + L], each bt ∈ E(t, γL ), t = S1 − 1, . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ γL , t = S1 , . . . , S2 − 1

160

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems 2 −1 2 and each ({bt }St =S )-program {xt }St =S which satisfies 1 1

f

f

ρ(xS1 , xS1 ) ≤ γ , ρ(xS2 , xS2 ) ≤ γ , S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + γ 1 1

we have f

ρ(xt , xt ) ≤  for all t = S1 , . . . , S2 . In the case of Theorem 5.2 set l2 = 1. By Lemmas 5.11 and 5.12 (with  = γ and M0 = 1), there exist γ¯ ∈ (0, γ ) and a natural number L > l1 + l2 + L0 such that the following properties hold: (ii) For each pair of integers S1 ≥ 0, S2 ≥ S1 + L, each bt ∈ E(t, γ¯ ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ γ¯ , t = S1 , . . . , S2 − 1, 2 −1 2 )-program {xt }St =S for which at least one of the following each ({bt }St =S 1 1 conditions holds: 2 −1 xS2 ∈ Y ({bt }St =S , S2 − l2 , S2 ) 2 −l2

and S 2 −1 t =S1 S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + 1; 1 1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 ) + 1 1 1

5.3 Proofs of Theorems 5.1 and 5.2

161

and each integer S ∈ [S1 , S2 − L] satisfying 1 −1 xS ∈ Y¯ ({bt }S+l , S, S + l1 ) t =S

we have f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L} ≤ γ . By property (i), there exists δ ∈ (0, γ¯ ) such that the following property holds: (iii) For each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + 2L + 4], each bt ∈ E(t, δ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ, t = S1 , . . . , S2 − 1 2 −1 2 )-program {yt }St =S which satisfies and each ({bt }St =S 1 1

f

f

ρ(yS1 , xS1 ) ≤ γ , ρ(yS2 , xS2 ) ≤ γ , S 2 −1 t =S1

2 −1 2 −1 ut (yt , yt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , yS1 , yS2 ) + γ 1 1

we have f

ρ(yt , xt ) ≤  for all t = S1 , . . . , S2 . Set L˜ = 2L.

(5.86)

Assume that T1 ≥ 0, T2 > T1 + 2L˜ are integers, bt ∈ E(t, δ), t = T1 , . . . , T2 − 1,

(5.87)

ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfy ut − ft  ≤ δ, t = T1 , . . . , T2 − 1

(5.88)

2 −1 2 )-program {xt }Tt =T satisfies and that a ({bt }Tt =T 1 1 1 +l1 −1 xT1 ∈ Y¯ ({bt }Tt =T , T1 , T1 + l1 ). 1

(5.89)

162

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

Moreover, in the case of Theorem 5.1, 2 −1 xT2 ∈ Y ({bt }Tt =T , T2 − l2 , T2 ), 2 −l2

T 2 −1 t =T1

(5.90)

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 , xT2 ) + δ 1 1

(5.91)

and in the case of Theorem 5.2, T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) + δ. 1 1

(5.92)

By induction, it follows from (5.8), (5.87)–(5.89), the inequality γ < λ¯ and property (ii) that there exists a finite sequence of integers Si , i = 0, . . . , q such that T1 ≤ S0 ≤ L + T1 , Sq ∈ [T2 − L, T2 ],

(5.93)

1 ≤ Si+1 − Si ≤ L, i = 0, . . . , q − 1,

(5.94)

f

ρ(xSi , xSi ) ≤ γ , i = 0, . . . , q.

(5.95)

Let an integer ˜ T2 − L] ˜ = [T1 + 2L, T2 − 2L]. t ∈ [T1 + L, In view of (5.93) and (5.94), there exists an integer i ∈ [0, q − 1] such that t ∈ [Si , Si+1 ].

(5.96)

Equations (5.94) and (5.96) imply that Si ≥ t − L ≥ T1 + L˜ − L ≥ T1 + L > L0 . By (5.87), (5.88), (5.91), (5.92), (5.94), (5.95), (5.97) and property (iii), f

ρ(xt , xt ) ≤ . This completes the proofs of Theorems 5.1 and 5.2.

(5.97)

5.4 Proofs of Theorems 5.3 and 5.4

163

5.4 Proofs of Theorems 5.3 and 5.4 We prove Theorems 5.3 and 5.4 simultaneously. By Lemma 5.10, there exist γ ∈ (0, min{λ¯ , , 1}) and a natural number L0 such that the following property holds: (i) For each natural number L ≥ 2 there exists γL ∈ (0, γ ) such that for each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + L], each bt ∈ E(t, γL ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ γL , t = S1 , . . . , S2 − 1 2 −1 2 )-program {xt }St =S which satisfies and each ({bt }St =S 1 1

f

f

ρ(xS1 , xS1 ) ≤ γ , ρ(xS2 , xS2 ) ≤ γ , S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + γ 1 1

we have f

ρ(xt , xt ) ≤  for all t = S1 , . . . , S2 . In the case of Theorem 5.4 set l2 = 1. By Lemmas 5.11 and 5.12 (with  = γ and M0 = M + 1), there exist γ˜ ∈ (0, γ ) and a natural number L1 > l1 + l2 + L0 such that the following properties hold: (ii) For each pair of integers S1 ≥ 0, S2 ≥ S1 + L1 , each bt ∈ E(t, γ˜ ), t = S1 , . . . , S2 − 1,

(5.98)

164

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ γ˜ , t = S1 , . . . , S2 − 1, 2 −1 2 )-program {xt }St =S for which at least one of the following each ({bt }St =S 1 1 conditions holds: 2 −1 xS2 ∈ Y ({bt }St =S , S2 − l2 , S2 ) 2 −l2

and S 2 −1 t =S1 S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + M + 1; 1 1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 ) + M + 1 1 1

and each integer S ∈ [S1 , S2 − 1] satisfying 1 −1 xS ∈ Y¯ ({bt }S+l , S, S + l1 ) t =S

we have f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L1 } ≤ γ . By property (i), there exists δ ∈ (0, γ˜ ) such that the following property holds: (iii) For each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + 2L1 + 4], each bt ∈ E(t, δ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ, t = S1 , . . . , S2 − 1 2 −1 2 )-program {yt }St =S which satisfies and each ({bt }St =S 1 1

f

f

ρ(yS1 , xS1 ) ≤ γ , ρ(yS2 , xS2 ) ≤ γ , S 2 −1 t =S1

2 −1 2 −1 ut (yt , yt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , yS1 , yS2 ) + γ 1 1

5.4 Proofs of Theorems 5.3 and 5.4

165

we have f

ρ(yt , xt ) ≤  for all t = S1 , . . . , S2 . Choose a natural number L > (4 + γ −1 (4 + M))(L0 + L1 + 1).

(5.99)

Assume that T1 ≥ 0, T2 > T1 + L are integers, bt ∈ E(t, δ), t = T1 , . . . , T2 − 1,

(5.100)

ut ∈ B(X × X), t = T1 , . . . , T2 − 1, ut − ft  ≤ δ, t = T1 , . . . , T2 − 1

(5.101)

2 −1 2 and that a ({bt }Tt =T )-program {xt }Tt =T satisfies 1 1 1 +l1 −1 xT1 ∈ Y¯ ({bt }Tt =T , T1 , T1 + l1 ). 1

(5.102)

Moreover, in the case of Theorem 5.3, assume that 2 −1 , T2 − l2 , T2 ), xT2 ∈ Y ({bt }Tt =T 2 −l2

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 , xT2 ) + M 1 1

(5.103) (5.104)

and in the case of Theorem 5.4, that T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) + M. 1 1

(5.105)

¯ δ < By induction, applying property (ii) and using the inequalities γ < λ, γ˜ , (5.8), (5.100)–(5.105), we obtain that there exists a finite sequence of integers Si , i = 0, . . . , q such that T1 ≤ S0 ≤ L1 + T1 , Sq ∈ (T2 − L1 T2 ],

(5.106)

1 ≤ Si+1 − Si ≤ L1 , i = 0, . . . , q − 1,

(5.107)

f

ρ(xSi , xSi ) ≤ γ , i = 0, . . . , q.

(5.108)

166

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

By (5.104) and (5.105), M≥

T 2 −1 t =T1

≥

2 −1 2 −1 ut (xt , xt +1 ) − U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 , xT2 ) 1 1

q−1 Si+1  −1 i=0

t =Si

 Si+1 −1 Si+1 −1 ut (xt , xt +1 ) − U ({ut }t =S , {b } , S , S , x , x ) . t i i+1 S S i i+1 t =S i i (5.109)

Clearly, there exists an integer j0 ≥ 0 such that T1 + L1 ∈ [Sj0 , Sj0 +1 ].

(5.110)

In view of (5.107) and (5.110), Sj0 ≤ L1 + T1 , Sj0 +1 ≤ Sj0 + L1 ≤ 2L1 + T1 .

(5.111)

Equations (5.99), (5.106), (5.107) and (5.111) imply that j0 + 1 < q − 1.

(5.112)

Set E = {i ∈ {j0 + 1, . . . , q − 1} : Si+1 −1



t =Si

S

−1

S

−1

i+1 i+1 ut (xt , xt +1 ) > U ({ut }t =S , {bt }t =S , Si , Si+1 , xSi , xSi+1 ) + γ }. i i

(5.113) By (5.109) and (5.113), M ≥ γ Card(E) and Card(E) ≤ γ −1 M.

(5.114)

j ∈ {j0 + 1, . . . , q − 1} \ E.

(5.115)

Let

5.5 Proof of Theorem 5.5

167

In view of (5.113) and (5.115), Sj+1 −1



t =Sj

S

−1

j+1 ut (xt , xt +1 ) ≤ U ({ut }t =S j

S

−1

j+1 , {bt }t =S j

, Sj , Sj +1 , xSj , xSj+1 ) + γ . (5.116)

It follows from (5.100), (5.101), (5.108), (5.110), (5.115), (5.116) and property (iii) that f

ρ(xt , xt ) ≤ , t = Sj , . . . , Sj +1 . This implies that f

{t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > } ⊂ {T1 , . . . , Sj0 +1 } ∪ {Sq , . . . , T2 } ∪ ({Si , . . . , Si+1 } : i ∈ E}. By the inclusion above, (5.99), (5.106), (5.111) and (5.114), ¯ > }) Card({t ∈ {T1 , . . . , T2 } : ρ(xt , x) ≤ 3L1 + 1 + (L1 + 1)Card(E) ≤ 3L1 + 1 + (L1 + 1)Mγ −1 < L. This completes the proofs of Theorems 5.3 and 5.4.

5.5 Proof of Theorem 5.5 By Lemma 5.10, there exists ¯ , 1}) γ ∈ (0, min{λ, and a natural number L0 such that the following property holds: (i) For each natural number L ≥ 2 there exists γL ∈ (0, γ ) such that for each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + L], each bt ∈ E(t, γL ), t = S1 , . . . , S2 − 1,

168

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ γL , t = S1 , . . . , S2 − 1 2 −1 2 )-program {xt }St =S which satisfies and each ({bt }St =S 1 1

f

f

ρ(xS1 , xS1 ) ≤ γ , ρ(xS2 , xS2 ) ≤ γ , S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + γ 1 1

we have f

ρ(xt , xt ) ≤  for all t = S1 , . . . , S2 . By Lemma 5.9 (with  = γ ), there exists a natural number L1 > L0 such that for each integer L2 ≥ L1 there exists γ0 ∈ (0, γ ) such that the following property holds: (ii) For each pair of integers S1 ≥ 0, S2 ∈ [S1 + L1 , S1 + L2 ], each bt ∈ E(t, γ0 ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ γ0 , t = S1 , . . . , S2 − 1 2 −1 2 and each ({bt }St =S )-program {xt }St =S satisfying 1 1

S 2 −1

ut (xt , xt +1 ) ≤

t =S1

S 2 −1

f

f

ut (xt , xt ) + M + 1

t =S1

we have f

min{ρ(xt , xt ) : t = S1 + 1, . . . , S2 } ≤ γ . Let an integer L2 ≥ L1 and let γ0 ∈ (0, γ ) be as guaranteed by property (ii). By property (i), there exists δ ∈ (0, γ0 ) such that the following property holds: (iii) For each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + 2L2 + 4], each bt ∈ E(t, δ), t = S1 , . . . , S2 − 1,

5.5 Proof of Theorem 5.5

169

each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ, t = S1 , . . . , S2 − 1 2 −1 2 )-program {yt }St =S which satisfies and each ({bt }St =S 1 1

f

f

ρ(yS1 , xS1 ) ≤ γ , ρ(yS2 , xS2 ) ≤ γ , S 2 −1 t =S1

2 −1 2 −1 ut (yt , yt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , yS1 , yS2 ) + γ 1 1

we have f

ρ(xt , xt ) ≤  for all t = S1 , . . . , S2 . Assume that T1 ≥ 0, T2 > T1 + 6L2

(5.117)

bt ∈ E(t, δ), t = T1 , . . . , T2 − 1,

(5.118)

is a pair of integers,

ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfy ut − ft  ≤ δ, t = T1 , . . . , T2 − 1

(5.119)

2 −1 2 )-program {xt }Tt =T and a finite sequence of integers {Si }i=0 and that a ({bt }Tt =T 1 1 satisfy

q

S0 = T1 , Si+1 − Si ∈ [L1 , L2 ], i = 0, . . . , q − 1,

(5.120)

Sq ∈ (T2 − L2 , T2 ],

(5.121)

Si+1 −1



Si+1 −1

ut (xt , xt +1 ) ≤

t =Si



t =Si

f

f

ut (xt , xt +1 ) + M

(5.122)

for each integer i ∈ [0, q − 1], Si+2 −1



t =Si

S

−1

S

−1

i+2 i+2 ut (xt , xt +1 ) ≤ U ({ut }t =S , {bt }t =S , Si , Si+2 , xSi , xSi+2 ) + γ i i

(5.123)

170

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

for each integer i ∈ [0, q − 2] and T 2 −1 t =Sq−2

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =S , {bt }Tt =S , Sq−2 , T2 , xSq−2 , xT2 ) + γ . q−2 q−2

(5.124) In view of (5.117), (5.120), (5.121) and (5.122), 6L2 < T2 − T1 ≤ Sq − S0 + L2 ≤ (q + 1)L2 , q ≥ 6.

(5.125)

Let an integer i ∈ [0, q − 1]. By (5.118)–(5.121), the inequality δ < γ0 , the choice of γ0 and property (ii), there exists an integer τi such that τi ∈ [Si + 1, Si+1 ] and ρ(xτi , xτfi ) ≤ γ .

(5.126)

In view of (5.120), (5.121) and (5.126), τ0 ≤ T1 + L2 , τq−1 > T2 − 2L2 .

(5.127)

Let an integer i ∈ [0, q − 2]. By (5.120) and (5.127), 1 ≤ τi+1 − τi ≤ 2L2 , τi , τi+1 ∈ [Si , Si+2 ].

(5.128)

By (5.128), τi+1 −1



t =τi

τ

−1

τ

i+1 ut (xt , xt +1 ) ≤ U ({ut }t =τ , {bt }τi+1 i i

−1

, τi , τi+1 , xτi , xτi+1 ) + γ .

(5.129)

q−1

Thus we have shown that there exists a finite sequence of integers {τi }i=0 such that ρ(xτi , xτfi ) ≤ γ , i = 0, . . . , q − 1,

(5.130)

1 ≤ τi+1 − τi ≤ 2L2 , i = 0, . . . , q − 2,

(5.131)

τ0 ≤ T1 + L2 , τq−1 > T2 − 2L2

(5.132)

and (5.129) holds for all integers i ∈ [0, q − 2]. It follows from (5.120), (5.125), and (5.126) that τ1 > S1 ≥ L1 + T2 > L0 .

(5.133)

5.6 Proofs of Theorems 5.6 and 5.7

171

Let an integer i satisfy 1 ≤ i < q − 1. By (5.118), (5.119), (5.129)–(5.131), (5.133) and property (iii), f

ρ(xt , xt ) ≤ , t = τi , . . . , τi+1 . This implies that f

ρ(xt , xt ) ≤ , t = τ1 , . . . , τq−1 . Together with (5.131) and (5.132) this implies that f

ρ(xt , xt ) ≤ , t = T1 + 3L2 , . . . , T2 − 2L2 and completes the proof of Theorem 5.5.

5.6 Proofs of Theorems 5.6 and 5.7 We prove Theorems 5.6 and 5.7 simultaneously. In the case of Theorem 5.7 set ¯ l2 = 1. We may assume without loss of generality that  < λ. By Lemma 5.10, there exist δ1 ∈ (0, ) and a natural number L0 such that the following property holds: (i) For each natural number L ≥ 2 there exists γL ∈ (0, δ1 ) such that for each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + L], each bt ∈ E(t, γL ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ γL , t = S1 , . . . , S2 − 1 2 −1 2 )-program {xt }St =S which satisfies and each ({bt }St =S 1 1

f

f

ρ(xS1 , xS1 ) ≤ δ1 , ρ(xS2 , xS2 ) ≤ δ1 , S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + δ1 1 1

we have f

ρ(xt , xt ) ≤  for all t = S1 , . . . , S2 .

172

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

By Theorem 5.3 and 5.4, there exist δ2 ∈ (0, δ1 ) and a natural number L1 > 2L0 + l1 + l2

(5.134)

such that the following property holds: (ii) For each pair of integers T1 ≥ 0, T2 > T1 + L1 , each bt ∈ E(t, δ2 ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ2 , t = T1 , . . . , T2 − 1 2 −1 2 and each ({bt }Tt =T )-program {xt }Tt =T which satisfies 1 1 1 +l1 −1 xT1 ∈ Y¯ ({bt }tT=T , T1 , T1 + l1 ) 1

and such that at least one of the following conditions holds: 2 −1 xT2 ∈ Y ({bt }Tt =T , T2 − l2 , T2 ), 2 −l2

T 2 −1 t =T1 T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 , xT2 ) + M + 1; 1 1

2 −1 2 −1 ut (xt , xt +1) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) + M + 1 1 1

we have f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > δ1 }) ≤ L1 . By property (i), there exists δ ∈ (0, δ2 ) such that the following property holds: (iii) For each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + 4L1 + 4], each bt ∈ E(t, δ), t = S1 , . . . , S2 − 1,

5.6 Proofs of Theorems 5.6 and 5.7

173

each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ, t = S1 , . . . , S2 − 1 2 −1 2 )-program {xt }St =S which satisfies and each ({bt }St =S 1 1

f

f

ρ(xS1 , xS1 ) ≤ δ1 , ρ(xS2 , xS2 ) ≤ δ1 , S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + δ1 1 1

we have f

ρ(xt , xt ) ≤  for all t = S1 , . . . , S2 . Choose a natural number L > 2L1 + 4.

(5.135)

Assume that T1 ≥ 0, T2 > T1 + 2L are integers, bt ∈ E(t, δ), t = T1 , . . . , T2 − 1,

(5.136)

ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfy ut − ft  ≤ δ, t = T1 , . . . , T2 − 1

(5.137)

2 −1 2 )-program {xt }Tt =T satisfies and that a ({bt }Tt =T 1 1 1 +l1 −1 xT1 ∈ Y¯ ({bt }Tt =T , T1 , T1 + l1 ), 1

τ +L−1  t =τ

(5.138)

+L−1 +L−1 ut (xt , xt +1 ) ≤ U ({ut }τt =τ , {bt }τt =τ , τ, τ + L, xτ , xτ +L ) + δ

(5.139) for each integer τ ∈ [T1 , T2 − L]. In the case of Theorem 5.6 we assume that 2 −1 xT2 ∈ Y ({bt }Tt =T , T2 − l2 , T2 ), 2 −l2

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 , xT2 ) + M 1 1

(5.140) (5.141)

174

5 Stability of the Turnpike Phenomenon for Nonautonomous Problems

and in the case of Theorem 5.7 assume that T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) + M. 1 1

(5.142)

Property (ii) and (5.135) imply that f

Card({t ∈ {T1 , . . . , T2 } : ρ(xt , xt ) > δ1 }) ≤ L1 .

(5.143)

Assume that τ ∈ [T1 + L, T2 − L]

(5.144)

is an integer. By (5.135), (5.143) and (5.144), there exist integers S1 ∈ [τ − L1 − 2, τ − 1], S2 ∈ [τ + 1, τ + L1 + 2]

(5.145)

f

(5.146)

such that ρ(xSi , xSi ) ≤ δ1 , i = 1, 2.

Equations (5.135), (5.144) and (5.145) imply that there exists an integer S such that [S1 , S2 ] ⊂ [S, S + L] ⊂ [T1 , T2 ].

(5.147)

In view of (5.139), S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + δ. 1 1

(5.148)

By (5.134), (5.144) and (5.145), S1 ≥ T1 + L − L1 − 2 > L0 .

(5.149)

Property (iii), (5.136), (5.137), (5.145)–(5.149) imply that f

ρ(xt , xt ) ≤  for all t = S1 , . . . , S2 and in particular ρ(xτ , xτf ) ≤  for all τ ∈ {T1 +L, . . . , T2 −L}. This completes the proofs of Theorems 5.6 and 5.7.

5.7 An Example

175

5.7 An Example We use the notation, definitions and assumptions of Sect. 5.1 and consider the example of Sect. 2.9. Recall that for each integer t ≥ 0, there exists a mapping at : X → 2X \ {∅} such that Ωt = {(x, y) ∈ X × X : y ∈ at (x)}. In addition to assumptions made in Sect. 2.9 we assume that the families of functions {Lt : t = 0, 1, . . . }, {πtt : t = 0, 1, . . . } are uniformly equicontinuous and that for each  > 0 there exists δ > 0 such that for each integer t ≥ 0 and each (x, y) ∈ X satisfying ρ(x, y) ≤ δ the inequality H (at (x), at (y)) ≤  holds. Then all the conclusions made in Sect. 5.1 hold for this example and all the results of this chapter hold too.

Chapter 6

Stability of the Turnpike Phenomenon for Nonautonomous Problems with Discounting

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. These problems are described by a sequence of nonempty closed sets Ωt ⊂ X × X, t = 0, 1, . . . which determines a class of admissible trajectories (programs) and by a bounded sequence of lower semicontinuous objective functions ft : X ×X → R 1 , t = 0, 1, . . . which determines an optimality criterion. We show the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint sets in the case with discounting.

6.1 Preliminaries and the Main Results We use the notation, definitions and assumptions introduced in Chap. 5. Note that all the results of Chaps. 2–5 hold. Denote by Bl (X×X) the set of all lower semicontinuous functions u ∈ B(X×X). Let t ≥ 0 be an integer. Denote by E(t) the set of all mappings a : X → 2X \ {∅} such that graph(a) = {(x, y) ∈ X × X : y ∈ a(x)} is a closed set and f

f

xt +1 ∈ a(xt ). It is not difficult to see that the following result holds.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_6

177

178

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

Proposition 6.1 Let l be a natural number, integers T1 , T2 satisfy 0 ≤ T1 ≤ T2 − l, 2 −1 {ut }Tt =T ⊂ Bl (X × X), 1 bt ∈ E(t), t = T1 , . . . , T2 − 1, 1 +l−1 x ∈ Y¯ ({bt }Tt =T , T1 , T1 + l). 1 2 −1 2 Then there exists a ({bt }Tt =T )-program {xt }Tt =T such that xT1 = x and 1 1

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) = U ({ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ). 1 1

In this chapter we prove the following result which shows the stability of the turnpike phenomenon in the case of discounting. Theorem 6.2 Let  ∈ (0, λ¯ ) and let l be a natural number. Then there exist δ ∈ (0, ), a natural number L > l and λ ∈ (0, 1) such that for each pair of integers T1 ≥ 0, T2 > T2 + 4L, each bt ∈ E(t, δ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ, t = T1 , . . . , T2 − 1, 2 −1 each sequence {αt }Tt =T ⊂ (0, 1] such that 1

αi αj−1 ≥ λ for each i, j ∈ {T1 , . . . , T2 − 1} satisfying |i − j | ≤ L 2 −1 2 and each ({bt }Tt =T )-program {xt }Tt =T which satisfies 1 1 1 +l−1 , T1 , T1 + l), xT1 ∈ Y¯ ({bt }Tt =T 1

T 2 −1 t =T1

2 −1 2 −1 αt ut (xt , xt +1) = U ({αt ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) 1 1

the following inequality holds: f

ρ(xt , xt ) ≤  for all t = T1 + 2L, . . . , T2 − L. −1 Roughly speaking, the turnpike property holds if discount coefficients {αt }Tt =0 ⊂ (0, 1] are changed rather slowly.

6.1 Preliminaries and the Main Results

179

Let S ≥ 0 be an integer, ut ∈ B(X × X), t = S, S + 1, . . . and let bt : X → 2X \ {∅}, t = S, S + 1, . . . . ∞ A sequence {xt }∞ t =S ⊂ X is called a ({bt }t =S )-program if xt +1 ∈ bt (xt ) for all integers t ≥ S. ∞ ∞ ∞ A ({bt }∞ t =S )-program {xt }t =S is called ({ut }t =S , {bt }t =S )-overtaking optimal if ∞ ∞ for each ({bt }t =S )-program {yt }t =S satisfying xS = yS , we have −1 T −1 

T lim sup ut (xt , xt +1 ) − ut (yt , yt +1 ) ≤ 0. T →∞

t =S

t =S

The following result establishes the stability of the turnpike phenomenon for overtaking optimal programs. ¯ and let l be a natural number. Then there exist δ ∈ Theorem 6.3 Let  ∈ (0, λ) (0, ), a natural number L and λ ∈ (0, 1) such that for each integer T1 ≥ 0, each bt ∈ E(t, δ), t = T1 , T1 + 1, . . . , each ut ∈ B(X × X), t = T1 , T1 + 1, . . . , satisfying ut − ft  ≤ δ, t = T1 , T1 + 1, . . . , each sequence {αt }∞ t =T1 ⊂ (0, 1] such that αi αj−1 ≥ λ for each pair of integers i, j ≥ T1 satisfying |i − j | ≤ L ∞ ∞ and each ({αt ut }∞ t =T1 , {bt }t =T1 )-overtaking optimal program {xt }t =T1 which satisfies 1 +l−1 xT1 ∈ Y¯ ({bt }Tt =T , T1 , T1 + l) 1

the following inequality holds: f

ρ(xt , xt ) ≤  for all integers t ≥ 2L + T1 . In this chapter we prove the following existence result. Theorem 6.4 Let l ≥ 1 be an integer,  = λ¯ /4 and let δ ∈ (0, ), an integer L > l and λ ∈ (0, 1) be as guaranteed by Theorem 6.2. Let ut ∈ Bl (X × X) and ut − ft  ≤ δ, t = 0, 1, . . . , bt ∈ E(t, δ), t = 0, 1, . . . ,

180

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

for any integer t ≥ 1, {(x, y) ∈ X × X : x ∈ bt (x)} be a closed subset of X × X, and let {αt }∞ t =0 ⊂ (0, 1] satisfy the relations lim αt = 0,

t →∞

αi αj−1 ≥ λ for all nonnegative integers i, j satisfying |i − j | ≤ L. Then for each integer T ≥ 0 and each +l−1 , T , T + l) z ∈ Y¯ ({bt }Tt =T (z,T ) ∞ there exists a ({bt }∞ }t =T such that xT(z,T ) = z and that the t =T )-program {xt following property holds: For each real number γ > 0 there exists an integer n0 ≥ 0 such that for each pair of integers T1 ≥ 0, T2 ≥ T1 + n0 and each point 1 +l−1 z ∈ Y¯ ({bt }Tt =T , T1 , T1 + l) 1

the inequality 2 −1 2 −1 , {bt }Tt =T , T1 , T2 , z) |U ({αt ut }Tt =T 1 1

−

T 2 −1 t =T1

1) αt ut (xt(z,T1) , xt(z,T +1 )| ≤ γ

holds. It is clear that Theorem 6.4 establishes the existence of an ({αt ut }∞ t =0 )-overtaking optimal program when the sequence of the discount coefficients {αt }∞ t =0 tends to zero slowly. Note that the existence of an ({αt ut }∞ t =0 )-overtaking optimal program when the discount coefficients {αt }∞ t =0 tends to zero rapidly is a well-known fact. Theorems 6.1–6.4 are new.

6.2 Auxiliary Results In the proofs of Theorems 6.2 and 6.3 we use the following two lemmas. ¯ and l be a natural number. Then there exist δ ∈ (0, ), Lemma 6.5 Let  ∈ (0, λ) a natural number L > l and λ ∈ (0, 1) such that for each pair of integers T1 ≥ 0,

6.2 Auxiliary Results

181

T2 ≥ T2 + L, each bt ∈ E(t, δ), t = T1 , . . . , T2 − 1 each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ δ for all integers t = T1 , . . . , T2 − 1, 2 −1 ⊂ (0, 1] such that each sequence {αt }Tt =T 1

αi αj−1 ≥ λ for each i, j ∈ {T1 , . . . , T2 − 1} satisfying |i − j | ≤ L, 2 −1 2 each ({bt }Tt =T )-program {xt }Tt =T which satisfies 1 1

T 2 −1 t =T1

2 −1 2 −1 αt ut (xt , xt +1 ) = U ({αt ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) 1 1

and each integer S ∈ [T1 , T2 − L] satisfying xS ∈ Y¯ ({bt }S+l−1 t =S , S, S + l) the inequality f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L} ≤  holds. ¯ and l be a natural number. Then there exist δ ∈ (0, ), Lemma 6.6 Let  ∈ (0, λ) a natural number L and λ ∈ (0, 1) such that for each integer T1 ≥ 0, each bt ∈ E(t, δ), t = T1 , T1 + 1, . . . , each ut ∈ B(X × X), t = T1 , T1 + 1, . . . satisfying ut − ft  ≤ δ for all integers t ≥ T1 , each sequence {αt }∞ t =T1 ⊂ (0, 1] such that αi αj−1 ≥ λ for each pair of integers i, j ≥ T1 satisfying |i − j | ≤ L,

182

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

∞ ∞ each ({αt ut }∞ t =T1 , {bt }t =T1 )-overtaking optimal program {xt }t =T1 and each integer S ≥ T1 satisfying

xS ∈ Y¯ ({bt }S+l−1 t =S , S, S + l) the inequality f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L} ≤  holds. Proof We prove Lemmas 6.5 and 6.6 simultaneously. We may assume that  < 1. By Lemma 5.9, there exists a natural number L0 such that the following property holds: (i) For each integer L¯ ≥ L0 there exists δ¯ > 0 such that for each pair of integers ¯ each S1 ≥ 0, S2 ∈ [S1 + L0 , S1 + L], ¯ t = S1 , . . . , S2 − 1, bt ∈ E(t, δ), each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ¯ t = S1 , . . . , S2 − 1 ut − ft  ≤ δ, 2 −1 2 )-program {xt }St =S satisfying and each ({bt }St =S 1 1

S 2 −1

ut (xt , xt +1 ) ≤

t =S1

S 2 −1

f

f

ut (xt , xt ) + 1

(6.1)

t =S1

the inequality f

min{ρ(xt , xt ) : t = S1 + 1, . . . , S2 } ≤ 

(6.2)

holds. Choose a natural number k0 such that k0 > 8(sup{ft  : t = 0, 1, . . . } + 1)(L0 + l + 1).

(6.3)

Set L = L0 k0 .

(6.4)

6.2 Auxiliary Results

183

By property (i), there exists δ ∈ (0, /4) such that the following property holds: (ii) For each pair of integers S1 ≥ 0, S2 ∈ [S1 + L0 , S1 + 8L + 8], each bt ∈ E(t, δ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ, t = S1 , . . . , S2 − 1 2 −1 2 )-program {xt }St =S satisfying (6.1) relation (6.2) is true. and each ({bt }St =S 1 1

Choose a number λ ∈ (0, 1) such that λl > 2−1 ,

(6.5)

8(sup{ft  : t = 0, 1, . . . } + 1)L0 |1 − λ| < 2−1 ,

(6.6)

λk0 +l+1 k0 > 8(sup{ft  : t = 0, 1, . . . } + 1)(L0 + l + 1)

(6.7)

(see (6.3)). We suppose that ∞ + x = ∞ for any x ∈ R 1 and that the sum over an empty set is zero. Let T1 ≥ 0 be an integer, T2 ∈ {1, 2 . . . , } ∪ {∞} and T2 ≥ T1 + L,

(6.8)

for all integers t satisfying T1 ≤ t < T2 , bt ∈ E(t, δ),

(6.9)

ut ∈ B(X × X) and ut − ft  ≤ δ,

(6.10)

αt ∈ (0, 1]

(6.11)

satisfy αi αj−1 ≥ λ for each pair of integes i, j ≥ T1 satisfying i, j < T2 and |i − j | ≤ L (6.12)

184

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

2 −1 2 and let a ({bt }Tt =T )-program {xt }Tt =T satisfy 1 1

T 2 −1 t =T1

2 −1 2 −1 αt ut (xt , xt +1 ) = U ({αt ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) 1 1

(6.13)

in the case of Lemma 6.5 and let ∞ ∞ {xt }∞ t =T1 be an ({αt ut }t =T1 , {bt }t =T1 )-overtaking optimal program

(6.14)

in the case of Lemma 6.6. Assume that an integer S satisfies S ≥ T1 , S ≤ T2 − L, xS ∈ Y¯ ({bt }S+l−1 t =S , S, S + l).

(6.15)

In order to complete the proof of Lemmas 6.5 and 6.6 it is sufficient to show that f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L} ≤ . Assume the contrary. Then f

ρ(xt , xt ) >  for all t = S + 1, . . . , S + L.

(6.16)

There are two cases: (1) There is an integer S0 > S such that S0 ≤ T2 and f

ρ(xS0 , xS0 ) ≤ ;

(6.17)

(2) f

ρ(xt , xt ) >  for all integers t satisfying S + 1 ≤ t ≤ T .

(6.18)

Assume that case (1) holds. Thus (6.17) holds. In view of (6.16) and (6.17), S0 > S + L.

(6.19)

We may assume without loss of generality that f

ρ(xt , xt ) >  for all t = S + 1, . . . , S0 − 1.

(6.20)

S+l In view of (6.15), there exists a ({bt }S+l−1 t =S )-program {yt }t =S such that f

yS = xS , yS+l = xS+l .

(6.21)

6.2 Auxiliary Results

185

Set f

yS0 = xS0 , yt = xt for all integers t satisfying S + l + 1 ≤ t < S0 .

(6.22)

¯ {yt }S0 is a ({bt }S0 −1 )By (5.8), (6.17), (6.21), (6.22) and the inequality  < λ, t =S t =S program. It follows from (6.3), (6.4), (6.10), (6.12)–(6.14), (6.21) and (6.22) that S 0 −1

αt ut (xt , xt +1 ) ≤

S 0 −1

t =S

≤

S 0 −1 t =S

≤

S 0 −1 t =S

αt ut (yt , yt +1 )

t =S f

f

αt ut (xt , xt +1 ) + 2 f

S+l−1 

αt ut  + 2uS0 −1 αS0 −1

t =S

f

αt ut (xt , xt +1 )

+ 2αS λ−1 l(sup{ft  : t = 0, 1, . . . } + 1) + 2αS0 −1 (sup{ft  : t = 0, 1, . . . } + 1).

(6.23)

There exists a natural number k such that S0 − S ∈ (kL0 , (k + 1)L0 ].

(6.24)

k ≥ k0 .

(6.25)

By (6.4), (6.19) and (6.24),

It follows from (6.9), (6.10), (6.20), (6.24) and property (ii) that for each integer j ∈ [0, k − 1], S+(j  +1)L0 −1

ut (xt , xt +1 ) >

t =S+j L0

S+(j  +1)L0 −1 t =S+j L0

f

f

ut (xt , xt +1 ) + 1.

(6.26)

Let an integer j ∈ [0, k − 1].

(6.27)

By (6.4)–(6.6), (6.12) and (6.26), S+(j  +1)L0 −1 t =S+j L0

αt ut (xt , xt +1 ) −

S+(j  +1)L0 −1 t =S+j L0

f

f

αt ut (xt , xt +1 )

186

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

= αS+j L0

+1)L0 −1

S+(j 

ut (xt , xt +1 ) −

S+(j  +1)L0 −1

t =S+j L0

t =S+j L0

f

f



ut (xt , xt +1 )

S+(j  +1)L0 −1 S+(j  +1)L0 −1

 −1 α u (x , x ) − ut (xt , xt +1 ) + αS+j L0 αS+j t t t t +1 L0 t =S+j L0

+ αS+j L0

+1)L0 −1

S+(j  t =S+j L0

> αS+j L0 − αS+j L0

f f −1 ut (xt , xt +1 ) − αS+j L0

+1)L0 −1

S+(j  t =S+j L0

− αS+j L0

+1)L0 −1

S+(j  t =S+j L0

t =S+j L0

S+(j  +1)L0 −1 t =S+j L0

f

f



αt ut (xt , xt +1 )

−1 |αt αS+j L0 − 1|(sup{ft  : t = 0, 1, . . . } + 1)

−1 |αt αS+j L0 − 1|(sup{ft  : t = 0, 1, . . . } + 1)





≥ αS+j L0 [1 − 2(sup{ft  : t = 0, 1, . . . } + 1)L0 |1 − λ|λ−1 ] ≥ αS+j L0 [1 − 4(sup{ft  : t = 0, 1, . . . } + 1)L0 |1 − λ|] ≥ 2−1 αS+j L0 . (6.28) By (6.23), (6.24) and (6.28), (2αS0 −1 + 2αS λ−1 l)(sup{ft  : t = 0, 1, . . . } + 1) ≥

S 0 −1

αt ut (xt , xt +1 ) −

t =S

=

t =S

+1)L0 −1 k−1 S+(j   j =0

S 0 −1

t =S+j L0

f

f

αt ut (xt , xt +1 )

αt ut (xt , xt +1 ) −

S+(j  +1)L0 −1 t =S+j L0

f

f



αt ut (xt , xt +1 )

− 2(sup{ft  : t = 0, 1, . . . } + 1)L0 max{αt : t = S + kL0 , . . . , S0 } ≥ 2−1

k−1 

αS+j L0 − 2(sup{ft  : t = 0, 1, . . . } + 1)L0 λ−1 αS+kL0 .

(6.29)

j =0

It follows from (6.4), (6.7), (6.12), (6.24) and (6.29) that k−1 

αS+j L0 ≤ 4(sup{ft  : t = 0, 1, . . . } + 1)[αS λ−1 l + αS0 −1 + L0 λ−1 αS+kL0 ]

j =0

≤ 4(sup{ft  : t = 0, 1, . . . } + 1)[αS λ−1 l + (L0 + 1)λ−1 αS+kL0 ].

6.2 Auxiliary Results

187

In view of the inequality above, (6.4), (6.12) and (6.25), 0≥

k−1 

αS+j L0 − 4(sup{ft  : t = 0, 1, . . . } + 1)[αS λ−1 l + (L0 + 1)λ−1 αS+kL0 ]

j =0

= 2−1

k−1 

αS+j L0 − 4(sup{ft  : t = 0, 1, . . . } + 1)αS λ−1 l

j =0

+ 2−1

k−1 

αS+j L0 − 4(sup{ft  : t = 0, 1, . . . } + 1)(L0 + 1)λ−1 αS+kL0

j =0

≥ 2−1

k 0 −1

αS λj − 4(sup{ft  : t = 0, 1, . . . } + 1)αS λ−1 l

j =0 k−1 

+ 2−1 αS+kL0

λk−j − 4(sup{ft  : t = 0, 1, . . . } + 1)(L0 + 1)λ−1 αS+kL0

j =k−k0

≥ αS [2−1 k0 λk0 − 4(sup{ft  : t = 0, 1, . . . } + 1)λ−1 l] + αS+kL0 [2−1 k0 λk0 − 4(sup{ft  : t = 0, 1, . . . } + 1)(L0 + 1)λ−1 ] > 0,

a contradiction. The contradiction we have reached proves that case (1) does not hold. Therefore case (2) holds and (6.18) is true. S+l In view of (6.15), there exists a ({bt }S+l−1 t =S )-program {yt }t =S such that f

yS = xS , yS+l = xS+l .

(6.30)

Set f

yt = xt for all integers t satisfying S + l < t ≤ T2 .

(6.31)

2 2 −1 By (5.8), (6.30) and (6.31), {yt }Tt =S is a ({bt }Tt =S )-program. It follows from (6.9), (6.10), (6.15), (6.18) and property (ii) that for each integer j ≥ 0 satisfying S + (j + 1)L0 ≤ T2 ,

S+(j  +1)L0 −1 t =S+j L0

ut (xt , xt +1 ) >

S+(j  +1)L0 −1 t =S+j L0

f

f

ut (xt , xt +1 ) + 1.

Arguing as in case (1) we show that for each integer j ≥ 0 satisfying S+(j +1)L0 ≤ T2 , (6.28) holds.

188

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

Consider the case of Lemma 6.6. By (6.3), (6.4), (6.7), (6.12), (6.14), (6.28), (6.30) and (6.31), S+kL  S+kL 0 −1 0 −1 αt ut (xt , xt +1 ) − αt ut (yt , yt ) 0 ≥ lim sup k→∞

t =S

t =S

S+kL S+kL  0 −1 0 −1 f f = lim sup αt ut (xt , xt +1 ) − αt ut (xt , xt +1 ) k→∞

+

t =S

S+l−1  t =S

t =S

f

f

αt ut (xt , xt +1 ) −

S+l−1 

αt ut (yt , yt +1 )

t =S

+1)L0 −1 S+(j  +1)L0 −1 k−1 S+(j 

  f f ≥ lim sup αt ut (xt , xt +1 ) − αt ut (xt , xt +1 ) k→∞

j =0

t =S+j L0

t =S+j L0

− 2(sup{ft  : t = 0, 1, . . . } + 1)lαS λ−1 ≥ lim sup

k−1 

k→∞ j =0

≥ 2−1

k 0 −1

2−1 αS+j L0 − 2(sup{ft  : t = 0, 1, . . . } + 1)lαS λ−1

αS λj − 2(sup{ft  : t = 0, 1, . . . } + 1))lαS λ−1

j =0

≥ αS (k0 λk0 − 2(sup{ft  : t = 0, 1, . . . } + 1)lλ−1 ) > 0, a contradiction. The contradiction we have reached proves Lemma 6.6. Let us complete the proof of Lemma 6.5. In view of (6.15), there exists a natural number k such that T2 − S ∈ [kL0 , (k + 1)L0 ).

(6.32)

k ≥ k0 .

(6.33)

By (6.4), (6.15) and (6.32),

By (6.3), (6.4), (6.7), (6.10), (6.12), (6.13), (6.30), (6.32), (6.33) and (6.28) which holds for each integer j ≥ 0 satisfying S + (j + 1)L0 ≤ T2 , 0≥

T 2 −1

αt ut (xt , xt+1 ) −

t=S

≥

T 2 −1 t=S

T 2 −1

αt ut (yt , yt+1 )

t=S

αt ut (xt , xt+1 ) −

T 2 −1 t=S

f

f

αt ut (xt , xt+1 )

6.3 Proofs of Theorems 6.2 and 6.3

+

S+l−1 

f

t=S

=

+



S+l−1 

αt ut (yt , yt+1 )

t=S

+1)L0 −1 k−1 S+(j   j =0

f

αt ut (xt , xt+1 ) −

189

αt ut (xt , xt+1 ) −

S+(j  +1)L0 −1

t=S+j L0

 f f αt ut (xt , xt+1 )

t=S+j L0 f

f

{αt ut (xt , xt+1 ) − αt ut (xt , xt+1 ) : t is an integer and S + kL0 ≤ t < T2 }

− 2(sup{ft  : t = 0, 1, . . . } + 1)lαS λ−l ≥ 2−1

k−1 

αS+j L0 − 2L0 (sup{ft  : t = 0, 1, . . . } + 1)αS+kL0 λ−1

j =0

− 2(sup{ft  : t = 0, 1, . . . } + 1)αS λ−l l ≥ 4−1 αS

k 0 −1

λj − 2(sup{ft  : t = 0, 1, . . . } + 1)αS λ−l l

j =0 k−1 

+ 4−1

αS+kL0 λk−j − 2L0 (sup{ft  : t = 0, 1, . . . } + 1)αS+kL0 λ−1

j =k−k0 −1

≥ αS (4

k0 λk0 − 2(sup{ft  : t = 0, 1, . . . } + 1)λ−1 l)

+ αS+kL0 (4−1 k0 λk0 − 2L0 (sup{ft  : t = 0, 1, . . . } + 1)λ−1 ) > 0,

a contradiction. The contradiction we have reached proves Lemmas 6.5 and 6.6.

6.3 Proofs of Theorems 6.2 and 6.3 We prove Theorems 6.2 and 6.3 simultaneously. ¯ , 1}) and a natural number L0 such By Lemma 5.10, there exist γ ∈ (0, min{λ, that the following property holds: (i) For each natural number L ≥ 2 there exists γL ∈ (0, γ ) such that for each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + L], each bt ∈ E(t, γL ), t = S1 , . . . , S2 − 1,

190

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ γL , t = S1 , . . . , S2 − 1 2 −1 2 )-program {xt }St =S which satisfies and each ({bt }St =S 1 1

f

f

ρ(xS1 , xS1 ) ≤ γ , ρ(xS2 , xS2 ) ≤ γ , S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , xS1 , xS2 ) + γ 1 1

we have f

ρ(xt , xt ) ≤  for all t = S1 , . . . , S2 . By Lemmas 6.5 and 6.6 (with  = γ ), there exists γ˜ ∈ (0, γ ), a natural number L > l + L0 + 4 and λ˜ ∈ (0, 1) such that the following properties hold: (ii) For each pair of integers T1 ≥ 0, T2 ≥ T2 + L, each bt ∈ E(t, γ˜ ), t = T1 , . . . , T2 − 1, each ut ∈ B(X × X), t = T1 , . . . , T2 − 1 satisfying ut − ft  ≤ γ˜ for all integers t = T1 , . . . , T2 − 1, 2 −1 each sequence {αt }Tt =T ⊂ (0, 1] such that 1

αi αj−1 ≥ λ˜ for each i, j ∈ {T1 , . . . , T2 − 1} satisfying |i − j | ≤ L, 2 −1 2 each ({bt }Tt =T )-program {xt }Tt =T which satisfies 1 1

T 2 −1 t =T1

2 −1 2 −1 αt ut (xt , xt +1 ) = U ({αt ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) 1 1

6.3 Proofs of Theorems 6.2 and 6.3

191

and each integer S ∈ [T1 , T2 − L] satisfying xS ∈ Y¯ ({bt }S+l−1 t =S , S, S + l)

(6.34)

the inequality f

min{ρ(xt , xt ) : t = S + 1, . . . , S + L} ≤ γ

(6.35)

holds; (iii) For each integer T1 ≥ 0, each bt ∈ E(t, γ˜ ), t = T1 , T1 + 1, . . . , each ut ∈ B(X × X), t = T1 , T1 + 1, . . . satisfying ut − ft  ≤ γ˜ for all integers t ≥ T1 , each sequence {αt }∞ t =T1 ⊂ (0, 1] such that αi αj−1 ≥ λ˜ for each pair of integers i, j ≥ T1 satisfying |i − j | ≤ L, ∞ ∞ each ({αt ut }∞ t =T1 , {bt }t =T1 )-overtaking optimal program {xt }t =T1 and each integer S ≥ T1 satisfying (6.34), equation (6.35) holds. Choose λ ∈ (0, 1) such that

˜ λ > λ, 2|1 − λ|λ−1 (sup{ft  : t = 0, 1, . . . } + 1)L < γ /2.

(6.36)

By property (i), there exists δ ∈ (0, γ˜ ) such that the following property holds: (iv) For each pair of integers S1 > L0 , S2 ∈ [S1 + 1, S1 + 2L + 4], each bt ∈ E(t, δ), t = S1 , . . . , S2 − 1, each ut ∈ B(X × X), t = S1 , . . . , S2 − 1 satisfying ut − ft  ≤ δ, t = S1 , . . . , S2 − 1 2 −1 2 and each ({bt }St =S )-program {yt }St =S which satisfies 1 1

f

f

ρ(yS1 , xS1 ) ≤ γ , ρ(yS2 , xS2 ) ≤ γ , S 2 −1 t =S1

2 −1 2 −1 ut (yt , yt +1 ) ≤ U ({ut }St =S , {bt }St =S , S1 , S2 , yS1 , yS2 ) + γ 1 1

192

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

we have f

ρ(yt , xt ) ≤  for all t = S1 , . . . , S2 . Assume that T1 ≥ 0 is an integer, in the case of Theorem 6.2, T2 > T1 + 4L is an integer and that in the case of Theorem 6.3, T2 = ∞. Assume that for all integers t satisfying T1 ≤ t < T2 , bt ∈ E(t, δ),

(6.37)

ut − ft  ≤ δ,

(6.38)

αt ∈ (0, 1]

(6.39)

ut ∈ B(X × X) satisfies

satisfies αi αj−1 ≥ λ for each pair of integers i, j satisfying T1 ≤ i, j < T2 , |i − j | ≤ L (6.40) 2 −1 2 and that a ({bt }Tt =T )-program {xt }Tt =T satisfies 1 1 1 +l−1 xT1 ∈ Y¯ ({bt }Tt =T , T1 , T1 + l). 1

(6.41)

Assume that in the case of Theorem 6.2 T 2 −1 t =T1

2 −1 2 −1 αt ut (xt , xt +1 ) = U ({αt ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ). 1 1

(6.42)

In the case of Theorem 6.3 assume that ∞ ∞ {xt }∞ t =T1 is an ({αt ut }t =T1 , {bt }t =T1 )-overtaking optimal program.

(6.43)

In the case of Theorem 6.3 using (5.8), (6.37)–(6.41), (6.43) the inequality γ < λ¯ and applying by induction property (iii) we obtain a sequence of integers Si , i = 0, 1, . . . , such that T1 + 1 ≤ S0 ≤ T1 + L

(6.44)

6.3 Proofs of Theorems 6.2 and 6.3

193

and for each integer i ≥ 0, 1 ≤ Si+1 − Si ≤ L,

(6.45)

f

ρ(xSi , xSi ) ≤ γ .

(6.46)

In the case of Theorem 6.2, using (5.8), (6.37)–(6.42), the inequality γ < λ¯ and applying by induction property (ii) we obtain a sequence of integers Si , i = 0, 1, . . . , q such that T1 + 1 ≤ S0 ≤ T1 + L, T2 − L < Sq ≤ T2 ,

(6.47)

1 ≤ Si+1 − Si ≤ L, i = 0, . . . , q − 1,

(6.48)

f

ρ(xSi , xSi ) ≤ γ , i = 0, . . . , q.

(6.49)

In the case of Theorem 6.3 set q = ∞. Let an integer i ≥ 0 satisfy i + 1 ≤ q. By (6.42) and (6.43), Si+1 −1



t =Si

S

−1

S

−1

i+1 i+1 αt ut (xt , xt +1 ) = U ({αt ut }t =S , {bt }t =S , Si , Si+1 , xSi , xSi+1 ). i i

S

−1

(6.50)

S

i+1 i+1 In view of (6.38), (6.40), (6.45) and (6.48), for each ({bt }t =S )-program {yt }t =S , i i

−1 Si+1 −1 Si+1      ut (yt , yt +1 ) − αS−1 α u (y , y )   t t t t +1 i t =Si

t =Si

Si+1 −1

≤ sup{ft  : t = 0, 1, . . . } + 1)



t =Si

|1 − αS−1 αt | i

≤ (sup{ft  : t = 0, 1, . . . } + 1)L|λ − 1|λ−1 .

(6.51)

It follows from (6.36), (6.50) and (6.51) that −1  Si+1    Si+1 −1 Si+1 −1 ut (xt , xt +1 ) − U ({ut }t =S , {b } , S , S , x , x )  t t =Si i i+1 Si Si+1  i t =Si

≤ 2(sup{ft  : t = 0, 1, . . . } + 1)L|λ − 1|λ−1 < γ .

(6.52)

Thus for each integer i satisfying 0 ≤ i ≤ q − 1, (6.52) holds. By (6.45) and (6.47), there exists an integer p ≥ 0 such that p ≤ q, Sp ≥ T1 + L

(6.53)

194

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

and if an integer i satisfies 0 ≤ i < p, then Si < T1 + L.

(6.54)

In view of (6.45), (6.47) and (6.54), Sp ≤ T1 + 2L.

(6.55)

p ≤ i ≤ q − 1.

(6.56)

Assume that an integer i satisfies

Equations (6.55) and (6.56) imply that Si > T1 + L0 .

(6.57)

By (6.37), (6.38), (6.45), (6.46), (6.48), (6.49), (6.52), (6.57) and property (iv), f

ρ(xt , xt ) ≤  for all integers t ∈ [Si , Si+1 ].

(6.58)

Thus (6.58) holds for all integers i satisfying (6.56). Together with (6.47) and (6.55) this implies that f

ρ(xt , xt ) ≤ , t = T1 + 2L, . . . , T2 − L in the case of Theorem 6.2 and f

ρ(xt , xt ) ≤  for all integers t ≥ T1 + 2L in the case of Theorem 6.3. Theorems 6.2 and 6.3 are proved.

6.4 Proof of Theorem 6.4 We recall that δ ∈ (0, min{, 1}), λ ∈ (0, 1) and the integer L > l are as guaranteed by Theorem 6.2, ut ∈ Bl (X × X), t = 0, 1, . . . ,

(6.59)

ut − ft  ≤ δ, t = 0, 1, . . . ,

(6.60)

bt ∈ E(t, δ), t = 0, 1, . . . ,

(6.61)

6.4 Proof of Theorem 6.4

195

for every integer t ≥ 1, the set {(x, y) ∈ X × X : y ∈ bt (x)} is closed and that {αt }∞ t =0 ⊂ (0, 1] satisfy the relations lim αt = 0,

(6.62)

t →∞

αi αj−1 ≥ λ for all nonnegative integers i, j satisfying |i − j | ≤ L.

(6.63)

In the proof we use the following auxiliary result. Lemma 6.7 Let γ ∈ (0, 1). Then there is a natural number n0 such that for each 2 −1 integer T1 ≥ 0 and each pair of integers T2 > S ≥ T1 + n0 and each ({bt }Tt =T )1 2 program {xt }Tt =T satisfying 1

1 +l−1 xT1 ∈ Y¯ ({bt }Tt =T , T1 , T1 + l), 1

T 2 −1 t =T1

2 −1 2 −1 αt ut (xt , xt +1 ) = U ({αt ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) 1 1

(6.64) (6.65)

the following inequality holds: S−1  t =T1

S−1 αt ut (xt , xt +1 ) ≤ U ({αt ut }S−1 t =T1 , {bt }t =T1 , T1 , S, xT1 ) + γ .

Proof Since limt →∞ αt = 0 there exists a natural number n0 > 8L + 8

(6.66)

such that for all integers t > n0 − L − 4, αt ≤ γ (8L + 8)−1 (sup{ft  : t = 0, 1, . . . } + 1)−1 .

(6.67)

2 −1 Assume that integers T1 ≥ 0, T2 > S ≥ T1 + n0 and that a ({bt }Tt =T )-program 1

2 {xt }Tt =T satisfies (6.64) and (6.65). By (6.59), (6.64), (6.66) and Proposition 6.1, 1 S there is a ({bt }S−1 t =T1 )-program {x˜ t }t =T1 such that

x˜T1 = xT1 , S−1  t =T1

S−1 αt ut (x˜t , x˜t +1 ) = U ({αt ut }S−1 t =T1 , {bt }t =T1 , T1 , S, xT1 ).

(6.68) (6.69)

196

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

By the choice of δ and L, Theorem 6.2, (6.58)–(6.61), (6.63)–(6.66) and (6.69), ρ(xt , xt ) ≤ λ¯ /4, t = T1 + 2L, . . . , T2 − L,

(6.70)

f ¯ t = T1 + 2L, . . . , S − L. ρ(x˜t , xt ) ≤ λ/4,

(6.71)

f

2 −1 2 By (5.8) and (6.71), there is a ({bt }Tt =T )-program {yt }Tt =T such that 1 1

f

yt = x˜t , t = T1 , . . . , S − L − 1, yS−L = xS−L ,

(6.72)

yt = xt , t = S − L + 1, . . . , T2 .

(6.73)

In view of (6.60), (6.65), (6.67)–(6.69), (6.72) and (6.73), 0≥

T 2 −1

αt ut (xt , xt +1 ) −

t =T1

=

S−L 

αt ut (yt , yt +1 )

t =T1

αt ut (xt , xt +1 ) −

t =T1

=

T 2 −1

S−L 

αt ut (yt , yt +1 )

t =T1

S−L−1 

αt ut (xt , xt +1 ) −

t =T1

S−L−1 

αt ut (x˜t , x˜t +1 )

t =T1

− 2αS−L (sup{ft  : t = 0, 1, . . . } + 1) ≥

S−1 

αt ut (xt , xt +1 ) −

t =T1

S−1 

αt (sup{ft  : t = 0, 1, . . . } + 1)

t =S−L

S−1 − U ({αt ut }S−1 t =T1 , {bt }t =T1 , T1 , S, xT1 )

−

S−1 

αt (sup{ft  : t = 0, 1, . . . } + 1)

t =S−L

− 2αS−L (sup{ft  : t = 0, 1, . . . } + 1) ≥

S−1  t =T1

S−1 αt ut (xt , xt +1 ) − U ({αt ut }S−1 t =T1 , {bt }t =T1 , T1 , S, xT1 ) − γ .

Lemma 6.7 is proved. Completion of the Proof of Theorem 6.4 Let T ≥ 0 be an integer and +l−1 , T , T + l). z ∈ Y¯ ({bt }Tt =T

(6.74)

6.4 Proof of Theorem 6.4

197

By Proposition 6.1, for each integer S > T + l there is a ({bt }S−1 t =T )-program (z,T ,S) S {xt }t =T such that xT(z,T ,S) = z, S−1 

(6.75)

(z,T ,S)

αt ut (xt

t =T

(z,T ,S)

, xt +1

S−1 ) = U ({αt ut }S−1 t =T , {bt }t =T , T , S, z).

(6.76)

Clearly, there exists a strictly increasing sequence of natural numbers {Sj }∞ j =1 such that for any integer t ≥ T there exists (z,T )

xt

(z,T ,Sj )

= lim xt j →∞

.

(6.77)

∞ Clearly, {xt(z,T ) }∞ t =T is a ({bt }t =T )- program and

xT(z,T ) = z,

(6.78)

Let γ > 0. By Lemma 6.7 there is a natural number n0 such that the following property holds: (i) For each integer T1 ≥ 0 and each pair of integers T2 > S ≥ T1 + n0 and each 2 −1 2 ({bt }Tt =T )-program {xt }Tt =T satisfying 1 1 1 +l−1 xT1 ∈ Y¯ ({bt }Tt =T , T1 , T1 + l), 1

T 2 −1 t =T1

2 −1 2 −1 αt ut (xt , xt +1 ) = U ({αt ut }Tt =T , {bt }Tt =T , T1 , T2 , xT1 ) 1 1

we have S−1  t =T1

S−1 αt ut (xt , xt +1 ) ≤ U ({αt ut }S−1 t =T1 , {bt }t =T1 , T1 , S, xT1 ) + γ .

Let T1 ≥ 0 and S ≥ T1 + n0 be integers and (6.74) hold. Property (i), (6.75) and (6.78) imply that for each natural number j satisfying Sj > S, S−1  t =T

(z,T ,Sj )

αt ut (xt

(z,T ,Sj )

, xt +1

S−1 ) ≤ U ({αt ut }S−1 t =T , {bt }t =T , T , S, z) + γ .

198

6 Stability of the Turnpike for Nonautonomous Problems with Discounting

Together with (6.77) this implies that S−1  t =T

) S−1 S−1 αt ut (xt(z,T ) , xt(z,T +1 ) ≤ U ({αt ut }t =T , {bt }t =T , T , S, z) + γ .

Theorem 6.4 is proved.

Chapter 7

Turnpike Properties for Autonomous Problems

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 which is a subset of a finite-dimensional Euclidean space. This control system is described by a nonempty closed set Ω ⊂ X ×X which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous function v : Ω → R 1 which determines an optimality criterion. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. Usually in economic dynamics turnpike properties were studied for systems such that all their good programs converge to a turnpike which was an interior point of Ω. In this chapter we prove turnpike results for a large class of control systems for which the turnpike is not necessarily an interior point of Ω. The Robinson–Solow–Srinivasan model is a particular case of the general model studied in the chapter.

7.1 Preliminaries and the Main Results Assume that the n-dimensional space R n with the Euclidean norm  ·  is ordered by n the cone R+ = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n} and that x  y, x > y, x ≥ y have their usual meaning. n n Let X ⊂ R+ be a compact subset of R+ , Ω be a nonempty closed subset of 1 X × X and let v : Ω → R be a bounded upper semicontinuous function. A sequence {xt }∞ t =0 ⊂ X is called an (Ω)-program (or just a program if the set Ω is understood) if (xt , xt +1 ) ∈ Ω for all integers t ≥ 0. A sequence {xt }Tt=0 where T is a natural number is called an (Ω)-program (or just a program if the set Ω is understood) if (xt , xt +1) ∈ Ω for all integers t ∈ [0, T − 1].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_7

199

200

7 Turnpike Properties for Autonomous Problems

In this chapter we study the problems T −1

−1 v(xi , xi+1 ) → max, {(xi , xi+1 )}Ti=0 ⊂ Ω, x0 = z1

(P1)

i=0

and T −1

v(xi , xi+1 ) → max,

(P2)

i=0 −1 {(xi , xi+1 )}Ti=0 ⊂ Ω, x0 = z1 , xT ≥ z2 ,

where T is a natural number and z1 , z2 ∈ X. Set v = sup{|v(x, y)| : (x, y) ∈ Ω}.

(7.1)

We may assume without loss of generality that v > 0. For every pair of points x, y ∈ X and every natural number T define σ (v, T , x) = sup

−1 T

 v(xi , xi+1 ) : {xi }Ti=0 is a program and x0 = x ,

(7.2)

i=0 −1  T v(xi , xi+1 ) : {xi }Ti=0 is a program and x0 = x, xT ≥ y , σ (v, T , x, y) = sup i=0

(7.3) −1 T  σ (v, T ) = sup v(xi , xi+1 ) : {xi }Ti=0 is a program .

(7.4)

i=0

(Here we use the convention that the supremum over an empty set is −∞.) We assume that there exist a point x¯ ∈ X and a positive constant c¯ such that the following assumptions hold: (A1) (x, ¯ x) ¯ ∈ Ω and the function v : Ω → R 1 is continuous at the point (x, ¯ x). ¯ (A2) σ (v, T ) ≤ T v(x, ¯ x) ¯ + c¯ for all natural numbers T . It is not difficult to see that for every integer T ≥ 1 and every program {xt }Tt=0 , we have T −1 t =0

v(xt , xt +1 ) ≤ σ (v, T ) ≤ T v(x, ¯ x) ¯ + c. ¯

(7.5)

7.1 Preliminaries and the Main Results

201

Inequality (7.5) implies the following useful proposition. Proposition 7.1 For every program {xt }∞ t =0 either the sequence −1 T

v(xt , xt +1 ) − T v(x, ¯ x) ¯

t =0

is bounded or limT →∞ [

T −1 t =0

∞ T =1

v(xt , xt +1 ) − T v(x, ¯ x)] ¯ = −∞.

We say that a program {xt }∞ t =0 is good if the sequence −1 T

v(xt , xt +1 ) − T v(x, ¯ x) ¯

t =0

∞ T =1

is bounded. We suppose that the following assumption holds: (A3) (the asymptotic turnpike property) For every good program {xt }∞ t =0 the equality limt →∞ xt − x ¯ = 0 is valid. Set v(x, y) = −v − 1, (x, y) ∈ (X × X) \ Ω. Evidently, v is a bounded upper semicontinuous function on the space X × X. We suppose that the following assumptions hold: (A4) If a point (x0 , x1 ) ∈ Ω and if a point y0 ∈ X satisfies the inequality y0 ≥ x0 , then there exists y1 ∈ X for which (y0 , y1 ) ∈ Ω, v(y0 , y1 ) ≥ v(x0 , x1 ) and 0 ≤ y1 − x1 ≤ y0 − x0 . (A5) There exists a positive number r¯ such that for every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ there exists a point y ∈ X satisfying the

inequality y ≥ y and the inclusion (x, y ) ∈ Ω. Moreover, for every positive number  there exists a positive number δ such that for every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ δ, there exists a point y ∈ X for which ¯ ≤ . y ≥ y, (x, y ) ∈ Ω and y − x We show in Sects. 7.7 and 7.8 that assumptions (A1)–(A5) hold for many models of economic dynamics. It is not difficult to see that assumption (A4) is a natural monotonicity property of the technology set Ω, while (A5) is a weakened version of a property which holds for the RSS model.

202

7 Turnpike Properties for Autonomous Problems

Assumption (A4) implies that if {xt }∞ t =0 is a program and if a point y0 ∈ X satisfies the inequality y0 ≥ x0 , then there exists a program {yt }∞ t =0 such that for every nonnegative integer t, we have v(yt , yt +1 ) ≥ v(xt , xt +1 ), yt ≥ xt and yt − xt ≤ y0 − x0 . The last two inequalities imply that if all xt , t = 0, 1, . . . and y0 are close to x, ¯ then yt is also close to x¯ for every nonnegative integer t. Assumption (A5) means that for every pair of points x, y which are close to x, ¯ if a state of the model at time t is x, then at moment t + 1 a state of the model can be y which is also close to x¯ and satisfies y ≥ y. For every positive number M denote by XM the set of all points x ∈ X such that there exists a program {xt }∞ t =0 satisfying x0 = x and T −1

v(xt , xt +1 ) − T v(x, ¯ x) ¯ ≥ −M

t =0

for every natural number T . It is clear that ∪{XM : M > 0} is the set of all z ∈ X such that there exists a good program from z. For every integer T ≥ 1 denote by Y¯T the set of all points x ∈ X such that there exists a program {xt }Tt=0 satisfying x0 = x, ¯ xT ≥ x. We say that a program {xt }∞ is overtaking optimal if for every program {yt }∞ t =0 t =0 such that y0 = x0 the inequality lim sup

T −1

T →∞ t =0

[v(yt , yt +1 ) − v(xt , xt +1 )] ≤ 0

is valid. In this section we present the results obtained in [118]. Theorem 7.2 Let , M > 0. Then there exist an integer L ≥ 1 and δ > 0 such that for every natural number T > 2L and every program {xt }Tt=0 satisfying x0 ∈ XM ,

T −1

v(xt , xt +1 ) ≥ σ (v, T , x0 ) − δ

(7.6)

t =0

¯ ≤  for all integers t = there exist integers τ1 , τ2 ∈ [0, L] such that xt − x τ1 , . . . , T − τ2 and if x0 − x ¯ ≤ δ, then τ1 = 0. The following result establishes the existence of an overtaking optimal program.

7.1 Preliminaries and the Main Results

203

Theorem 7.3 Assume that x ∈ X and that there exists a good program {xt }∞ t =0 such that x0 = x. Then there exists an overtaking optimal program {xt∗ }∞ t =0 such that x0∗ = x. The next theorem is a refinement of Theorem 7.2. In view of Theorem 7.2, we have τ2 ≤ L, where the constant L depends on M and . The next theorem shows that τ2 ≤ L0 , where the constant L0 depends only on . Theorem 7.4 Let  be a positive number. Then there exists an integer L0 ≥ 1 such that for every positive number M there exist an integer L > L0 and a positive number δ for which the following assertion holds: For every natural number T > 2L and every program {xt }Tt=0 satisfying (7.6) there exist integers τ1 ∈ [0, L], τ2 ∈ [0, L0 ] such that xt − x ¯ ≤  for all integers t = τ1 , . . . , T − τ2 and if x0 − x ¯ ≤ δ, then τ1 = 0. The following result provides necessary and sufficient conditions for overtaking optimality. Theorem 7.5 Let {xt }∞ t =0 be a program satisfying x0 ∈ ∪{XM : M ∈ (0, ∞)}. Then the program {xt }∞ t =0 is overtaking optimal if and only if the following conditions hold: (i) limt →∞ xt − x ¯ = 0; (ii) for every integer  T−1≥ 1 and every program {yt }Tt=0 such that y0 = x0 , yT ≥ xT the inequality Tt =0 v(yt , yt +1 ) ≤ T −1 v(x , x ) is valid. t t +1 t =0 The next theorem claims that if {xt }∞ t =0 is an overtaking optimal program and the initial state x0 is close to x, ¯ then xt is close to x¯ for every nonnegative integer t. Theorem 7.6 Let  be a positive number. Then there exists a positive number δ such ¯ ≤ δ the that for every overtaking optimal program {xt }∞ t =0 which satisfies x0 − x inequality xt − x ¯ ≤  is valid for every nonnegative integer t. The following result shows the uniform convergence of overtaking optimal programs to the turnpike x. ¯ Theorem 7.7 Let M,  be positive numbers. Then there exists an integer L ≥ 1 such that for every overtaking optimal program {xt }∞ t =0 which satisfies x0 ∈ XM the inequality xt − x ¯ ≤  is valid for every natural number t ≥ L. The next result establishes the turnpike phenomenon for problem (P2). Theorem 7.8 Let , M0 > 0 and let L0 ≥ 1 be an integer. Then there exist an integer L and δ > 0 such that for every natural number T > 2L, every point z0 ∈ XM0 and every point z1 ∈ Y¯L0 , the value σ (v, T , z0 , z1 ) is finite and that for every program {xt }Tt=0 satisfying x0 = z0 , xT = z1 ,

T −1 t =0

v(xt , xt +1 ) ≥ σ (v, T , z0 , z1 ) − δ

204

7 Turnpike Properties for Autonomous Problems

there exist integers τ1 , τ2 ∈ [0, L] such that xt − x ¯ ≤ , t = τ1 , . . . , T − τ2 . ¯ ≤ δ, then τ1 = 0 and if xT − x ¯ ≤ δ. then τ2 = 0. Moreover, if x0 − x The chapter is organized as follows. Section 7.2 contains auxiliary results. Theorem 7.2 is proved in Sect. 7.3, Theorem 7.3 is proved in Sect. 7.4, Theorems 7.4 and 7.5 are proved in Sect. 7.5, while proofs of Theorems 7.6, 7.7 and 7.8 are given in Sect. 7.6. In Sects. 7.7 and 7.8 we consider classes of models of economic dynamics satisfying (A1)–(A5).

7.2 Auxiliary Results For every integer T ≥ 1 denote by YT the set of all x ∈ X for which there exists a program {xt }∞ t =0 which satisfies x0 = x, xT ≥ x. ¯

(7.7)

In view of (7.7), (A1) and (A4) (used with x0 = x1 = x¯ and y0 ≥ x), ¯ we have YT ⊂ YT +1 for every natural number T .

(7.8)

By (7.7) and assumptions (A1) and (A4), if T is a natural number and x ∈ YT , then there exists a good program {xt }∞ t =0 satisfying x0 = x. Assumptions (A3) and (A5) imply that if a program {xt }∞ is t =0 good, then there exists an integer T ≥ 1 such that x0 ∈ YT . The next proposition follows from the boundedness of v and assumptions (A1) and (A4). Proposition 7.9 Let T be a natural number. Then there exists a positive number M for which YT ⊂ XM . Lemma 7.10 Let , M0 be positive numbers. Then there exists an integer T ≥ 1 such that for every program {xt }Tt=0 satisfying T −1

v(xt , xt +1 ) ≥ T v(x, ¯ x) ¯ − M0

t =0

the inequality min{xi − x ¯ : 1, . . . , T } ≤  is valid.

7.2 Auxiliary Results

205

Proof Assume that the lemma does not hold. Then for every integer k ≥ 1 there exists a program {xt(k)}kt=0 such that k−1  t =0

(k)

(k)

v(xt , xt +1 ) ≥ kv(x, ¯ x) ¯ − M0 ,

xt(k) − x ¯ > , t = 1, . . . , k.

(7.9) (7.10)

By (7.5) and (7.9), for each natural number k and each pair of integers S1 , S2 ∈ {0, . . . , k} satisfying S1 < S2 , we have S 2 −1 t =S1

=

(k)

k−1  t =0

−

(k)

v(xt , xt +1 ) (k)

(k)

v(xt , xt +1 ) −





(k)

(k)

{v(xt , xt +1 ) : t ∈ {0, . . . , S1 } \ {S1 }}

{v(xt(k) , xt(k) +1 ) : t ∈ {S2 , . . . , k} \ {S2 }}

≥ kv(x, ¯ x) ¯ − M0 − S1 v(x, ¯ x) ¯ − c − (k − S2 )v(x, ¯ x) ¯ −c = (S2 − S1 )v(x, ¯ x) ¯ − M0 − 2c.

(7.11)

Extracting subsequences and using a diagonalization process, we obtain a strictly increasing sequence of natural numbers {ki }∞ i=1 such that for every nonnegative integer t there exists (kp )

xt = lim xt p→∞

.

(7.12)

Evidently, the sequence {xt }∞ t =0 is a program. By (7.11) and (7.12), for every pair of nonnegative integers S1 < S2 , we have S 2 −1

v(xt , xt +1 ) ≥ (S2 − S1 )v(x, ¯ x) ¯ − M0 − 2c.

t =S1

Thus {xt }∞ t =0 is a good program and ¯ lim xt = x.

t →∞

There exists a natural number τ such that xt − x ¯ ≤ /4 for all integers t ≥ τ.

206

7 Turnpike Properties for Autonomous Problems

Together with (7.12) this implies that for all large natural numbers p, (kp )

xτ

− x ¯ ≤ /2.

This contradicts (7.10). The contradiction we have reached proves Lemma 7.10. Let r¯ ∈ (0, 1) be as guaranteed by assumption (A5). In view of assumption (A5), the choice of the constant r¯ and assumption (A4), for every natural number T and every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ the value σ (v, T , x, y) is finite. Lemma 7.11 Let  be a positive number. Then there exists a number δ ∈ (0, r¯ ) such that for every natural number T and every program {xt }Tt=0 satisfying ¯ xT − x ¯ ≤ δ, x0 − x, T −1

v(xt , xt +1 ) ≥ σ (v, T , x0 , xT ) − δ

(7.13) (7.14)

t =0

¯ ≤  is valid for all integers t = 0, . . . , T . the inequality xt − x Proof In view of (A1), for every integer k ≥ 1, there exists a positive number k < 16−1 min{2−2k r¯ , 2−2k }

(7.15)

such that for every (x, y) ∈ Ω which satisfies x − x, ¯ y − x ¯ ≤ 4k

(7.16)

|v(x, y) − v(x, ¯ x)| ¯ ≤ 2−2k−1 .

(7.17)

we have

In view of (A5), for every integer k ≥ 1 there exists a positive number δk < 4−1 k

(7.18)

such that the following property holds: (P1) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − y ¯ ≤ 4δk there exists a point y ∈ X for which y ≥ y, (x, y ) ∈ Ω and y − x  ≤ k .

7.2 Auxiliary Results

207

We may assume without loss of generality that δk+1 < δk for every natural number k.

(7.19)

Assume that the lemma does not hold. Then for every integer k ≥ 1 there exist a k which satisfies natural number Tk and a program {xt(k) }Tt =0 x0(k) − x, ¯ xT(k) − x ¯ ≤ δk , k T k −1 t =0

(7.20)

(k) (k) v(xt(k) , xt(k) +1 ) ≥ σ (v, Tk , x0 , xTk ) − δk ,

(7.21)

max{xt(k) − x ¯ : t = 0, . . . , Tk } > .

(7.22)

Evidently, Tk > 1 for every natural number k.

(7.23)

Let k be a natural number. Set (k)

(k)

x¯0 = x0 .

(7.24) (k)

It follows from (7.20), (7.24) and property (P1) that there exists x¯1 ∈ X for which (k)

(k)

(k)

x¯1 ≥ x, ¯ (x¯0 , x¯1 ) ∈ Ω,

(7.25)

(k)

¯ ≤ k . x¯1 − x In view of (7.18), (7.20), (7.24), (7.25) and the choice of k (see (7.15)–(7.17)), |v(x¯0 , x¯1 ) − v(x, ¯ x)| ¯ ≤ 2−2k−1 . (k)

(k)

(7.26) (k) T −1

k Assumptions (A1) and (A4) and (7.25) imply that there exists a program {x¯t }t =0 such that for all natural numbers t satisfying 1 ≤ t < Tk ,

0 ≤ x¯t(k) − x¯ ≤ x¯1(k) − x¯

(7.27)

and that for each integer t satisfying 1 ≤ t < Tk − 1 (k)

(k)

v(x¯t , x¯t +1 ) ≥ v(x, ¯ x). ¯

(7.28)

It follows from (7.20) and (P1) that there exists a point z ∈ X which satisfies , z − x ¯ ≤ k , (x, ¯ z) ∈ Ω. z ≥ xT(k) k

(7.29)

208

7 Turnpike Properties for Autonomous Problems

In view of (7.29) and the choice of k (see (7.15)–(7.17)), |v(x, ¯ z) − v(x, ¯ x)| ¯ ≤ 2−2k−1 .

(7.30)

∈ X which satisfies By (7.29), (7.23), (A4) and (7.27), there exists a point x¯T(k) k (k)

(k)

(k)

(x¯Tk −1 , x¯Tk ) ∈ Ω, x¯Tk ≥ z,

(7.31)

v(x¯T(k) , x¯T(k) ) ≥ v(x, ¯ z). k −1 k

(7.32)

k Relations (7.24), (7.29) and (7.31) imply that {x¯t(k) }Tt =0 is a program,

x¯0(k) = x0(k), x¯ T(k) ≥ xT(k) . k k

(7.33)

It follows from (7.26), (7.28), (7.30) and (7.32) that T k −1 t =0

v(x¯t(k) , x¯t(k) ¯ x) ¯ − 2−2k . +1 ) ≥ Tk v(x,

(7.34)

In view of (7.15), (7.18), (7.21), (7.23), and (7.24), T k −1 t =0

(k)

(k)

(k)

(k)

v(xt , xt +1 ) ≥ σ (v, Tk , x0 , xTk ) − δk ≥

T k −1 t =0

(k)

(k)

v(x¯t , x¯t +1 ) − δk

≥ Tk v(x, ¯ x) ¯ − δk − 2−2k ≥ Tk v(x, ¯ x) ¯ − 2−k+1 . Hence T k −1 t =0

(k)

(k)

v(xt , xt +1 ) ≥ Tk v(x, ¯ x) ¯ − 2k+1 for every natural numbers k.

(7.35)

Set S0 = 0, Sk =

k  (Ti + 1) − 1 for all natural numbers k.

(7.36)

i=1 S

k By induction for all natural numbers k we construct a program {xt }t =0 . Define

(1)

xt = xt , t = 0, . . . , T1 = S1 .

(7.37)

7.2 Auxiliary Results

209 S

k Assume that k is a natural number, the sequence {xt }t =0 has been defined such that (7.37) is valid and that for every integer i = 0, . . . , k − 1 we have defined a point zi ∈ R n such that

zi ≥ 0, i = 0, . . . , k − 1, z0 = 0, zi  ≤ 2

i 

j for all integers i satisfying 1 ≤ i ≤ k − 1

(7.38)

j =1

and if a natural number k ≥ 2, then for every natural number i ∈ {1, . . . , , k − 1}, we have 0 ≤ xSi +1+t − xt(i+1) ≤ z(i) , t = 0, . . . , Ti+1 , v(xSi , xSi +1 ) − v(x, ¯ x) ¯ ≥ −2 v(xSi +1+t , xSi +2+t ) ≥

−2i−1

(7.39) (7.40)

,

v(xt(i+1 , xt(i+1 +1 ),

t = 0, . . . , Ti+1 − 1.

(7.41)

Set . z˜ (k) = xSk − xT(k) k

(7.42)

It follows from (7.37), (7.39) and (7.42) that 0 ≤ z˜ (k) .

(7.43)

(k+1)

(7.44)

In view of (7.19) and (7.20), (k)

¯ ≤ δk , x0 xTk − x

− x ¯ ≤ δk+1 < δk .

By (P1) and (7.44), there exists ξ (k) ∈ R n such that (k)

(k+1)

ξ (k) ≥ 0, (xTk , x0 (k+1)

x0

+ ξ (k) ) ∈ Ω,

(7.45)

+ ξ (k) − x ¯ ≤ k .

By (7.18), (7.44), (7.45) and the choice of k (see (7.15)–(7.17)), |v(xT(k) , x0(k+1) + ξ (k) ) − v(x, ¯ x)| ¯ ≤ 2−2k−1 . k

(7.46)

It follows from (7.44) and (7.45) that (k+1)

ξ (k)  ≤ ξ (k) + x0

(k+1)

− x ¯ + x¯ − x0

 ≤ k + δk+1 ≤ 2k .

(7.47)

210

7 Turnpike Properties for Autonomous Problems

By (7.42), (7.43), (7.45) and (A4) there exists a point xSk +1 ∈ X for which (xSk , xSk +1 ) ∈ Ω, v(xSk , xSk +1 ) ≥ v(xT(k) , x0(k+1) + ξ (k) ), k

(7.48)

0 ≤ xSk +1 − (x0(k+1) + ξ (k) ) ≤ xSk − xT(k) ≤ z˜ (k). k

(7.49)

By (7.43), (7.45) and (7.49), we have 0 ≤ xSk +1 − x0(k+1) ≤ z˜ (k) + ξ (k) .

(7.50)

In view of (7.36), (7.48), (7.50) and assumption (A4), there exists a finite sequence Sk+1 {xt }t =S ⊂ X such that for all integers t ∈ {Sk , . . . , Sk+1 − 1}, we have k +1 (xt , xt +1 ) ∈ Ω,

(7.51)

for all integers t = 0, . . . , Tk+1 , (k+1)

0 ≤ xSk +1+t − xt

(k+1)

≤ xSk +1 − x0

≤ z˜ (k) + ξ (k)

(7.52)

and that for all integers t satisfying 0 ≤ t < Tk+1 , we have v(xSk +1+t , xSk +2+t ) ≥ v(xt(k+1) , xt(k+1) +1 ).

(7.53)

z(k) = z˜ (k) + ξ (k) .

(7.54)

Set

It follows from (7.39), (7.43), (7.45), (7.54) that 0 ≤ z˜ (k) ≤ z(k−1)

(7.55)

and by (7.38), (7.47), z(k)  ≤ z(k−1)  + ξ (k)  ≤

k 

2i .

(7.56)

i=1 S +1

k is a program. It follows from (7.43), (7.46), In view of (7.48) and (7.51), {xt }t =0 (7.48), (7.52), (7.54), (7.55), (7.53) that the assumptions made for k also hold for (i) ∈ k + 1. Hence, by induction we have constructed the program {xt }∞ t =0 and z n R+ , i = 0, 1, 2, . . . such that (7.37) is valid, for every natural number i > 1 the

7.2 Auxiliary Results

211

inequality z(i)  ≤ 2

i 

j

j =1

is true and for every natural number i relations (7.35), (7.39), (7.40) and (7.41) are valid. It follows from (7.37), (7.40) and (7.41) that for every integer k ≥ 2, we have S k −1

[v(xt , xt +1 ) − v(x, ¯ x)] ¯

t =0

=

1 −1

S



v(xt , xt +1) − v(x, ¯ x) ¯ +

t =0

≥

S 1 −1 t =0

+

p=1 (1)

v(xt , xt +1 ) − v(x, ¯ x) ¯



t =Sp

(1)

[v(xt , xt +1 ) − v(x, ¯ x)] ¯

k−1 

Tp+1 −1

−2−2p−1 +

p=1

≥ −2

k−1 Sp+1  −1

−1



(p+1)

(v(xt

(p+1)

, xt +1 ) − v(x, ¯ x)) ¯



i=0

+

k−1 

[−2−2p−1 − 2−p−1 ]

p=1

≥ −2−1 −

k−1 

2−p ≥ −2.

p=1

Together with Proposition 7.1 the relation above implies that {xt }∞ t =0 is a good program. In view of assumption (A3), we have lim xt = x. ¯

t →∞

(7.57)

It follows from (7.15), (7.18), (7.20) and (7.22) that for every natural number k there exists an integer τk ∈ {1, . . . , Tk − 1} for which − x ¯ > . xτ(k) k

(7.58)

212

7 Turnpike Properties for Autonomous Problems

By (7.15), (7.38), (7.39) and (7.58), for every natural number i ≥ 2, we have 0 ≤ xSi +1+τi+1 − xτ(i+1) ≤ z(i) , i+1 xSi +1+τi+1 − x ¯ ≥ xτ(i+1) − x ¯ − xτ(i+1) − xSi +1+τi+1  i+1 i+1 >  − z(i)  ≥  − 2

∞ 

p ≥  − 16−1 

p=1

∞ 

2−p > /2.

p=1

This contradicts (7.57). The contradiction we have reached completes the proof of Lemma 7.11. Lemma 7.10 and (A1) imply the following result. Lemma 7.12 Let , M0 > 0. Then there exists an integer T0 ≥ 1 such that for every natural number T ≥ T0 and every program {xt }Tt=0 satisfying T −1

v(xt , xt +1 ) ≥ T v(x, ¯ x) ¯ − M0

t =0

and every integer S ∈ [0, T − T0 ] the following inequality is valid: min{xi − x ¯ : i = S + 1, . . . , S + T0 } ≤ . Lemma 7.13 Let M0 be a positive number and L0 ≥ 1 be an integer. Then there exist an integer T0 ≥ 1 and a number M1 > 0 such that for every natural number T ≥ T0 , every point z0 ∈ XM0 and every z1 ∈ Y¯L0 , σ (v, T , z0 , z1 ) ≥ T v(x, ¯ x) ¯ − M1 . Proof In view of Lemma 7.12 and (A5), there exists an integer L1 ≥ 1 for which XM0 ⊂ YL1 .

(7.59)

T0 = L0 + L1 + 1, M1 = 2T0 v + 1.

(7.60)

Set

Assume that a natural number T ≥ T0 , z0 ∈ XM0 , z1 ∈ Y¯L0 .

(7.61)

By (7.59) and (7.61), we have z0 ∈ YL1 .

(7.62)

7.2 Auxiliary Results

213

1 In view of (7.62), there exists a program {xt }L t =0 for which

x0 = z0 , xL1 ≥ x. ¯

(7.63)

−L0 Assumption (A4) and (7.63) imply that there exists a program {xt }Tt =L satisfying 1

xt ≥ x¯ for every t = L1 , . . . , T − L0 , v(xt , xt +1 ) ≥ v(x, ¯ x) ¯ for every integer t satisfying L1 ≤ t < T − L0 .

(7.64) (7.65)

In view of (7.61), (7.64) and (A4), there exists a program {xt }Tt=T −L0 such that xT ≥ z1 .

(7.66)

Evidently, the finite sequence {xt }Tt=0 is a program. It follows from (7.60), (7.63), (7.66) and (7.65) that σ (v, T , z0 , z1 ) ≥

T −1

v(xt , xt +1 ) ≥ v(x, ¯ x)(T ¯ − L0 − L1 − 1) − v(L1 + L0 )

t =0

= T v(x, ¯ x)−(L ¯ ¯ x)|) ¯ ≥ T v(x, ¯ x)−2T ¯ ¯ x)−M ¯ 1 +L0 +1)(v+|v(x, 0 v = T v(x, 1. Lemma 7.13 is proved. The next lemma easily follows from (A4). Lemma 7.14 Let z0 , z1 ∈ X, M be a positive number, T be a natural number, σ (v, T , z0 , z1 ) be finite and let a program {xt }Tt=0 satisfy T −1

v(xt , xt +1 ) ≥ σ (v, T , z0 , z1 ) − M0 .

t =0

Then for every pair of integers S1 , S2 which satisfy 0 ≤ S1 < S2 ≤ T the inequality S 2 −1 t =S1

holds.

v(xt , xt +1 ) ≥ σ (v, S2 − S1 , xS1 , xS2 ) − M

214

7 Turnpike Properties for Autonomous Problems

7.3 Proof of Theorem 7.2 Let r¯ ∈ (0, 1) be as guaranteed by assumption (A5). Namely, the following property holds: (P2) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ there exists a point y ∈ X for which y ≥ y and (x, y ) ∈ Ω. We may assume without loss of generality that  < r¯ /2, M > 4(v + 1).

(7.67)

Lemma 7.11 implies that there exists a positive number δ <  such that the following property holds: (P3) For every natural number τ and every program {xt }τt=0 satisfying x0 − x, ¯ xτ − x ¯ ≤ δ and

τ −1 

v(xt , xt +1 ) ≥ σ (v, x0 , xτ , τ ) − δ

t =0

¯ ≤  holds for all t = 0, . . . , τ . the inequality xt − x Lemma 7.10 implies that there exists an integer L ≥ 1 such that the following property holds: (P4) For every program {xt }L t =0 satisfying L−1 

v(xt , xt +1 ) ≥ Lv(x, ¯ x) ¯ − M − c¯ − 1

t =0

¯ : t = 1, . . . , L} ≤ δ is valid. the inequality min{xt − x Assume that T > 2L is a natural number and that a program {xt }Tt=0 satisfies x0 ∈ XM ,

T −1

v(xt , xt +1 ) ≥ σ (v, T , x0 ) − δ.

(7.68)

t =0

In view of (7.68) and the definition of XM , T −1 t =0

v(xt , xt +1 ) ≥ T v(x, ¯ x) ¯ − M − 1.

(7.69)

7.4 Proof of Theorem 7.3

215

Assumption (A1) and (7.69) imply that for all integers S ∈ {1, . . . , T − 1}, we have S−1 

v(xt , xt +1 ) ≥ Sv(x, ¯ x) ¯ − M − 1 − c, ¯

t =0 T −1

v(xt , xt +1 ) ≥ (T − S)v(x, ¯ x) ¯ − M − 1 − c. ¯

(7.70)

t =S

Property (P4) and (7.70) imply that there exist τ1 ∈ {0, . . . , L}, τ2 ∈ {T −L, . . . , T } for which xτ1 − x, ¯ xτ2 − x ¯ ≤ δ.

(7.71)

¯ ≤ δ, then we set τ1 = 0. It follows from (7.68) and Evidently, if x0 − x Lemma 7.14 that τ 2 −1

v(xt , xt +1 ) ≥ σ (v, xτ1 , xτ2 , τ2 − τ1 ) − δ.

(7.72)

t =τ1

¯ ≤  for all integers t = τ1 , . . . , τ2 . This In view of (P3), (7.71) and (7.72), xt − x completes the proof of Theorem 7.2.

7.4 Proof of Theorem 7.3 Let {Tk }∞ k=1 be a strictly increasing sequence of natural numbers with T1 > 4. For k every integer k ≥ 1 there exists a program {xt(k)}Tt =0 which satisfies (k)

x0 = x,

T k −1 t =0

(k)

(k)

v(xt , xt +1 ) = σ (v, x, Tk ).

(7.73)

Extracting a subsequence and re-indexing if necessary we may assume without loss of generality that for every nonnegative integer t there exists (k)

xt = lim xt . k→∞

By (7.73), there exists M˜ > 0 such that T k −1 t =0

(k) (k) ˜ k = 1, 2, . . . . v(xt , xt +1 )) ≥ Tk v(x, ¯ x) ¯ − M,

(7.74)

216

7 Turnpike Properties for Autonomous Problems

Together with (A1) this implies that for each natural number k, and each pair of integers S1 , S2 ∈ {0, . . . , k} satisfying S1 < S2 , S 2 −1 t =S1

v(xt , xt +1 ) ≥ (S2 − S1 )v(x, ¯ x) ¯ − M˜ − 2c. ¯ (k)

(k)

Together with (7.74) this implies that for each pair of integers S2 > S1 ≥ 0, S 2 −1 t =S1

v(xt(k) , xt(k) ¯ x) ¯ − M˜ − 2c. ¯ +1 ) ≥ (S2 − S1 )v(x,

Thus {xt }∞ t =0 is a good program and ¯ = 0. lim xt − x

t →∞

(7.75)

We claim that {xt }∞ t =0 is overtaking optimal program. Assume the contrary. Then there exist a program {zt }∞ t =0 and a number Δ > 0 such that z0 = x, lim sup

T −1

T →∞ t =0

[v(zt , zt +1 ) − v(xt , xt +1 )] ≥ Δ.

(7.76)

It is clear that the sequence {zt }∞ t =0 is a good program. Assumption (A3) implies that lim zt − x ¯ = 0.

t →∞

(7.77)

In view of assumption (A1) we choose a positive number  such that  < Δ/2

(7.78)

and that for every point (x, y) ∈ Ω which satisfies x − x, ¯ y − x ¯ ≤ 2 the inequality |v(x, y) − v(x, ¯ x)| ¯ ≤ Δ/4

(7.79)

holds. In view of (A5), there exists a number δ ∈ (0, /2) for which the following property holds:

7.4 Proof of Theorem 7.3

217

(P5) For every pair x, y ∈ X which satisfies x − x, ¯ y − x ¯ ≤δ ¯ ≤ . there exists a point y ∈ X such that y ≥ y, (x, y ) ∈ Ω and y − x It follows from (7.75) and (7.77) there exists a natural number τ0 ≥ 4 such that zt − x, ¯ xt − x ¯ ≤ δ/4 for every integer t ≥ τ0 .

(7.80)

In view of (7.76), there exists a natural number τ1 ≥ 4(τ0 + 4) such that τ 1 −1

[v(zt , zt +1 ) − v(xt , xt +1 )] ≥ (3/4)Δ.

(7.81)

t =0

By (7.74) and upper semicontinuity of the function v, there exists an integer k ≥ 1 such that Tk ≥ 4(τ1 + 4),

(7.82)

v(xt , xt +1 ) − v(xt(k) , xt(k) +1 )

≥ −(Δ(16(τ1 + 1))−1 , t = 0, . . . , τ1 ,

(k)

xt − xt  ≤ δ/4, t = 0, 1, . . . , 4(τ1 + 4).

(7.83) (7.84)

Equations (7.80) and (7.84) imply that (k)

xt

− x ¯ ≤ δ/2 for all t = τ0 , . . . , , 4(τ1 + 4).

(7.85)

Set x¯t = zt , t = 0, . . . , τ1 .

(7.86)

It follows from (7.80), (7.85) and (7.86) that x¯τ1 − x ¯ ≤ δ/4, xτ(k) − x ¯ ≤ δ/2. 1 +1

(7.87)

Property (P5) and (7.87) imply that there exists x¯τ1 +1 ∈ X such that , (x¯τ1 , x¯τ1 +1 ) ∈ Ω, x¯τ1 +1 − x ¯ ≤ . x¯τ1 +1 ≥ xτ(k) 1 +1

(7.88)

k which satisfies In view of (7.88), there exists a program {x¯t }Tt =τ 1 +1

(k)

x¯t ≥ xt , t = τ1 + 1, . . . , Tk , (k)

(k)

v(x¯t , x¯t +1 ) ≥ v(xt , xt +1 ), t = τ1 + 1, . . . , Tk − 1.

(7.89)

218

7 Turnpike Properties for Autonomous Problems T

k It is clear that {x¯t }t =0 is a program. By (7.76) and (7.86), we have

x¯0 = x.

(7.90)

Equations (7.81), (7.83), (7.86) and (7.89) imply that T k −1

v(x¯t , x¯t +1 ) −

T k −1

t =0

=

t =0

τ 1 −1 t =0

=

v(xt(k) , xt(k) +1 ) ≥ (k)

τ1  t =0

τ1  t =0

v(xt(k) , xt(k) +1 )

(k)

(k)

[v(zt , zt +1 ) − v(xt , xt +1 )] + v(zτ1 , x¯τ1 +1 ) − v(xτ(k) , xτ1 +1 ) 1

τ 1 −1

τ 1 −1

t =0

t =0

[v(zt , zt +1 ) − v(xt , xt +1 )] +

+

v(x¯t , x¯t +1 ) −

[v(xt , xt +1 ) − v(xt(k) , xt(k) +1 )]

v(zτ1 , x¯τ1 +1 ) − v(xτ(k) , xτ(k) ) 1 1 +1

≥ (3/4)Δ − Δ(16(τ1 + 1))−1 τ1 + v(zτ1 , x¯τ1 +1 ) − v(xτ(k) , xτ1 +1 ). 1 (k)

(7.91)

In view of (7.85), the choice of  (see (7.79)), (7.80) and (7.88), , xτ(k) ) − v(x, ¯ x)| ¯ ≤ Δ/4, |v(xτ(k) 1 1 +1 |v(zτ1 , x¯τ1 +1 ) − v(x, ¯ x)| ¯ ≤ Δ/4.

(7.92)

Equations (7.85) and (7.92) imply that T k −1 t =0

v(x¯t , x¯t +1 ) −

T k −1 t =0

(k)

(k)

v(xt , xt +1 ) ≥ (3/4)Δ − Δ/16 − Δ/2 ≥ Δ/8.

The relation above contradicts (7.73) and (7.90). The contradiction we have reached completes the proof of Theorem 7.3.

7.5 Proofs of Theorems 7.4 and 7.5 Theorem 7.4 follows directly by using Theorem 7.2, (A5) and (A4). Proof of Theorem 7.5 It is easy to see that if a program {xt }∞ t =0 is overtaking optimal, then conditions (i) and (ii) hold. Assume that conditions (i) and (ii) hold. We claim that the program {xt }∞ t =0 is overtaking optimal. Assume the contrary. Then there exist a program {yt }∞ , a t =0 positive number Δ and a strictly increasing sequence

7.5 Proofs of Theorems 7.4 and 7.5

219

of natural numbers {Tk }∞ k=1 such that for every natural number k, we have T k −1

v(yt , yt +1 ) ≥

t =0

T k −1

v(xt , xt +1) + Δ, y0 = x0 .

(7.93)

t =0

In view of (A1), there exists  ∈ (0, Δ/2) such that for every pair of points (x, y) ∈ Ω which satisfy x − x, ¯ y − x ¯ ≤ 2 the following inequality holds: |v(x, y) − v(x, ¯ x)| ¯ ≤ Δ/4.

(7.94)

In view of assumption (A5), the following property holds: (P6) There exists a number δ ∈ (0, /2) such that for every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ 2δ there exists a point y ∈ X satisfying y ≥ y, (x, y ) ∈ Ω, y − x ¯ ≤ . Proposition 7.1, conditions (i), (ii), property (P6) and (7.93) imply that {xt }∞ t =0 , {yt }∞ t =0 are good programs and in view of assumption (A3) there exists an integer L¯ ≥ 1 such that ¯ xt − x, ¯ yt − x ¯ ≤ δ/2 for every natural number t ≥ L.

(7.95)

Choose an integer k ≥ 1 such that Tk > L¯ + 4 and set x¯t = yt for all integers t = 0, . . . , Tk − 1.

(7.96)

In view of (7.95) and (7.96), ¯ ≤ δ/2, xTk − x ¯ ≤ δ/2. x¯Tk −1 − x

(7.97)

Property (P6) and (7.97) imply that there exists a point x¯Tk ∈ X for which ¯ ≤ . x¯Tk ≥ xTk , (x¯Tk −1 , x¯Tk ) ∈ Ω, x¯Tk − x

(7.98)

220

7 Turnpike Properties for Autonomous Problems T

k By (7.96) and (7.98), {x¯t }t =0 is a program. It follows from (7.93), (7.96) and (7.98) that

x¯0 = x0 , x¯Tk ≥ xTk .

(7.99)

Equations (7.93) and (7.96) imply that T k −1

v(x¯t , x¯t +1 ) −

t =0

=

T k −1

T k −1

v(xt , xt +1 )

t =0

v(yt , yt +1 ) −

t =0

T k −1

v(xt , xt +1 ) +

t =0

T k −1

v(x¯t , x¯t +1) −

t =0

T k −1

v(yt , yt +1 )

t =0

≥ Δ + v(yTk −1 , x¯Tk ) − v(yTk −1 , yTk ).

(7.100)

In view of (7.95), (7.98) and the choice of  (see (7.94)), we have ¯ x)| ¯ ≤ Δ/4, |v(yTk −1 , x¯Tk ) − v(x, ¯ x)| ¯ ≤ Δ/4. |v(yTk −1 , yTk ) − v(x,

(7.101)

It follows from (7.100) and (7.101) that T k −1

v(x¯t , x¯t +1 ) −

t =0

T k −1

v(xt , xt +1 ) ≥ Δ/2.

t =0

Combined with (7.99) the relation above contradicts condition (ii). The contradiction we have reached completes the proof of Theorem 7.5.

7.6 Proofs of Theorems 7.6, 7.7 and 7.8 It is not difficult to verify in a straightforward manner that the following result is true. Lemma 7.15 Let M0 be a positive number. Then there exists a positive number M1 such that for every overtaking optimal program {xt }∞ t =0 satisfying x0 ∈ XM0 the inequality T −1

[v(xt , xt +1 ) − v(x, ¯ x)] ¯ ≥ −M1

t =0

holds for every natural number T . Lemmas 7.12 and 7.15 imply the following auxiliary result.

7.6 Proofs of Theorems 7.6, 7.7 and 7.8

221

Lemma 7.16 Let M0 ,  be positive numbers. Then there exists an integer L ≥ 1 such that for every overtaking optimal program {xt }∞ t =0 satisfying x0 ∈ XM0 the inequality min{xi − x ¯ : i = 1, . . . , L} ≤  is valid. Proof of Theorem 7.8 Let r¯ ∈ (0, 1) be as guaranteed by assumption (A5). Lemma 7.13 implies that there exist an integer T0 ≥ 1 and a positive number M1 such that for every natural number T ≥ T0 , every z0 ∈ XM0 and every z1 ∈ Y¯L0 , we have σ (v, T , z0 , z1 ) ≥ T v(x, ¯ x) ¯ − M1 .

(7.102)

In view of Lemma 7.11 there is a positive number δ < min{¯r , }

(7.103)

such that the following property holds: (P7) For every natural number T and every program {yt }Tt=0 satisfying ¯ yT − x ¯ ≤ δ, y0 − x,

T −1

v(yt , yt +1 ) ≥ σ (v, T , y0 , yT ) − δ

t =0

the inequality ¯ ≤ yt − x holds for all t = 0, . . . , T . Lemma 7.12 implies that there exists an integer T1 ≥ 1 such that the following property holds: (P8) For every natural number T ≥ T1 , every program {yt }Tt=0 satisfying T −1

v(yt , yt +1 ) ≥ T v(x, ¯ x) ¯ − M1 − r¯ − 1

t =0

and every integer s ∈ [0, T − T1 ] the inequality min{yt − x ¯ : t = s + 1, . . . , s + T1 } ≤ δ is valid.

222

7 Turnpike Properties for Autonomous Problems

Set L = 8(2T0 + 2T1 + 4).

(7.104)

Assume that T > 2L is an integer and that z0 ∈ XM0 , z1 ∈ Y¯L0 .

(7.105)

In view of (7.104), (7.105) and the choice of M1 and T0 , relation (7.102) is valid. Assume that a program {xt }Tt=0 satisfies x0 = z0 , xT = z1 ,

T −1

v(xt , xt +1) ≥ σ (v, T , z0 , z1 ) − δ.

(7.106)

t =0

It follows from (7.102), (7.103) and (7.106) that T −1

v(xt , xt +1 ) ≥ T v(x, ¯ x) ¯ − M1 − r¯ .

(7.107)

t =0

Property (P8), (7.104) and (7.107) imply that there exists a sequence of nonnegative q integers {Si }i=0 ⊂ [0, T ] such that S0 ∈ [0, T1 + 1], Si+1 − Si ∈ [1, T1 + 1] for every integer i ∈ [0, q − 1], Sq + T1 > T ,

(7.108)

xSi − x ¯ ≤ δ, i = 0, . . . , q. If x0 − x ¯ ≤ δ, then we may assume that S0 = 0 and if xT − x ¯ ≤ δ, then we may assume that Sq = T . In view of (7.106) and (A4), for i = 0, . . . , q − 1, we have Si+1 −1



v(xi , xi+1 ) ≥ σ (v, Si+1 − Si , xSi , xSi+1 ) − δ

t =Si

and combined with (7.108) and property (P7) this implies that for integers i = ¯ ≤  is true for all integers t = Si , . . . , Si+1 . 0, . . . , q − 1 the inequality xt − x This implies that the inequality xt − x ¯ ≤  holds for all integers t = S0 , . . . , Sq . This completes the proof of Theorem 7.8. Theorem 7.6 follows from Theorem 7.8 and (A5), while Theorem 7.7 follows from Theorem 7.6 and Lemma 7.16.

7.7 The Robinson–Solow–Srinivasan Model

223

7.7 The Robinson–Solow–Srinivasan Model We consider the RSS model studied in Chap. 1. n Let ei , i = 1, . . . , n be the ith unit vector in R n and e be an element of R+ , n n all of whose coordinates are unity. For any x, y ∈ R , let xy = x y . Let i i i=1 a = (a1 , . . . , an )  0, b = (b1 , . . . , bn )  0, d ∈ (0, 1), ci = bi /(1 + dai ), i = 1, . . . , n. We assume the following: There exists σ ∈ {1, . . . , n} such that for all i ∈ {1, . . . , n} \ {σ }, cσ > ci . Let a function w : [0, ∞) → R 1 be strictly increasing concave and differentiable. Set n n Ω˜ = {(x, x ) ∈ R+ × R+ : x − (1 − d)x ≥ 0 and a(x − (1 − d)x) ≤ 1}. n We have a correspondence Λ : Ω˜ → R+ given by n : 0 ≤ y ≤ x and ey ≤ 1 − a(x − (1 − d)x)}, (x, x ) ∈ Ω. Λ(x, x ) = {y ∈ R+

For any (x, x ) ∈ Ω˜ define v(x, x ) = max{w(by) : y ∈ Λ(x, x )}. n A golden-rule stock is x ∈ R+ such that ( x, x ) is a solution to the problem:

˜ It was shown in maximize v(x, x ) subject to (i) x ≥ x, (ii) (x, x ) ∈ Ω. Chapter 2 of [138] that there exists a unique golden-rule stock

x = (1/(1 + daσ ))e(σ ). For i = 1, . . . , n set i = w (b x ) qi . qi = ai bi /(1 + dai ), p For every point (x, x ) ∈ Ω˜ and every point y ∈ Λ(x, x ) set (x − x ) − (w(by) − w(b x )). δ(x, y, x ) = p It was shown in Chapter 2 of [138] that δ(x, y, x ) ≥ 0 for every point (x, x ) ∈ ˜ and every y ∈ Λ(x, x ). Ω, Fix M0 > max{(ai d)−1 : i = 1, . . . , n}.

224

7 Turnpike Properties for Autonomous Problems

Lemma 5.3 of [138] implies that ˜ x ≤ M0 e, then x ≤ M0 e. if (x, x ) ∈ Ω, Set n : x ≤ M0 e} X = {x ∈ R+

and Ω = {(x, x ) ∈ Ω˜ : x ≤ M0 e}. Clearly, Ω is closed subset of X × X and the function v is bounded and upper semicontinuous on Ω. It is easy to see that assumption (A1) holds with x¯ = x . By the definition of the function δ(·, ·, ·), for every natural number T and every program {xt }Tt=0 we have T −1

[w(b x ) − v(xt , xt +1 )] ≥ p (xT − x0 ) ≥ −2 pM0 n.

t =0

Hence assumption (A2) holds with x¯ = x and c¯ = 2 pM0 n + 1. It is not difficult to see that assumption (A4) holds. Proposition 3.13 of [138] implies that assumption (A5) holds. In Chap. 1 (see Theorems 1.23 and 1.24) we showed that if at least one of the following assumptions holds: (i) the function w is strictly concave; (ii) 1 − d − (1/aσ ) = −1, then assumption (A3) holds and all the assumptions made in Sect. 7.1 hold for the Robinson–Solow–Srinivasan model.

7.8 A Model of Economic Dynamics  Let n be a natural number, xy = ni=1 xi yi , x, y ∈ R n be a scalar product in R n and let V = (vi,j )ni,j =1 be a n × n diagonal matrix such that vi,i ∈ [0, 1), i = 1, . . . , n. For every point x ∈ R n put x∞ = max{|xi | : i = 1, . . . , n}. n ) the interior of R n and set Denote by int(R+ +

v˜ = max{vi,i : i = 1, . . . , n} and e = (1, 1, . . . , 1).

(7.109)

7.8 A Model of Economic Dynamics

225

n

n Let T : R+ → 2R+ \ {∅} be a set-valued mapping such that the set T (x) is n bounded for every point x ∈ R+ and that the graph n n × R+ : y ∈ T (x)} graph(T ) = {(x, y) ∈ R+

is a convex closed set. Assume that n n n ) = ∅ for every x ∈ R+ \ int(R+ ), T (x) ∩ int(R+

(7.110)

lim sup r −1 sup{z∞ : z ∈ T (re)} < 1 − v, ˜

(7.111)

r→∞

n satisfying x ≤ y, for every pair of points x, y ∈ R+

T (x) ⊂ T (y)

(7.112)

n , we have and that for every point x ∈ R+ n n ) ∩ R+ = T (x). (T (x) − R+

(7.113)

n 1 be a continuous concave function such that Let w : R+ → R+

w(z) > 0 if and only if z  0, if z1 , z2 ∈ if z1 , z2 ∈

n R+ n R+

(7.114)

satisfy z1 ≤ z2 , then w(z1 ) ≤ w(z2 ), satisfy 0  z1  z2 , then w(z1 ) < w(z2 ).

We assume that there exists a positive number c0 such that c0 e − c0 V e is an interior point of T (c0 e)

(7.115)

and that n n × R+ , yi ∈ T (xi ), i = 1, 2, if (x1 , y1 ), (x2 , y2 ) ∈ R+

x1  0, x2  0, y1  0, y2  0, x1 = x1 , then 2−1 (y1 + y2 ) is an interior point of T (2−1 (x1 + x1 )).

(7.116)

In view of (7.109) and (7.111), there exist M0 > c0 + 1, Δ0 ∈ (0, 1 − v) ˜

(7.117)

226

7 Turnpike Properties for Autonomous Problems

such that sup{z∞ : z ∈ T (re)}r −1 ≤ Δ0 for all r ≥ M0 .

(7.118)

Choose M > M0

(7.119)

and set n : z ≤ Me}, X = {z ∈ R+

Ω = {(x, y) ∈

n R+

n × R+

: x ∈ X, y ≥ V x, y − V x ∈ T (x)}.

(7.120) (7.121)

Proposition 7.17 Ω ⊂ X × X. Proof Let (x, y) ∈ Ω be given. In view of (7.120) and (7.121), we have x ≤ Me, y ≥ V x, y − V x ∈ T (x).

(7.122)

It follows from (7.112) and (7.122) that y − V x ∈ T (Me) and together with (7.109), (7.117)–(7.119) and (7.122) this implies that y − V x∞ M −1 ≤ Δ0 < (1 − v), ˜ y − V x ≤ M(1 − v)e, ˜ y ≤ M(1 − v)e ˜ + Mve ≤ Me and that y ∈ X. This completes the proof of Proposition 7.17. It is not difficult to see that Ω is a nonempty closed subset of X × X. For every point (x, y) ∈ Ω define n v(x, y) = sup{w(z) : z ∈ R+ , y − V x + z ∈ T (x)}.

(7.123)

It is clear that v : Ω → R 1 is bounded upper semicontinuous concave function. Consider the problem v(x, x) → max, (x, x) ∈ Ω. n Evidently, this problem has a solution (x, ¯ x) ¯ ∈ Ω and there exists a point z¯ ∈ R+ for which

x¯ − V x¯ + z¯ ∈ T (x), ¯ w(¯z) = v(x, ¯ x). ¯

(7.124)

In view of (7.114), (7.115), (7.117), (7.119)–(7.121), (7.123) and (7.124), we have w(¯z) = v(x, ¯ x) ¯ ≥ v(c0 e, c0 e) > 0.

(7.125)

7.8 A Model of Economic Dynamics

227

Lemma 7.18 Let (x1 , y1 ) ∈ Ω, (x2 , y2 ) ∈ Ω, x1 = x2 , v(xi , yi ) > 0, i = 1, 2. Then v(2−1 (x1 + x2 ), 2−1 (y1 + y2 )) > 2−1 v(x1 , y1 ) + 2−1 v(x2 , y2 ). n , i = 1, 2 such that for Proof In view of (7.123), there exist a pair of points zi ∈ R+ i = 1, 2,

0 < v(xi , yi ) = w(zi ), yi − V xi + zi ∈ T (xi ).

(7.126)

By (7.110), (7.114), (7.121) and (7.126), for i = 1, 2, we have zi  0, xi  0.

(7.127)

Equations (7.116), (7.121), (7.126) and (7.127) imply that 2−1 (y1 − V x1 + z1 + y2 − V x2 + z2 ) is an interior point of the set T (2−1 (x1 + x2 )). Thus there exists a point ξ ∈ R n satisfying ξ  0 and 2−1 (y1 − V x1 + y2 − V x2 ) + 2−1 (z1 + z2 ) + ξ ∈ T (2−1 (x1 + x2 )).

(7.128)

Combined with (7.114), (7.125), and concavity of the function w this implies that v(2−1 (x1 + x2 ), 2−1 (y1 + y2 )) ≥ w(2−1 (z1 + z2 ) + ξ ) > w(2−1 (z1 + z2 )) ≥ 2−1 v(x1 , y1 ) + 2−1 v(x2 , y2 ).

This completes the proof of Lemma 7.18. Lemma 7.19 Let (xi , yi ) ∈ Ω, i = 1, 2, v(x2 , y2 ) > 0 and x1 = x2 . Then for every number α ∈ (0, 1) the following inequality holds: v(α(x1 , y1 ) + (1 − α)(x2 , y2 )) > αv(x1 , y1 ) + (1 − α)v(x2 , y2 ). Proof Let α ∈ (0, 1) be given. Fix a positive number  such that  < min{α, 1 − α} and set β1 = α + , β2 = α − . Evidently, β1 , β2 ∈ (0, 1) and β1 = β2 . Since the function v is concave we obtain that v(β1 (x1 , y1 ) + (1 − β1 )(x2 , y2 )) ≥ β1 v(x1 , y1 ) + (1 − β1 )v(x2 , y2 ) > 0, v(β2 (x1 , y1 ) + (1 − β2 )(x2 , y2 )) ≥ β2 v(x1 , y1 ) + (1 − β2 )v(x2 , y2 ) > 0, β1 x1 + (1 − β1 )x2 = β2 x1 + (1 − β2 )x2 .

228

7 Turnpike Properties for Autonomous Problems

Combined with Lemma 7.18 the relations above imply that v(α(x1 , y1 ) + (1 − α)(x2 , y2 )) = v(2−1 (β1 + β2 )(x1 , y1 ) + (1 − 2−1 (β1 + β2 ))(x2 , y2 )) > 2−1 v(β1 (x1 , y1 ) + (1 − β1 )(x2 , y2 )) + 2−1 v(β2 (x1 , y1 ) + (1 − β2 )(x2 , y2 )) ≥ 2−1 (β1 + β2 )v(x1 , y1 ) + (1 − 2−1 (β1 + β2 ))v(x2 , y2 ) = αv(x1 , y1 ) + (1 − α)v(x2 , y2 ). This completes the proof of Lemma 7.19. By (7.110), (7.113), (7.114), (7.124), (7.125) and Lemma 7.19, z¯  0, x¯  0.

(7.129)

Lemma 7.19, (7.125) and (7.129) imply the following auxiliary result. Proposition 7.20 If (x, x) ∈ Ω and v(x, ¯ x) ¯ = v(x, x) then x = x. ¯ Proposition 7.21 (x, ¯ x) ¯ is an interior point of Ω. Proof Set ˜ r0 = min{min{x¯i : i = 1, . . . , n}, min{¯zi : i = 1, . . . , n}}(1 − v)/4.

(7.130)

Fix a number r¯ > 0 such that r¯ < r0 /8, 4¯r e + 4¯r min{x¯i : i = 1, . . . , n}−1 x¯ + 4¯r min{x¯i : i = 1, . . . , n}x¯  2−1 z¯ .

(7.131)

x, y ∈ R n , x − x, ¯ y − x ¯ ≤ r¯ .

(7.132)

Assume that

We claim that (x, y) ∈ Ω. In view of (7.130) and (7.132), for all integers i = 1, . . . , n, we have r0 − r¯ ≤ x¯i − r¯ ≤ xi , yi ≤ x¯i + r¯ .

(7.133)

By (7.113), (7.121), (7.124) and (7.130), ¯ (x, ¯ x¯ + r0 e) ∈ Ω. x¯ − V x¯ + r0 e ∈ T (x),

(7.134)

7.8 A Model of Economic Dynamics

229

It follows from Proposition 7.17 and (7.134) that x¯ + r0 e ∈ X.

(7.135)

Equations (7.120), (7.131), (7.133) and (7.135) imply that x, y ≤ x¯ + r0 e, x, y ∈ X, x¯ − r¯ e, x¯ + r¯ e ∈ X.

(7.136)

In view of (7.121) and (7.136), in order to prove that a point (x, y) ∈ Ω it is sufficient to show that y ≥ V x, y − V x ∈ T (x).

(7.137)

In view of (7.109), (7.130), (7.131) and (7.133), we have y − V x ≥ x¯ − r¯ e − V (x¯ + r¯ e) ≥ x¯ − V x¯ − 2¯r e ≥ (1 − v) ˜ x¯ − 2¯r e ≥ 0.

(7.138)

It follows from (7.112), (7.113), (7.130)–(7.133) that ¯ T (x) ⊃ T (x¯ − r¯ e) ⊃ T (x¯ − r¯ (min{x¯i : i = 1, . . . , n})−1 x) = T ((1 − r¯ (min{x¯i : i = 1, . . . , n})−1 x) ¯ ⊃ (1 − r¯ (min{x¯i : i = 1, . . . , n})−1 )T (x) ¯  (1 − r¯ (min{x¯i : i = 1, . . . , n})−1 )(x¯ − V x¯ + z¯ ) ≥ x¯ − V x¯ − r¯ (min{xi : i = 1, . . . , n})−1 (x¯ − V x) ¯ + 2−1 z¯ ≥ x¯ − V x¯ − r¯ (min{x¯i : i = 1, . . . , n})−1 x¯ + 2−1 z¯ ≥ y − V x − (y − x) ¯ − (V x¯ − V x) − r¯ (min{x¯i : i = 1, . . . , n})−1 x¯ + 2−1 z¯ ≥ y − V x − 2¯r e − r¯ (min{x¯i : i = 1, . . . , n})−1 x¯ + 2−1 z¯ ≥ y − V x + 4−1 z¯ . Combined with (7.113) and (7.138) the relation above implies that y − V x ∈ T (x). This completes the proof of Proposition 7.21. Since the function v is concave Proposition 7.21 implies that the function v is continuous at (x, ¯ x). ¯ Therefore assumption (A1) holds. In view of Proposition 7.21, it is well-known fact of convex analysis [73, 82] that there exists a vector l ∈ R n for which v(x, y) ≤ v(x, ¯ x) ¯ + l(x − y) for all (x, y) ∈ Ω.

(7.139)

230

7 Turnpike Properties for Autonomous Problems

Equation (7.139) implies assumption (A2). By Proposition 7.21, assumption (A5) holds. It is clear that assumption (A4) holds too. We claim that assumption (A3) holds. Set for every point (x, y) ∈ Ω, L(x, y) = v(x, ¯ x) ¯ + l(x − y) − v(x, y).

(7.140)

Evidently, the function L is convex on the set Ω. In view of (7.139) and (7.140), L(x, y) ≥ 0 for every point (x, y) ∈ Ω and L(x, ¯ x) ¯ = 0. Lemma 7.19, (7.124), (7.125), (7.139), (7.140) imply that if (x, y) ∈ Ω satisfies L(x, y) = 0, then x = x. ¯

(7.141)

Evidently, the function L is lower semicontinuous. Assume that {xi }∞ t =0 is a good ∞ program. In view of (7.140), we have t =0 L(xt , xt +1 ) < ∞. This implies that every limit point (z1 , z2 ) of the sequence {(xt , xt +1 )}∞ t =1 satisfies L(z1 , z2 ) = 0. and by (7.141), z1 = x. ¯ This implies that limt →∞ xt = x¯ and (A3) holds. Therefore (A1)–(A5) hold and all the results stated in Sect. 7.1 are true for the model considered in this section.

7.9 Equivalence of Optimality Criteria We continue to use the notation, definitions and assumptions introduced in Sect. 7.1. In particular, we assume that (A1)–(A5) hold. ∞ A program {xt∗ }∞ t =0 is called weakly optimal if for every program {yt }t =0 which ∗ satisfies y0 = x0 the inequality lim inf T →∞

T −1 t =0

[v(yt , yt +1 ) − v(xt∗ , xt∗+1 )] ≤ 0

is valid. A program {xt∗ }∞ t =0 is called weakly maximal if for every natural number T and every program {yt }Tt=0 which satisfies y0 = x0∗ , yT ≥ xT∗ the inequality T −1 t =0

is valid.

v(yt , yt +1 ) ≤

T −1 t =0

v(xt∗ , xt∗+1 )

7.10 Proof of Theorem 7.22

231

A program {xt∗ }∞ t =0 is called agreeable if for every integer T0 ≥ 1 and every positive number  there exists a natural number T > T0 such that for every natural number T ≥ T there exists a program {xt }Tt=0 which satisfies xt = xt∗ , t = 0, . . . , T0 and T −1

v(xt , xt +1 ) ≥ σ (v, T , x0∗ ) − .

t =0

The notion of agreeable programs is well-known in the economic literature [34– 36]. We prove the following theorem obtained in [136]. Theorem 7.22 Let {xt∗ }∞ t =0 be a program and assume that there exists a good ∗

program {xt }∞ t =0 such that x0 = x0 . Then the following properties are equivalent: (i) (ii) (iii) (iv) (v) (vi)

the program {xt∗ }∞ t =0 is overtaking optimal; the program {xt∗ }∞ t =0 is weakly optimal; the program {xt∗ }∞ t =0 is weakly maximal and good; the program {xt∗ }∞ ¯ t =0 is weakly maximal and satisfies limt →∞ xt = x; the program {xt∗ }∞ is weakly maximal and satisfies lim inf x − x ¯ = 0; t →∞ t t =0 the program {xt∗ }∞ is agreeable. t =0

7.10 Proof of Theorem 7.22 Evidently, (i) implies (ii). We claim that (ii) implies (iii). Assume that the program {xt∗ }∞ t =0 is weakly optimal. Then lim inf T →∞

T −1 t =0

[v(xt , xt +1 ) − v(xt∗ , xt∗+1 )] ≤ 0.

(7.142)

Since the program {xt }∞ t =0 is good we have −1   T   v(xt , xt +1 ) − T v(x, ¯ x) ¯  : T = 1, 2, . . . < ∞. sup 

(7.143)

t =0

In view of (7.142) and (7.143), we have lim sup[ T →∞

T −1 t =0

v(xt∗ , xt∗+1 ) − T v(x, ¯ x)] ¯ > −∞.

(7.144)

232

7 Turnpike Properties for Autonomous Problems

It follows from Proposition 7.1 and (7.144) that {xt∗ }∞ t =0 is good. Assumption (A3) implies that lim x ∗ t →∞ t

= x. ¯

(7.145)

We show that the program {xt∗ }∞ t =0 is weakly maximal. Assume the contrary. Then T0 there exist a positive number Δ, an integer T0 ≥ 1 and a program {yt }t =0 which satisfies y0 = x0∗ , yT0 ≥ xT∗0 , T 0 −1

v(yt , yt +1 ) ≥

t =0

(7.146)

T 0 −1 t =0

v(xt∗ , xt∗+1 ) + Δ.

(7.147)

In view of assumption (A4) and (7.146), there exist yt ∈ X for all integers t > T0 ∗ such that the sequence {yt }∞ t =0 is a program, yt ≥ xt for all integers t ≥ T0 and that for all integers t ≥ T0 the inequality v(yt , yt +1 ) ≥ v(xt∗ , xt∗+1 ) holds. Combined with (7.147) this implies that −1 T −1

T  lim inf v(yt , yt +1 ) − v(xt∗ , xt∗+1 ) T →∞

≥

T 0 −1

t =0

t =0

v(yt , yt +1 ) −

t =0

T 0 −1 t =0

v(xt∗ , xt∗+1 ) ≥ Δ > 0

and {xt∗ }∞ t =0 is not weakly optimal. The contradiction we have reached proves that {xt∗ }∞ is weakly maximal. t =0 In view of assumption (A3), (iii) implies (iv). It is clear that (iv) implies (v). We claim that (v) implies (iii). Assume that {xt∗ }∞ t =0 is a weakly maximal program and that lim inf xt∗ − x ¯ = 0. t →∞

(7.148)

We claim that the sequence {xt∗ }∞ t =0 is a good program. By assumption (A1), there exists a number 0 ∈ (0, 1) such that |v(x, y) − v(x, ¯ x)| ¯ ≤ 1 for each (x, y) ∈ Ω satisfying x − x, ¯ y − x ¯ ≤ 0 . (7.149)

7.10 Proof of Theorem 7.22

233

It follows from assumption (A5) that there exists a number δ0 ∈ (0, ) such that the following property holds: (a) For every point (x, y) ∈ X for which x − x, ¯ y − x ¯ ≤ δ0 there exists a point y ∈ X satisfying ¯ ≤ 0 . y ≥ y, (x, y ) ∈ Ω, y − x There exists a positive number M0 such that for every integer T ≥ 1, we have −1 T    v(xt , xt +1 ) − T v(x, ¯ x) ¯  ≤ M0 . 

(7.150)

t =0

In view of assumption (A3), we have lim x

t →∞ t

= x. ¯

(7.151)

Equations (7.148) and (7.151) imply that there exists a strictly increasing sequence of natural numbers {Tk }∞ k=1 such that T1 ≥ 10, xT∗k − x ¯ ≤ δ0 , k = 1, 2, . . . ,

(7.152)

¯ ≤ δ0 for all integers t ≥ T1 − 1. xt − x

(7.153)

Let k be a natural number. By (7.152) and (7.153), we have ¯ ≤ δ0 , xT k −1 − x ¯ ≤ δ0 . xT∗k − x

(7.154)

By property (a) and (7.154), there exists a point z ∈ X for which z ≥ xT∗k , (xT k −1 , z) ∈ Ω, z − x ¯ ≤ 0 .

(7.155)

yt = xt , t = 0, . . . , Tk − 1, yTk = z.

(7.156)

Set

T

k is a program which satisfies Equations (7.155) and (7.156) imply that {yt }t =0

y0 = x0∗ , yTk ≥ xT∗k .

(7.157)

234

7 Turnpike Properties for Autonomous Problems

By (7.150), (7.156) and (7.157), we have T k −1 t =0

v(xt∗ , xt∗+1 ) ≥

T k −1

v(yt , yt +1 ) ≥

t =0

T k −1 t =0

v(xt , xt +1 ) − 2v

¯ x). ¯ ≥ −2v − M0 + Tk v(x,

(7.158)

Proposition 7.1 and (7.158) imply that {xt∗ }∞ t =0 is a good program and (iii) holds. By Theorem 7.5, (iv) implies (i). Thus properties (i)–(v) are equivalent. We claim that (vi) implies (iv). Assume that the program {xt∗ }∞ t =0 is agreeable.

= x ∗ there exists a positive number M Since the program {xt }∞ is good and x t =0 0 0 such that x0∗ ∈ XM . We show that the program {xt∗ }∞ t =0 is weakly maximal. Assume the contrary. Then T0 there exist a positive number Δ, an integer T0 ≥ 1 and a program {yt }t =0 which satisfy y0 = x0∗ , yT0 ≥ xT∗0 , T 0 −1

v(yt , yt +1 ) ≥

t =0

T 0 −1 t =0

v(xt∗ , xt∗+1 ) + Δ.

(7.159)

(7.160)

Since {xt∗ }∞ t =0 is an agreeable program there exist an integer T1 > T0 + 8 and a 1 program {xt }Tt =0 such that xt = xt∗ , t = 0, . . . , T0 , T 1 −1

v(xt , xt +1 ) ≥ σ (v, T1 , x0∗ ) − Δ/4.

(7.161) (7.162)

t =0

In view of assumption (A4) and (7.159), there exist yt , t = T0 + 1, . . . , T1 such that 1 {yt }Tt =0 is a program, y t ≥ x t , t = T0 , . . . , T1 ,

(7.163)

v(yt , yt +1 ) ≥ v(xt , xt +1 ), t = T0 , . . . , T1 − 1.

(7.164)

7.10 Proof of Theorem 7.22

235

It follows from (7.159), (7.160), (7.162) and (7.164) that Δ/4 ≥ σ (v, T1 , x0∗ ) −

T 1 −1

v(xt , xt +1 ) ≥

t =0

≥

T 0 −1

v(yt , yt +1 ) −

t =0

T 1 −1

v(yt , yt +1 ) −

t =0

T 0 −1

T 1 −1

v(xt , xt +1 )

t =0

v(xt , xt +1 ) > Δ,

t =0

a contradiction. The contradiction we have reached proves that {xt∗ }∞ t =0 is a weakly maximal program. Now we show that limt →∞ xt∗ − x ¯ = 0. Let  > 0 be given. By Theorem 7.2, there exist an integer L0 ≥ 1 and δ ∈ (0, ) such that the following property holds: (b) For every natural number T > 2L0 and every program {xt }Tt=0 satisfying x0 = x0∗ ,

T −1

v(xt , xt +1 ) ≥ σ (v, T , x0 ) − δ

t =0

we have xt − x ¯ ≤  for all integers t = L0 , . . . , T − L0 . Let an integer S > 2L0 be given. Since {xt∗ }∞ t =0 is an agreeable program there exist an integer T > S + L0 and a program {xt }Tt=0 satisfying xt = xt∗ , t = 0, . . . , S, T −1

v(xt , xt +1 ) ≥ σ (v, T1 , x0∗ ) − δ.

(7.165) (7.166)

t =0

It follows from property (b), (7.165) and (7.166) that ¯ ≤ , t = L0 , . . . , T − L0 . xt − x By (7.165) and (7.167), we have ¯ ≤ , t = L0 , . . . , S. xt∗ − x This implies that ¯ ≤  for all interges t ≥ L0 xt − x and limt →∞ xt∗ − x ¯ = 0. Hence (iv) holds.

(7.167)

236

7 Turnpike Properties for Autonomous Problems

We show that (iv) implies (vi). Assume that {xt∗ }∞ t =0 is a weakly maximal program and that lim xt∗ − x ¯ = 0.

t →∞

(7.168)

Let T0 ≥ 1 be an integer and let  ∈ (0, 1) be given. By assumption (A1), there exists a number δ0 ∈ (0, /4) such that |v(x, y) − v(x, ¯ x)| ¯ ≤ /8

(7.169)

for every point (x, y) ∈ Ω such that x − x, ¯ y − x ¯ ≤ 2δ0. In view of assumption (A5), there exists a number δ ∈ (0, δ0 ) such that the following property holds: (c) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤δ there exists a point y ∈ X for which ¯ ≤ δ0 . y ≥ y, (x, y ) ∈ Ω, y − x Evidently, there exists a positive number M such that x0∗ ∈ XM

(7.170)

and there exists an integer T1 ≥ 1 such that xt − x ¯ ≤ δ for all integers t ≥ T1 .

(7.171)

By Theorem 7.2 and (7.179), there exists an integer L0 ≥ 1 such that the following property holds: (d) For every natural number T > 2L0 and every program {zt }Tt=0 satisfying z0 = x0∗ ,

T −1

v(zt , zt +1 ) = σ (v, T , x0∗ )

t =0

¯ ≤ δ holds for all integers t = L0 , . . . , T − L0 . the inequality zt − x Let an integer T > T0 + 2L0 + T1 + 4.

(7.172)

7.10 Proof of Theorem 7.22

237

There exists a program {yt }Tt=0 such that y0 = x0∗ ,

T −1

v(yt , yt +1) = σ (v, T , x0∗ ).

(7.173)

t =0

By property (d), (7.172) and (7.173), we have yt − x ¯ ≤ δ, t = L0 , . . . , T − L0 .

(7.174)

Equations (7.171), (7.172) and (7.174) imply that ¯ ≤ δ, yT −L0 − x ¯ ≤ δ, yT −L0 −1 − x

(7.175)

xT∗ −L0 −1 − x ¯ ≤ δ, xT∗ −L0 − x ¯ ≤ δ.

(7.176)

By property (c), (7.175) and (7.176), there exists a point y ∈ X such that ¯ ≤ δ0 , (xT∗ −L0 −1 , y ) ∈ Ω. y ≥ yT −L0 , y − x

(7.177)

Assumption (A4) and (7.177) imply that there exists a program { yt }Tt=0 such that yt = xt∗ , t = 0, . . . , T − L0 − 1,

(7.178)



yt ≥ yt , t = T − L0 , . . . , T , yT −L0 = y ,

(7.179)

v( yt , yt +1 ) ≥ v(yt , yt +1 ), t = T − L0 , . . . , T − 1.

(7.180)

By of (7.173) and (7.178), we have T −1

v( yt , yt +1 ) ≤ σ (v, T , x0∗ ) =

t =0

T −1

v(yt , yt +1 ).

t =0

In view of (7.169), (7.171), (7.176)–(7.180), we have T −1

v( yt , yt +1 ) −

t =0

≥

T −1

v(yt , yt +1 )

t =0

T −L 0 −1 

v( yt , yt +1 ) −

t =0

=

T −L 0 −1  t =0

T −L 0 −1 

v(yt , yt +1 )

t =0

v(xt∗ , xt∗+1 ) −

T −L 0 −1  t =0

v(yt , yt +1 )

(7.181)

238

7 Turnpike Properties for Autonomous Problems

− v(xT∗ −L0 −1 , xT∗ −L0 ) + v(xT∗ −L0 −1 , y ) ≥

T −L 0 −1  t =0

v(xt∗ , xt∗+1 ) −

T −L 0 −1 

v(yt , yt +1 ) − /4.

(7.182)

t =0

By property (c), (7.175) and (7.176), there exists a point z ∈ X such that z ≥ xT∗ −L0 , z − x ¯ ≤ δ0 , (yT∗ −L0 −1 , z ) ∈ Ω. T −L0

In view of (7.172) and (7.183), there exists a program { zt }t =0 zT −L0 = z . zt = yt , t = 0, . . . , T − L0 − 1,

(7.183) such that (7.184)

Equations (7.173), (7.183) and (7.184) imply that T −L 0 −1  t =0

v(xt∗ , xt∗+1 )

≥

T −L 0 −1 

v( zt , zt +1 ).

(7.185)

t =0

It follows from (7.169), (7.175) and (7.183)–(7.185) that 0≤

T −L 0 −1 

∗ v(xt∗ , xt+1 )−

t=0

=

≤

T −L 0 −1 

v( zt , zt+1 )

t=0

T −L 0 −1 

∗ v(xt∗ , xt+1 )−

T −L 0 −1 

t=0

t=0

T −L 0 −1 

T −L 0 −1 

∗ v(xt∗ , xt+1 )−

t=0

v(yt , yt+1 ) + v(yT −L0 −1 , yT −L0 ) − v(yT −L0 −1 , z )

v(yt , yt+1 ) + /4.

(7.186)

t=0

By (7.186), we have T −L 0 −1  t =0

v(xt∗ , xt∗+1 ) −

T −L 0 −1 

v(yt , yt +1 ) ≥ −/4.

t =0

Equations (7.182) and (7.187) imply that T −1 t =0

v( yt , yt +1 ) −

T −1 t =0

v(yt , yt +1 ) ≥ −/2.

(7.187)

7.11 Weak Turnpike Theorems

239

In view of the relation above, (7.172), (7.173) and (7.178), we have T −1

v( yt , yt +1 ) ≥ σ (v, T , x0∗ ) − /2,

t =0

yt = xt∗ , t = 0, . . . , T0 . Thus {xt∗ }∞ t =0 is an agreeable program, and (iv) implies (vi). Theorem 7.22 is proved.

7.11 Weak Turnpike Theorems We continue to use the notation, definitions and assumptions introduced in Sect. 7.1. In particular, we assume that assumptions (A1)–(A5) hold. Recall that we denote by Card(A) the cardinality of the set A. We prove the following results. Theorem 7.23 Let , M1 be positive numbers. Then there exist a natural number L and a natural number Q such that for each integer T > 2L and for each program {xt }Tt=0 which satisfies T −1

v(xt , xt +1 ) ≥ T v(x, ¯ x) ¯ − M1

t =0

the following inequality holds: ¯ > }) ≤ Q. Card({t ∈ {0, . . . , T } : xt − x Theorem 7.24 Let , M0 , M1 be positive numbers and let L0 be a natural number. Then there exists a natural number L and a natural number Q such that for each integer T > 2L, each z0 ∈ XM0 and each z1 ∈ Y¯L0 , σ (v, T , z0 , z1 ) is finite and for each program {xt }Tt=0 which satisfies x0 = z1 , xT ≥ z2 ,

T −1

v(xt , xt +1) ≥ σ (v, T , z0 , z1 ) − M1

t =0

the following inequality holds: Card({t ∈ {0, . . . , T } : xt − x ¯ > }) ≤ Q. Theorem 7.25 Let , M0 , M1 be positive numbers. Then there exists a natural number L and a natural number Q such that for each integer T > 2L and each

240

7 Turnpike Properties for Autonomous Problems

z0 ∈ XM0 , σ (v, T , z0 ) is finite and for each program {xt }Tt=0 which satisfies x0 = z0 ,

T −1

v(xt , xt +1 ) ≥ σ (v, T , z0 ) − M1

t =0

the following inequality holds: ¯ > }) ≤ Q. Card({t ∈ {0, . . . , T } : xt − x Theorem 7.23 is proved in the next section. Theorems 7.24 and 7.25 follow from Theorem 7.23 and Lemma 7.13.

7.12 Proof of Theorem 7.23 Let r¯ ∈ (0, 1) be as guaranteed by (A5). Namely, the following property holds: (i) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ , there exists a point y ∈ X for which y ≥ y, (x, y ) ∈ Ω. We may assume without loss of generality that  < r¯ /4.

(7.188)

By Lemma 7.11, there exists a number δ ∈ (0, ) such that the following property holds: (ii) For every natural number τ and every program {xt }τt=0 satisfying x0 − x, ¯ xτ − x ¯ ≤ δ, T −1

v(xt , xt +1 ) ≥ σ (v, τ, x0 , xτ ) − δ

t =0

¯ ≤  is valid for all integers t = 0, . . . , τ . the inequality xt − x By Lemma 7.12, there exists an integer L0 ≥ 1 such that the following property holds: (iii) For every natural number τ ≥ L0 and every program {yt }τt=0 satisfying τ −1  t =0

v(yt , yt +1 ) ≥ τ v(x, ¯ x) ¯ − M1 − 2

7.12 Proof of Theorem 7.23

241

and every integer S ∈ [0, τ − L0 ] the following inequality is valid: min{yt − x ¯ : t = S + 1, . . . , S + L0 } ≤ δ. Set L = 2L0 + 8

(7.189)

¯ Q > (8 + δ −1 (M1 + c))(L 0 + 2) + 4.

(7.190)

and choose a natural number

Assume that an integer T > 2L and that a program {xt }Tt=0 satisfies T −1

v(xt , xt +1 ) ≥ T v(x, ¯ x) ¯ − M1 .

(7.191)

t =0

It follows from property (iii), (7.189) and (7.191) that there exists a sequence of q nonnegative integers {Si }i=0 ⊂ [0, T ] such that S0 ∈ [0, L0 + 1],

(7.192)

Si+1 − Si ∈ [1, L0 ] for each integer i ∈ [0, q − 1],

(7.193)

Sq + L0 > T ,

(7.194)

xSi − x ¯ ≤ δ, i = 0, . . . , q.

(7.195)

Set Si+1 −1

E = {i ∈ {0, . . . , q − 1} :



v(xt , xt +1 ) < σ (v, Si+1 − Si , xSi , xSi+1 ) − δ}.

t =Si

(7.196) We show that Card(E) ≤ δ −1 (M1 + c) ¯ + 4. We may assume without loss of generality that Card(E) ≥ 4. Let E = {i1 , . . . , ip },

(7.197)

242

7 Turnpike Properties for Autonomous Problems

where {i1 , . . . , ip } is a finite strictly increasing sequence of integers. In view (k) Si

+1

k such that of (7.196), for each k ∈ {1, . . . , p}, there exists a program {xt }t =S i k

xS(k) = xSik , xS(k) ≥ xSik +1 , i i +1 k

(7.198)

k

Sik +1 −1



t =Sik

Sik +1 −1

v(xt(k) , xt(k) +1 )



−δ >

v(xt , xt +1 ).

(7.199)

t =Sik

Set (1)

x˜t = xt , t = 0, . . . , Si1 , x˜t = xt , t = Si1 + 1, . . . , Si1 +1 . Si

(7.200)

+1

1 Clearly, {x˜t }t =0 is a program.

Si

+1

k Assume that k ∈ {1, . . . , p − 1} and that we have defined a program {x˜t }t =0 such that (7.200) holds, for every integer j = i1 , . . . , ik + 1,

x˜Sj ≥ xSj ,

(7.201)

for every j = 1, . . . , k, (j )

x˜t ≥ xt , t = Sij , . . . , Sij +1 , (j )

(7.202)

(j )

v(x˜t , x˜t +1 ) ≥ v(xt , xt +1), t = Sij , . . . , Sij +1 − 1,

(7.203)

and if k > 1, then for every j = 1, . . . , k − 1, x˜t ≥ xt , t = Sij +1 , . . . , Sij+1 ,

(7.204)

v(x˜t x˜t +1 ) ≥ v(xt , xt +1 ), t = Sij +1 , . . . , Sij+1 − 1.

(7.205)

(Note that by (7.198), our assumption holds for k = 1.) In view of (7.202), (k)

x˜ Sik +1 ≥ xSi

k +1

≥ xSik +1 .

(7.206)

Assumption (A4) and (7.206) imply that there exist x˜t ∈ X, t ∈ {Sik +1 + Si

k+1 1, . . . , Sik+1 } such that {x˜t }t =0 is a program,

x(t) ˜ ≥ x(t), t = Sik +1 , . . . , Sik+1 ,

(7.207)

v(x˜t x˜t +1 ) ≥ v(xt , xt +1 ), t = Sik +1 , . . . , Sik+1 \ {Sik+1 }.

(7.208)

7.12 Proof of Theorem 7.23

243

Assumption (A4), (7.207) and (7.208) imply that there exist x˜t ∈ X, t ∈ Si

k+1 {Sik+1 , . . . , Sik+1 +1 } \ {Sik+1 } such that {x˜t }t =0

+1

is a program,

x(t) ˜ ≥ x(t), t = Sik+1 , . . . , Sik+1 +1 , v(x˜t , x˜t +1 ) ≥ v(xt(k+1) , xt(k+1) +1 ), t = Sik+1 , . . . , Sik+1 +1 \ {Sik+1 +1 }. It is clear that the assumption made for k also holds for k + 1. By induction we showed that our assumption holds for k = p. Therefore we constructed the program Si

+1

p {x˜t }t =0 such that (7.200) holds, for every integer j = i1 , . . . , ip + 1, (7.201) is valid, for all integers j = 1, . . . , p, (7.202) and (7.203) are true and for every j = 1, . . . , p − 1 (7.204) and (7.205) hold. Assumption (A4), (7.198) and (7.202) imply that there exist x˜t ∈ X, t ∈ {Sip +1 + 1, . . . , T } \ {Sip +1 } such that {x˜t }Tt=0 is a program,

x(t) ˜ ≥ x(t), t = Sip +1 , . . . , T ,

(7.209)

v(x˜t , x˜t +1 ) ≥ v(xt , xt +1), t ∈ {Sip +1 , . . . , T } \ {T }.

(7.210)

Assumption (A2), (7.199), (7.200), (7.203), (7.205), (7.209) and (7.210) imply that ¯ x) ¯ + c¯ − (T v(x, ¯ x) ¯ − M1 ) c¯ + M1 = T v(x, ≥

T −1

v(x˜t , x˜t +1 ) −

t =0

T −1

v(xt , xt +1 ) ≥ (p − 1)δ,

t =0

p ≤ 1 + δ −1 (c¯ + M1 ). Thus (7.197) holds. By (7.195), for all i ∈ {0, . . . , q − 1} \ E, xt − x ¯ ≤ , t = Si , . . . , Si+1 . This inequality implies that 

¯ ≤ } {{Si , . . . , Si+1 } : i ∈ {0, . . . , q − 1} \ E} ⊂ {t ∈ {0, . . . , T } : xt − x

and ¯ > } {t ∈ {0, . . . , T } : xt − x ⊂ [0, S0 ] ∪ [Sq , T ] ∪ {{Si , . . . , Si+1 } : i ∈ E}.

(7.211)

244

7 Turnpike Properties for Autonomous Problems

By (7.190), (7.192)–(7.194), (7.197) and (7.211), Card({t ∈ {0, . . . , T } : xt − x ¯ > }) ≤ 2(L0 + 2) + Card(E)(L0 + 2) ≤ (L0 + 2)(δ −1 (M1 + c) ¯ + 8) < Q. Theorem 7.23 is proved.

Chapter 8

Autonomous Problems with Perturbed Objective Functions

In this chapter we continue to study the class of autonomous optimal control problems considered in Chap. 7. For these optimal control problems the turnpike is a singleton. We show the stability of the turnpike phenomenon under small perturbations of the objective functions. The Robinson–Solow–Srinivasan model is a particular case of the general model studied in the chapter.

8.1 Preliminaries and the Main Results Assume that the n-dimensional space R n with the Euclidean norm  ·  is ordered by n the cone R+ = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n} and that x  y, x > y, x ≥ y have their usual meaning. n n Let X ⊂ R+ be a compact subset of R+ , Ω be a nonempty closed subset of 1 X × X and let v : Ω → R be a bounded upper semicontinuous function. 2 Let T2 > T1 be nonnegative integers. A sequence {xt }Tt =T is called an (Ω)1 program (or just a program if the set Ω is understood) if (xt , xt +1 ) ∈ Ω for all integers t ∈ [T1 , T2 − 1]. A sequence {xt }∞ t =T1 ⊂ X is called an (Ω)-program (or just a program if the set Ω is understood) if (xt , xt +1 ) ∈ Ω for all integers t ≥ T1 . Set v = sup{|v(x, y)| : (x, y) ∈ Ω}.

(8.1)

We may assume without loss of generality that v > 0.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_8

245

246

8 Autonomous Problems with Perturbed Objective Functions

For every pair of points x, y ∈ X and every natural number T define −1  T v(xi , xi+1 ) : {xi }Ti=0 is a program and x0 = x , σ (v, T , x) = sup i=0

σ (v, T , x, y) = sup

−1 T

 v(xi , xi+1 ) : {xi }Ti=0 is a program and x0 = x, xT ≥ y ,

i=0

σ (v, T ) = sup

−1 T

 v(xi , xi+1 ) : {xi }Ti=0 is a program .

i=0

(Here we use the convention that the supremum over an empty set is −∞.) We suppose that all the assumptions introduced in Sect. 7.1 hold. Namely, we assume that there exist a point x¯ ∈ X and a positive constant c¯ such that the following assumptions hold. (A1) (A2)

(x, ¯ x) ¯ ∈ Ω and the function v : Ω → R 1 is continuous at the point (x, ¯ x). ¯ σ (v, T ) ≤ T v(x, ¯ x) ¯ + c¯ for all natural numbers T .

Recall that a program {xt }∞ t =0 is good if the sequence −1 T

v(xt , xt +1 ) − T v(x, ¯ x) ¯

t =0

∞ T =1

is bounded. We suppose that the following assumption holds. (A3)

(the asymptotic turnpike property) For every good program {xt }∞ t =0 the equality limt →∞ xt − x ¯ = 0 is valid.

Set v(x, y) = −v − 1, (x, y) ∈ (X × X) \ Ω. Evidently, v is a bounded upper semicontinuous function on the space X × X. We also suppose that the following assumptions hold. (A4)

If a point (x0 , x1 ) ∈ Ω and if a point y0 ∈ X satisfies the inequality y0 ≥ x0 , then there exists y1 ∈ X for which (y0 , y1 ) ∈ Ω, v(y0 , y1 ) ≥ v(x0 , x1 ) and 0 ≤ y1 − x1 ≤ y0 − x0 .

(A5)

There exists a positive number r¯ such that for every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ there exists a point y ∈ X satisfying

the inequality y ≥ y and the inclusion (x, y ) ∈ Ω. Moreover, for every

8.1 Preliminaries and the Main Results

247

positive number  there exists a positive number δ such that for every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ δ, there exists a point y ∈ X for which ¯ ≤ . y ≥ y, (x, y ) ∈ Ω and y − x For every positive number M denote by XM the set of all points x ∈ X such that there exists a program {xt }∞ t =0 satisfying x0 = x and T −1

v(xt , xt +1 ) − T v(x, ¯ x) ¯ ≥ −M

t =0

for every natural number T . For every integer T ≥ 1 denote by Y¯T the set of all points x ∈ X such that there exists a program {xt }Tt=0 satisfying x0 = x, ¯ xT ≥ x. For every integer T ≥ 1 denote by YT the set of all points x ∈ X such that there exists a program {xt }Tt=0 satisfying x0 = x, xT ≥ x. ¯ For every bounded function φ : Ω → R 1 set φ = sup{|φ(z) : z ∈ Ω}. For each pair of points z0 , z1 ∈ X, each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 and each finite sequence of bounded functions ut : Ω → R 1 , t = T1 , . . . , T2 − 1 set 2 −1 T  2 −1 2 σ ({ut }Tt =T ) = sup ut (xt , xt +1 ) : {xt }Tt =T is a program , 1 1

t =T1

2 −1  T T2 2 −1 , σ ({ut }Tt =T , z ) = sup u (x , x ) : {x } is a program and x = z 0 t t t +1 t T 0 1 t =T 1 1

t =T1

2 −1 σ ({ut }Tt =T , z0 , z1 ) 1

2 −1 T = sup ut (xt , xt +1 ) :

t =T1

 2 . {xt }Tt =T is a program and x = z , x ≥ z T 0 T 1 1 2 1 (Here we assume that the supremum over an empty set is −∞.) Denote by M the set of all bounded functions u : Ω → R 1 such that the following assumption holds.

248

(A6)

8 Autonomous Problems with Perturbed Objective Functions

If a point (x0 , x1 ) ∈ Ω and if a point y0 ∈ X satisfies the inequality y0 ≥ x0 , then there exists y1 ∈ X for which y1 ≥ x1 , (y0 , y1 ) ∈ Ω, u(y0 , y1 ) ≥ u(x0 , x1 ).

We prove the following three theorems which describe the structure of approximate solutions of our discrete-time control system with perturbed objective functions. Theorem 8.1 Let M0 > 0,  > 0 and let L0 ≥ 1 be an integer. Then there exist a positive number δ <  and a natural number Q such that for each integer T > Q, each sequence of bounded functions ut ∈ M, t = 0, . . . , T − 1, satisfying ut − v ≤ δ, t = 0, . . . , T − 1 and each program {xt }Tt=0 which satisfies x0 ∈ XL0 , xT ∈ Y¯L0 , T −1 t =0

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , x0 , xT ) − M0

the following inequality holds: ¯ > }) ≤ Q. Card({t ∈ {0, . . . , T } : xt − x Theorem 8.2 Let M0 > 0,  > 0 and let L0 ≥ 1 be an integer. Then there exist a positive number δ <  and a natural number Q such that for each integer T > Q, each sequence of bounded functions ut ∈ M, t = 0, . . . , T − 1, satisfying ut − v ≤ δ, t = 0, . . . , T − 1 and each program {xt }Tt=0 which satisfies x0 ∈ XL0 , T −1 t =0

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , x0 ) − M0

the following inequality holds: ¯ > }) ≤ Q. Card({t ∈ {0, . . . , T } : xt − x

8.1 Preliminaries and the Main Results

249

Theorem 8.3 Let  > 0, M0 > 0 and let L0 be a natural number. Then there exist δ ∈ (0, ) and a natural number L such that for each integer T > 2L, each sequence of bounded functions ut ∈ M, t = 0, . . . , T − 1, satisfying ut − v ≤ δ, t = 0, . . . , T − 1, and each program {xt }Tt=0 which satisfies x0 ∈ XL0 , xT ∈ Y¯L0 , T −1 t =0

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , x0 , xT ) − M0

and τ +L−1  t =τ

+L−1 ut (xt , xt +1 ) ≥ σ ({ut }τt =τ , xτ , xτ +L ) − δ

for each integer τ ∈ [0, T − L] there exist integers τ1 ∈ [0, L], τ2 ∈ [T − L, T ] such that xt − x ¯ ≤ , t = τ1 , . . . , τ2 . Moreover, if x0 − x ¯ ≤ δ, then τ1 = 0 and if xT − x ¯ ≤ δ, then τ2 = T . Theorem 8.4 Let  > 0, M0 > 0 and let L0 be a natural number. Then there exist δ ∈ (0, ) and a natural number L such that for each integer T > 2L, each sequence of bounded functions ut ∈ M, t = 0, . . . , T − 1, satisfying ut − v ≤ δ, t = 0, . . . , T − 1, and each program {xt }Tt=0 which satisfies x0 ∈ XL0 , T −1 t =0

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , x0 ) − M0

and τ +L−1  t =τ

+L−1 ut (xt , xt +1 ) ≥ σ ({ut }τt =τ , xτ , xτ +L ) − δ

250

8 Autonomous Problems with Perturbed Objective Functions

for each integer τ ∈ [0, T − L] there exist integers τ1 ∈ [0, L], τ2 ∈ [T − L, T ] such that xt − x ¯ ≤ , t = τ1 , . . . , τ2 . ¯ ≤ δ, then τ1 = 0 and if xT − x ¯ ≤ δ, then τ2 = T . Moreover, if x0 − x All the results of this chapter are new.

8.2 Auxiliary Results Assumption (A6) implies the following auxiliary result. Lemma 8.5 Let T2 > T1 ≥ 0 be integers, M > 0, ut ∈ M, t = T1 , . . . , T2 − 1 2 and let a program {xt }Tt =T satisfy 1 T 2 −1 t =T1

2 −1 u( xt , xt +1 ) ≥ U ({ut }Tt =T , xT1 , xT2 ) − M. 1

Then for each pair of integers S1 , S2 ∈ {T1 , . . . , T2 } satisfying S1 < S2 , S 2 −1 t =S1

2 −1 u( xt , xt +1 ) ≥ U ({ut }St =S , xS1 , xS2 ) − M. 1

Lemma 8.6 Let M > 0. Then there exists a natural number L0 such that XM ⊂ YL0 . Proof By (A5), there exists  ∈ (0, 1) such that the following property holds: (i) For each x, x ∈ X satisfying x − x, ¯ x − x ¯ ≤ there exists y ∈ X such that y ≥ x , (x, y) ∈ Ω. By Lemma 7.10, there exists an integer L ≥ 1 such that the following property holds: (ii) For every program {xt }L t =0 satisfying L−1  t =0

v(xt , xt +1 ) ≥ Lv(x, ¯ x) ¯ −M

8.2 Auxiliary Results

251

we have ¯ : 1, . . . , L} < . min{xi − x

Set L0 = L + 2. Let x ∈ XM .

(8.2)

In view of (8.2), there exists a program {xt }∞ t =0 such that x0 = x T −1

v(xt , xt +1 ) ≥ T v(x, ¯ x) ¯ −M

(8.3)

t =0

for all integer T ≥ 1. Property (ii) and (8.3) imply that there exists an integer S ∈ {1, . . . , L} such that xS − x ¯ ≤ .

(8.4)

yt = xt , t = 0, . . . , S.

(8.5)

Set

Property (i), (8.4) and (8.5) imply that there exists yS+1 ∈ X such that yS+1 ≥ x, ¯ (yS , yS+1 ) ∈ Ω.

(8.6)

0 By (A4) and (8.6), there exist yt ∈ X, t = S + 2, . . . , L0 such that {yt }L t =0 is a program and that

yt ≥ x, ¯ t = S + 1, . . . , L0 . Therefore x ∈ YL0 . Lemma 8.6 is proved. Lemma 8.7 Let M, M0 > 0,  > 0 and let L0 ≥ 1 be an integer. Then there exist a positive number δ and a natural number L > L0 such that for each integer T ≥ L, each sequence of bounded functions ut : Ω → R 1 , t = 0, . . . , T − 1

(8.7)

252

8 Autonomous Problems with Perturbed Objective Functions

satisfying ut − v ≤ δ, t = 0, . . . , T − 1

(8.8)

and each program {xt }Tt=0 which satisfies at least one of the following two conditions: (a) x0 ∈ XM , xT ∈ Y¯L0 , T −1 t =0

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , x0 , xT ) − M0 ;

(8.9) (8.10)

(b) x0 ∈ XM , T −1 t =0

(8.11)

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , x0 ) − M0

(8.12)

there exists an integer S ∈ {1, . . . , T − 1} for which ¯ ≤ . xS − x Proof Lemma 8.6 implies that there exists a natural number L1 such that XM ⊂ YL1 .

(8.13)

By Lemma 7.10, there exists an integer L2 ≥ 1 such that the following property holds: 2 (i) For every program {yt }L t =0 satisfying

L 2 −1

v(xt , xt +1 ) ≥ L2 v(x, ¯ x) ¯ −4

t =0

we have ¯ : 1, . . . , L2 } ≤ . min{yt − x Choose a positive number δ < (32L2 )−1

(8.14)

8.2 Auxiliary Results

253

a natural number q ≥ 4 + 4M0 + 8(L0 + L1 + L2 )(v + 1)

(8.15)

L = 2q(2 + L0 + L1 + L2 ).

(8.16)

and set

Let T ≥ L be an integer, ut : Ω → R 1 , t = 0, . . . , T − 1 be bounded functions satisfying (8.8) and let a program {xt }Tt=0 satisfy at least one of conditions (a), (b). In order to complete the proof it is sufficient to show the existence of an integer S ∈ {1, . . . , T − 1} for which ¯ ≤ . xS − x Assume the contrary. Then xS − x ¯ > , S = 1, . . . , T − 1.

(8.17)

1 Properties (a) and (b) and (8.13) imply that there exists a program {yt }L t =0 such that

y0 = x0 , yL1 ≥ x. ¯

(8.18)

Assume that property (a) holds. Property (a) and (8.9) imply that there exists a program {ξt }TT −L0 such that ¯ ξT ≥ xT . ξT −L0 = x,

(8.19)

Assumption (A4), (8.18) and (8.19) imply that there exist yt ∈ X, t = L1 +1, . . . , T such that {yt }Tt=0 is a program, ¯ t = L1 + 1, . . . , T − L0 , yt ≥ x,

(8.20)

¯ x), ¯ t = L1 , . . . , T − L0 − 1, v(yt , yt +1) ≥ v(x,

(8.21)

yt ≥ ξt , t = L − L0 + 1, . . . , T ,

(8.22)

v(yt , yt +1) ≥ v(ξt , ξt +1 ), t = L − L0 , . . . , T − 1.,

(8.23)

yT ≥ xT .

(8.24)

254

8 Autonomous Problems with Perturbed Objective Functions

In view of (8.21), T −1

v(yt , yt +1 ) ≥ v(x, ¯ x)(T ¯ − L0 − L1 ) − (L0 + L1 )v

t =0

≥ T v(x, ¯ x) ¯ − 2(L0 + L1 )v.

(8.25)

Assume that property (b) holds. Then in view of (8.13), there exists a program 1 {yt }L 0 such that ¯ y0 = x0 , yL1 ≥ x.

(8.26)

Assumption (A4) and (8.26) imply that there exist yt ∈ X, t = L1 + 1, . . . , T such that {yt }Tt=0 is a program, ¯ t = L1 + 1, . . . , T , yt ≥ x,

(8.27)

¯ x), ¯ t = L1 , . . . , T − 1. v(yt , yt +1 ) ≥ v(x,

(8.28)

In view of (8.28), T −1

v(yt , yt +1 ) ≥ v(x, ¯ x)(T ¯ − L1 ) − L1 v

t =0

≥ T v(x, ¯ x) ¯ − 2L1 v. Thus in both cases T −1

v(yt , yt +1 ) ≥ T v(x, ¯ x) ¯ − 2(L0 + L1 )v.

(8.29)

t =0

We consider the two cases simultaneously. Equations (8.8) and (8.29) imply that T −1

ut (yt , yt +1 ) ≥

T −1

t =0

v(yt , yt +1 ) − δT

t =0

≥ T v(x, ¯ x) ¯ − 2(L0 + L1 )v − δT .

(8.30)

By properties (a) and (b), the construction of the program {yt }Tt=0 , (8.10), (8.12), (8.18), (8.24), (8.26) and (8.30), T −1 t =0

ut (xt , xt +1 ) ≥

T −1

ut (yt , yt +1 ) − M0

t =0

≥ T v(x, ¯ x) ¯ − 2(L0 + L1 )v − δT − M0 .

(8.31)

8.2 Auxiliary Results

255

Property (i) and (8.17) imply that for every integer τ ∈ [0, T − 1 − L2 ], τ +L 2 −1 

vt (xt , xt +1 ) ≤ L2 v(x, ¯ x) ¯ − 4.

(8.32)

t =τ

In view of (8.16), T L−1 2  > q. There exists an integer q0 > q

(8.33)

q0 L2 ≤ T < (q0 + 1)L2 .

(8.34)

such that

By (8.32), for all integers i = 0, . . . , q0 − 1, (i+1)L 2 −1

v(xt , xt +1 ) ≤ L2 v(x, ¯ x) ¯ − 4.

t =iL2

Together with (8.34) this implies that T −1

v(xt , xt +1 ) ≤ (L2 v(x, ¯ x) ¯ − 4)q0 + 2L2 v.

t =0

Combined with (8.8) and (8.34) this implies that T −1

ut (xt , xt +1 ) ≤ (L2 v(x, ¯ x) ¯ − 4)q0 + 2L2 v + δT

t =0

≤ δT + 2L2 v − 4q0 + T v(x, ¯ x) ¯ + L2 v.

(8.35)

By (8.31) and (8.35), δT + 4L2 v − 4q0 ≥

T −1

ut (xt , xt +1 ) − T v(x, ¯ x) ¯

t =0

≥ −2(L0 + L1 )v − δT − M0 .

(8.36)

256

8 Autonomous Problems with Perturbed Objective Functions

It follows from (8.14), (8.33), (8.34) and (8.36) that q0 ≤ 2δT + M0 + 4(L0 + L1 + L2 )v ≤ M0 + 4(L0 + L1 + L2 )v + 4δq0 L2 ≤ M0 + 4(L0 + L1 + L2 )v + 8−1 q0 and 2−1 q ≤ 2−1 q0 ≤ M0 + 4(L0 + L1 + L2 )v. This contradicts (8.15). The contradiction we have reached completes the proof of Lemma 8.7.

8.3 Proofs of Theorems 8.1 and 8.2 We prove Theorems 8.1 and 8.2 simultaneously. Let r¯ ∈ (0, 1) be as guaranteed by (A5). Namely, the following property holds: (i) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ , there exists a point y ∈ X for which y ≥ y, (x, y ) ∈ Ω. We may assume without loss of generality that  < r¯ /4, M0 , L0 > 4(v + 1).

(8.37)

By Lemma 7.11, there exists a number δ0 ∈ (0, ) such that the following property holds: (ii) For every natural number τ and every program {xt }τt=0 satisfying x0 − x, ¯ xτ − x ¯ ≤ δ0 , T −1

v(xt , xt +1) ≥ σ (v, τ, x0 , xτ ) − δ0

t =0

the inequality xt − x ¯ ≤  is valid for all integers t = 0, . . . , τ . By Lemma 8.7, there exist a positive number δ1 < δ0

8.3 Proofs of Theorems 8.1 and 8.2

257

and a natural number L1 > L0 such that the following property holds: (iii) For each integer T ≥ L1 , each sequence of bounded functions ut : Ω → R 1 , t = 0, . . . , T − 1 satisfying ut − v ≤ δ1 , t = 0, . . . , T − 1 and each program {ξt }Tt=0 which satisfies at least one of the following two conditions: (a) ξ0 ∈ XL0 , ξT ∈ Y¯L0 , T −1 t =0

−1 ut (ξt , ξt +1 ) ≥ σ ({ut }Tt =0 , ξ0 , ξT ) − M0 ;

(b) ξ0 ∈ XL0 , T −1 t =0

−1 ut (ξt , ξt +1 ) ≥ σ ({ut }Tt =0 , ξ0 ) − M0

there exists an integer S ∈ {1, . . . , T − 1} for which xS − x ¯ ≤ δ0 . Fix a positive number δ ≤ δ1 (4L1 )−1

(8.38)

and a natural number Q > 2(L0 + L1 + 2) + 2(L1 + 1)(5 + 4M0 δ0−1 ).

(8.39)

Assume that T > Q is an integer, a sequence of bounded functions ut ∈ M, t = 0, . . . , T − 1

(8.40)

ut − v ≤ δ, t = 0, . . . , T − 1

(8.41)

satisfies

258

8 Autonomous Problems with Perturbed Objective Functions

and {xt }Tt=0 is a program which satisfies at least one of the following conditions: (a) x0 ∈ XL0 , xT ∈ Y¯L0 , T −1 t =0

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , x0 , xT ) − M0 ;

(b) x0 ∈ XL0 , T −1 t =0

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , x0 ) − M0 .

In order to complete the proofs of Theorems 8.1 and 8.2 it is sufficient to show that ¯ > }) ≤ Q. Card({t ∈ {0, . . . , T } : ρ(xt , x) Property (iii), conditions (a) and (b), (8.38), (8.39) and (8.41) imply that there exists S0 ∈ {1, . . . , T − 1}

(8.42)

¯ ≤ δ0 . xS0 − x

(8.43)

such that

We may assume without loss of generality that {t ∈ {1, . . . , S0 } \ {S0 } : xt − x ¯ ≤ δ0 } = ∅.

(8.44)

We show that S0 ≤ L1 . Assume the contrary. Then S0 > L1 .

(8.45)

8.3 Proofs of Theorems 8.1 and 8.2

259

Conditions (a), (b) and (8.40) imply that S 0 −1 t =0

0 −1 ut (xt , xt +1 ) ≥ σ ({ut }St =0 , x0 , xS0 ) − M0 .

By (8.37) and (8.43), xS0 ∈ Y¯1 .

(8.46)

Property (iii), conditions (a), (b), (8.41) and (8.45)–(8.46) imply that there exists S ∈ {1, . . . , S0 − 1} such that ¯ ≤ δ0 . xS − x

(8.47)

This contradicts (8.44). The contradiction we have reached proves that S0 ≤ L1 .

(8.49)

Assume that k ≥ 0 is an integer and that we defined a strictly increasing sequence of integers {Si }ki=0 such that S0 ∈ {1, . . . , L1 ],

(8.50)

Sk ≤ T ,

(8.51)

¯ ≤ δ0 , xSi − x

(8.52)

1 ≤ Si+1 − Si ≤ L1 .

(8.53)

for each integer i = 1, . . . , k,

if i ∈ {0, . . . , k − 1}, then

(Note that by (8.42), (8.43), and (8.47), our assumption holds for k = 0.) If Sk + L1 > T , then the construction is completed. Assume that Sk + L1 ≤ T .

(8.54)

¯ ≤ δ0 . xSk − x

(8.55)

In view of (8.52),

260

8 Autonomous Problems with Perturbed Objective Functions

Property (i), (8.37) and (8.55) imply that there exists x˜ ∈ X such that ˜ ∈ Ω. x˜ ≥ x, ¯ (xSk , x)

(8.56)

By (8.37) and (8.56), xSk ∈ X2v ⊂ XL0 . In view of (8.54) in the case of condition (a), T −1 t =Sk

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =S , xSk , xT ) − M0 k

(8.57)

and in the case of condition (b), T −1 t =Sk

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =S , xSk ) − M0 . k

(8.58)

Property (iii), conditions (a), (b), (8.38), (8.41), (8.54), (8.57) and (8.58) imply that there exists Sk+1 ∈ {Sk + 1, . . . , T − 1}

(8.59)

¯ ≤ δ0 . xSk+1 − x

(8.60)

such that

We may assume without loss of generality that ¯ ≤ δ0 } = ∅. {t ∈ {Sk + 1, . . . , Sk+1 } \ {Sk+1 } : xt − x

(8.61)

We show that Sk+1 ≤ Sk + L1 . Assume the contrary. Then Sk+1 > Sk + L1 .

(8.62)

8.3 Proofs of Theorems 8.1 and 8.2

261

Conditions (a), (b), (8.40) and (8.62) imply that Sk+1 −1



t =Sk

S

−1

k+1 ut (xt , xt +1 ) ≥ σ ({ut }t =S , xSk , xSk+1 ) − M0 . k

(8.63)

Assumption (A4), property (i), (8.37) and (8.60) imply that xSk+1 ∈ Y¯L0 .

(8.64)

By property (iii), (8.41), (8.56), (8.62) and (8.63), there exists S ∈ {Sk + 1, . . . , Sk+1 − 1} such that ¯ ≤ δ0 . xS − x This contradicts (8.61). The contradiction we have reached proves that Sk+1 ≤ Sk + L1 . Now it is easy to see that the assumption made for k also holds for k + 1. Thus by q induction we have constructed the strictly increasing sequence of integers {Si }i=0 such that S0 ∈ {1, . . . , L1 ], Sq ≤ T , Sq > T − L1 ,

(8.65)

for each integer i = 0, . . . , q, ¯ ≤ δ0 , xSi − x

(8.66)

1 ≤ Si+1 − Si ≤ L1 .

(8.67)

for all i ∈ {0, . . . , q − 1}, then

Set Sj+1 −1

E = {j ∈ {0, . . . , q − 1} :



v(xt , xt +1 ) < σ (v, Sj +1 − Sj , xSj , xSj+1 ) − δ0 }.

t =Sj

(8.68)

262

8 Autonomous Problems with Perturbed Objective Functions

We show that Card(E) ≤ 4δ0−1 M0 + 4. We may assume without loss of generality that Card(E) > 4. Let j ∈ E.

(8.69)

By (8.38), (8.41), (8.67) and (8.68), Sj+1 −1



Sj+1 −1

ut (xt , xt+1 ) ≤

t=Sj



v(xt , xt+1 ) + δL1

t=Sj S

j+1 ≤ σ (v, Sj +1 − Sj , xSj , xSj+1 ) + δL1 − δ0 ≤ σ ({ut }t=S j

S

j+1 ≤ σ ({ut }t=S j

−1

−1

, xSj , xSj+1 ) + 2δL1 − δ0

, xSj , xSj+1 ) − δ0 /2.

(8.70) (j ) S

j+1 such that By (8.70), for every j ∈ E there exists a program {yt }t =S j

(j )

(j )

ySj = xSj , ySj+1 ≥ xSj+1 , Sj+1 −1



t =Sj

(8.71)

Sj+1 −1 (j ) (j ) ut (yt , yt +1 )

≥



ut (xt , xt +1 ) − δ0 /4.

(8.72)

t =Sj

Let E = {j1 , . . . , jp }, where {j1 , . . . , jp } is a finite strictly increasing sequence of integers. Set (j1 )

x˜t = xt , t = 0, . . . , Sj1 , x˜ t = yt Sj

, t = Sj1 + 1, . . . , Sj1 +1 .

(8.73)

+1

1 is a program. Clearly, {x˜t }t =0

Sj

+1

q Assume that q ∈ {1, . . . , p − 2} and that we have defined a program {x˜t }t =0 such that (8.73) holds, for every integer k = 1, . . . , q,

(jk )

x˜t ≥ yt

, t = Sjk , . . . , Sjk +1 ,

(8.74)

(jk )

(8.75)

ut (x˜t , x˜t +1 ) ≥ ut (yt

(j )

k , yt +1 ), t = Sjk , . . . , Sjk +1 − 1,

8.3 Proofs of Theorems 8.1 and 8.2

263

for every k ∈ {1, . . . , q} \ {q}, and every integer j satisfying jk < j < jk+1 , x˜t ≥ xt , t = Sj , . . . , Sj +1 ,

(8.76)

ut (x˜t , x˜t +1 ) ≥ ut (xt , xt +1 ), t = Sj , . . . , Sj +1 − 1.

(8.77)

(Note that by (8.73), our assumption holds for q = 1.) In view of (8.71) and (8.74), (j )

x˜Sjq +1 ≥ ySjq +1 ≥ xSjq +1 .

(8.78)

q

Equations (8.70) and (8.78) imply that there exist x˜t ∈ X, t ∈ {Sjq +1 +1, . . . , Sjq+1 } Sj

q+1 is a program, such that {x˜t }t =0

x(t) ˜ ≥ x(t), t = Sjq +1 , . . . , Sjq+1 ,

(8.79)

ut (x˜t , x˜t +1 ) ≥ ut (xt , xt +1 ), t ∈ {Sjq +1 , . . . , Sjq+1 } \ {Sjq+1 }.

(8.80)

In view of (8.79), x˜ Sjq+1 ≥ xSjq+1 . Assumption (A6), (8.40), (8.71), (8.80) and the relation above imply that there exist Sj

q+1 x˜t ∈ X, t ∈ {Sjq+1 , . . . , Sjq+1 +1 } \ {Sjq+1 } such that {x˜t }t =0

(jq+1 )

x˜t ≥ yt

+1

is a program,

, t = Sjq+1 , . . . , Sjq+1 +1 , (jq+1 )

ut (x˜t , x˜t +1 ) ≥ ut (yt

(j

)

q+1 , yt +1 ), t = Sjq+1 , . . . , Sjq+1 +1 − 1.

It is clear that the assumption made for q also holds for q + 1. By induction, we Sj

+1

p−1 such that (8.73) holds, for every integer k = constructed the program {x˜t }t =0 1, . . . , p − 1, (8.74) and (8.75) are valid, for all integers k = 1, . . . , p − 2 and every integer j satisfying jk < j < jk+1 , (8.76) and (8.77) hold. By (8.71), (8.73) and (8.74),

x˜0 = x0 , x˜Sjp−1 +1 ≥ xSjp−1 +1 .

(8.81)

Assumption (A6), conditions (a), (b), (8.40), (8.72), (8.73), (8.75), (8.77) and (8.81), Sjp−1 +1 −1

M0 ≥



Sjp−1 +1 −1

ut (x˜t , x˜t +1 ) −

t =0

≥

S +1 −1 p−1  jk k=1

t =Sjk



ut (xt , xt +1 )

t =0 Sjk +1 −1

ut (x˜t , x˜t +1 ) −



t =Sjk

 ut (xt , xt +1 )

264

8 Autonomous Problems with Perturbed Objective Functions

≥

S +1 −1 p−1  jk k=1

Sjk +1 −1 (jk )

ut (yt

t =Sjk

(j )

k , yt +1 )−



 ut (xt , xt +1 )

t =Sjk

≥ 4−1 pδ0 and p ≤ 4M0 δ0−1 . Thus Card(E) ≤ p ≤ 4M0 δ0−1 + 4.

(8.82)

j ∈ {0, . . . , q − 1} \ E.

(8.83)

v(xt , xt +1 ) < σ (v, Sj +1 − Sj , xSj , xSj+1 ) − δ0 .

(8.84)

Let

By (8.68) and (8.83), Sj+1 −1



t =Sj

Property (ii), (8.66) and (8.84) imply that xt − x ¯ ≤ , t = Sj , . . . , Sj +1 . This inequality implies that ¯ > } ⊂ [0, S0 ] ∪ [Sq , T ] ∪ {{Sj , . . . , Sj +1 } : j ∈ E}. {t ∈ {0, . . . , T } : xt − x Together with (8.39), (8.65), (8.67) and (8.82), ¯ > }) Card({t ∈ {0, . . . , T } : xt − x ≤ 2(L1 + 1) + Card(E)(L1 + 1) ≤ 2(L1 + 1)(4δ0−1 M0 + 5) < Q. Theorems 8.1 and 8.2 are proved.

8.4 Proofs of Theorems 8.3 and 8.4

265

8.4 Proofs of Theorems 8.3 and 8.4 We prove Theorems 8.3 and 8.4 simultaneously. By Lemma 7.11, there exists a number δ0 ∈ (0, ) such that the following property holds: (i) For every natural number τ and every program {xt }τt=0 satisfying x0 − x, ¯ xτ − x ¯ ≤ δ0 , T −1

v(xt , xt +1) ≥ σ (v, τ, x0 , xτ ) − δ0

t =0

the inequality xt − x ¯ ≤  is valid for all integers t = 0, . . . , τ . By Theorems 8.1 and 8.2, there exist a positive number δ1 < δ0 and a natural number Q such that the following property holds: (ii) For each integer T > Q, each sequence of bounded functions ut ∈ M, t = 0, . . . , T − 1, satisfying ut − v ≤ δ1 , t = 0, . . . , T − 1 and each program {xt }Tt=0 for which at least one of the following conditions hold: x0 ∈ XL0 , xT ∈ Y¯L0 , T −1 t =0

−1 ut (xt , xt +1) ≥ σ ({ut }Tt =0 , x0 , xT ) − M0 ;

x0 ∈ XL0 , T −1 t =0

−1 ut (xt , xt +1) ≥ σ ({ut }Tt =0 , x0 ) − M0

we have ¯ > δ0 }) ≤ Q. Card({t ∈ {0, . . . , T } : xt − x Fix a positive number δ < 8−1 δ1 (2Q + 8)−1

(8.85)

266

8 Autonomous Problems with Perturbed Objective Functions

and a natural number L > 4Q + 4.

(8.86)

Assume that T > 2L is an integer, a sequence of bounded functions ut ∈ M, t = 0, . . . , T − 1

(8.87)

ut − v ≤ δ, t = 0, . . . , T − 1

(8.88)

satisfies

and a program {xt }Tt=0 satisfies x0 ∈ XL0

(8.89)

and at least one of the following conditions: xT ∈ Y¯L0 ,

T −1 t =0

T −1 t =0

−1 ut (xt , xt +1) ≥ σ ({ut }Tt =0 , x0 , xT ) − M0 ;

−1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , x0 ) − M0

(8.90)

(8.91)

and τ +L−1  t =τ

+L−1 ut (xt , xt +1 ) ≥ σ ({ut }τt =τ , xτ , xτ +L ) − δ

(8.92)

for each integer τ ∈ [0, T − L]. Property (ii), (8.85), (8.86) and (8.88)–(8.91) imply that Card({t ∈ {0, . . . , T } : xt − x ¯ > δ0 }) ≤ Q.

(8.93)

This implies that there exists a finite strictly increasing sequence of integers S0 , . . . , Sq ∈ [0, T ] such that S0 ≤ Q,

8.4 Proofs of Theorems 8.3 and 8.4

267

for each i = 0, . . . , q − 1, Si+1 − Si ≤ Q + 1,

(8.94)

Sq + Q + 1 > T , xSi − x ¯ ≤ δ0 , i = 0, . . . , q.

(8.95)

Moreover, if x0 − x ¯ ≤ δ, then we may assume that S0 = 0 and if xT − x ¯ ≤ δ, then we may assume that Sq = T . Let i ∈ {0, . . . , q − 1}. By (8.86) and (8.94), there exists an integer τ ≥ 0 such that [Si , Si+1 ] ⊂ [τ, τ + L].

(8.96)

Assumption (A6), (8.87), (8.92) and (8.96) imply that Si+1 −1



t =Si

S

−1

i+1 ut (xt , xt +1 ) ≥ σ ({ut }t =S , xSi , xSi+1 ) − δ. i

(8.97)

By (8.85), (8.88), (8.94) and (8.97), Si+1 −1



Si+1 −1

v(xt , xt +1 ) ≥

t =Si



ut (xt , xt +1 ) − δ(Q + 2)

t =Si S

−1

i+1 ≥ σ ({ut }t =S , xSi , xSi+1 ) − δ(Q + 3) i

≥ σ (v, Si+1 − Si , xSi , xSi+1 ) − δ(2Q + 5) ≥ σ (v, Si+1 − Si , xSi , xSi+1 ) − δ0 . (8.98) Property (i), (8.95) and (8.98) imply that xt − x ¯ ≤ , t = Si , . . . , Si+1 , i = 0, . . . , q − 1. Therefore xt − x ¯ ≤ , t = S0 , . . . , Sq . Theorems 8.3 and 8.4 are proved.

268

8 Autonomous Problems with Perturbed Objective Functions

8.5 Optimal Control Systems with Discounting Denote by M0 the set of all upper semicontinuous functions u ∈ M. It is easy to see that the following proposition holds. 2 −1 Proposition 8.8 Let integers T1 , T2 satisfy 0 ≤ T1 < T2 , {ut }Tt =T ⊂ M0 and 1 2 such that xT1 = x x ∈ ∪{XM : M ∈ (0, ∞)}. Then there exists a program {xt }Tt =T 1 and

T 2 −1 t =T1

2 −1 ut (xt , xt +1 ) = σ ({ut }Tt =T , T1 , T2 , x). 1

The following result shows that the turnpike phenomenon is stable under perturbations of objective function for problems with discounting. Theorem 8.9 Let  ∈ (0, 1) and L0 ≥ 1 be an integer. Then there exist an integer L > L0 , real numbers δ ∈ (0, ) and λ ∈ (0, 1) such that for each integer T > 2L, −1 each sequence of functions {ut }Tt =0 ⊂ M0 satisfying ut − v ≤ δ, t = 0 . . . , T − 1, −1 each sequence {αt }Tt =0 ⊂ (0, 1] such that for each integer τ ∈ [0, T − L],

αi αj−1 ≥ λ for each i, j ∈ {τ, . . . , τ + L} and each program {xt }Tt=0 which satisfies the inclusion x0 ∈ YL0 and the equality T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 )

there is an integer τ0 ∈ {0, . . . , L} such that xt − x ¯ ≤ , t = τ0 , . . . , T − L. Moreover, if x0 − x ¯ ≤ δ, then τ0 = 0. −1 Roughly speaking, the turnpike property holds if discount coefficients {αt }Tt =0 ⊂ (0, 1] are changed rather slowly.

8.6 An Auxiliary Result for Theorems 8.9 and 8.10

269

∞ ∞ Let {wt }∞ t =0 ⊂ M0 be given. A program {xt }t =0 is called ({wt }t =0 )-overtaking ∞ optimal if for each program {yt }t =0 satisfying x0 = y0 , we have −1 T −1

T  wt (yt , yt +1 ) − wt (xt , xt +1 ) ≤ 0. lim sup T →∞

t =0

t =0

The following result establishes the turnpike property for overtaking optimal programs, under perturbations of objective function, for problems with discounting. Theorem 8.10 Let  ∈ (0, 1) and L0 ≥ 1 be an integer. Then there exist an integer L > L0 , real numbers δ ∈ (0, ) and λ ∈ (0, 1) such that for each sequence of functions {ut }∞ t =0 ⊂ M0 satisfying ut − v ≤ δ, t = 0, 1, . . . , each {αt }∞ t =0 ⊂ (0, 1] such that αi αj−1 ≥ λ for each i, j ∈ {0, 1, . . . , } satisfying |i − j | ≤ L ∞ and each ({αt ut }∞ t =0 )-overtaking optimal program {xt }t =0 which satisfies the inclusion

x0 ∈ YL0 the following inequality holds: xt − x ¯ ≤  for all integers t ≥ L.

8.6 An Auxiliary Result for Theorems 8.9 and 8.10 Lemma 8.11 Let  ∈ (0, 1) and L0 ≥ 1 be an integer. Then there exist an integer L ≥ L0 , real numbers δ ∈ (0, ) and λ ∈ (0, 1) such that for each integer T ≥ L, −1 each sequence of functions {ut }Tt =0 ⊂ M0 satisfying ut − v ≤ δ, t = 0 . . . , T − 1, −1 each sequence {αt }Tt =0 ⊂ (0, 1] such that for each integer τ ∈ [0, T − L],

αi αj−1 ≥ λ for each i, j ∈ {τ, . . . , τ + L}

270

8 Autonomous Problems with Perturbed Objective Functions

and each program {xt }Tt=0 for which at least one of the following conditions holds: (a) x0 ∈ YL0 and T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 );

(b) x0 ∈ YL0 , xT ∈ Y¯L0 and T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 , xt )

there is an integer S ∈ {1, . . . , T − 1} such that ¯ ≤ . xS − x Lemma 8.12 Let  ∈ (0, 1) and L0 ≥ 1 be an integer. Then there exist an integer L ≥ L0 , real numbers δ ∈ (0, ) and λ ∈ (0, 1) such that for each sequence of functions {ut }∞ t =0 ⊂ M0 satisfying ut − v ≤ δ, t = 0, 1, . . . , each {αt }∞ t =0 ⊂ (0, 1] such that αi αj−1 ≥ λ for each i, j ∈ {0, 1, . . . , } satisfying |i − j | ≤ L ∞ and each ({αt ut }∞ t =0 )-overtaking optimal program {xt }t =0 which satisfies the inclusion

x0 ∈ YL0 there is an integer S ≥ 1 such that xS − x ¯ ≤ .

8.6 An Auxiliary Result for Theorems 8.9 and 8.10

271

Proof of Lemmas 8.11 and 8.12 We prove Lemmas 8.11 and 8.12 simultaneously. By Lemma 7.10, there exists an integer L1 ≥ 1 such that the following property holds: 1 (i) For every program {yt }L t =0 satisfying

L 1 −1

v(xt , xt +1 ) ≥ L1 v(x, ¯ x) ¯ −4

t =0

we have ¯ : t, . . . , L1 } ≤ . min{yt − x Choose a positive number δ < 32−1 L−1 1 ,

(8.99)

q > 16(v + 1)(L0 + L1 + 2),

(8.100)

L ≥ 2q(L0 + L1 + 2) + 2

(8.101)

natural numbers

and choose λ ∈ [2−1 , 1) such that 4L(1 − λ)λ−1 (v + 1) ≤ 1, q 

λi > 4(1 + λ−1 )(v + 1)(L0 + L1 + 2)λ−1 .

(8.102) (8.103)

i=0

Let us first prove Lemma 8.11. Assume that T ≥ L is an integer, −1 ⊂ M0 {ut }Tt =0

(8.104)

ut − v ≤ δ, t = 0 . . . , T − 1,

(8.105)

satisfies

−1 {αt }Tt =0 ⊂ (0, 1], for each integer τ ∈ [0, T − L],

αi αj−1 ≥ λ for each i, j ∈ {τ, . . . , τ + L}

(8.106)

272

8 Autonomous Problems with Perturbed Objective Functions

and that a program {xt }Tt=0 satisfies at least one of the following conditions: (a) x0 ∈ YL0 , and T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 );

(b) x0 ∈ YL0 , xT ∈ Y¯L0 and T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 , xT ).

In order to complete the proof of Lemma 8.11 we need to show that there is an integer S ∈ {1, . . . , T − 1} such that ¯ ≤ . xS − x Assume the contrary. Then ¯ >  for all integers t = 1, . . . , T − 1. xt − x

(8.107)

Property (i) and (8.107) imply that for every integer τ ∈ {0, . . . , T − 1 − L1 }, τ +L 1 −1 

v(yt , yt +1 ) < L1 v(x, ¯ x) ¯ − 4.

(8.108)

t =τ

Let τ ∈ {0, . . . , T − 1 − L1 }.

(8.109)

8.6 An Auxiliary Result for Theorems 8.9 and 8.10

273

By (8.99), (8.105) and (8.108), τ +L 1 −1 

ut (xt , xt +1 ) ≤

τ +L 1 −1 

t =τ

v(xt , xt +1 ) + δL1

t =τ

≤ δL1 + L1 v(x, ¯ x) ¯ −4 ≤ δL1 − 4 +

τ +L 1 −1 

ut (x, ¯ x) ¯ + δL1 ≤

t =τ

τ +L 1 −1 

ut (x, ¯ x) ¯ − 3.

(8.110)

t =τ

It follows from (8.102), (8.105), (8.106) and (8.110) that τ +L 1 −1 

αt ut (xt , xt +1 )

t =τ τ +L 1 −1 

≤ αt

t =τ

≤ ατ

τ +L 1 −1 

ut (xt , xt +1 ) +

τ +L 1 −1 

ατ (αt ατ−1 − 1)ut (xt , xt +1 )

t =τ

ut (xt , xt +1 ) + ατ L1 (v + 1)(1 − λ)λ−1

t =τ

≤ ατ

τ +L 1 −1 

ut (x, ¯ x) ¯ − 3ατ + ατ L1 (v + 1)(1 − λ)λ−1

t =τ

≤ ατ

τ +L 1 −1 

ut (x, ¯ x) ¯ − ατ (3 − 1/4)

t =τ

≤

τ +L 1 −1 

αt ut (x, ¯ x) ¯ − ατ (3 − 1/4)

t =τ

+ L1 (v + 1)ατ max{|1 − αt ατ−1 | : t = τ, . . . , τ + L1 − 1} ≤

τ +L 1 −1 

ut (x, ¯ x) ¯ − ατ (3 − 1/4) + L1 (v + 1)ατ (1 − λ)λ−1

t =τ

≤

τ +L 1 −1 

αt ut (x, ¯ x) ¯ − ατ (3 − 1/2).

(8.111)

t =τ

There exists a natural number q0 such that q0 L1 ≤ T − 1 < L1 (q0 + 1).

(8.112)

274

8 Autonomous Problems with Perturbed Objective Functions

In view of (8.101) and (8.112), q0 ≥ 2q.

(8.113)

By (8.105), (8.106), (8.112) and (8.111) which holds for every integer τ satisfying (8.109), T −1

αt ut (xt , xt +1 )

t =0

=

q 0 −1 L1 (i+1)−1  t =L1 i

i=0

≤

αt ut (xt , xt +1 ) − 2αiL1 + (v + 1)

t =L1 i

L1 q0 −1

T −1

αt ut (xt , xt +1 ) 

T −1

αt

t =L1 q0

αt ut (x, ¯ x) ¯ −2

t =0

≤

T −1 t =L1 q0

q 0 −1L1 (i+1)−1  i=0

≤

αt ut (xt , xt +1 ) +

q 0 −1

αiL1 + (v + 1)(L1 + 1)αL1 q0 λ−1

i=0

αt ut (x, ¯ x) ¯ + 2αq0 L1 (L1 + 1)(v + 1)λ−1 − 2

t =0

q 0 −1

αL1 i .

(8.114)

i=0

Assumption (A6) and (8.104) imply that there exists a program {yt }Tt=0 such that in the case of condition (b) y0 = x0 , yL0 ≥ x, ¯ t = L0 , . . . , T − L0 .

(8.115)

yt ≥ x, ¯ ut (yt , yt +1 ) ≥ ut (x, ¯ x), ¯ t = L0 , . . . , T − L0 − 1,

(8.116)

yT −L0 ≥ x, ¯ yT ≥ xT

(8.117)

and in the case of condition (a), ¯ t = L0 , . . . , T , y0 = x0 , yL0 ≥ x,

(8.118)

¯ ut (yt , yt +1 ) ≥ ut (x, ¯ x), ¯ t = L0 , . . . , T − 1. yt ≥ x,

(8.119)

It is clear that in both cases T −1 t =0

αt ut (yt , yt +1 ) ≤

T −1 t =0

αt ut (xt , xt +1 ).

(8.120)

8.6 An Auxiliary Result for Theorems 8.9 and 8.10

275

By (8.104), (8.105) (8.106), (8.116) and (8.119), T −1

αt ut (yt , yt +1 ) ≥

T −1

t =0

αt ut (x, ¯ x) ¯

t =0

− 2(L0 + 1)(v + 1)α0 λ−1 − 2(L0 + 1)(v + 1)αT λ−1 .

(8.121)

It follows from (8.103), (8.106), (8.113), (8.114), (8.120) and (8.121) that 0≤

T −1

αt ut (xt , xt +1 ) −

t =0

≤

T −1

αt ut (yt , yt +1 )

t =0

T −1

αt ut (x, ¯ x) ¯ + 2(L1 + 1)(v + 1)αL1 q0 λ−1

t =0

− 2

q 0 −1

αL1 i −

T −1

αt ut (x, ¯ x) ¯

t =0

i=0

+ 2(L0 + 1)(v + 1)α0 λ−1 + 2(L0 + 1)(v + 1)αT λ−1 ≤ −2

q 0 −1

αL1 i + 2(L1 + 1)(v + 1)αL1 q0 λ−1

i=0

+ 2(L0 + 1)(v + 1)αT λ−1 + 2(L0 + 1)(v + 1)α0 λ−1 ≤−

q0 

αiL1 + 2αL1 q0 (1 + λ−1 )(v + 1)(L1 + L2 + 2)λ−1

i=0

−

q0 

αiL1 + 2α0 (v + 1)(L0 + 1)λ−1

i=0

≤ −αq0 L1

q 

λi − 2(1 + λ−1 )(v + 1)(L1 + L0 + 2)λ−1



i=0

− α0

q 

 λi − 2(v + 1)(L0 + 1)λ−1 < 0.

i=0

The contradiction we have reached proves that there exists S ∈ {1, . . . , T − 1} such that xS − x ¯ ≤ and Lemma 8.11 is proved.

276

8 Autonomous Problems with Perturbed Objective Functions

Let us complete the proof of Lemma 8.12. Assume that {ut }∞ t =0 ⊂ M0 ,

(8.122)

ut − v ≤ δ, t = 0, 1, . . . ,

(8.123)

{αt }∞ t =0 ⊂ (0, 1], αi αj−1 ≥ λ for each i, j ∈ {0, 1, . . . , } satisfying |i − j | ≤ L

(8.124)

∞ and that an ({αt ut }∞ t =0 )-overtaking optimal program {xt }t =0 satisfies the inclusion

x0 ∈ YL0 .

(8.125)

In order to prove Lemma 8.12 it is sufficient to show that there is an integer S ≥ 1 such that ¯ ≤ . xS − x Assume the contrary. Then ¯ >  for all integers t ≥ 1. xt − x

(8.126)

Property (i) and (8.126) imply that for every integer τ ≥ 0, τ +L 1 −1 

v(yt , yt +1 ) < L1 v(x, ¯ x) ¯ − 4.

t =τ

Let τ ≥ 0 be an integer. Arguing as in the proof of Lemma 8.11 we obtain that τ +L 1 −1 

ut (xt , xt +1 ) ≤

τ +L 1 −1 

t =τ τ +L 1 −1 

ut (x, ¯ x) ¯ − 3,

t =τ

αt ut (xt , xt +1 ) ≤

τ +L 1 −1 

t =τ

αt ut (x, ¯ x) ¯ − 5ατ /2.

t =τ

By (8.127), for every natural number k, kL 1 −1 

αt ut (xt , xt +1 )

t =0

=

k−1 L1 (i+1)−1   i=0

t =L1 i

αt ut (xt , xt +1 )

(8.127)

8.6 An Auxiliary Result for Theorems 8.9 and 8.10 k−1 L1 (i+1)−1  

≤

αt ut (x, ¯ x) ¯ − 2αL1 i



t =L1 i

i=0 kL 1 −1 

=

277

αt ut (x, ¯ x) ¯ −2

t =0

k−1 

αL1 i .

(8.128)

i=0

Assumption (A6), (8.122) and (8.125) imply that there exists a program {yt }∞ t =0 such that y0 = x0 , yL0 ≥ x, ¯

(8.129)

yt ≥ x, ¯ u(yt , yt +1 ) ≥ u(x, ¯ x). ¯

(8.130)

for all integers t ≥ L0 ,

By (8.101) (8.124) and (8.130), for every natural number k > k0 , kL 1 −1 

αt ut (yt , yt +1 ) ≥

kL 1 −1 

t =0

αt ut (x, ¯ x) ¯ − 2(L0 + 1)(v + 1)α0 λ−1 .

(8.131)

t =0

It follows from (8.124), (8.128) and (8.131) that for every integer k > k0 , kL 1 −1 

αt ut (yt , yt +1 ) −

t =0

≥

kL 1 −1 

αt ut (xt , xt +1 )

t =0

kL 1 −1 

αt ut (x, ¯ x) ¯ − 2(L0 + 1)(v + 1)α0 λ−1

t =0

−

kL 1 −1 

αt ut (x, ¯ x) ¯ + 2α0

t =0

= 2α0

k−1 

k−1 

λi

i=0

 λi − (v + 1)(L0 + 1) .

(8.132)

i=0

By (8.103) and (8.132), for every integer k > L0 + q, kL 1 −1  t =0

αt ut (yt , yt +1 ) −

kL 1 −1 

αt ut (xt , xt +1 ) ≥ 4α0 .

t =0

This contradicts (8.129) and the overtaking optimality of {xt }∞ t =0 . The contradiction we have reached proves Lemma 8.12.

278

8 Autonomous Problems with Perturbed Objective Functions

8.7 Proof of Theorem 8.9 Let r¯ ∈ (0, 1) be as guaranteed by (A5). Namely, the following property holds: (i) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ , there exists a point y ∈ X for which y ≥ y, (x, y ) ∈ Ω. We may assume without loss of generality that  < r¯ /4. By Lemma 7.11, there exists a number δ0 ∈ (0, ) such that the following property holds: (ii) For every natural number τ and every program {xt }τt=0 satisfying x0 − x, ¯ xτ − x ¯ ≤ δ0 , T −1

v(xt , xt +1 ) ≥ σ (v, τ, x0 , xτ ) − δ0

t =0

the inequality xt − x ¯ ≤  is valid for all integers t = 0, . . . , τ . By Lemma 8.11, there exist an integer L1 > L0 , real numbers δ1 ∈ (0, δ0 ) and λ1 ∈ (0, 1) such that the following property holds: −1 (iii) For each integer T ≥ L1 , each sequence of functions {ut }Tt =0 ⊂ M0 satisfying ut − v ≤ δ1 , t = 0 . . . , T − 1, −1 each sequence {αt }Tt =0 ⊂ (0, 1] such that for each integer τ ∈ [0, T − L1 ],

αi αj−1 ≥ λ1 for each i, j ∈ {τ, . . . , τ + L1 } and each program {xt }Tt=0 for which at least one of the following conditions holds: (a) x0 ∈ YL0 , and T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 );

8.7 Proof of Theorem 8.9

279

(b) x0 ∈ YL0 , xT ∈ Y¯L0 and T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 , xT )

there is an integer S ∈ {1, . . . , T − 1} such that ¯ ≤ δ0 . xS − x Choose a positive number δ ≤ min{δ0 , δ1 }(8L1 )−1 ,

(8.133)

L > 2(L0 + L1 + 2)

(8.134)

a natural number

and λ ∈ (λ1 , 1) such that 2(1 − λ)λ−1 (L1 + 1)(v + 1) < δ0 /2.

(8.135)

Assume that T > 2L is an integer, a sequence of functions −1 {ut }Tt =0 ⊂ M0

(8.136)

ut − v ≤ δ, t = 0 . . . , T − 1,

(8.137)

satisfies

−1 ⊂ (0, 1], for each integer τ ∈ [0, T − L], {αt }Tt =0

αi αj−1 ≥ λ for each i, j ∈ {τ, . . . , τ + L}

(8.138)

and a program {xt }Tt=0 satisfies the inclusion x0 ∈ YL0

(8.139)

280

8 Autonomous Problems with Perturbed Objective Functions

and T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 ).

(8.140)

Property (iii), (8.134), (8.136)–(8.140) imply that there is an integer S0 ∈ {1, . . . , T − 1} such that xS0 − x ¯ ≤ δ0 .

(8.141)

If x0 − x ¯ ≤ δ, then we may assume that S0 = 0. We may assume without loss of generality that {t ∈ {0, . . . , S0 } \ {S0 } : xt − x ¯ ≤ δ0 } = ∅.

(8.142)

We show that S0 ≤ L1 . Assume the contrary. Then S0 > L1 .

(8.143)

Assumption (A6), (8.136) and (8.141) imply that xS0 ∈ Y¯L0 . Property (iii), (8.136)–(8.140) and (8.143) imply that there exists S ∈ {1, . . . , S0 − 1} such that ¯ ≤ δ0 . xS − x This contradicts (8.142). The contradiction we have reached proves that S0 ≤ L1 .

(8.144)

8.7 Proof of Theorem 8.9

281

Assume that k ≥ 0 is an integer and that we defined a strictly increasing sequence of nonnegative Si ≤ T , i = 0, . . . , k such that if x0 − x ¯ ≤ δ, then S0 , S0 ≤ L1 ,

(8.145)

for each integer i = 0, . . . , k, xSi − x ¯ ≤ δ0 ,

(8.146)

1 ≤ Si+1 − Si ≤ L1 .

(8.147)

if i ∈ {0, . . . , k} \ {k}, then

(Clearly, our assumption holds for k = 0.) If Sk + L1 > T , then the construction is completed. Assume that Sk + L1 ≤ T .

(8.148)

Assumption (A6), (8.136) and (8.146) imply that xSk ∈ YL0 .

(8.149)

Property (iii) applied to the sequence {xt }Tt=Sk , (8.136)–(8.140), (8.148) and (8.149) imply that there exists Sk+1 ∈ {Sk + 1, . . . , T − 1}

(8.150)

¯ ≤ δ0 . xSk+1 − x

(8.151)

such that

We may assume without loss of generality that ¯ ≤ δ0 } = ∅. {t ∈ {Sk + 1, . . . , Sk+1 } \ {Sk+1 } : xt − x

(8.152)

Assume that Sk+1 > Sk + L1 .

(8.153)

By (8.136)–(8.138), (8.140), (8.149), (8.151) and property (iii) applied to the Sk+1 sequence {xt }t =S there exists k S ∈ {Sk + 1, . . . , Sk+1 − 1}

282

8 Autonomous Problems with Perturbed Objective Functions

such that xS − x ¯ ≤ δ0 . This contradicts (8.152). The contradiction we have reached proves that Sk+1 ≤ Sk + L1 . Now it is easy to see that the assumption made for k also holds for k + 1. Thus by p induction we have constructed the strictly increasing sequence of integers {Si }i=0 ⊂ [0, T ] such that S0 ≤ L1 ,

(8.154)

¯ ≤ δ, then S0 = 0, if x0 − x Sp + L1 > T ,

(8.155)

for each integer i = 0, . . . , p, ¯ ≤ δ0 , xSi − x

(8.156)

Si < Si+1 ≤ Si + L1 .

(8.157)

for all i ∈ {0, . . . , p − 1},

Let i ∈ {0, . . . , p − 1}. Assumption (A6), (8.136) and (8.140) imply that Si+1 −1



t =Si

S

−1

i+1 αt ut (xt , xt +1 ) = σ ({αt ut }t =S , xSi , xSi+1 ). i

(8.158)

In view of (8.158), Si+1 −1



t =Si

−1

i+1 αS−1 αt ut (xt , xt +1 ) = σ ({αS−1 αt ut }t =S , xSi , xSi+1 ). i i i

S

(8.159)

It follows from (8.137), (8.138) and (8.157) that for all t ∈ {Si , . . . , Si+1 − 1}, we have αt ut − ut  ≤ ut |αS−1 αt − 1| ≤ (v + 1)(1 − λ)λ−1 . αS−1 i i

(8.160)

8.8 Proof of Theorem 8.10

283

By (8.135), (8.151), (8.159) and (8.160), Si+1 −1



t =Si

S

−1

S

−1

i+1 ut (xt , xt +1 ) ≥ σ ({ut }t =S , xSi , xSi+1 ) − 2(L1 + 1)(v + 1)(1 − λ)λ−1 i

i+1 ≥ σ ({ut }t =S , xSi , xSi+1 ) − δ0 /2. i

By the relation above, (8.133), (8.137) and (8.157), Si+1 −1



v(xt , xt +1 ) ≥ σ (v, Si+1 − Si , xSi , xSi+1 ) − δ0 /2 − 2δ(L1 + 1)

t =Si

≥ σ (v, Si+1 − Si , xSi , xSi+1 ) − δ0 .

(8.161)

Property (ii), (8.150) and (8.161) imply that xt − x ¯ ≤ , t = Si , . . . , Si+1 . This implies that ¯ ≤ , t = S0 , . . . , Sp . xt − x Theorem 8.9 is proved.

8.8 Proof of Theorem 8.10 Let r¯ ∈ (0, 1) be as guaranteed by (A5). Namely, the following property holds: (i) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ , there exists a point y ∈ X for which y ≥ y, (x, y ) ∈ Ω. We may assume without loss of generality that  < r¯ /4. By Lemma 7.11, there exists a number δ0 ∈ (0, ) such that the following property holds:

284

8 Autonomous Problems with Perturbed Objective Functions

(ii) For every natural number τ and every program {xt }τt=0 satisfying ¯ xτ − x ¯ ≤ δ0 , x0 − x, T −1

v(xt , xt +1 ) ≥ σ (v, τ, x0 , xτ ) − δ0

t =0

the inequality xt − x ¯ ≤  is valid for all integers t = 0, . . . , τ . By Lemmas 8.11 and 8.12, there exist an integer L1 ≥ L0 , real numbers δ1 ∈ (0, δ0 ) and λ1 ∈ (0, 1) such that the following property holds: −1 (iii) For each integer T ≥ L1 , each sequence of functions {ut }Tt =0 ⊂ M0 satisfying ut − v ≤ δ1 , t = 0 . . . , T − 1, −1 ⊂ (0, 1] such that for each integer τ ∈ [0, T − L1 ], each sequence {αt }Tt =0

αi αj−1 ≥ λ for each i, j ∈ {τ, . . . , τ + L1 } and each program {xt }Tt=0 satisfying x0 ∈ YL0 , xT ∈ Y¯L0 and T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 , xT )

there is an integer S ∈ {1, . . . , T − 1} such that ¯ ≤ δ0 . xS − x (iv) For each sequence of functions {ut }∞ t =0 ⊂ M0 satisfying ut − v ≤ δ1 , t = 0, 1 . . . each sequence {αt }∞ t =0 ⊂ (0, 1] such that for each pair i, j ∈ {0, 1, . . . } satisfying |i − j | ≤ L1 , αi αj−1 ≥ λ1 ∞ and each ({αt ut }∞ t =0 )-overtaking optimal program {xt }t =0 satisfying

x0 ∈ YL0

8.8 Proof of Theorem 8.10

285

there is an integer S ≥ 1 such that xS − x ¯ ≤ δ0 . Choose a positive number δ ≤ min{δ0 , δ1 }(8L1 )−1 ,

(8.162)

L > 2(L0 + L1 + 3)

(8.163)

a natural number

and λ ∈ (λ1 , 1) such that (1 − λ)λ−1 L1 (v + 1) < δ0 /8.

(8.164)

Assume that {ut }∞ t =0 ⊂ M0 ,

(8.165)

ut − v ≤ δ, t = 0, 1, . . . ,

(8.166)

{αt }∞ t =0 ⊂ (0, 1], αi αj−1 ≥ λ for each i, j ∈ {0, 1, . . . , } satisfying |i − j | ≤ L

(8.167)

∞ and that an ({αt ut }∞ t =0 )-overtaking optimal program {xt }t =0 satisfies

x0 ∈ YL0 .

(8.168)

Property (iv), (8.165) and (8.168) imply that there is a natural number S0 such that ¯ ≤ δ0 . xS0 − x

(8.169)

If x0 − x ¯ ≤ δ, then we may assume that S0 = 0. We may assume without loss of generality that {t ∈ {0, . . . , S0 } \ {S0 } : xt − x ¯ ≤ δ0 } = ∅.

(8.170)

286

8 Autonomous Problems with Perturbed Objective Functions

We show that S0 ≤ L1 . Assume the contrary. Then S0 > L1 .

(8.171)

xS0 ∈ Y¯L0 .

(8.172)

Property (i) and (8.169) imply that

Property (iii), (8.165)–(8.168), (8.171) and (8.172) imply that there exists S ∈ {1, . . . , S0 − 1} such that xS − x ¯ ≤ δ0 . This contradicts (8.170). The contradiction we have reached proves that S0 ≤ L1 .

(8.173)

Assume that k ≥ 0 is an integer and that we defined a strictly increasing sequence of nonnegative integers Si , i = 0, . . . , k such that (8.173) is true, ¯ ≤ δ, then S0 = 0, if x0 − x for each integer i = 0, . . . , k, ¯ ≤ δ0 xSi − x

(8.174)

1 ≤ Si+1 − Si ≤ L1 .

(8.175)

and for all i ∈ {0, . . . , k} \ {k},

(Clearly, our assumption holds for k = 0.) Property (i) and (8.174) imply that xSk ∈ YL0 .

(8.176)

Property (iv) applied to the sequence {xt }∞ t =Sk , (8.165)–(8.167) and (8.176) imply that there exists an integer Sk+1 > Sk

8.8 Proof of Theorem 8.10

287

such that xSk+1 − x ¯ ≤ δ0 .

(8.177)

We may assume without loss of generality that ¯ ≤ δ0 } = ∅. {t ∈ {Sk + 1, . . . , Sk+1 } \ {Sk+1 } : xt − x

(8.178)

Assume that Sk+1 > Sk + L1 .

(8.179)

By (8.165)–(8.167), (8.176), (8.177), (8.179), property (i) and property (iii) applied Sk+1 , there exists to the sequence {xt }t =S k S ∈ {Sk + 1, . . . , Sk+1 − 1} such that ¯ ≤ δ0 . xS − x This contradicts (8.178). The contradiction we have reached proves that Sk+1 ≤ Sk + L1 . Now it is easy to see that the assumption made for k also holds for k + 1. Thus by induction we have constructed the strictly increasing sequence of integers Si ≥ 0, i = 0, 1, . . . such that S0 ≤ L1 ,

(8.180)

if x0 − x ¯ ≤ δ, then S0 = 0, xSi − x ¯ ≤ δ0 , i = 0, 1, . . . ,

(8.181)

Si < Si+1 ≤ Si + L1 , i = 0, 1, . . . .

(8.182)

Let i ≥ 0 be an integer. It is clear that Si+1 −1



t =Si Si+1 −1



t =Si

S

−1

i+1 αt ut (xt , xt +1 ) = σ ({αt ut }t =S , xSi , xSi+1 ), i

−1

i+1 αS−1 αt ut (xt , xt +1 ) = σ ({αS−1 αt ut }t =S , xSi , xSi+1 ). i i i

S

(8.183)

288

8 Autonomous Problems with Perturbed Objective Functions

It follows from (8.166), (8.182) that for all t ∈ {Si , . . . , Si+1 − 1}, we have αS−1 αt ut − ut  ≤ ut |αS−1 αt − 1| ≤ (v + 1)(1 − λ)λ−1 i i and αS−1 αt ut − v ≤ (v + 1)(1 − λ)λ−1 + δ. i

(8.184)

By (8.183) and (8.184), Si+1 −1



v(xt , xt +1 ) − σ (v, Si+1 − Si , xSi , xSi+1 )

t =Si

≥ 2((v + 1)(1 − λ)λ−1 L1 ) + L1 δ) ≥ 4−1 δ0 − 4−1 δ0 .

(8.185)

Property (i) and (8.185) imply that xt − x ¯ ≤ , t = Si , . . . , Si+1 . This implies that ¯ ≤  for all integers t ≥ S0 . xt − x Theorem 8.10 is proved.

8.9 Existence of Overtaking Optimal Programs Let r¯ ∈ (0, 1) be as guaranteed by (A5). Namely, the following property holds: (i) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ , there exists a point y ∈ X for which y ≥ y, (x, y ) ∈ Ω. Let L0 ≥ 1 be an integer,  = r¯ /4 and let δ ∈ (0, r¯ /4), an integer L > L0 and λ ∈ (0, 1) be as guaranteed by Theorem 8.9. Let ut ∈ M0 and ut − v ≤ δ, t = 0, 1, . . . ,

(8.186)

8.9 Existence of Overtaking Optimal Programs

289

and let {αt }∞ t =0 ⊂ (0, 1] satisfy the relations lim αt = 0,

t →∞

αi αj−1 ≥ λ for all nonnegative integers i, j satisfying |i − j | ≤ L.

(8.187) (8.188)

(z) Theorem 8.13 For each z ∈ Y¯L0 there is a program {xt(z)}∞ t =0 ⊂ X such that x0 = z and that the following property holds: For each real number γ > 0 there exists an integer n0 ≥ 1 such that for each integer T ≥ n0 and each point z ∈ YL0 the inequality

−1 , 0, T , z) − |σ ({αt ut }Tt =0

T −1 t =0

αt ut (xt(z), xt(z) +1 )| ≤ γ

holds. It is clear that Theorem 8.13 establishes the existence of an ({αt ut }∞ t =0 )overtaking optimal program when (8.186)–(8.188) hold. Roughly speaking, an ({αt ut }∞ t =0 )-overtaking optimal program exists if the objective functions ut , t = 0, 1, . . . belong to the δ-neighborhood of v in the topology of the uniform convergence on the set Ω and the sequence of the discount coefficients {αt }∞ t =0 tends to zero slowly. Note that the existence of an ({αt ut }∞ t =0 )-overtaking optimal program when the discount coefficients {αt }∞ t =0 tends to zero rapidly is a well-known fact. In the proof we use the following auxiliary result. Lemma 8.14 Let γ be a positive number. Then there exists an integer n0 ≥ 1 such that for each pair of integers T > S ≥ n0 and each program {xt }Tt=0 satisfying the inclusion x0 ∈ YL0

(8.189)

and the inequality T −1 t =0

−1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , x0 )

(8.190)

the following inequality holds: S−1  t =0

αt ut (xt , xt +1 ) ≥ σ ({αt ut }S−1 t =0 , x0 ) − γ .

(8.191)

290

8 Autonomous Problems with Perturbed Objective Functions

Proof Since limt →∞ αt = 0 (see (8.187)) there exists a natural number n0 ≥ 4L + 4

(8.192)

such that for all integers t > n0 − L − 4, we have αt ≤ γ (8L + 8)−1 (v + 1)−1 .

(8.193)

Assume that integers T > S ≥ n0 and that a program {xt }Tt=0 satisfies relations (8.189) and (8.190). There exists a program {x˜t }St=0 such that x˜0 = x0 , S−1  t =0

(8.194)

αt ut (x˜t , x˜t +1 ) = σ ({αt ut }S−1 t =0 , x0 ).

(8.195)

In view of the choice of the numbers δ, λ and L, Theorem 8.9, (8.186), (8.188), (8.189), (8.192), (8.194) and (8.195), we have ¯ ≤ r¯ /4, t = L, . . . , T − L, xt − x

(8.196)

x˜t − x ¯ ≤ r¯ /4, t = L, . . . , S − L.

(8.197)

It follows from (A6), (8.186), (8.196) and (8.197) that there exists a program {yt }Tt=0 such that yt = x˜t , t = 0, . . . , S − L,

(8.198)

yt ≥ xt , t = S − L + 1, . . . , T ,

(8.199)

ut (yt , yt +1 ) ≥ ut (xt , xt +1 ), t = S − L + 1, . . . , T − 1.

(8.200)

By (8.186), (8.190), (8.193)–(8.195), (8.198) and (8.200), we have 0≤

T −1

αt ut (xt , xt +1 ) −

t =0

=

S−L 

αt ut (yt , yt +1 )

t =0

αt ut (xt , xt +1 ) −

t =0

≤

T −1

S−L−1  t =0

S−L 

αt ut (yt , yt +1 )

t =0

αt ut (xt , xt +1 ) −

S−L−1  t =0

αt ut (x˜t , x˜t +1 ) + 2αS−L (v + 1)

8.9 Existence of Overtaking Optimal Programs

≤

S−1 

291 S−1 

αt ut (xt , xt +1 ) + (v + 1)

t =0

− σ ({αt ut }S−1 t =0 , x0 ) + (v + 1) ≤

S−1  t =0

αt + 2αS−L (v + 1)

t =S−L S−1 

αt

t =S−L

αt ut (xt , xt +1 ) − σ ({αt ut }S−1 t =0 , x0 ) + γ .

Lemma 8.14 is proved. Completion of the Proof of Theorem 8.13 Let a point z ∈ YL0 be given. For each natural number T there exists a program {xt(z,T ) }Tt=0 such that (z,T )

x0

T −1 t =0

= z,

(8.201)

) T −1 αt ut (xt(z,T ) , xt(z,T +1 ) = σ ({αt ut }t =0 , x0 ).

(8.202)

Evidently, there exists a strictly increasing sequence of natural numbers {Tj }∞ j =1 such that for any nonnegative integer t there exists a limit (z,Tj )

xt(z) = lim xt j →∞

.

(8.203)

It is clear that {xt(z)}∞ t =0 is a program and x0(z) = z. Let γ > 0 be given. It follows from Lemma 8.14 that there exists an integer n0 ≥ 1 such that the following property holds: (iv) For each pair of integers T > S ≥ n0 and each program {xt }Tt=0 satisfying (8.189) and (8.190) Eq. (8.191) holds. Let S ≥ n0 be an integer. In view of property (iv), (8.201) and (8.202) for each natural number j satisfying Tj > S, we have S−1  t =0

) S−1 αt ut (xt(z,T ) , xt(z,T +1 ) ≥ σ ({αt ut }t =0 , z) − γ .

292

8 Autonomous Problems with Perturbed Objective Functions

Combined with (8.203) this inequality implies that S−1  t =0

S−1 αt ut (xt(z) , xt(z) +1 ) ≥ σ ({αt ut }t =0 , z) − γ

for all integers S ≥ n0 and z ∈ YL0 . Theorem 8.13 is proved.

Chapter 9

Stability Results for Autonomous Problems

In this chapter we continue to study the class of autonomous optimal control problems considered in Chaps. 7 and 8. For these optimal control problems the turnpike is a singleton. We show the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint maps.

9.1 Preliminaries and the Main Results We continue to consider the class of optimal control problems studied in Chaps. 7 and 8. Assume that the n-dimensional space R n with the Euclidean norm  ·  is ordered n by the cone R+ = {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n} and that x  y, x > y, x ≥ y have their usual meaning. Let e(i), i = 1, . . . , n, be the ith n unit vector in R n , and e be an element of R+ , all of whose coordinates are unity. n n Let X ⊂ R+ be a compact subset of R+ , Ω be a nonempty closed subset of X × X. 2 Let T2 > T1 be nonnegative integers. Recall that a sequence {xt }Tt =T ⊂ X is 1 called an (Ω)-program (or just a program if the set Ω is understood) if (xt , xt +1 ) ∈ Ω for all integers t ∈ [T1 , T2 − 1]. A sequence {xt }∞ t =T1 ⊂ X is called an (Ω)program (or just a program if the set Ω is understood) if (xt , xt +1 ) ∈ Ω for all integers t ≥ T1 . Let v : X × X → R 1 be a bounded upper semicontinuous function. Set v = sup{|v(x, y)| : (x, y) ∈ Ω}.

(9.1)

We may assume without loss of generality that v > 0.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_9

293

294

9 Stability Results for Autonomous Problems

For every pair of points x, y ∈ X and every natural number T define −1  T v(xi , xi+1 ) : {xi }Ti=0 is a program and x0 = x , σ (v, T , x) = sup i=0

σ (v, T , x, y) = sup

−1 T

 v(xi , xi+1 ) : {xi }Ti=0 is a program and x0 = x, xT ≥ y ,

i=0

σ (v, T ) = sup

−1 T

 v(xi , xi+1 ) : {xi }Ti=0 is a program .

i=0

(Here we use the convention that the supremum over an empty set is −∞.) We suppose that all the assumptions introduced in Sect. 7.1 hold. Namely, we assume that there exist a point x¯ ∈ X and a positive constant c¯ such that the following assumptions hold: (A1) (x, ¯ x) ¯ ∈ Ω and the function v : Ω → R 1 is continuous at the point (x, ¯ x). ¯ (A2) σ (v, T ) ≤ T v(x, ¯ x) ¯ + c¯ for all natural numbers T . Recall that a program {xt }∞ t =0 is good if the sequence −1 T t =0

v(xt , xt +1 ) − T v(x, ¯ x) ¯

∞ T =1

is bounded. We suppose that the following assumptions hold: (A3) (the asymptotic turnpike property) For every good program {xt }∞ t =0 the equality limt →∞ xt − x ¯ = 0 is valid. (A4) If a point (x0 , x1 ) ∈ Ω and if a point y0 ∈ X satisfies the inequality y0 ≥ x0 , then there exists y1 ∈ X for which (y0 , y1 ) ∈ Ω, v(y0 , y1 ) ≥ v(x0 , x1 ) and 0 ≤ y1 − x1 ≤ y0 − x0 . For every positive number M denote by XM the set of all points x ∈ X such that there exists a program {xt }∞ t =0 satisfying x0 = x and T −1

v(xt , xt +1 ) − T v(x, ¯ x) ¯ ≥ −M

t =0

for every natural number T . For every integer T ≥ 1 denote by Y¯T the set of all points x ∈ X such that there exists a program {xt }Tt=0 satisfying x0 = x, ¯ xT ≥ x.

9.1 Preliminaries and the Main Results

295

For every integer T ≥ 1 denote by YT the set of all points x ∈ X such that there exists a program {xt }Tt=0 satisfying x0 = x, xT ≥ x. ¯ In this chapter instead of assumption (A5), used in Chaps. 7 and 8, we use its following strong version. (A5)’ For every positive number  there exists a positive number δ such that for every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤δ there exists a point y ∈ X for which y ≥ y + δe, (x, y ) ∈ Ω and y − x ¯ ≤ . We also suppose that the following assumption holds: (B1) For every positive number  there exists a positive number δ such that for every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ δ, y ≥ x, we have |v(x, y) − v( ¯ x, ¯ x) ¯ ≤ . For each x ∈ X and each nonempty set C ⊂ X set ρ(x, C) = inf{x − y : y ∈ C}. For each x ∈ X and each r > 0 set B(x, r) = {y ∈ X : x − y ≤ r}. We equip the space X × X with the metric ρ1 defined by ρ1 ((x1 , x2 ), (y1 , y2 )) = x1 − y1  + x2 − y2 , x1 , x2 , y1 , y2 ∈ X. For each (x1 , x2 ) ∈ X × X and each nonempty set C ⊂ X × X set ρ1 ((x1 , x2 ), C) = inf{ρ1 ((x1 , x2 ), (y1 , y2 )) : (y1 , y2 ) ∈ C}. Denote by M the set of all bounded functions u : X × X → R 1 for which the following assumption holds: (A6) If a point (x0 , x1 ) ∈ Ω and if a point y0 ∈ X satisfies the inequality y0 ≥ x0 , then there exists y1 ∈ X for which (y0 , y1 ) ∈ Ω, y1 ≥ x1 , u(y0 , y1 ) ≥ u(x0 , x1 ).

296

9 Stability Results for Autonomous Problems

For each w ∈ M set w = sup{|w(x, y)| : (x, y) ∈ X × X}.

(9.2)

λ¯ ∈ (0, 1).

(9.3)

Fix

For each λ > 0 denote by E(λ) the collection of all nonempty sets Ω ⊂ X × X such that (x, ¯ x) ¯ ∈ Ω , ρ1 (z, Ω) ≤ λ for each z ∈ Ω

(9.4)

and that for each (x, y) ∈ Ω ∩ (B(x, ¯ λ¯ ), B(x, ¯ λ¯ )), we have ρ(y, {z ∈ X : (x, ¯ z) ∈ Ω }) ≤ λ, ρ(y, {z ∈ X : (x, z) ∈ Ω }) < λ.

(9.5)

Let integers T1 , T2 satisfy 0 ≤ T1 < T2 and let Ωt , t = T1 , . . . , T2 − 1 be nonempty subsets of X × X. 2 2 −1 A sequence {xt }Tt =T ⊂ X is called an ({Ωt }Tt =T )-program if (xt , xt +1 ) ∈ Ωt for 1 1 all integers t ∈ [T1 , T2 − 1]. 2 −1 ⊂ M set For each x, y ∈ X and each finite sequence {ut }Tt =T 1 2 −1 T T2 −1 T2 −1 σ ({ut }t =T , {Ωt }t =T , x) = sup ut (xt , xt +1 ) : 1

1

t =T1

 T2 2 −1 is an ({Ωt }Tt =T )-program and xT1 = x , {xt }t =T 1

1

(9.6) 2 −1 T T2 −1 T2 −1 σ ({ut }t =T , {Ωt }t =T , x, y) = sup ut (xt , xt +1 ) : 1

1

t =T1

T2 {xt }t =T 1



2 −1 is an ({Ωt }Tt =T )-program, xT1 = x and xT2 = y , 1

(9.7) 2 −1 2 −1 σ ({ut }Tt =T , {Ωt }Tt =T ) 1 1

2 −1 T = sup ut (xt , xt +1 ) :

t =T1

 T2 −1 2 {xt }Tt =T is an ({Ω } )-program . t t =T1 1

9.1 Preliminaries and the Main Results

297

(Here we use the convention that the supremum of an empty set is −∞.) Denote by A the set of all pairs (u, Ω ) such that u ∈ M, Ω ⊂ X × X is a nonempty set and that the following property holds: (B2) For each point (x, y) ∈ Ω , each x ∈ X satisfying the inequality x ≥ x, there exists y ∈ X for which (x , y ) ∈ Ω , y ≥ y, u(x , y ) ≥ u(x, y). By (A5)’, there exists ¯ r¯ ∈ (0, λ) such that the following property holds: (B3) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ r¯ there exists a point y ∈ X for which ¯ ≤ λ¯ . y ≥ y + r¯ e, (x, y ) ∈ Ω and y − x Proposition 9.1 Let λ ∈ (0, r¯ ), Ω0 ⊂ X × X be a nonempty set such that Ω0 ∈ E(λ).

(9.8)

Then there exists y˜ ∈ X such that y˜ ≥ x, ¯ (x, ¯ y) ˜ ∈ Ω0 . Proof By (B3), there exists a point y ∈ X for which ¯ (ex, ¯ y ) ∈ Ω and y − x ¯ ≤ λ,

(9.9)

y ≥ x¯ + r¯ e.

(9.10)

¯ ¯ × B(x, ¯ λ)). (x, ¯ y ) ∈ Ω ∩ ({x}

(9.11)

In view of (9.9) and (9.10),

It follows from (9.5), (9.8), and (9.11) that ¯ z) ∈ Ω0 }) < λ < r¯ . ρ(y , {z ∈ X : (x,

298

9 Stability Results for Autonomous Problems

This implies that there exists y˜ ∈ X such that (x, ¯ y) ˜ ∈ Ω0 , y − y ˜ < r¯ . Together with (9.10) this implies y˜ ≥ x. ¯ Proposition 9.1 is proved. Proposition 9.1 and (B2) imply the following result. Proposition 9.2 Let λ ∈ (0, r¯ ) and T2 > T1 ≥ 0 be integers, Ωt ⊂ X × X, t = T1 , . . . , T2 − 1 be nonempty sets satisfying Ωt ∈ E(λ), t = T1 , . . . , T2 − 1 and let ut ∈ M, t = T1 , . . . , T2 − 1 satisfy (ut , Ωt ) ∈ A, t = T1 , . . . , T2 − 1. 2 −1 2 Then there exists an ({Ωt }Tt =T )-program {xt }Tt =T such that xT1 = x¯ and 1 1

¯ t = T1 , . . . , T2 − 1. xt ≥ x, Let T2 > T1 ≥ 0 be integers and let Ωt , t = T1 , . . . , T2 − 1 be nonempty subsets 2 −1 , T1 , T2 ) the set of all x ∈ X for which there exists of X × X. Denote by Y ({Ωt }Tt =T 1 2 −1 2 an ({Ωt }Tt =T )-program {xt }Tt =T such that xT1 = x and xT2 ≥ x. ¯ 1 1 T2 −1 ¯ Denote by Y ({Ωt }t =T1 , T1 , T2 ) the set of all x ∈ X for which there exists an

2 −1 2 ({Ωt }Tt =T )-program {xt }Tt =T such that xT1 = x¯ and xT2 = x. 1 1 For sufficiently small positive numbers δ, we study the structure of approximate solutions of the problems

T −1

ui (xi , xi+1 ) → max,

i=0 −1 {xi }Ti=0 is an ({Ωt }Tt =0 )-program and x0 = y,

and T −1

ui (xi , xi+1 ) → max,

i=0 −1 {xi }Ti=0 is an ({Ωt }Tt =0 )-program and x0 = y, xT ≥ z,

9.1 Preliminaries and the Main Results

299

where T ≥ 1 is an integer, y, z ∈ X and for all t = 0, . . . , T − 1, we have Ωt ∈ E(δ), ut ∈ M and ut − v ≤ δ. Denote by Card(B) the cardinality of a set B. In this chapter we prove the following two stability results which are new. Theorem 9.3 Let , M be positive numbers and let l1 , l2 be natural numbers. Then there exist δ > 0 and a natural number Q ≥ l1 + l2 such that for each integer T > Q, each Ωt ∈ E(δ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ, t = 0, . . . , T − 1, (ut , Ωt ) ∈ A, t = 0, . . . , T − 1 −1 and each ({Ωt }Tt =0 )-program {xt }Tt=0 for which at least one of the following conditions holds:

(a) −1 −1 , 0, l1 ), xT ∈ Y¯ ({Ωt }Tt =T x0 ∈ Y ({Ωt }tl1=0 −l2 , T − l2 , T ), −1 −1 σ ({ut }Tt =0 , {Ωt }Tt =0 , x0 , xT ) ≤

T −1

ut (xt , xt +1 ) + M;

t =0

(b) −1 , 0, l1 ), x0 ∈ Y ({Ωt }lt1=0 −1 −1 , {Ωt }Tt =0 , x0 ) ≤ σ ({ut }Tt =0

T −1

ut (xt , xt +1 ) + M

t =0

the inequality ¯ > }) ≤ Q Card({t ∈ {0, . . . , T } : xt − x holds. Theorem 9.4 Let , M be positive numbers and let l1 , l2 be natural numbers. Then there exist δ > 0 and a natural number L > l1 + l2 such that for each integer T > 2L, each Ωt ∈ E(δ), t = 0, . . . , T − 1,

300

9 Stability Results for Autonomous Problems

each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ, t = 0, . . . , T − 1, (ut , Ωt ) ∈ A, t = 0, . . . , T − 1 −1 )-program {xt }Tt=0 which satisfies and each ({Ωt }Tt =0 τ +L−1  t =τ

+L−1 +L−1 ut (xt , xt +1 ) ≥ σ ({ut }τt =τ , {Ωt }τt =τ , xτ , xτ +L ) − δ

for each integer τ ∈ [0, T −L] and for which at least one of the following conditions holds: (a) −1 −1 x0 ∈ Y ({Ωt }tl1=0 , 0, l1 ), xT ∈ Y¯ ({Ωt }Tt =T −l2 , T − l2 , T ), −1 −1 σ ({ut }Tt =0 , {Ωt }Tt =0 , x0 , xT ) ≤

T −1

ut (xt , xt +1 ) + M;

t =0

(b) −1 x0 ∈ Y ({Ωt }lt1=0 , 0, l1 ), −1 −1 σ ({ut }Tt =0 , {Ωt }Tt =0 , x0 ) ≤

T −1

ut (xt , xt +1 ) + M

t =0

there exist integers τ1 ∈ [0, L], τ2 ∈ [T − L, T ] such that ¯ ≤  for all t = τ1 , . . . , τ2 . xt − x Moreover, if x0 − x ¯ ≤ δ, then τ1 = 0 and if xT − x ¯ ≤ δ, then τ2 = T .

9.2 Examples In Sects. 7.7 and 7.8 we considered two classes of optimal control problems related to some models of economic growth. It turns out that all the assumptions posed in this chapter hold for them. First we consider the RSS model discussed in Sect. 7.7 using the notation introduced there. It is easy to see that (A1) and (A2) hold with x¯ = x , (A3) holds if w is strictly concave or d + aσ−1 = 2, (A4) holds too and (A5)

9.2 Examples

301

is equivalent to (A5)’. Consider the function v : Ω → R 1 defined in Sect. 7.7. For all (x, y) ∈ (X × X) \ Ω set v(x, y) = −v − 1. The following result shows that (B1) holds for the RSS model. Proposition 9.5 Let r0 = (1 + daσ )−1 (2a + 1)−1 n and x, y ∈ R+ satisfy

y ≥ x, x − x  ≤ r0 , y − x  ≤ r0 . Then (x, y) ∈ Ω. Proof It is easy to see that y ≥ x, y − (1 − d)x ≥ 0, a(y − (1 − d)x) = a(y − x ) + a( x − (1 − d) x ) + a(1 − d)( x − x) = aσ d(1 + daσ )−1 + 2ar0 ≤ 1 and (x, y) ∈ Ω. Proposition 9.5 is proved. Now we consider the second example discussed in Sect. 7.8 using the notation introduced there. It was shown there that (A1)–(A5) holds. Set v(x, y) = −v − 1, (x, y) ∈ (X × X) \ Ω. Note the (A5) implies (A5)’. We need only to show that (B1) holds. It was shown in Sect. 7.8 that v(x, ¯ x) ¯ > 0, z¯  0, x¯  0, x¯ − V x¯ + z¯ ∈ T (x). ¯ This implies that x¯ − V x¯ + min{¯zi : i = 1, . . . , n}e ∈ T (x). ¯ By Proposition 7.21, (x, ¯ x) ¯ is an interior point of Ω. Therefore v is continuous at (x, ¯ x) ¯ and (B1) holds.

302

9 Stability Results for Autonomous Problems

9.3 Auxiliary Results Lemma 9.6 Let  be a positive number and let L be a natural number. Then there exists δ > 0 such that for each Ωt ∈ E(δ), t = 0, . . . , L − 1, each ut ∈ M, t = 0, . . . , L − 1 satisfying ut − v ≤ δ, t = 0, . . . , L − 1 L L and each ({Ωt }L−1 t =0 )-program {xt }t =0 there exists an (Ω)-program {yt }t =0 such that

xt − yt  ≤  for all t = 0, . . . , L and L−1 

v(yt , yt +1 ) ≥

t =0

L−1 

ut (xt , xt +1 ) − .

t =0

Proof Assume that the lemma does not hold. Then for each natural number k there exist sets (k)

Ωt

∈ E(k −1 ), t = 0, . . . , L − 1,

(9.12)

functions (k)

ut

∈ M, t = 0, . . . , L − 1

(9.13)

satisfying −1 u(k) t − v ≤ k , t = 0, . . . , L − 1 (k)

(k)

(9.14)

L and an ({Ωt }L−1 t =0 )-program {xt }t =0 such that the following property holds:

(i) If an (Ω)-program {yt }L t =0 satisfies xt(k) − yt  ≤  for all t = 0, . . . , L, then L−1  t =0

v(yt , yt +1 )
k0 such that 16k1−1 L < .

(9.18)

Assume that an integer k ≥ k1 . Then (9.16) and (9.17) hold. In view of (9.17) and (9.18), L−1 

v(xt , xt +1 ) ≥

t =0

≥

L−1  t =0

≥

L−1  t =0

L−1  t =0

v(xt(k) , xt(k) +1 ) − /4

(k) (k) (k) u(k) t (xt , xt +1 ) − /4 − L max{ut − v : t = 0, . . . , L − 1}

ut (xt , xt +1 ) − /4 − Lk1−1 ≥ (k)

(k)

(k)

L−1  t =0

(k)

(k)

(k)

ut (xt , xt +1 ) − /2.

304

9 Stability Results for Autonomous Problems

When combined with (9.16) this contradicts property (i). The contradiction we have reached proves Lemma 9.6. Assumption (B2) and (9.19) easily imply the following result. Lemma 9.7 Let T2 > T1 ≥ 0 be integers, Ωt ⊂ X × X, t = T1 , . . . , T2 − 1 be nonempty sets, ut ∈ M, t = T1 , . . . , T2 − 1 satisfy (ut , Ωt ) ∈ A, t = T1 , . . . , T2 − 1,

(9.19)

2 −1 2 S2 > S1 ≥ T1 be integers, S2 ≤ T2 , M > 0 and an ({Ωt }Tt =T )-program {xt }Tt =T 1 1 satisfy

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =T , {Ωt }Tt =T , xT1 , xT2 ) − M. 1 1

Then S 2 −1 t =S1

2 −1 2 −1 ut (xt , xt +1 ) ≥ σ ({ut }St =S , {Ωt }St =S , xS1 , xS2 ) − M. 1 1

Lemma 9.8 Let , M be positive numbers. Then there exists a natural number L such that for each integer L¯ ≥ L there exists δ > 0 such that the following assertion holds. ¯ each For each integer T ∈ [L, L], Ωt ∈ E(δ), t = 0, . . . , T − 1,

(9.20)

each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ, t = 0, . . . , T − 1

(9.21)

−1 and each ({Ωt }Tt =0 )-program {xt }Tt=0 satisfying T −1 t =0

ut (xt , xt +1 ) ≥

T −1

ut (x, ¯ x) ¯ −M

(9.22)

t =0

the inequality min{ρ(xt , x) ¯ : t = 1, . . . , T } ≤  holds. Proof We may assume without loss of generality that  < 1. By Lemma 7.12, there exists a natural number L such that the following property holds:

9.3 Auxiliary Results

305

(i) For each integer T ≥ L, each (Ω)-program {xt }Tt=0 which satisfies T −1

v(xt , xt +1 ) ≥ T v(x, ¯ x) ¯ −M −4

t =0

and each integer S ∈ [0, T − L], we have min{xt − x ¯ : t = S + 1, . . . , S + L} ≤ /4. Let an integer L¯ ≥ L. By Lemma 9.6, there exists δ0 > 0 such that the following property holds: ¯ each (ii) For each integer T ∈ [L, L], Ωt ∈ E(δ0 ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ0 , t = 0, . . . , T − 1 −1 )-program {zt }Tt=0 there exists an (Ω)-program {yt }L and each ({Ωt }Tt =0 t =0 such that

zt − yt  ≤ /4 for all t = 0, . . . , T and T −1

v(yt , yt +1 ) ≥

t =0

L−1 

ut (zt , zt +1 ) − /4.

t =0

Set δ = min{δ0 ,  L¯ −1 }.

(9.23)

¯ (9.20) holds, functions ut ∈ M, t = Assume that an integer T ∈ [L, L], −1 0, . . . , T − 1 satisfy (9.21) and ({Ωt }Tt =0 )-program {xt }Tt=0 satisfies (9.22). By (9.20), (9.21), and (9.23) and property (ii), there exists an (Ω)-program {yt }Tt=0 such that xt − yt  ≤ /4 for all t = 0, . . . , T

(9.24)

and T −1 t =0

v(yt , yt +1 ) ≥

T −1 t =0

ut (xt , xt +1 ) − /4.

(9.25)

306

9 Stability Results for Autonomous Problems

In view of (9.21)–(9.23) and (9.25), T −1

v(yt , yt +1 ) ≥

t =0

≥

T −1

T −1

ut (xt , xt +1 ) − /4

t =0

ut (x, ¯ x) ¯ − M − /4 ≥ T v(x, ¯ x) ¯ − T δ − M − /4

t =0

≥ T v(x, ¯ x) ¯ − M − 2 ≥ T v(x, ¯ x) ¯ − M − 2.

(9.26)

It follows from (9.26) and property (i) that min{yt − x ¯ : t = 1, . . . , L} ≤ /4. When combined with (9.24) this implies that ¯ : t = 1, . . . , L} ≤ /2. min{xt − x Lemma 9.8 is proved. Assumptions (A3) and (A4) imply the following result. Lemma 9.9 For each natural number T , σ (v, T , x, ¯ x) ¯ = T v(x, ¯ x). ¯ Lemma 9.10 Let  ∈ (0, 1). Then there exists δ > 0 such that for each natural number T and each z0 , z1 ∈ X satisfying zi − x ¯ ≤ δ, i = 0, 1

(9.27)

the value σ (v, T , z0 , z1 ) is finite and the inequality ¯ x)| ¯ ≤ |σ (v, T , z0 , z1 ) − T v(x, holds. Proof In view of (A1) and (A5)’, there exists a positive number δ <  such that the following property holds: (i) For each x, y ∈ X satisfying x − x, ¯ y − x ¯ ≤ δ,

9.3 Auxiliary Results

307

there exists y ∈ X such that ¯ ≤ , y ≥ y, (x, y ) ∈ Ω, y − x ¯ x)| ¯ ≤ /4; |v(x, y ) − v(x, (ii) For each (x, y) ∈ Ω satisfying x − x, ¯ y − x ¯ ≤ δ, we have |v(x, y) − v(x, ¯ x)| ¯ ≤ . Assume that T is a natural number and that z0 , z1 ∈ X satisfy (9.27). We show that σ (v, T , z0 , z1 ) ≥ T v(x, ¯ x) ¯ − /2. We may assume that T ≥ 2. Set y0 = z0 .

(9.28)

Property (i), (9.27) and (9.28) imply that there exists y1 ∈ X such that ¯ y1 ≥ x,

(9.29)

(y0 , y1 ) ∈ Ω,

(9.30)

¯ x)| ¯ ≤ /4. |v(y0 , y1 ) − v(x,

(9.31)

Assumption (A4) and (9.29) imply that for all integers t satisfying 2 ≤ t < T there −1 exist yt ∈ X such that {yt }Tt =0 is an (Ω)-program satisfying ¯ t ∈ {2, . . . , T } \ {T }, yt ≥ x, v(yt , yt +1 ) ≥ v(x, ¯ x) ¯ for all integers t satisfying 1 ≤ t < T − 1.

(9.32) (9.33)

In view of (9.29) and (9.32), ¯ yT −1 ≥ x. Property (i) and (9.27) imply that there exists ξ ∈ X such that ¯ ξ ) ∈ Ω, |v(x, ¯ ξ ) − v(x, ¯ x)| ¯ ≤ /4. ξ ≥ z1 , (x,

(9.34)

308

9 Stability Results for Autonomous Problems

Assumption (A4) and (9.34) imply that there exists yT ∈ X such that (yT −1 , yT ) ∈ Ω, yT ≥ ξ, ¯ ξ ). v(yT −1 , yt ) ≥ v(x, Clearly, {yt }Tt=0 is an (Ω)-program and yT ≥ z1 . It follows from (9.28), (9.31), and (9.33) and the relations above that σ (v, T , z0 , z1 ) ≥

T −1

v(yt , yt +1 )

t =0

≥ v(y0 , y1 ) + v(yT −1 , yT ) +

 {v(yt , yt +1 ) : t ∈ {1, . . . , T − 1} \ {T − 1}}

≥ v(x, ¯ x) ¯ − /4 + v(x, ¯ x) ¯ − /4 + v(x, ¯ x)Card({t ¯ ∈ {1, . . . , T − 1} \ {T − 1}) ≥ T v(x, ¯ x) ¯ − /2.) Let us show that σ (v, T , z0 , z1 ) ≤ T v(x, ¯ x) ¯ + /2. (Here we do not assume that T ≥ 2.) Assume that an (Ω)-program {yt }Tt=0 satisfies y0 = z0 , yT ≥ z1 .

(9.35)

¯ y0 = x.

(9.36)

Set

Property (i), (9.27), (9.35), and (9.36) imply that there exists y1 ∈ X such that y1 ≥ z0 = y0 , ( y0 , y1 ) ∈ Ω,

(9.37)

y1 ) − v(x, ¯ x)| ¯ ≤ /4. |v( y0 ,

(9.38)

Assumption (A4) and (9.37) imply that for all integers t = 2, . . . , T + 1 there exist +1 yt ∈ X such that { yt }Tt =0 is an (Ω)-program satisfying yt ≥ yt −1, t ∈ {2, . . . , T + 1},

(9.39)

v( yt , yt +1 ) ≥ v(yt −1 , yt ), t = 1, . . . , T .

(9.40)

9.3 Auxiliary Results

309

In view of (9.35) and (9.39), yT +1 ≥ yT ≥ z1 .

(9.41)

Property (i) and (9.27) imply that there exists ξ ∈ X such that ξ ≥ x, ¯ (z1 , ξ ) ∈ Ω,

(9.42)

|v(z1 , ξ ) − v(x, ¯ x)| ¯ ≤ /4.

(9.43)

Assumption (A4), (9.41) and (9.42) imply that there exists yT +2 ∈ X such that yT +2 ≥ ξ ≥ x, ¯ ( yT +1 , yT +2 ) ∈ Ω,

(9.44)

yT +2 ) ≥ v(z1 , ξ ). v( yT +1 ,

(9.45)

+2 Clearly, { yt }Tt =0 is an (Ω)-program. Lemma 9.9, (9.36), (9.38), (9.43)–(9.45) imply that

(T + 2)v(x, ¯ x) ¯ = σ (v, T + 2, x, ¯ x) ¯ ≥

T +1

v( yt , yt +1 )

t =0

≥ v(x, ¯ x) ¯ − /2 +

T −1

v(yt , yt +1 ) + v(x, ¯ x) ¯ − /4

t =0

and T −1

v(yt , yt +1 ) ≤ T v(x, ¯ x) ¯ + /2

t =0

for every (Ω)-program {yt }Tt=0 satisfying (9.35). Thus ¯ x) ¯ + /2. σ (v, T , z0 , z1 ) ≤ T v(x, This completes the proof of Lemma 9.10. Lemma 9.11 Let  be a positive number. Then there exists δ0 ∈ (0, ) such that for each integer T ≥ 2, each ξ0 , ξ1 ∈ X satisfying ¯ ≤ δ0 , i = 0, 1, ξi − x

(9.46)

310

9 Stability Results for Autonomous Problems

each Ωt ∈ E(δ0 ), t = 0, . . . , T − 1

(9.47)

and each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ0 , t = 0, . . . , T − 1,

(9.48)

(ut , Ωt ) ∈ A, t = 0, . . . , T − 1

(9.49)

−1 )-program {xt }Tt=0 such that there exists an ({Ωt }Tt =0

x0 = ξ0 , xT ≥ ξ1 , T −1

ut (xt , xt +1 ) ≥

t =0

(9.50) T −1

ut (x, ¯ x) ¯ − .

t =0

Proof Choose a positive number 0 < min{(16T )−1 , 8−1 λ¯ }.

(9.51)

By (B1), there exists a positive number 1 < 8−1 0 such that the following property holds: (i) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ 21 , y ≥ x, we have |v(x, y) − v( ¯ x, ¯ x) ¯ ≤ 0 /4. By (A5)’, there exists a positive number γ0 < 1 /4 such that the following property holds: (ii) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ γ0

(9.52)

9.3 Auxiliary Results

311

there exists a point y ∈ X for which ¯ ≤ 1 . y ≥ y + γ0 e, (x, y ) ∈ Ω and y − x Choose a positive number δ0 < γ0 /8.

(9.53)

Assume that ξ0 , ξ1 ∈ X satisfy (9.46), T ≥ 2 is an integer, (9.47) holds and ut ∈ M, t = 0, . . . , T − 1 satisfy (9.48) and (9.49). Set x0 = ξ0 .

(9.54)

Property (ii), (9.46) and (9.53) imply that there exists ξ2 ∈ X such that ξ2 − x ¯ ≤ 1 , (ξ0 , ξ2 ) ∈ Ω,

(9.55)

ξ2 ≥ x¯ + γ0 e.

(9.56)

By (9.46), (9.47) and (9.51)–(9.56), there exists x1 ∈ X such that (x0 , x1 ) ∈ Ω0 , x1 − ξ2  ≤ δ0 .

(9.57)

In view of (9.46), (9.54), (9.56), and (9.57), x1 ≥ ξ2 − δ0 e ≥ x¯ + γ0 e − δ0 e ≥ x0 − δ0 e + γ0 e − δ0 e ≥ x0 + 2−1 γ0 e,

(9.58)

x1 ≥ x¯ + 2−1 γ0 e.

(9.59)

Equations (9.53), (9.55), and (9.57) imply that x1 − x ¯ ≤ δ0 + ξ2 − x ¯ ≤ 1 + δ0 ≤ 21 .

(9.60)

Property (i), (9.46), (9.54), (9.58), and (9.60) imply that ¯ x)| ¯ ≤ 0 /4. |v(x0 , x1 ) − v(x,

(9.61)

¯ x)| ¯ ≤ 4−1 0 + 2δ0 . |u0 (x0 , x1 ) − u0 (x,

(9.62)

By (9.48) and (9.61),

312

9 Stability Results for Autonomous Problems

It follows from (B2), (9.4), (9.42), (9.49), and (9.59) that there exist xt ∈ X, t ∈ −1 −2 {1, . . . , T − 1} \ {1} such that {xt }Tt =0 is an ({Ωt }Tt =0 )-program, xt ≥ x, ¯ t = 1, . . . , T − 1,

(9.63)

ut (xt , xt +1 ) ≥ ut (x, ¯ x), ¯ t ∈ {1, . . . , T − 1} \ {1}.

(9.64)

Property (ii) applied with x = x¯ and y = ξ1 , (9.46) and (9.53) imply that there exists η0 ∈ X such that (x, ¯ η0 ) ∈ Ω, η0 ≥ ξ1 + γ0 e, η0 − x ¯ ≤ 1 .

(9.65)

By (9.5), (9.47), and (9.65), there exists η1 ∈ X such that (x, ¯ η1 ) ∈ ΩT −1 , η0 − η1  ≤ δ0 .

(9.66)

Equations (9.65) and (9.66) imply that ¯ ≤ η1 − η0  + η0 − x ¯ ≤ δ0 + 1 < 21 . η1 − x

(9.67)

In view of (9.46), (9.65), and (9.66), η1 ≥ η0 − δ0 e ≥ ξ1 + γ0 e − δ0 e ≥ x¯ − δ0 e + γ0 e − δ0 e.

(9.68)

Property (ii), (9.53), (9.67), and (9.68) imply that |v(x, ¯ η1 ) − v(x, ¯ x)| ¯ ≤ 4−1 0 .

(9.69)

¯ η1 ) − uT −1 (x, ¯ x)| ¯ ≤ 2δ0 + 4−1 0 . |uT −1 (x,

(9.70)

By (9.48) and (9.69),

Assumption (B2), (9.49), (9.63), and (9.66) imply that there exists xT ∈ X such that (xT −1 , xT ) ∈ ΩT −1 ,

(9.71)

xT ≥ η1 ,

(9.72)

¯ η1 ). uT −1 (xT −1 , xT ) ≥ uT −1 (x,

(9.73)

−1 In view of (9.71), {xt }Tt=0 is an ({Ωt }Tt =0 )-program. By (9.54), (9.68), and (9.72),

x0 = ξ0 , xT ≥ η1 ≥ ξ1 .

(9.74)

9.3 Auxiliary Results

313

It follows from (9.62), (9.64), (9.70), and (9.73) that T −1

ut (xt , xt+1 )

t=0

≥ u0 (x0 , x1 ) +



{ut (xt , xt+1 ) : t ∈ {1, . . . , T − 1} \ {T − 1}} + uT −1 (xT −1 , xT )

≥ u0 (x, ¯ x) ¯ − 4−1 0 − 2δ0  + {ut (x, ¯ x) ¯ : t ∈ {1, . . . , T − 1} \ {T − 1}} + uT −1 (x, ¯ x) ¯ − 4−1 0 − 2δ0 =

T −1

ut (x, ¯ x) ¯ − 2−1 0 − 4δ0 .

(9.75)

t=0

Equations (9.74) and (9.75) imply that Lemma 9.11 is true. Lemma 9.12 Let  be a positive number. Then there exists δ ∈ (0, ) such that for each natural number L there exists δ0 ∈ (0, δ)

(9.76)

such that the following assertion holds. For each natural number T ≤ L,

(9.77)

Ωt ∈ E(δ0 ), t = 0, . . . , T − 1,

(9.78)

each

each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ0 , t = 0, . . . , T − 1,

(9.79)

(ut , Ωt ) ∈ A, t = 0, . . . , T − 1

(9.80)

−1 and each ({Ωt }Tt =0 )-program {xt }Tt=0 satisfying

¯ xT − x ¯ ≤δ x0 − x,

(9.81)

and T −1 t =0

−1 −1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =0 , {Ωt }Tt =0 , x0 , xT ) − δ

the inequality xt − x ¯ ≤  holds for all t = 0, . . . , T .

(9.82)

314

9 Stability Results for Autonomous Problems

Proof By Lemma 7.11, there exists a positive number γ < min{, 1}/4

(9.83)

such that the following property holds: (i) For each integer T ≥ 1 and each (Ω)-program {xt }Tt=0 which satisfies x0 − x, ¯ xT − x ¯ ≤ 2γ , T −1

v(xt , xt +1 ) ≥ σ (v, T , x0 , xT ) − 2γ

t =0

¯ ≤ /4 holds for all t = 0, . . . , T . the inequality xt − x By Lemmas 9.10 and 9.11, there exists δ ∈ (0, γ /8) such that the following properties hold: (ii) For each integer T ≥ 2, ξi ∈ X, i = 0, 1 satisfying ξi − x ¯ ≤ δ, i = 0, 1, each Ωt ∈ E(δ), t = 0, . . . , T − 1 and each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ, t = 0, . . . , T − 1, (ut , Ωt ) ∈ A, t = 0, . . . , T − 1 −1 there exists an ({Ωt }Tt =0 )-program {xt }Tt=0 such that

x0 = ξ0 , xT ≥ ξ1 , T −1 t =0

ut (xt , xt +1 ) ≥

T −1

ut (x, ¯ x) ¯ − γ /8;

t =0

(iii) For each natural number T and each z0 , z1 ∈ X satisfying zi − x ¯ ≤ 2δ, i = 1, 2, we have that σ (v, T , z0 , z1 ) is finite and ¯ x)| ¯ ≤ γ /16. |σ (v, T , z0 , z1 ) − T v(x,

9.3 Auxiliary Results

315

Let L be a natural number. By Lemma 9.6, there exists δ1 ∈ (0, δ) such that the following property holds: (iv) For each integer T ∈ [1, L], each Ωt ∈ E(δ1 ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ1 , t = 0, . . . , T − 1 −1 and each ({Ωt }Tt =0 )-program {xt }Tt=0 there exists an (Ω)-program {yt }Tt=0 such that

xt − yt  ≤ δ/4 for all t = 0, . . . , T and T −1

v(yt , yt +1 ) ≥

t =0

T −1

ut (xt , xt +1 ) − δ/4.

t =0

Set δ0 = (2L)−1 δ1 .

(9.84)

Assume that an integer T ∈ [1, L], (9.78) holds, functions ut ∈ M, t = −1 0, . . . , T − 1 satisfy (9.79), (9.80) and that an ({Ωt }Tt =0 )-program {xt }Tt=0 satisfies (9.81) and (9.82). We may assume without loss of generality that T ≥ 2. By (9.78), (9.79), and (9.84) and property (iv), there exists an (Ω)-program {yt }Tt=0 such that xt − yt  ≤ δ/4 for all t = 0, . . . , T

(9.85)

and T −1

v(yt , yt +1 ) ≥

t =0

T −1

ut (xt , xt +1 ) − δ/4.

(9.86)

t =0

In view of (9.82) and (9.86), T −1 t =0

−1 −1 v(yt , yt +1 ) ≥ σ ({ut }Tt =0 , {Ωt }Tt =0 , x0 , xT ) − δ0 − δ/4.

(9.87)

316

9 Stability Results for Autonomous Problems

Property (ii) applied with ξ0 = x0 , ξ1 = xT , (9.78)–(9.81) and (9.84) imply that −1 −1 , {Ωt }Tt =0 , x0 , xT ) ≥ σ ({ut }Tt =0

T −1

ut (x, ¯ x) ¯ − γ /8.

(9.88)

t =0

By (9.79), (9.84), (9.87), and (9.88), T −1

v(yt , yt +1 ) ≥

t =0

T −1

ut (x˜t , x˜t +1 ) − 2δ − γ /8

t =0

≥ −2δ − γ /8 + T v(x, ¯ x) ¯ − Lδ0 ≥ T v(x, ¯ x) ¯ − γ /2.

(9.89)

In view of (9.81) and (9.85), for i = 0, T , yi − x ¯ ≤ yi − xi  + xi − x ¯ ≤ δ/4 + δ.

(9.90)

By (9.90) and property (iii), |σ (v, T , y0 , yT ) − T v(x, ¯ x)| ¯ ≤ γ /16.

(9.91)

It follows from (9.89) and (9.91) that T −1

v(yt , yt +1) ≥ σ (v, T , y0 , yT ) − γ /2 − γ /16.

(9.92)

t =0

In view of (9.90), (9.92) and property (i), yt − x ¯ ≤ /4 for all t = 0, . . . , T .

(9.93)

By (9.83), (9.85), and (9.93), for all t = 0, . . . , T , xt − x ¯ ≤ xt − yt  + yt − x ¯ ≤ δ/4 + /4 < . This completes the proof of Lemma 9.12. Lemma 9.13 Let , M be positive numbers and l1 , l2 be natural numbers. Then there exists δ > 0 and a natural number L > l1 + l2 such that for each integer T ≥ L, each Ωt ∈ E(δ), t = 0, . . . , T − 1,

(9.94)

each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ, t = 0, . . . , T − 1,

(9.95)

(ut , Ωt ) ∈ A, t = 0, . . . , T − 1

(9.96)

9.3 Auxiliary Results

317

−1 and each ({Ωt }Tt =0 )-program {xt }Tt=0 for which at least one of the following conditions hold:

(a) −1 x0 ∈ Y ({Ωt }lt1=0 , 0, l1 ), xT ∈ Y¯ ({Ωt }Tt =T −l2 , T − l2 , T ), T −1

−1 −1 σ ({ut }Tt =0 , {Ωt }Tt =0 , x0 , xT ) ≤

ut (xt , xt +1 ) + M

t =0

(b) x0 ∈ Y ({Ωt }lt1=0, 0, l1 ), −1 −1 , {Ωt }Tt =0 , x0 ) ≤ σ ({ut }Tt =0

T −1

ut (xt , xt +1 ) + M

t =0

there exists an integer S ∈ [1, T − 1] satisfying ¯ ≤ . xS − x Proof By Lemma 9.8, there exists a natural number L0 and δ0 ∈ (0, 1) such that the following property holds: (i) For each integer T ∈ [L0 , 2L0 ], each Ωt ∈ E(δ0 ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ0 , t = 0, . . . , T − 1 −1 )-program {xt }Tt=0 satisfying and each ({Ωt }Tt =0 T −1 t =0

ut (xt , xt +1 ) ≥

T −1

ut (x, ¯ x) ¯ −4

t =0

the inequality ¯ : t = 1, . . . , T } ≤  min{xt − x holds.

318

9 Stability Results for Autonomous Problems

Choose a natural number q ≥ 4 + 4M + 8(v + 1)

(9.97)

and set L = 2q(8 + 8L0 )(M + 2(v + 1)(l1 + l2 ) + 4 + 2L0 ).

(9.98)

By (B1), there exists a positive number γ < min{λ¯ /2, 1}

(9.99)

such that the following property holds: (ii) For every pair of points x, y ∈ X which satisfy x − x, ¯ y − x ¯ ≤ γ , y ≥ x, we have |v(x, y) − v( ¯ x, ¯ x) ¯ ≤ (16L0 )−1 . By (A5)’ there exist a positive number δ1 < 1 and y¯ ∈ X such that y¯ ≥ x¯ + δ1 e, (x, ¯ y) ¯ ∈ Ω and y¯ − x ¯ ≤ γ /2.

(9.100)

Choose a positive number δ < min{δ0 , δ1 /8, γ /8, (16L0)−1 }. Assume that an integer T ≥ L, (9.94) holds, ut ∈ M, t = 0, . . . , T − 1 −1 satisfy (9.95) and (9.96) and that an ({Ωt }Tt =0 )-program {xt }Tt=0 satisfies at least one of conditions (a), (b). In order to complete the proof of the lemma it is sufficient to show that there exists S ∈ {1, . . . , T − 1} such that ¯ ≤ . xS − x Assume the contrary. Then xS − x ¯ > , S = 1, . . . , T − 1.

(9.101)

It follows from property (i), (9.94), (9.95), and (9.101) that for every integer τ ∈ [0, T − 1 − L0 ], τ +L 0 −1  t =τ

ut (xt , xt +1 )
(4 + 4L)(5 + Mγ −1 ).

(9.125)

and a natural number

Assume that T > Q is an integer, Ωt ∈ E(δ), t = 0, . . . , T − 1,

(9.126)

ut ∈ M, t = 0, . . . , T − 1 satisfy ut − v ≤ δ, t = 0, . . . , T − 1,

(9.127)

(ut , Ωt ) ∈ A, t = 0, . . . , T − 1,

(9.128)

324

9 Stability Results for Autonomous Problems

−1 and that an ({Ωt }Tt =0 )-program {xt }Tt=0 satisfies at least one of the conditions (a), (b) with

li = li , i = 1, 2. In order to complete the proof of the theorem it is sufficient to show that Card({t ∈ {0, . . . , T } : xt − x ¯ > }) ≤ Q. Property (ii) and (9.124)–(9.128) imply that there exists S0 ∈ {1, . . . , T − 1} such that ¯ ≤ γ. xS0 − x

(9.129)

We may assume without loss of generality that ¯ ≤ γ } = ∅. {t ∈ {1, . . . , S0 } \ {S0 } : xt − x

(9.130)

We show that S0 ≤ L. Assume the contrary. Then (9.131)

S0 > L. Conditions (a), (b) and (9.128) imply that S 0 −1 t =0

S −1

S

0 0 ut (xt , xt +1 ) ≥ σ ({ut }t =0 , {Ωt }t =0 , x0 , xS0 ) − M.

(9.132)

By the choice of γ and (9.123), γ <  < r¯ /4.

(9.133)

In view of (B3), there exists y˜ ∈ X such that ¯ y˜ ≥ x¯ + r¯ e. (x, ¯ y) ˜ ∈ Ω, y˜ − x ¯ ≤ λ,

(9.134)

Equations (9.5), (9.124), (9.126), and (9.134) imply that ˜ < 2δ. (x, ¯ y) ¯ ∈ ΩS0 −1 , y¯ − y

(9.135)

9.4 Proof of Theorem 9.3

325

It follows from the choice of γ , (9.123), (9.124), (9.129), and (9.133)–(9.135) that y¯ ≥ y˜ − 2δe ≥ x¯ + r¯ e − 2δe ≥ xS0 − γ e + r¯ e − 2δe ≥ xS0 + (¯r − γ − 2δ)e ≥ xS0 .

(9.136)

By (9.135) and (9.136), S −1

0 xS0 ∈ Y¯ ({Ωt }t =S , S0 − 1, S0 ). 0 −1

Conditions (a), (b), property (ii), (9.124), (9.126)–(9.128), and (9.137) imply that there exists

(9.137) (9.131), (9.132),

S ∈ {1, . . . , S0 − 1} such that xS − x ¯ ≤ γ. This contradicts (9.130). The contradiction we have reached proves that S0 ≤ L.

(9.138)

Assume that k ≥ 0 is an integer and that we defined a strictly increasing sequence of integers {Si }ki=0 such that S0 ∈ {1, . . . , L], Sk ≤ T ,

(9.139)

for each integer i = 1, . . . , k, ¯ ≤ γ, xSi − x

(9.140)

1 ≤ Si+1 − Si ≤ L.

(9.141)

and if i ∈ {0, . . . , k} \ {k}, then

(Note that by (9.129) (9.138), our assumption holds for k = 0.) If Sk + L > T , then the construction is completed. Assume that Sk + L ≤ T . In view of (9.140), xSk − x ¯ ≤ γ.

(9.142)

326

9 Stability Results for Autonomous Problems

By (B3), the choice of r¯ and γ , (9.123) and (9.142), there exists η0 ∈ X such that ¯ (xSk , η0 ) ∈ Ω, η0 − x ¯ ≤ λ, η0 ≥ x¯ + r¯ e.

(9.143)

Equations (9.5), (9.126), (9.142), and (9.143) imply that there exists η1 ∈ X such that (xSk , η1 ) ∈ ΩSk , η0 − η1  < 2δ.

(9.144)

It follows from (9.123), (9.124), (9.143), and (9.144) that ¯ η1 ≥ η0 − 2δe ≥ x¯ + r¯ e − 2δe ≥ x.

(9.145)

In view of (9.144) and (9.145), xSk ∈ Y ({ΩSk }, Sk , Sk + 1).

(9.146)

By (9.128), if (a) holds, then T −1 t =Sk

−1 −1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =S , {Ωt }Tt =S , x Sk , x T ) − M k k

(9.147)

and if (b) holds, then T −1 t =Sk

−1 −1 ut (xt , xt +1 ) ≥ σ ({ut }Tt =S , {Ωt }Tt =S , xSk ) − M. k k

(9.148)

Property (ii), conditions (a), (b), (9.126)–(9.128) and (9.146)–(9.148) imply that there exists Sk+1 ∈ {Sk + 1, . . . , T − 1}

(9.149)

¯ ≤ γ. xSk+1 − x

(9.150)

such that

We may assume without loss of generality that ¯ ≤ γ } = ∅. {t ∈ {Sk + 1, . . . , Sk+1 } \ {Sk+1 } : xt − x

(9.151)

9.4 Proof of Theorem 9.3

327

We show that Sk+1 ≤ Sk + L. Assume the contrary. Then Sk+1 > Sk + L.

(9.152)

Equations (9.128), (9.147), and (9.148) imply that Sk+1 −1



t =Sk

S

−1

S

−1

k+1 k+1 ut (xt , xt +1 ) ≥ σ ({ut }t =S , {Ωt }t =S , xSk , xSk+1 −1 ) − M. k k

(9.153)

In view of (B3), there exists y˜ ∈ X satisfying (9.134). Equations (9.5), (9.124), (9.126), and (9.134) imply that there exists yk+1 ∈ X such that (x, ¯ yk+1 ) ∈ ΩSk+1 −1 , yk+1 − y ˜ < 2δ.

(9.154)

It follows from the choice of γ , (9.123), (9.124), (9.134), (9.150), and (9.154) that yk+1 ≥ y˜ − 2δe ≥ x¯ + r¯ e − 2δe ≥ xSk+1 − γ e + r¯ e − 2δe ≥ xSk+1 + (¯r − γ − 2δ)e ≥ xSk+1 .

(9.155)

xSk+1 ∈ Y¯ ({ΩSk+1 −1 }, Sk+1 − 1, Sk+1 ).

(9.156)

By (9.154) and (9.155),

Property (ii), (9.126)–(9.128), (9.146), (9.152), (9.153), and (9.156) imply that there exists S ∈ {Sk + 1, . . . , Sk+1 − 1} such that ¯ ≤ γ. xS − x This contradicts (9.151). The contradiction we have reached proves that Sk+1 ≤ Sk + L.

(9.157)

In view of (9.149), (9.156), and (9.157), the assumption made for k also holds for k + 1. Thus by induction we have constructed the strictly increasing sequence of q integers {Si }i=0 such that S0 ∈ {1, . . . , L], Sq ≤ T , Sq > T − L,

(9.158)

328

9 Stability Results for Autonomous Problems

for each integer i = 0, . . . , q, xSi − x ¯ ≤ γ,

(9.159)

1 ≤ Si+1 − Si ≤ L.

(9.160)

for all i ∈ {0, . . . , q − 1},

Set E = {j ∈ {0, . . . , q − 1} : Sj+1 −1



t =Sj

S

−1

j+1 ut (xt , xt +1 ) < σ ({ut }t =S j

S

−1

j+1 , {Ωt }t =S j

, xSj , xSj+1 −1 ) − γ }.

(9.161)

We show that Card(E) ≤ 4 + γ −1 M. We may assume without loss of generality that Card(E) > 4. S

j+1 By (9.161), for each j ∈ E there exists an ({Ωt }t =S j that

(j )

−1

(j ) S

j+1 )-program {yt }t =S such j

(j )

ySj = xSj , ySj+1 ≥ xSj+1 , Sj+1 −1



t =Sj

(9.162)

Sj+1 −1 (j )

(j )

ut (yt , yt +1 ) ≥



ut (xt , xt +1 ) + γ .

(9.163)

t =Sj

Let E = {j1 , . . . , jp },

(9.164)

where {j1 , . . . , jp } is a finite strictly increasing sequence of integers. Set (j1 )

x˜t = xt , t = 0, . . . , Sj1 , x˜t = yt

, t = Sj1 + 1, . . . , Sj1 +1 .

(9.165)

9.4 Proof of Theorem 9.3

329

Assume that q ∈ {1, . . . , p − 2} and that we have defined x˜t , t = 0, . . . , Sjq +1 Sjq +1

S

q+1 such that (9.165) holds, {x˜t }t =0 is an ({Ωt }t =0 1, . . . , q,

(jk )

x˜t ≥ yt

−1

)-program, for every integer k =

, t = Sjk , . . . , Sjk +1 ,

(9.166)

(jk )

(9.167)

ut (x˜t , x˜t +1 ) ≥ ut (yt

(j )

k , yt +1 ), t = Sjk , . . . , Sjk +1 − 1,

for every k ∈ {1, . . . , q} \ {q}, and every integer j satisfying jk < j < jk+1 , x˜t ≥ xt , t = Sj , . . . , Sj +1 ,

(9.168)

ut (x˜t , x˜t +1) ≥ ut (xt , xt +1 ), t = Sj , . . . , Sj +1 − 1.

(9.169)

(Note that by (9.165), our assumption holds for q = 1.) In view of (9.162) and (9.166), (j )

x˜ Sjq +1 ≥ ySjq +1 ≥ xSjq +1 .

(9.170)

q

Equation (9.170) and (B2) imply that there exist x˜t ∈ X, t ∈ {Sjq +1 , . . . , Sjq+1 } \ {Sjq+1 } Sj

S

q+1 q+1 such that {x˜t }t =0 is an ({Ωt }t =0

−1

)-program,

x(t) ˜ ≥ x(t), t = Sjq +1 , . . . , Sjq+1 ,

(9.171)

ut (x˜t , x˜t +1 ) ≥ ut (xt , xt +1), t = Sjq +1 , . . . , Sjq+1 − 1,

(9.172)

x˜Sjq+1 ≥ xSjq+1 .

(9.173)

Assumption (B2), (9.162) and (9.173) imply that there exist x˜t ∈ X, t ∈ {Sjq+1 + Sj

q+1 1, . . . , Sjq+1 +1 } such that {x˜t }t =0

(jq+1 )

x˜t ≥ yt

+1

Sj

q+1 is an ({Ωt }t =0

+1 −1

)-program,

, t = Sjq+1 , . . . , Sjq+1 +1 , (jq+1 )

ut (x˜t , x˜t +1 ) ≥ ut (yt

(j

)

q+1 , yt +1 ), t = Sjq+1 , . . . , Sjq+1 +1 − 1.

It is clear that the assumption made for q also holds for q + 1. By induction, we Sj

+1

p−1 such that (9.165) holds for every integer k = constructed the program {x˜t }t =0 1, . . . , p − 1, (9.166) and (9.167) are valid, for all integers k = 1, . . . , p − 2 and every integer j satisfying jk < j < jk+1 , (9.168) and (9.169) hold.

330

9 Stability Results for Autonomous Problems

By (9.162), (9.165) and (9.166), x˜0 = x0 , x˜Sjp−1 +1 ≥ xSjp−1 +1 .

(9.174)

Assumption (B2), conditions (a), (b), (9.163), (9.165), (9.167), (9.169) and (9.174) imply that Sjp−1 +1 −1

M≥



Sjp−1 +1 −1



ut (x˜t , x˜t +1 ) −

t =0

≥

≥

t =0

S +1 −1 p−1  jk k=0

Sjk +1 −1

t =Sjk



ut (x˜t , x˜t +1 ) −

t =Sjk

S +1 −1 p−1  jk k=0

ut (xt , xt +1 )  ut (xt , xt +1 )

t =Sjk Sjk +1 −1



(j ) (jk ) ut (yt k , yt +1 )−

 ut (xt , xt +1)

t =Sjk

≥ pγ and p ≤ Mγ −1 . Thus Card(E) ≤ p ≤ 4 + Mγ −1 .

(9.175)

j ∈ {0, . . . , q − 1} \ E.

(9.176)

Let

By (9.124), (9.159), (9.161) and (9.176), Sj+1 −1



t =Sj

S

−1

j+1 ut (xt , xt +1 ) ≥ σ ({ut }t =S j

S

−1

j+1 , {Ωt }t =S j

, xSj , xSj+1 ) − γ .

(9.177)

Property (i), (9.126)–(9.128), (9.160) and (9.177) imply that ¯ ≤ , t = Sj , . . . , Sj +1 . xt − x This inequality implies that {t ∈ {0, . . . , T } : xt − x ¯ > } ⊂ [0, S0 ] ∪ [Sq , T ] ∪ {{Sj , . . . , Sj +1 } : j ∈ E}.

9.5 Proof of Theorem 9.4

331

Together with (9.125), (9.158), (9.168) and (9.175) this implies that Card({t ∈ {0, . . . , T } : xt − x ¯ > }) ≤ 2(L + 1) + Card(E)(2L + 1) ≤ 2(L + 1)(5 + γ −1 M) < Q. Theorem 9.3 is proved.

9.5 Proof of Theorem 9.4 By Lemma 9.12, there exists γ ∈ (0, ) such that the following property holds: (i) For each natural number L there exists γL ∈ (0, γ ) such that for each natural number T ≤ L, each Ωt ∈ E(γL ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ γL , t = 0, . . . , T − 1, (ut , Ωt ) ∈ A, t = 0, . . . , T − 1 −1 and each ({Ωt }Tt =0 )-program {ξt }Tt=0 satisfying

ξ0 − x, ¯ ξT − x ¯ ≤γ and T −1 t =0

−1 −1 ut (ξt , ξt +1 ) ≥ σ ({ut }Tt =0 , {Ωt }Tt =0 , ξ0 , ξT ) − γ

we have ¯ ≤ , t = 0, . . . , T . ξt − x

332

9 Stability Results for Autonomous Problems

By Theorem 9.3, there exist δ0 ∈ (0, ) and a natural number Q0 ≥ l1 + l2 such that the following property holds: (ii) For each integer T > Q0 , each Ωt ∈ E(δ0 ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ0 , t = 0, . . . , T − 1, (ut , Ωt ) ∈ A, t = 0, . . . , T − 1 −1 )-program {xt }Tt=0 for which at least one of the following and each ({Ωt }Tt =0 conditions holds:

(a) −1 −1 x0 ∈ Y ({Ωt }lt1=0 , 0, l1 ), xT ∈ Y¯ ({Ωt }Tt =T −l2 , T − l2 , T ), −1 −1 σ ({ut }Tt =0 , {Ωt }Tt =0 , x0 , xT ) ≤

T −1

ut (xt , xt +1 ) + M;

t =0

(b) −1 , 0, l1 ), x0 ∈ Y ({Ωt }lt1=0 −1 −1 σ ({ut }Tt =0 , {Ωt }Tt =0 , 0, T , x0 ) ≤

T −1

ut (xt , xt +1 ) + M

t =0

the inequality ¯ > γ }) ≤ Q0 Card({t ∈ {0, . . . , T } : xt − x holds. Let γ2Q0 ∈ (0, γ ) be as guaranteed by property (i) with L = 2Q0 . Choose a positive number δ ≤ 8−1 min{γ2Q0 , δ0 }

(9.178)

9.5 Proof of Theorem 9.4

333

and a natural number L > 4 + 4Q0 .

(9.179)

Assume that for each integer T > 2L, Ωt ∈ E(δ), t = 0, . . . , T − 1,

(9.180)

ut ∈ M, t = 0, . . . , T − 1 satisfy ut − v ≤ δ, t = 0, . . . , T − 1,

(9.181)

(ut , Ωt ) ∈ A, t = 0, . . . , T − 1

(9.182)

−1 )-program {xt }Tt=0 satisfies and that an ({Ωt }Tt =0 τ +L−1  t =τ

+L−1 +L−1 ut (xt , xt +1 ) ≥ σ ({ut }τt =τ , {Ωt }τt =τ , xτ , xτ +L ) − δ

(9.183)

for each integer τ ∈ [0, T − L] and for which at least one of conditions (a), (b) holds. Property (ii), conditions (a), (b) and (9.179)–(9.182) imply that ¯ > γ }) ≤ Q0 . Card({t ∈ {0, . . . , T } : xt − x This implies that there exists a finite strictly increasing sequence of integers S0 , . . . , Sq ∈ [0, T ] such that S0 ≤ Q0 , for each i = 0, . . . , q − 1, 1 ≤ Si+1 − Si ≤ Q0 ,

(9.184)

Sq + Q0 > T , xSi − x ¯ ≤ γ , i = 0, . . . , q.

(9.185)

Moreover, if x0 − x ¯ ≤ δ, then we may assume that S0 = 0 and if xT − x ¯ ≤ δ, then we may assume that Sq = T . Let i ∈ {0, . . . , q − 1}. By (9.179) and (9.184), there exists an integer τ ≥ 0 such that [Si , Si+1 ] ⊂ [τ, τ + L].

(9.186)

334

9 Stability Results for Autonomous Problems

Conditions (a), (b), (9.182) and (9.186) imply that Si+1 −1



t =Si

S

−1

S

−1

i+1 i+1 ut (xt , xt +1 ) ≥ σ ({ut }t =S , {Ωt }t =S , xSi , xSi+1 ) − δ. i i

(9.187)

By (9.178), (9.180)–(9.182), (9.185), (9.187), the choice of γ2Q0 and property (i) with L = 2Q0 , xt − x ¯ ≤ , t = Si , . . . , Si+1 , i = 0, . . . , q − 1. Therefore ¯ ≤ , t = S0 , . . . , Sq . xt − x Theorem 9.4 is proved.

9.6 Optimal Control Problems with Discounting It is not difficult to see that the following result holds. 2 −1 Proposition 9.14 Let integers T1 , T2 satisfy 0 ≤ T1 < T2 , {ut }Tt =T ⊂ M be a 1 sequence of upper semicontinuous functions, for any integer t ∈ {T1 , . . . , T2 − 1}, 2 2 −1 be an ({Ωt }Tt =T )let Ωt be a nonempty closed subset of X × X and let {zt }Tt =T 1 1 2 −1 2 )-program {xt }Tt =T such that xT1 = zT1 program. Then there exists an ({Ωt }Tt =T 1 1 and

T 2 −1 t =T1

2 −1 2 −1 ut (xt , xt +1 ) = σ ({ut }Tt =T , {Ωt }Tt =T , zT1 ). 1 1

We prove the following new result which shows the stability of the turnpike phenomenon in the case of discounting. Theorem 9.15 Let  > 0 and let l be a natural number. Then there exist δ ∈ (0, ), a natural number L > l and λ ∈ (0, 1) such that for each integer T > 2L, each Ωt ∈ E(δ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ, t = 0, . . . , T − 1, (ut , Ωt ) ∈ A, t = 0, . . . , T − 1,

9.7 An Auxiliary Result

335

−1 each sequence {αt }Tt =0 ⊂ (0, 1] such that

αi αj−1 ≥ λ for each i, j ∈ {0, . . . , T − 1} satisfying |i − j | ≤ L −1 )-program {xt }Tt=0 which satisfies and each ({Ωt }Tt =0

x0 ∈ Y ({Ωt }l−1 t =0 , 0, l), −1 −1 σ ({αt ut }Tt =0 , {Ωt }Tt =0 , x0 ) =

T −1

αt ut (xt , xt +1 )

t =0

there exist integers τ0 ∈ [0, L] such that xt − x ¯ ≤  for all t = τ0 , . . . , T − L. ¯ ≤ δ, then τ0 = 0. Moreover, if x0 − x −1 ⊂ Roughly speaking, the turnpike property holds if discount coefficients {αt }Tt =0 (0, 1] are changed rather slowly.

9.7 An Auxiliary Result In the proof of Theorem 9.15 we use the following lemma. Lemma 9.16 Let  > 0 and l1 , l2 be natural numbers. Then there exist δ ∈ (0, ), a natural number L > l1 + l2 and λ ∈ (0, 1) such that for each integer T ≥ L, each Ωt ∈ E(δ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ, t = 0, . . . , T − 1, (ut , Ωt ) ∈ A, t = 0, . . . , T − 1, −1 each finite sequence {αt }Tt =0 ⊂ (0, 1] which satisfies

αi αj−1 ≥ λ for each i, j ∈ {0, 1, . . . , T − 1} satisfying |i − j | ≤ L,

336

9 Stability Results for Autonomous Problems

−1 and each ({Ωt }Tt =0 )-program {xt }Tt=0 which satisfies at least one of the following conditions:

(a) −1 , 0, l1 ), x0 ∈ Y ({Ωt }lt1=0 T −1 t =0

−1 −1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , {Ωt }Tt =0 , x0 );

(b) −1 −1 , 0, l1 ), xT ∈ Y¯ ({Ωt }Tt =T x0 ∈ Y ({Ωt }tl1=0 −l2 , T − l2 , T ), T −1 t =0

−1 −1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , {Ωt }Tt =0 , x0 , xT )

there exists S ∈ {1, . . . , T − 1} such that ¯ ≤ . xS − x Proof By Lemma 9.8, there exist a natural number L0 and δ0 ∈ (0, 1) such that the following property holds: (i) For each integer T ∈ [L0 , 2L0 ], each Ωt ∈ E(δ0 ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ0 , t = 0, . . . , T − 1 −1 and each ({Ωt }Tt =0 )-program {xt }Tt=0 satisfying T −1 t =0

ut (xt , xt +1 ) ≥

T −1

ut (x, ¯ x) ¯ −4

t =0

the inequality ¯ : t = 1, . . . , T } ≤  min{xt − x holds.

9.7 An Auxiliary Result

337

By (B1), there exists a positive number ¯ γ < min{λ/2, 1} such that the following property holds: (ii) For every point x ∈ X which satisfies x − x ¯ ≤ γ , x ≥ x, ¯ we have |v(x, ¯ x) − v( ¯ x, ¯ x) ¯ ≤ (16L0 )−1 . By (A5)’ there exist a positive number δ1 < 1 and y¯ ∈ X such that ¯ y) ¯ ∈ Ω and y¯ − x ¯ ≤ γ /2. y¯ ≥ x¯ + δ1 e, (x,

(9.188)

Choose a positive number δ < min{δ0 , δ1 /8, γ /8}(16L0)−1 ,

(9.189)

q > 16(v + 1)(l1 + l2 + L0 + 2)2 ,

(9.190)

natural numbers

L ≥ 4q(L0 + l1 + l2 + 2)(2v + 2)(l1 + l2 + 1) + 4L0 + 2,

(9.191)

and choose λ ∈ [2−1 , 1) such that 4L(1 − λ)λ−1 (v + 1) ≤ 1, λ

q 

λi > 16(1 + v)(L0 + l1 + l2 + 2)2 .

(9.192) (9.193)

i=0

Assume that an integer T ≥ L,

(9.194)

Ωt ∈ E(δ), t = 0, . . . , T − 1,

(9.195)

ut ∈ M, t = 0, . . . , T − 1, ut − v ≤ δ, t = 0, . . . , T − 1,

(9.196)

(ut , Ωt ) ∈ A, t = 0, . . . , T − 1,

(9.197)

338

9 Stability Results for Autonomous Problems

−1 {αt }Tt =0 ⊂ (0, 1] satisfies

αi αj−1 ≥ λ for each i, j ∈ {0, 1, . . . , T − 1} satisfying |i − j | ≤ L,

(9.198)

−1 )-program {xt }Tt=0 satisfies at least one of conditions (a), (b). and an ({Ωt }Tt =0 In order to complete the proof of the lemma it is sufficient to show that there exists S ∈ {1, . . . , T − 1} such that

xS − x ¯ ≤ . Assume the contrary. Then ¯ > , t = 1, . . . , T − 1. xt − x

(9.199)

Let an integer S ∈ [L0 , 2L0 ] and an integer τ ∈ {0, . . . , T − S − 1}. By (9.195)–(9.197) and (9.199), τ +S−1 

ut (xt , xt +1 )
2q + 2.

(9.202)

9.7 An Auxiliary Result

339

It follows from (9.192), (9.200) and (9.201) that for all i = 0, . . . , T L−1 0  − 2, (i+1)L 0−1

αt ut (xt , xt +1 )

t =iL0

≤

(i+1)L 0−1

αt ut (x, ¯ x) ¯ + 2αiL0 (v + 1)(1 − λ)λ−1 L0 − 4αiL0

t =iL0

≤

(i+1)L 0−1

αt ut (x, ¯ x) ¯ − 2αiL0 ,

(9.203)

t =iL0 T −1

αt ut (xt , xt +1 )

t =(T L−1 0 −1)L0

≤

T −1 t =(T L−1 0 −1)L0

αt ut (x, ¯ x) ¯ − 4α(T L−1 −1)L0 0

+ 4α(T L−1 −1)L0 (v + 1)(1 − λ)λ−1 L0 0

≤

T −1 t =(T L−1 0 −1)L0

αt ut (x, ¯ x) ¯ − 2α(T L−1 −1)L0 .

(9.204)

0

Let t ∈ {0, . . . , T }. In view of (9.188), ¯ y¯ ∈ B(x, ¯ γ /2) ⊂ B(x, ¯ λ/4),

(9.205)

(x, ¯ y) ¯ ∈ Ω.

(9.206)

By (9.5), (9.195), (9.205), and (9.206), ρ(y, ¯ {z ∈ X : (x, ¯ z) ∈ Ωt }) ≤ δ. Therefore there exists y¯t ∈ X such that (x, ¯ y¯t ) ∈ Ωt , y¯ − yt  < 2δ.

(9.207)

In view of (9.189), (9.205), and (9.207), ¯ + y¯ − y¯t  < γ /2 + 2δ < 3 · 4−1 γ . x¯ − y¯t  ≤ x¯ − y

(9.208)

340

9 Stability Results for Autonomous Problems

Equations (9.188), (9.189), and (9.207) imply that y¯t ≥ y¯ − 2δe ≥ x¯ + δ1 e − 2δe ≥ x¯ + 2−1 δ1 e.

(9.209)

−1 Conditions (a) and (b) imply that there exists an ({Ωt }tl1=0 )-program {ξt }lt1=0 such that (1)

(1)

ξ0

(1)

= x0 , ξl1 ≥ x. ¯

(9.210)

−1 In the case when condition (b) holds there exists an ({Ωt }Tt =T −l2 )-program (2)

{ξt }Tt=T −l2 such that (2)

(2)

¯ ξT ≥ xT2 . ξT −l2 = x,

(9.211)

In view of (9.197), (9.207), (9.209), (9.210) and (B2), in the case when condition −1 )-program {zt }Tt=0 such that (b) holds there exists an ({Ωt }Tt =0 zt = ξt(1) , t = 0, . . . , l1 ,

(9.212)

for all integers t = l1 + 1, . . . , T − l2 , zt ≥ y¯t ≥ x¯ + 2−1 δ1 e,

(9.213)

and for all integers t = l1 , . . . , T − l2 − 1, ut (zt , zt +1 ) ≥ ut (x, ¯ y¯t +1 ), (2)

zt ≥ ξt , t = T − l2 , . . . , T .

(9.214) (9.215)

−1 In the case when condition (a) holds there exists an ({Ωt }Tt =0 )-program {zt }Tt=0 such that

zt = ξt(1) , t = 0, . . . , l1 ,

(9.216)

and for all integers t = l1 + 1, . . . , T , zt ≥ y¯t ≥ x¯ + 2−1 δ,

(9.217)

9.7 An Auxiliary Result

341

for all integers t = l1 , . . . , T − 1, ut (zt , zt +1 ) ≥ ut (x, ¯ y¯t +1 ).

(9.218)

In both cases T −1

αt ut (zt , zt +1 ) ≤

t =0

T −1

(9.219)

αt ut (xt , xt +1 ).

t =0

By (9.191), (9.196), (9.198), (9.214) and (9.218), T −1

αt ut (zt , zt+1 )

t=0

≥ −l1 α0 (v + 1)(1 − λ)λ−1 +

T −l 2 −1 

αt ut (zt , zt+1 ) − αT (1 − λ)λ−1 l2 (v + 1)

t=l1 T −l 2 −1 

≥

αt ut (x, ¯ y¯t+1 ) − (αT l2 + α0 l1 )(1 − λ)λ−1 (v + 1).

(9.220)

t=l1

Property (ii), (9.189), (9.196), (9.208) and (9.209) imply that for all integers t = l1 , . . . , T − l2 − 1, |ut (x, ¯ yt +1 ) − ut (x, ¯ x)| ¯ ≤ 2δ + |v(x, ¯ yt +1 ) − v(x, ¯ x)| ¯ ≤ 2δ + (16L0 )−1 ≤ (12L0 )−1 .

(9.221)

It follows from (9.196), (9.198), (9.220) and (9.221) that T −1

αt ut (zt , zt +1 )

t =0

≥

T −l 2 −1 

−1

αt ut (x, ¯ x) ¯ − (αT l2 + α0 l1 )(1 − λ)λ

(v + 1) − (12L0 )

−1

t =l1

≥

T −1 t =0

αt ut (x, ¯ x) ¯ − 2(αT l2 + α0 l1 )(1 − λ)λ−1 (v + 1) − (12L0 )−1

T −l 2 −1 

αt

t =l1 T −1

αt .

t =0

(9.222)

342

9 Stability Results for Autonomous Problems

By (9.203), (9.204) and (9.222), T −1

T L−1 0 −2



αt ut (x, ¯ x) ¯ −2

t =0

≥

αiL0 − 2α(T L−1 −1)L0 0

i=0

T −1

αt ut (xt , xt +1 )

t =0

≥

T −1

αt ut (x, ¯ x) ¯ − 2(αT l2 + α0 l1 )(1 − λ)λ−1 (v + 1) − (12L0 )−1

t =0

T −1

αt .

t =0

Together with (9.198) this implies that (12L0 )−1

T −1

αt + 2(αT l2 + α0 l1 )(1 − λ)λ−1 (v + 1)

t =0 T L−1 0 −1

≥2



αiL0 ≥ 2λ(2L0 )

−1

αt

t =0

i=0

≥ λL−1 0

T −1

T −1

αt ≥ (4L0 )−1

t =0

T −1

αt .

t =0

Combined with (9.191), (9.194) and (9.198) this implies that 2(αT l2 + α0 l1 )(1 − λ)λ−1 (v + 1) ≥ (8L0 )−1

T −1

αt

t =0

≥ (8L0 )−1

q−1 

αt +

t =0

T −1

 αt

t =T −q

q−1 q     ≥ (8L0 )−1 α0 λt + αT λt t =0

t =1

≥ 16(v + 1)(l1 + l2 + L0 + 2)2 λ−1 ((8L0 )−1 α0 + (8L0 )−1 (αT )) ≥ 2(v + 1)(l1 + l2 + L0 + 2)2 λ−1 (α0 + αT ), a contradiction. The contradiction we have reached completes the proof of Lemma 9.16.

9.8 Proof of Theorem 9.15

343

9.8 Proof of Theorem 9.15 We may assume without loss of generality that  < r¯ /4. By Lemma 9.12, there exists γ ∈ (0, ) such that the following property holds: (i) For each natural number L there exists γL ∈ (0, γ ) such that for each natural number T ≤ L, each Ωt ∈ E(γL ), t = 0, . . . , T − 1, each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ γL , t = 0, . . . , T − 1, (ut , Ωt ) ∈ A, t = 0, . . . , T − 1 −1 and each ({Ωt }Tt =0 )-program {ξt }Tt=0 satisfying

ξ0 − x, ¯ ξT − x ¯ ≤γ and T −1 t =0

−1 −1 ut (ξt , ξt +1 ) ≥ σ ({ut }Tt =0 , {Ωt }Tt =0 , ξ0 , ξT ) − γ

we have ¯ ≤ , t = 0, . . . , T . ξt − x By Lemma 9.16, there exist δ0 ∈ (0, ), a natural number L1 > 2l + 2 and λ0 ∈ (0, 1) such that the following property holds: (ii) For each integer T ≥ L1 , each Ωt ∈ E(δ0 ), t = 0, . . . , T − 1,

344

9 Stability Results for Autonomous Problems

each ut ∈ M, t = 0, . . . , T − 1 satisfying ut − v ≤ δ0 , t = 0, . . . , T − 1, (ut , Ωt ) ∈ A, t = 0, . . . , T − 1, −1 each finite sequence {αt }Tt =0 ⊂ (0, 1] which satisfies

αi αj−1 ≥ λ0 for each i, j ∈ {0, 1, . . . , T − 1} satisfying |i − j | ≤ L, −1 each natural number l ≤ l and each ({Ωt }Tt =0 )-program {xt }Tt=0 which satisfies at least one of the following conditions:

(a)

−1 , 0, l ), x0 ∈ Y ({Ωt }lt =0 T −1 t =0

−1 −1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , {Ωt }Tt =0 , x0 );

(b)

−1 , 0, l ), xT ∈ Y¯ ({ΩT −1 }, T − 1, T ), x0 ∈ Y ({Ωt }lt =0

T −1 t =0

−1 −1 αt ut (xt , xt +1 ) = σ ({αt ut }Tt =0 , {Ωt }Tt =0 , x0 , xT )

there exists S ∈ {1, . . . , T − 1} such that ¯ ≤ γ. xS − x Let γL1 ∈ (0, γ ) be as guaranteed by property (i) with L = L1 . Choose a positive number δ ≤ min{δ0 , γL1 }(8L1 )−1 ,

(9.223)

L ≥ 4L1 + 4

(9.224)

a natural number

9.8 Proof of Theorem 9.15

345

and choose λ ∈ (λ0 , 1)) such that 8L(1 − λ)λ−1 (L1 + 1)(v + 1) < min{δ0 , γ }.

(9.225)

Assume that an integer T > 2L,

(9.226)

Ωt ∈ E(δ), t = 0, . . . , T − 1,

(9.227)

and ut ∈ M0 , t = 0, . . . , T − 1 satisfy ut − v ≤ δ, t = 0, . . . , T − 1,

(9.228)

(ut , Ωt ) ∈ A, t = 0, . . . , T − 1,

(9.229)

−1 a sequence {αt }Tt =0 ⊂ (0, 1] satisfies

αi αj−1 ≥ λ for each i, j ∈ {0, . . . , T − 1} satisfying |i − j | ≤ L

(9.230)

−1 )-program {xt }Tt=0 satisfies and an ({Ωt }Tt =0

x0 ∈ Y ({Ωt }l−1 t =0 , 0, l),

(9.231)

−1 −1 σ ({αt ut }Tt =0 , {Ωt }Tt =0 , x0 ) =

T −1

αt ut (xt , xt +1 ).

(9.232)

t =0

Property (ii) and (9.223)–(9.232) imply that there exists S1 ∈ {1, . . . , T − 1}

(9.233)

¯ ≤ γ. xS1 − x

(9.234)

t ∈ {0, . . . , T − 1}

(9.235)

¯ ≤ γ. xt − x

(9.236)

such that

Let

and

346

9 Stability Results for Autonomous Problems

In view of the choice of , γ , γ <  < r¯ /4.

(9.237)

By (B3), there exists y˜ ∈ X such that ¯ y˜ ≥ x¯ + r¯ e. ¯ ≤ λ, (x, ¯ y) ˜ ∈ Ωt , y˜ − x

(9.238)

In view of (9.5), (9.227) and (9.238), there exists y¯ ∈ X such that ˜ < 2δ. (x, ¯ y) ¯ ∈ Ωt −1, y¯ − y

(9.239)

It follows from the choice of γ , (9.233), (9.236), (9.238) and (9.239) that y¯ ≥ y˜ − 2δe ≥ x¯ + r¯ e − 2δe ≥ xt − γ e + r¯ e − 2δe ≥ xt + (¯r − γ − 2δ)e ≥ xt .

(9.240)

Equations (9.239) and (9.240) imply that xt ∈ Y¯ ({Ωt −1 }, t − 1, t).

(9.241)

In view of (B3), the choice of γ and (9.236), there exists η ∈ X such that (xt , η) ∈ Ω,

(9.242)

¯ η ≥ x¯ + r¯ e. η − x ¯ ≤ λ,

(9.243)

By (9.5), (9.227), (9.242) and (9.243), there exists η¯ ∈ X such that ¯ ∈ Ωt , η − η ¯ < 2δ. (xt , η) Equations (9.223), (9.243) and (9.244) imply that η¯ ≥ η − 2δe ≥ x¯ + r¯ e − 2δe ≥ x. ¯ Hence xt ∈ Y ({Ωt }, t, t + 1). Therefore we have shown that the following property holds: ¯ ≤ γ , then (iii) If an integer t ∈ {1, . . . , T − 1} satisfies xt − x xt ∈ Y ({Ωt }, t, t + 1) ∩ Y¯ ({Ωt −1 }, t − 1, t).

(9.244)

9.8 Proof of Theorem 9.15

347

We may assume without loss of generality that {t ∈ {1, . . . , S1 } \ {S1 } : xt − x ¯ ≤ γ } = ∅.

(9.245)

We show that S1 ≤ L1 . Assume the contrary. Then S1 > L1 .

(9.246)

In view of (9.229) and (9.232), S 1 −1 t =0

1 −1 1 −1 αt ut (xt , xt +1 ) = σ ({αt ut }St =0 , {Ωt }St =0 , x0 , xS1 ).

(9.247)

Properties (ii) and (iii), (9.226)–(9.231), (9.234), (9.246) and (9.247) imply that there exists S ∈ {1, . . . , S1 − 1} such that ¯ ≤ γ. xS − x This contradicts (9.245). The contradiction we have reached proves that S1 ≤ L1 .

(9.248)

Assume that k ≥ 1 is an integer and that we defined a strictly increasing sequence of integers {Si }ki=1 such that S1 ∈ {1, . . . , L1 },

(9.249)

Sk ≤ T , for each integer i = 1, . . . , k, ¯ ≤ γ, xSi − x

(9.250)

1 ≤ Si+1 − Si ≤ L1 .

(9.251)

if i ∈ {1, . . . , k} \ {k}, then

(Note that by (9.233), (9.234) and (9.248), our assumption holds for k = 1.) If Sk + L1 > T , then the construction is completed. Assume that Sk + L1 ≤ T .

348

9 Stability Results for Autonomous Problems

In view of (9.250), xSk − x ¯ ≤ γ.

(9.252)

By property (iii), (9.249) and (9.252), xSk ∈ Y ({ΩSk }, Sk , Sk + 1).

(9.253)

Equations (9.229) and (9.232) imply that T −1 t =Sk

−1 −1 αt ut (xt , xt +1) = σ ({αt ut }Tt =S , {Ωt }Tt =S , xSk ). k k

(9.254)

Property (ii), (9.227)–(9.230), (9.233), (9.253) and (9.254) imply that there exists Sk+1 ∈ {Sk + 1, . . . , T − 1}

(9.255)

xSk+1 − x ¯ ≤ γ.

(9.256)

such that

We may assume without loss of generality that ¯ ≤ γ } = ∅. {t ∈ {Sk + 1, . . . , Sk+1 } \ {Sk+1 } : xt − x

(9.257)

It follows from (9.229) and (9.232) that Sk+1 −1



t =Sk

S

−1

S

−1

k+1 k+1 αt ut (xt , xt +1 ) = σ ({αt ut }t =S , {Ωt }t =S , xSk , xSk+1 ). k k

(9.258)

We show that Sk+1 ≤ Sk + L1 . Assume the contrary. Then Sk+1 > Sk + L1 .

(9.259)

By (9.253), (9.255) and (9.256), xSk ∈ Y ({ΩSk }, Sk , Sk + 1), xSk+1 ∈ Y¯ ({ΩSk+1 −1 }, Sk+1 − 1, Sk+1 ). Property (ii), (9.227)–(9.230) and (9.258)–(9.260) imply that there exists S ∈ {Sk + 1, . . . , Sk+1 − 1}

(9.260)

9.8 Proof of Theorem 9.15

349

such that xS − x ¯ ≤ γ. This contradicts (9.267). The contradiction we have reached proves that Sk+1 ≤ Sk + L1 . It is clear that the assumption made for k also holds for k + 1. Thus by induction we q have constructed the strictly increasing sequence of integers {Si }i=1 such that S1 ∈ {1, . . . , L1 ], Sq ≤ T , Sq > T − L1 ,

(9.261)

for each integer i = 1, . . . , q, ¯ ≤ γ, xSi − x

(9.262)

1 ≤ Si+1 − Si ≤ L1 .

(9.263)

and for all i ∈ {1, . . . , q} \ {q},

¯ ≤ δ, set S0 = 0. If x0 − x Let i ∈ {0, . . . , q} \ {q} and if x0 − x ¯ ≤ δ, then i ∈ {0, . . . , q − 1}. Equations (9.229) and (9.230) imply that Si+1 −1



S

−1

S

−1

i+1 i+1 αt ut (xt , xt +1 ) = σ ({αt ut }t =S , {Ωt }t =S , xSi , xSi+1 ). i i

t =Si

(9.264)

In view of (9.224), (9.228), (9.230), (9.263) and (9.264), Si+1 −1



Si+1 −1



ut (xt , xt +1 ) ≥

t =Si

≥

t =Si

αS−1 αt ut (xt , xt +1 ) − (v + 1)(Si+1 − Si )(1 − λ)λ−1 i

Si+1 −1 Si+1 −1 σ ({αS−1 αt ut }t =S , {Ωt }t =S , xSi , xSi+1 ) i i i S

−1

S

−1

S

−1

S

−1

− (v + 1)(Si+1 − Si )(1 − λ)λ−1

i+1 i+1 ≥ σ ({ut }t =S , {Ωt }t =S , xSi , xSi+1 ) − 4(v + 1)L1 (1 − λ)λ−1 i i i+1 i+1 ≥ σ ({ut }t =S , {Ωt }t =S , xSi , xSi+1 ) − γ . i i

(9.265)

It follows from property (i), the choice of γL1 , (9.223), (9.227)–(9.230), (9.262), (9.263) and (9.265) that ¯ ≤ , t = Si , . . . , Si+1 . xt − x

350

9 Stability Results for Autonomous Problems

This implies that xt − x ¯ ≤ , t = S1 , . . . , Sq and if x0 − x ¯ ≤ δ, then xt − x ¯ ≤ , t = 0, . . . , Sq Theorem 9.15 is proved.

Chapter 10

Models with Unbounded Endogenous Economic Growth

In this chapter we study optimal control problems related to a model of knowledgebased endogenous economic growth. We establish the existence of trajectories of unbounded economic growth and provide estimates for the growth rate.

10.1 Introduction The origin of endogenous growth literature can be traced back to Arrow’s learningby-doing approach [2], Uzawa’s 2-sector model with an education sector [99] and Shell’s treatment of education investment as consuming income [91]. The recent literature follows [1, 30, 62, 83, 84]. These models establish balanced (exponential) growth without resorting to exogenous processes by imposing particular structures on the underlying economy. In this chapter, which is based on [98], we consider a general economic structure, by imposing minimal constraints, and study the conditions under which unbounded growth is feasible. The underlying technology is loosely specified (requiring it to satisfy very mild conditions) and the growth mechanism considered is that of labor-augmenting human capital (as in [62]) that competes with ordinary capital and consumption for the available resources (output). When unbounded growth is feasible, we provide a lower bound on the long-run rate of growth in terms of properties of the labor-augmenting and production functions. The economy is characterized by a production technology and by a budget constraint. The technology is very loosely defined, imposing minimal structure. The budget constraint is just a material balance requirement, identifying feasible investments (in both types of capital) and consumption actions. The state of the economy at time t (here t is a nonnegative integer) is a triplet (Kt , ht , Lt ), where Kt is capital, ht is human-capital and Lt is labor. Then the production value at moment t + 1 is F (Kt , A(ht )Lt ), where F (·, ·) is a production © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7_10

351

352

10 Models with Unbounded Endogenous Economic Growth

function satisfying F (λx, λy) = λF (x, y) for all x, y, λ ≥ 0 and the function A(·) measures the influence of the human capital. To simplify we assume constant labor Lt = L0 (a standard assumption in the theory of the economic growth), so per capita output takes the form −1 −1 −1 yt = L−1 0 F (Kt , A(ht )L0 ) = F (Kt L0 , A(ht )) = A(ht )f (Kt L0 A(ht ) ), (10.1)

where f (x) = F (x, 1) for every nonnegative real number x. In other words, for every nonnegative real number x, f (x) is the value production at the moment t, when the capital at the moment t is x and the human capital ht and labor Lt satisfy A(ht )Lt = 1. Let v ∈ [0, 1) represent the depreciation rate of K (we assume human capital does not depreciate), so Kt +1 ≥ vKt , ht +1 ≥ ht .

(10.2)

We suppose that the following assumptions on the functions f and A hold. Assumption (A) f : [0, ∞) → [0, ∞) is strictly increasing, continuous, concave, satisfies f (0) = 0 and there exists a positive number x ∗ such that f (x) > (1 − v)x for every x ∈ (0, x ∗ )

(10.3)

f (x) < (1 − v)x for every x > x ∗ ,

(10.4)

and

A : [0, ∞) → [0, ∞) is increasing and A(h) > 0 for every h > 0.

(10.5)

The economy’s budget constraint at time t is Kt +1 − vKt + Ct + L0 (ht +1 − ht ) ≤ F (Kt , A(ht )L0 ), where Ct ≥ 0 denotes aggregate consumption. In terms of the per capita quantities −1 kt = L−1 0 Kt and ct = L0 Ct , the budget constraint can be rendered as kt +1 − vkt + ct + ht +1 − ht ≤ A(ht )f (kt A(ht )−1 ).

(10.6)

10.2 Existence of Unbounded Growth and Balanced Growth Estimates

353

We proceed now to study growth prospects of an economy characterized by the functions f (·) and A(·), satisfying assumption (A). It should be mentioned that the concavity assumption on f is a usual assumption in the economic growth theory.

10.2 Existence of Unbounded Growth and Balanced Growth Estimates Let T1 ≥ 0 and T2 > T1 be integers. The triplet 2 2 2 −1 ({kt }Tt =T , {ht }Tt =T , {ct }Tt =T ) 1 1 1

is called a trajectory if kt ≥ 0, t = T1 , . . . , T2 , ht > 0, t = T1 , . . . , T2 , ct ≥ 0, t = T1 , . . . , T2 − 1 and for every integer t satisfying T1 ≤ t < T2 , the budget constraint (10.6) and kt +1 ≥ vkt , ht +1 ≥ ht

(10.7)

∞ ∞ are satisfied. A triplet ({kt }∞ t =0 , {ht }t =0 , {ct }t =0 ) is called a trajectory if the triplet −1 ({kt }Tt=0 , {ht }Tt=0 , {ct }Tt =0 )

is a trajectory for every integer number T ≥ 1. The question that concerns us is the following: can a technology, characterized by the mild conditions stated in assumption (A), support unbounded growth? Or, more precisely, does there exist a trajectory ∞ ∞ ({kt }∞ t =0 , {ht }t =0 , {ct }t =0 )

such that limt →∞ kt = ∞, limt →∞ ct = ∞ and limt →∞ ht = ∞? We will answer this question in the affirmative and provide bounds for the growth rate. The underlying driving force, of course, is human capital. We thus consider first the case in which human capital is constant, i.e., the classical one-sector model. Clearly, in view of (10.3) and (10.4), f (x ∗ ) = (1 − v)x ∗ .

(10.8)

g(x) = vx + f (x), x ∈ [0, ∞),

(10.9)

Set

g 0 (x) = x, x ∈ [0, ∞)

(10.10)

354

10 Models with Unbounded Endogenous Economic Growth

and g 1 = g, g t +1 = g t ◦ g for all integers t ≥ 0.

(10.11)

The following result easily follows from Eqs. (10.3), (10.4) and (10.8)–(10.11). Proposition 10.1 For every positive real number x, lim g t (x) = x ∗

t →∞

∗ and the sequence {g t (x)}∞ t =0 is strictly decreasing if x > x and strictly increasing ∗ if x ∈ (0, x ).

Suppose that human capital is constant at h0 > 0 and A(h0 ) = 1. For this ∞ submodel, trajectories are pairs of sequences ({kt }∞ t =0 , {ct }t =0 ) such that, for every nonnegative integer t, we have kt ≥ 0, ct ≥ 0, kt +1 ≥ vkt and kt +1 − vkt + ct ≤ f (kt ). Proposition 10.1, then, implies that all trajectories of this submodel are bounded, in that the long-run upper limit of consumption and capital does not exceed x ∗ . We now relax the assumption that h is constant and state the following result. Theorem 10.2 For every pair of positive numbers h0 and k0 , there exists a trajectory ∞ ∞ ({kt }∞ t =0 , {ht }t =0 , {ct }t =0 )

such that lim ht = ∞.

t →∞

Moreover, if limx→∞ A(x) = ∞, then lim kt = ∞ and lim ct = ∞.

t →∞

t →∞

The next result provides lower bounds on the growth rate of capital and consumption under certain conditions.

10.2 Existence of Unbounded Growth and Balanced Growth Estimates

355

Theorem 10.3 Assume that h0 , k0 is a pair of positive real numbers and that  sup

 A(t + 1) : t ∈ [h0 , ∞) < ∞. A(t)

Then there exist an integer q0 ≥ 1, c¯ > 0 and a trajectory ∞ ∞ ({kt }∞ t =0 , {ht }t =0 , {ct }t =0 )

such that for every integer t ≥ q0 , −1 −1 ht ≥ tq0−1 , kt +1 ≥ cA(tq ¯ ¯ 0 ), ct ≥ cA(tq 0 ).

If A(·) is bounded below by a function that increases at the rate α > 0, then the following result can be proved. Corollary 10.4 Let the assumptions of Theorem 10.3 hold, for some α > 0 A(x) ≥ x α for all x > 0 and let a natural number q0 , a positive number c¯ and a trajectory ∞ ∞ ({kt }∞ t =0 , {ht }t =0 , {ct }t =0 )

be as guaranteed by Theorem 10.3. Then for all natural numbers t ≥ q0 , kt +1 ≥ c(tq ¯ 0−1 )α , ct ≥ c(tq ¯ 0−1 )α . Theorems 10.2 and 10.3 follow, respectively, from Theorems 10.6 and 10.5, which are stated below. These theorems, in addition to establishing existence, provide uniform estimates on the growth rate of capital and consumption for all ¯ k0 ≥ A(h ¯ 0 )Δ, where h, ¯ Δ are arbitrary positive numbers. We initial states h0 ≥ h, 1 use the following notation: If x ∈ R , then x = max{i : i is an integer, i ≤ x}. Theorem 10.5 Let h¯ ∈ (0, 1), Λ0 > 1 A(t + 1)A(t)−1 ≤ Λ0 for all t ≥ h¯ and let Δ ∈ (0, 1) satisfy Δ < x ∗ /8, f (Δ), Δ < (8Λ0 )−1 f (x ∗ /8).

356

10 Models with Unbounded Endogenous Economic Growth

Assume that a natural number τ0 > 3 satisfies g τ0 −1 (Δ) > x ∗ /2, an integer q ≥ 1 satisfies ¯ (x ∗ /2) ≥ 1 4−1 qA(h)(f and that ¯ k0 ≥ A(h)Δ. ¯ h0 ≥ h, ∞ ∞ Then there exists a trajectory ({kt }∞ t =0 , {ht }t =0 , {ct }t =0 ) such that for every integer j ≥ 1,

kj qτ0 ≥ A(h¯ + j )Δ, hj qτ0 ≥ h0 + j, ct = 2−1 A(h¯ + j − 1)(f (Δ) − (1 − v)Δ), t = (j − 1)qτ0 , . . . , j qτ0 − 1, kt ≥ 2−1 A(h¯ + j − 1)f (Δ), t = (j − 1)qτ0 + 1, . . . , j qτ0 . Moreover, for every nonnegative integer t, ht ≥ h0 + [t (qτ0 )−1 ], kt +1 ≥ 2−1 A(h¯ + [t (qτ0)−1 ])f (Δ), ct = 2−1 A(h¯ + [t (qτ0 )−1 ])(f (Δ) − (1 − v)Δ) and for every integer t ≥ qτ0 , ht ≥ t (qτ0 )−1 − 1, kt +1 ≥ 2−1 A(t (qτ0)−1 − 1)f (Δ), ct ≥ 2−1 A(t (qτ0)−1 − 1)(f (Δ) − (1 − v)Δ). Theorem 10.6 Let h¯ ∈ (0, 1), Δ ∈ (0, 1), Δ < x ∗ /8, f (Δ), Δ < 8−1 f (x ∗ /8). Assume that ¯ k0 ≥ A(h0 )Δ. h0 ≥ h,

10.3 Auxiliary Results

357

∞ ∞ Then there exists a trajectory ({kt }∞ t =0 , {ht }t =0 , {ct }t =0 ) and a strictly increasing ∞ sequence of integers {Ts }s=0 such that T0 = 0, for every nonnegative integers j ,

hTj ≥ h0 + j, kTj ≥ A(hTj )Δ and for all integers t = Tj , . . . , Tj +1 − 1, ct ≥ 2−1 A(hTj )Δ, kt ≥ 2−1 A(hTj ) min{Δ, f (Δ) − (1 − v)Δ, f (f (Δ)) − (1 − v)f (Δ)}, t = Tj + 1, . . . , Tj +1 .

10.3 Auxiliary Results We use the notation, definitions and assumptions introduced in Sects. 10.1 and 10.2. Assume that h¯ ∈ (0, 1), Δ ∈ (0, 1)

(10.12)

Δ < x ∗ /8, f (Δ), Δ < f (x ∗ /8)/8.

(10.13)

satisfy

Lemma 10.7 Assume that for a nonnegative integer s, ¯ ks ≥ A(h¯ s )Δ hs ≥ h¯ s ≥ h,

(10.14)

and that for every integer t ≥ s, ht = hs , ct = 2−1 A(h¯ s )(f (Δ) − (1 − v)Δ),

(10.15)

kt +1 = vkt + A(hs )f (kt A(hs )−1 ) − ct .

(10.16)

Then for every integer t ≥ s the inequality (kt +1 − vkt )(A(h¯ s ))−1 ≥ 2−1 f (g t −s (Δ))

(10.17)

k1,s = ks , k2,s = ks

(10.18)

is true. Proof Put

358

10 Models with Unbounded Endogenous Economic Growth

and for every integer t ≥ s put k1,t +1 = vk1,t + A(hs )f (k1,t A(hs )−1 ) − 2ct ,

(10.19)

k2,t +1 = vk2,t + A(hs )f (k2,t A(hs )−1 ).

(10.20)

In view of Eqs. (10.5), (10.12), and (10.14) and concavity of the function f , for every nonnegative number x, we have A(hs )f (xA(hs )−1 ) = A(hs )f (xA(h¯ s )−1 (A(h¯ s )A(hs )−1 )) ≥ A(hs )A(h¯ s )A(hs )−1 f (xA(h¯ s )−1 ) = A(h¯ s )f (xA(h¯ s )−1 ).

(10.21)

Equations (10.19)–(10.21) imply that for every integer t ≥ s, we have k1,t +1 ≥ vk1,t + A(h¯ s )f (k1,t A(h¯ s )−1 ) − 2ct ,

(10.22)

k2,t +1 ≥ vk2,t + A(h¯ s )f (k2,t A(h¯ s )−1 ).

(10.23)

Assume that an integer t ≥ s satisfies k1,t ≥ A(h¯ s )Δ.

(10.24)

(Note that in view of Eqs. (10.14), (10.18), and (10.24) holds with t = s.) It follows from (10.15), (10.23) and (10.24) that k1,t +1 ≥ vA(h¯ s )Δ + A(h¯ s )f (Δ) − A(h¯ s )(f (Δ) − (1 − v)Δ) = A(h¯ s )Δ. Therefore Eq. (10.24) is valid for every integer t ≥ s. In view of (10.15), (10.22) and (10.24), for every integer t ≥ s, we have k1,t +1 − vk1,t ≥ A(h¯ s )f (Δ) − A(h¯ s )(f (Δ) − (1 − v)Δ ≥ A(h¯ s )(1 − v)Δ.

(10.25)

k1,t +1 − vk1,t + 2ct ≤ A(hs )f (k1,t A(hs )−1 ).

(10.26)

By (10.19), we have

We show that for every integer t ≥ s the inequality kt ≥ 2−1 (k1,t + k2,t )

(10.27)

is valid. In view of (10.18), Eq. (10.27) is true for t = s. Assume that t ≥ s is an integer and that Eq. (10.27) is valid. In view of Eqs. (10.16), (10.19), (10.20), and (10.27) and monotonicity and concavity of the

10.3 Auxiliary Results

359

function f , we have kt +1 − 2−1 (k1,t +1 + k2,t +1) = vkt + A(hs )f (kt A(hs )−1 ) − ct − 2−1 (vk1,t + vk2,t ) − 2−1 A(hs )f (k1,t A(hs )−1 ) + ct − 2−1 A(hs )f (k2,t A(hs )−1 ) ≥ A(hs )[f (kt A(hs )−1 ) − 2−1 f (k1,t A(hs )−1 ) − 2−1 f (k2,t A(hs )−1 )] ≥ A(hs )[f (2−1 (k1,t + k2,t )A(hs )−1 ) − 2−1 f (k1,t A(hs )−1 ) − 2−1 f (k2,t A(hs )−1 )] ≥ 0. Hence (10.27) is true for every integer t ≥ s. It follows from (10.15), (10.16), (10.21), (10.24), and (10.27) and monotonicity and concavity of the function f that for every integer t ≥ s, we have kt +1 − vkt ≥ A(hs )f (2−1 A(hs )−1 (k1,t + k2,t )) − ct ≥ 2−1 A(hs )f (A(hs )−1 k1,t ) + 2−1 A(hs )f (A(hs )−1 k2,t ) − ct ≥ 2−1 A(h¯ s )f (A(h¯ s )−1 k1,t ) + 2−1 A(h¯ s )f (A(h¯ s )−1 k2,t ) − ct ≥ 2−1 A(h¯ s )f (Δ) − 2−1 A(h¯ s )(f (Δ) − (1 − v)Δ) + 2−1 A(h¯ s )f (A(h¯ s )−1 k2,t ) ≥ 2−1 A(h¯ s )(1 − v)Δ + 2−1 A(h¯ s )f (A(h¯ s )−1 k2,t ).

(10.28)

We show that for every integer t ≥ s Eq. (10.17) is valid. In view of (10.20), for every integer t ≥ s, we have A(h¯ s )−1 k2,t +1 ≥ vA(h¯ s )−1 k2,t + f (A(h¯ s )−1 k2,t ).

(10.29)

We claim that for every integer t ≥ s the inequality A(h¯ s )−1 k2,t ≥ g t −s (Δ)

(10.30)

is true. In view of Eqs. (10.14), (10.18), and (10.30) is true with t = s. Assume that t ≥ s is an integer and that Eq. (10.30) is true. It follows from 10.29) and (10.30) that A(h¯ s )−1 k2,t +1 ≥ vg t −s (Δ) + f (g t −s (Δ)) = g t +1−s (Δ). Hence Eq. (10.30) is valid for every integer t ≥ s. Combined with Eq. (10.28) and monotonicity of the function f this implies that for every integer t ≥ s, we have kt +1 − vkt ≥ 2−1 A(h¯ s )f (g t −s (Δ)).

360

10 Models with Unbounded Endogenous Economic Growth

This completed the proof of Lemma 10.7. Proposition 10.1 implies that there exists an integer τ0 > 3 for which g τ0 −1 (Δ) > x ∗ /2.

(10.31)

Lemma 10.8 Let s be a nonnegative integer, hs ≥ h¯ s ≥ h¯

(10.32)

ks ≥ A(h¯ s )Δ.

(10.33)

and let

s+τ

s+τ

s+τ −1

Then there exists a trajectory ({kt }t =s 0 , {ht }t =s 0 , {ct }t =s 0

) satisfying

ct = 2−1 A(h¯ s )(f (Δ) − (1 − v)Δ), t = s, . . . , s + τ0 − 1, ht = hs , t = s, . . . , s + τ0 − 1, hs+τ0 = hs + 4−1 A(h¯ s )f (x ∗ /2), kt ≥ 2−1 A(h¯ s )f (Δ), t = s + 1, . . . , s + τ0 − 1, ks+τ0 ≥ 4−1 A(h¯ s )f (x ∗ /2). Proof Lemma 10.7 and Eq. (10.31) imply that there exists a trajectory s+τ s+τ s+τ −1 ({k˜t }t =s 0 , {h˜ t }t =s 0 , {c˜t }t =s 0 )

for which k˜s = ks ,

(10.34)

h˜ t = hs , t = s, . . . , s + τ0 ,

(10.35)

c˜t = 2−1 A(h¯ s )(f (Δ) − (1 − v)Δ), t = s, . . . , s + τ0 − 1,

(10.36)

k˜t ≥ 2−1 A(h¯ s )f (Δ), t = s + 1, . . . , s + τ0 ,

(10.37)

k˜s+τ0 − v k˜s+τ0 −1 ≥ 2−1 A(h¯ s )f (g τ0 −1 (Δ)) ≥ 2−1 A(h¯ s )f (x ∗ /2).

(10.38)

By Eq. (10.35), we have k˜s+τ0 − v k˜s+τ0 −1 + c˜s+τ0 −1 ≤ A(h˜ s+τ0 −1 )f (k˜s+τ0 −1 A(hs )−1 ).

(10.39)

10.3 Auxiliary Results

361

Define kt = k˜t , t = s, . . . , s + τ0 − 1, ct = c˜t , t = s, . . . , s + τ0 − 1,

(10.40)

ht = h˜ t = hs , t = s, . . . , s + τ0 − 1,

(10.41)

ks+τ0 = vks+τ0 −1 + 4−1 A(h¯ s )f (x ∗ /2),

(10.42)

hs+τ0 = hs + 4−1 A(h¯ s )f (x ∗ /2).

(10.43)

s+τ0 s+τ0 −1 0 We show that the sequence ({kt }s+τ ) is a trajectory. t =s , {ht }t =s , {ct }t =s In order to meet this goal it is sufficient to show that

ks+τ0 − vks+τ0 −1 + cs+τ0 −1 + hs+τ0 − hs+τ0 −1 ≤ A(hs+τ0 −1 )f (ks+τ0 −1 A(hs+τ0 −1 )−1 ).

(10.44)

In view of Eqs. (10.35) and (10.38)–(10.43), we have ks+τ0 − vks+τ0 −1 + cs+τ0 −1 + hs+τ0 − hs+τ0 −1 = 2−1 A(h¯ s )f (x ∗ /2) + c˜s+τ0 −1 ≤ k˜s+τ0 − v k˜s+τ0 −1 + c˜s+τ0 −1 ≤ A(h˜ s+τ0 −1 )f (k˜s+τ0 −1 A(h˜ s+τ0 −1 )−1 ) = A(hs+τ0 −1 )f (ks+τ0 −1 A(hs+τ0 −1 )−1 ). Hence Eq. (10.44) is true. Lemma 10.8 is proved. Fix a natural number q > 1 for which ¯ (x ∗ /2) ≥ 1. 4−1 qA(h)f

(10.45)

Lemma 10.9 Let p be a nonnegative integer, ¯ hp ≥ h¯ p ≥ h, kp ≥ A(h¯ p )Δ. p+qτ

p+qτ

(10.46) p+qτ0 −1

Then there exists a trajectory ({kt }t =p 0 , {ht }t =p 0 , {ct }t =p

) which satisfies

ct = 2−1 A(h¯ p )(f (Δ) − (1 − v)Δ), t = p, . . . , p + qτ0 − 1,

(10.47)

hp+qτ0 ≥ hp + 1,

(10.48)

kp+j τ0 ≥ 4−1 A(h¯ p )f (x ∗ /2), j = 1, . . . , q,

(10.49)

kt ≥ 2

−1

A(h¯ p )f (Δ), t = p + 1, . . . , p + qτ0 .

(10.50)

362

10 Models with Unbounded Endogenous Economic Growth

Proof Applying Lemma 10.8 with s = p, we obtain a trajectory p+τ

p+τ

p+τ −1

({kt }t =p 0 , {ht }t =p 0 , {ct }t =p 0

)

which satisfies ct = 2−1 A(h¯ p )(f (Δ) − (1 − v)Δ)

(10.51)

for every integer t = p, . . . , p + τ0 − 1, hp+τ0 = hp + 4−1 A(h¯ p )f (x ∗ /2),

(10.52)

kt ≥ 2−1 A(h¯ p )f (Δ), t = p + 1, . . . , p + τ0 − 1,

(10.53)

kp+τ0 ≥ 4−1 A(h¯ p )f (x ∗ /2).

(10.54)

Assume that j < q is a natural number and that we defined a trajectory p+j τ

p+j τ

p+j τ0 −1

({kt }t =p 0 , {ht }t =p 0 , {ct }t =p

)

such that Eq. (10.51) is valid for all integers t = p, . . . , p + j τ0 − 1, Eq. (10.53) is true for all integers t = p + 1 . . . , p + j τ0 − 1 and that hp+j τ0 = hp + 4−1 j A(h¯ p )f (x ∗ /2),

(10.55)

kp+j τ0 ≥ 4−1 A(h¯ p )f (x ∗ /2).

(10.56)

(By Eqs. (10.51)–(10.54) this assumption holds with j = 1.) Define h¯ p+j τ0 = h¯ p .

(10.57)

In view of (10.46) and (10.55), we have ¯ hp+j τ0 ≥ h¯ p+j τ0 ≥ h.

(10.58)

By Eqs. (10.13), (10.56) and (10.57), we have kp+j τ0 ≥ A(h¯ p+j τ0 )Δ.

(10.59)

Using (10.57)–(10.59), we apply Lemma 10.8 with s = p + j τ0 and obtain p+(j +1)τ p+(j +1)τ p+(j +1)τ −1 a trajectory ({kt }t =p+j τ0 0 , {ht }t =p+j τ0 0 , {ct }t =p+j τ0 0 ) such that Eq. (10.51) is valid for all integers t = p + j τ0 , . . . , p + (j + 1)τ0 − 1, Eq. (10.53) is true for all

10.3 Auxiliary Results

363

integers t = p + j τ0 , . . . , p + (j + 1)τ0 − 1 and that kp+(j +1)τ0 ≥ 4−1 A(h¯ p )f (x ∗ /2), hp+(j +1)τ0 = hp+j τ0 + 4−1 A(h¯ p )f (x ∗ /2) = hp + 4−1 (j + 1)A(h¯ p )f (x ∗ /2). (The last equality follows from Eq. (10.55).) Evidently, the assumption made for j also holds for j + 1. Therefore by induction we constructed the trajectory p+qτ

p+qτ

p+qτ0 −1

({kt }t =p 0 , {ht }t =p 0 , {ct }t =p

)

such that Eqs. (10.47), (10.49) and (10.50) are valid and that hp+qτ0 = hp + 4−1 qA(h¯ p )f (x ∗ /2) ≥ hp + 1. (The last inequality follows from Eq. (10.45).) This completes the proof of Lemma 10.9. Lemma 10.10 Assume that Λ0 > 1, ¯ A(t + 1)A(t)−1 ≤ Λ0 for all t ≥ h,

(10.60)

0 < Δ < (4Λ0 )−1 f (x ∗ /2).

(10.61)

Let p be a nonnegative integer, ¯ kp ≥ A(h¯ p )Δ hp ≥ h¯ p ≥ h, p+qτ

p+qτ

p+qτ0 −1

and let a trajectory ({kt }t =p 0 , {ht }t =p 0 , {ct }t =p Lemma 10.9. Then

(10.62) ) be as guaranteed by

kp+qτ0 ≥ A(h¯ p + 1)Δ. Proof In view of Lemma 10.9, (10.49) and (10.60)–(10.62), we have kp+qτ0 ≥ 4−1 A(h¯ p )f (x ∗ /2) = A(h¯ p + 1)Δ(f (x ∗ /2)Δ−1 )4−1 A(h¯ p )A(h¯ p + 1)−1 ¯ ≥ A(h¯ p + 1)Δ(4−1 f (x ∗ /2)Δ−1 Λ−1 0 ) ≥ A(hp + 1)Δ. This completes the proof of Lemma 10.10.

364

10 Models with Unbounded Endogenous Economic Growth

Lemma 10.11 Assume that p, is a nonnegative integer, ¯ hp ≥ h¯ p ≥ h,

(10.63)

kp ≥ A(h¯ p )Δ.

(10.64)

Then there exist an integer τ ≥ p + qτ0 and a trajectory −1 ({kt }τt=p , {ht }τt=p , {ct }τt =p )

such that hτ ≥ hp + 1, kτ ≥ A(hτ )Δ, for all integers t = p, . . . , τ − 1, ct ≥ 2−1 A(h¯ p ) min{f (Δ) − (1 − v)Δ, f (f (Δ)) − (1 − v)f (Δ)} and for all integers t = p + 1, . . . , τ, kt ≥ 2−1 A(h¯ p ) min{Δ, f (Δ)}. Proof In view of Lemma 10.9, (10.63) and (10.64), there exists a trajectory p+qτ

p+qτ

p+qτ0 −1

({kt }t =p 0 , {ht }t =p 0 , {ct }t =p

)

which satisfies ct = 2−1 A(h¯ p )(f (Δ) − (1 − v)Δ), t = p, . . . , p + qτ0 − 1,

(10.65)

hp+qτ0 ≥ hp + 1,

(10.66)

kp+qτ0 ≥ 4

−1

A(h¯ p )f (x /2), ∗

kt ≥ 2−1 A(h¯ p )f (Δ), t = p + 1, . . . , p + qτ0 .

(10.67) (10.68)

Define k1,p+qτ0 = kp+qτ0 , k2,p+qτ0 = kp+qτ0

(10.69)

and for every integer t ≥ p + qτ0 define ht = hp+qτ0 ,

(10.70)

ct = 2−1 A(h¯ p )(f (f (Δ)) − (1 − v)f (Δ)),

(10.71)

10.3 Auxiliary Results

365

k1,t +1 = vk1,t + A(hp+qτ0 )f (k1,t A(hp+qτ0 )−1 ) − 2ct ,

(10.72)

k2,t +1 = vk2,t + A(hp+qτ0 )f (k2,t A(hp+qτ0 )−1 ),

(10.73)

kt +1 = vkt + A(hp+qτ0 )f (kt A(hp+qτ0 )−1 ) − ct .

(10.74)

In view of Eqs. (10.63) and (10.66), concavity of the function f and monotonicity of the function A for every nonnegative number x, we have A(hp+qτ0 )f (xA(hp+qτ0 )−1 ) = A(hp+qτ0 )f (xA(h¯ p )−1 (A(h¯ p )A(hp+qτ0 )−1 )) ≥ A(h¯ p )f (xA(h¯ p )−1 ).

(10.75)

It follows from (10.72), (10.73), and (10.75) that for all integers t ≥ p + qτ0 , we have k1,t +1 ≥ vk1,t + A(h¯ p )f (k1,t A(h¯ p )−1 ) − 2ct ,

(10.76)

k2,t +1 ≥ vk2,t + A(h¯ p )f (k2,t A(h¯ p )−1 ).

(10.77)

We show that for every integer t ≥ p + qτ0 , we have k1,t ≥ A(h¯ p )f (Δ).

(10.78)

In view of Eqs. (10.67), (10.13), and (10.78) is valid with t = p + qτ0 . Assume that t ≥ p + qτ0 is an integer and that Eq. (10.78) is valid. It follows from (10.65), (10.71), (10.76), (10.78) and monotonicity of the function f that k1,t +1 ≥ vA(h¯ p )f (Δ) + A(h¯ p )f (f (Δ)) − A(h¯ p )(f (f (Δ)) − (1 − v)f (Δ)) = A(h¯ p )f (Δ). Hence Eq. (10.78) is true for every integer t ≥ p + qτ0 . In view of (10.71), (10.76) and (10.78) for every integer t ≥ p + qτ0 , k1,t +1 − vk1,t ≥ A(h¯ p )f (f (Δ)) − A(h¯ p )(f (f (Δ)) − (1 − v)f (Δ)) = A(h¯ p )(1 − v)f (Δ).

(10.79)

By (10.72), for every integer t ≥ p + qτ0 , we have k1,t +1 − vk1,t + 2ct ≤ A(hp+qτ0 )f (k1,t A(hp+qτ0 )−1 ).

(10.80)

We show that for every integer t ≥ p + qτ0 , we have kt ≥ 2−1 (k1,t + k2,t ). In view of (10.69) and (10.81) is valid for t = p + qτ0 .

(10.81)

366

10 Models with Unbounded Endogenous Economic Growth

Assume that t ≥ p + qτ0 is an integer and that Eq. (10.81) is true. In view of (10.72)–(10.74), (10.81) and monotonicity and concavity of the function f , kt +1 − 2−1 (k1,t +1 + k2,t +1) = vkt + A(hp+qτ0 )f (kt A(hp+qτ0 )−1 ) − ct − 2−1 (vk1,t + vk2,t ) − 2−1 A(hp+qτ0 )(f (k1,t A(hp+qτ0 )−1 ) + f (k2,t A(hp+qτ0 )−1 ) + ct ≥ A(hp+qτ0 )[f (kt A(hp+qτ0 )−1 ) − 2−1 f (k1,t A(hp+qτ0 )−1 ) − 2−1 f (k2,t A(hp+qτ0 )−1 )] ≥ 0. Hence Eq. (10.81) is true for every integer t ≥ p + qτ0 . Equations (10.71), (10.74), (10.75), (10.78), and (10.81) and concavity of the function f imply that for every integer t ≥ p + qτ0 , we have kt +1 − vkt ≥ A(hp+qτ0 )f (A(hp+qτ0 )−1 kt ) − ct ≥ 2−1 A(hp+qτ0 )f (A(hp+qτ0 )−1 k1,t ) + 2−1 A(hp+qτ0 )f (A(hp+qτ0 )−1 k2,t ) − ct ≥ 2−1 A(h¯ p )f (A(h¯ p )−1 k1,t ) + 2−1 A(h¯ p )f (A(h¯ p )−1 k2,t ) − ct ≥ 2−1 A(h¯ p )f (f (Δ)) − 2−1 A(h¯ p )(f (f (Δ)) − (1 − v)f (Δ)) + 2−1 A(h¯ p )f (A(h¯ p )−1 k2,t ) ≥ 2−1 (1 − v)f (Δ)A(h¯ p ) + 2−1 A(h¯ p )f (A(h¯ p )−1 k2,t ).

(10.82)

In view of Eq. (10.73), for every integer t ≥ p + qτ0 , we have A(hp+qτ0 )−1 k2,t +1 = vA(hp+qτ0 )−1 k2,t + f (A(hp+qτ0 )−1 k2,t ).

(10.83)

Proposition 10.1, (10.67), (10.69) and (10.83) imply that there exists an integer τ > p + qτ0 for which k2,τ > 2−1 x ∗ A(hp+qτ0 ).

(10.84)

Now it is easy to see that Eqs. (10.67), (10.71), (10.74) and (10.82) imply that −1 ({kt }τt=p , {ht }τt=p , {ct }τt =p )

is a trajectory. In view of (10.70), (10.81) and (10.84), we have kτ > 2−1 k2,τ ≥ 4−1 x ∗ A(hp+qτ0 ) ≥ ΔA(hτ ). This completes the proof of Lemma 10.11.

10.4 Proof of Theorem 10.5

367

10.4 Proof of Theorem 10.5 Let ¯ h¯ 0 = h.

(10.85)

In view of (10.85) and Lemmas 10.9 and 10.10 applied with p = 0, there exists qτ0 qτ0 qτ0 −1 a trajectory ({kt }t =0 , {ht }t =0 , {ct }t =0 ) which satisfies ¯ (Δ) − (1 − v)Δ), t = 0, . . . , qτ0 − 1, ct = 2−1 A(h)(f

(10.86)

hqτ0 ≥ h0 + 1,

(10.87)

kqτ0 ≥ A(h¯ + 1)Δ,

(10.88)

¯ (Δ), t = 1, . . . , qτ0 . kt ≥ 2−1 A(h)f

(10.89)

Assume that s is a natural number and we constructed a trajectory sqτ

sqτ

sqτ −1

({kt }t =00 , {ht }t =00 , {ct }t =00

)

such that for every integer j = 1, . . . , s, we have kj qτ0 ≥ A(h¯ + j )Δ, hj qτ0 ≥ h0 + j,

(10.90)

ct = 2−1 A(h¯ + j − 1)(f (Δ) − (1 − v)Δ), t = (j − 1)qτ0 , . . . , j qτ0 − 1, (10.91) kt ≥ 2−1 A(h¯ + j − 1)f (Δ), t = (j − 1)qτ0 + 1, . . . , j qτ0 .

(10.92)

(By Eqs. (10.86)–(10.89) our assumptions hold for j = 1.) Let p = sqτ0 , h¯ sqτ0 = h¯ p = h¯ + s.

(10.93)

¯ we have By (10.90), (10.93) and the inequality h0 ≥ h, hp = hsqτ0 ≥ h0 + s ≥ h¯ sqτ0 = h¯ p .

(10.94)

Equations (10.90) and (10.93) imply that kp = ksqτ0 ≥ A(h¯ + s)Δ = A(h¯ p )Δ.

(10.95)

368

10 Models with Unbounded Endogenous Economic Growth

Applying Lemmas 10.9 and 10.10 and using Eqs. (10.93), (10.94) and (10.95) we obtain that there exists a trajectory (s+1)qτ

(s+1)qτ

(s+1)qτ0 −1

({kt }t =sqτ0 0 , {ht }t =sqτ0 0 , {ct }t =sqτ0

)

which satisfies h(s+1)qτ0 ≥ hsqτ0 + 1 ≥ h0 + s + 1, k(s+1)qτ0 ≥ A(h¯ p + 1)Δ = A(h¯ + s + 1)Δ, ct = 2−1 A(h¯ + s)(f (Δ) − (1 − v)Δ), t = sqτ0 , . . . , (s + 1)qτ0 − 1, kt ≥ 2−1 A(h¯ + s)f (Δ), t = sqτ0 + 1, . . . , (s + 1)qτ0 . By the equations above, the assumption made for s also holds for s + 1. Thus by ∞ ∞ induction we constructed a trajectory ({kt }∞ t =0 , {ht }t =0 , {ct }t =0 ) such that for every integer j ≥ 1, Eqs. (10.90)–(10.92) are true. Assume that t is a nonnegative integer. There exists an integer j ≥ 1 such that (j − 1)qτ0 ≤ t < j qτ0 . Clearly, j − 1 = [tq −1 τ0−1 ], j = [tq −1 τ0−1 ] + 1.

(10.96)

In view of (10.90) and (10.96), we have ht ≥ h(j −1)qτ0 ≥ h0 + (j − 1) = h0 + [tq −1 τ0−1 ].

(10.97)

By (10.92) and (10.96), kt +1 ≥ 2−1 A(h¯ + j − 1)f (Δ) = 2−1 A(h¯ + [tq −1 τ0−1 ])f (Δ). It follows from (10.91) and (10.96) that −1 −1 ¯ ¯ −1)(f (Δ)−(1−v)Δ) = 2−1 A(h+[tq τ0 ])(f (Δ)−(1−v)Δ). ct = 2−1 A(h+j

This completes the proof of Theorem 10.5.

10.5 Proof of Theorem 10.6 Set Δ0 = min{Δ, f (Δ) − (1 − v)Δ, f (f (Δ)) − (1 − v)f (Δ)}.

10.5 Proof of Theorem 10.6

369

Let h¯ 0 = h0 , T0 = 0.

(10.98)

In view of Lemma 10.11 with p = 0 there exist a natural number T1 > T0 and a trajectory 1 1 1 −1 , {ht }Tt =0 , {ct }Tt =0 ) ({kt }Tt =0

which satisfies hT1 ≥ hT0 + 1 = h0 + 1, kT1 ≥ A(hT1 )Δ,

(10.99)

for all integers t = 0, . . . , T (1) − 1, ct ≥ 2−1 A(hT0 )Δ0 ,

(10.100)

and for all integers t = T0 + 1, . . . , T1 , kt ≥ 2−1 A(hT0 )Δ0 .

(10.101)

Assume that s ≥ 1 is an integer and that we constructed a strictly increasing sequence of integers {Tj }sj =0 such that T0 = 0 and a trajectory s s s −1 ({kt }Tt =0 , {ht }Tt =0 , {ct }Tt =0 )

such that for every integer j = 0, . . . , s, hTj ≥ h0 + j, kTj ≥ A(hTj )Δ,

(10.102)

for every integer j = 0, . . . , s − 1, ct ≥ 2−1 A(hTj )Δ0 , t = Tj , . . . , Tj +1 − 1

(10.103)

kt ≥ 2−1 A(hTj )Δ0 , t = Tj + 1, . . . , Tj +1 .

(10.104)

and

(In view of Eqs. (10.99) and (10.100) our assumption hold for s = 1.) Using (10.102) and applying Lemma 10.11 with p = Ts , h¯ p = hTs we obtain that there Ts+1 Ts+1 Ts+1 −1 exist a natural number Ts+1 > Ts and a trajectory ({kt }t =T , {ht }t =T , {ct }t =T ) s s s satisfying hTs+1 ≥ hTs + 1, kTs+1 ≥ A(hTs+1 )Δ,

370

10 Models with Unbounded Endogenous Economic Growth

for all integers t = Ts , . . . , Ts+1 − 1, ct ≥ 2−1 A(hTs )Δ0 , and for all integers t = Ts + 1, . . . , Ts+1 , kt ≥ 2−1 A(hTs )Δ0 . It is not difficult to see that the assumption made for s also holds for s +1. Therefore by induction we constructed the trajectory and Theorem 10.6 is proved.

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© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7

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Index

A Admissible program, 199 Admissible trajectory, 1 Agreeable program, 231 Approximate solution, 1, 3 Asymptotic turnpike property, 2, 6, 42 Autonomous optimal control problem, 35 Average turnpike property, 22

B Bad program, 21 Baire category, 3, 75

C Cardinality of a set, 16 Compact metric space, 1, 5 Complete metric space, 38 Concave function, 20 Constraint map, 36 Convex set, 2

D Differentiable function, 20 Discrete-time problem, 1 Dynamic optimization problem, 1, 35

E Euclidean space, 2, 17

G Golden-rule stock, 21, 27 Good program, 2, 15, 21, 42 Gross investment sequence, 18

H Hausdorff metric, 35

I Increasing function, 20 Infinite horizon problem, 14, 39 Inner product, 17 Interior point, 5

L Lower semicontinuous function, 37

M Metric space, 1 Minimal program, 43

N Norm, 7

O Objective function, 2 Overtaking optimal program, 6, 24, 43

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 A. J. Zaslavski, Optimal Control Problems Arising in Mathematical Economics, Monographs in Mathematical Economics 5, https://doi.org/10.1007/978-981-16-9298-7

377

378 P Program, 1, 8, 19 S Scalar product, 7 Set-valued mapping, 35, 225 Strict contraction, 35 Strictly concave function, 2 T Trajectory, 1 Turnpike, 2 Turnpike phenomenon, 2 Turnpike property, 2, 11, 40

Index U Uniform equicontinuity, 175 Upper semicontinuous function, 1, 5 Utility function, 30

V von Neumann facet, 24

W Weak controllability property, 35 Weakly maximal program, 230 Weakly optimal program, 21