Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model (Monographs in Mathematical Economics, 4) 9811622515, 9789811622519

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Monographs in Mathematical Economics 4

Alexander J. Zaslavski

Optimal Control Problems Related to the Robinson–Solow– Srinivasan Model

Monographs in Mathematical Economics Volume 4

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

More information about this series at http://www.springer.com/series/13278

Alexander J. Zaslavski

Optimal Control Problems Related to the Robinson– Solow–Srinivasan Model

123

Alexander J. Zaslavski Department of Mathematics Technion - Israel Institute of Technology Haifa, Israel

ISSN 2364-8279 ISSN 2364-8287 (electronic) Monographs in Mathematical Economics ISBN 978-981-16-2251-9 ISBN 978-981-16-2252-6 (eBook) https://doi.org/10.1007/978-981-16-2252-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 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

The growing importance of the turnpike theory and infinite horizon optimal control has been recognized in recent years. This is due not only to impressive theoretical developments, but also because of numerous applications to engineering, economics, life sciences, etc. This book is devoted to the study of large classes of discrete-time optimal control problems arising in mathematical economics, related to the Robinson–Solow–Srinivasan (RSS) model. In the 1960s, this model was introduced by Robinson, Solow and Srinivasan and was studied by Robinson, Okishio and Stiglitz. In 2005, it was revisited by Ali Khan and Mitra and is still an important and interesting topic in mathematical economics. Many turnpike results on the RSS model are collected in [126]. In the RSS model, an economy produces a finite number n of alternative types of machines. For every i ¼ 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) input-coefficients vector, a ¼ ða1 ; . . .; an Þ, and an output-coefficients vector, b ¼ ðb1 ; . . .; bn Þ. We assume that all machines depreciate at a rate d 2 ð0; 1Þ. For each nonnegative integer t, let xðtÞ ¼ ðx1 ðtÞ; . . .; xn ðtÞÞ  0 denote the amounts of the n types of machines that are available in time-period t, and let zðt þ 1Þ ¼ ðz1 ðt þ 1Þ; . . .; zn ðt þ 1ÞÞ  0 be the gross investments in the n types of machines during period t þ 1. Hence, zðt þ 1Þ ¼ ðxðt þ 1Þ  xðtÞÞ þ dxðtÞ, the sum of net investment and of depreciation. Let yðtÞ ¼ ðy1 ðtÞ; . . .; yn ðtÞÞ be the amounts of the n types of machines used for the production of the consumption good, byðtÞ, during period t þ 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, zðt þ 1Þ representing the production of new machines of the various types, requires azðt þ 1Þ units of labor in period t. Also yðtÞ representing the use of available machines for the manufacture of the consumption good requires eyðtÞ units of labor in period t, where e is an n-dimensional vector all of whose coordinates are unity. Thus, the availability of labor constrains employment in the consumption and investment sectors which is described by azðt þ 1Þ þ eyðtÞ  1. The preferences of the planner v

vi

Preface

are represented by a continuous strictly increasing concave and differentiable function w : ½0; 1Þ ! R1 . A formal description of this technological structure is given in Sect. 1.5. In our books [121, 123], we study a class of discrete-time optimal control problems which describe many models of economic dynamics except for the RSS model. This happens because some assumptions posed in [121, 123], which are true for many models of economic dynamics, do not hold for the RSS model. Namely, for many models of economic dynamics, the turnpike is a singleton and a local controllability property holds in a neighborhood of the turnpike. For the RSS model, the turnpike is a singleton too, but the local controllability property does not hold. This makes the situation more difficult and less understood. Nevertheless, it was shown in [126] that the turnpike theory presented in [121] is extended for the RSS model. This is possible because the model possesses some interesting and important features. Namely, its technological map is a strict contraction and has the so-called monotonicity property. These two properties play a crucial role in the analysis of the turnpike phenomenon for the RSS model. This understanding leads us to the idea of developing a turnpike theory for large classes of optimal control problems possessing the strict contraction property and the monotonicity property which includes the RSS model as a particular case. This theory is developed in this book. The turnpike theory presented in [126] was developed for the original RSS model, which is a simple growth model with linear technologies and concave utilities. Most of the turnpike results of [126] were obtained for stationary models. For non-stationary models, the turnpike phenomenon was investigated there only for one-dimensional models. In the present book, we extend the turnpike theory to large classes of models with nonlinear technologies and nonconcave utilities, studying both autonomous and nonautonomous multi-dimensional cases. Our present book also contains a chapter on the Robinson–Shinkai–Leontief model and a chapter on discrete dispersive dynamical systems, which are not considered in [126]. In Chap. 1, we discuss turnpike properties for a large class of discrete-time optimal control problems studied in [121] and for the the RSS model. In Chap. 2, we study infinite horizon optimal control problems with nonautonomous optimality criteria. The utility functions, which determine the optimality criterion, are nonconcave. The class of models contains, as a particular case, the RSS model. We establish the existence of good programs and optimal programs. Chapter 3 is devoted to the analysis of the one-dimensional nonautonomous concave RSS model. We show the stability of the turnpike phenomenon under small perturbations of objective functions and study the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals. In Chap. 4, we consider a subclass of the class of discrete-time optimal problems studied in Chap. 2. Problems belonging to this subclass are autonomous, multi-dimensional and nonconcave. We obtain turnpike conditions and establish the stability of the turnpike phenomenon. Chapter 5 contains the study of the class of

Preface

vii

nonautonomous discrete-time optimal problems analyzed in Chap. 2. We obtain turnpike conditions and establish the stability of the turnpike phenomenon. In Chap. 6, we continue to study the class of one-dimensional nonautonomous discrete-time optimal problems, which determine the one-dimensional RSS model. This class of problems is identified with a complete space of sequences. We show the existence of a set in this space which is a countable intersection of open and everywhere dense sets such that for every element the corresponding model has the turnpike property. In Chap. 7, we discuss the Robinson–Shinkai–Leontief model. We are interested in the turnpike phenomenon and in the existence of solutions to the corresponding infinite horizon problems. Chapter 8 contains turnpike results for a discrete dispersive dynamical system generated by set-valued mappings which were introduced by A. M. Rubinov in 1980. This dispersive dynamical system has a prototype in mathematical economics. The main results of Chaps. 3–6 are new. The author believes that this book will be useful to researchers interested in the turnpike theory, infinite horizon optimal control and their applications. Rishon LeTsiyon, Israel October 2020

Alexander J. Zaslavski

Contents

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1 1 3 5 6 8 13 16 24

2 Infinite Horizon Nonautonomous Optimization Problems . 2.1 The Model Description and Main Results . . . . . . . . . 2.2 Upper Semicontinuity of Cost Functions . . . . . . . . . . 2.3 The Nonstationary RSS Model . . . . . . . . . . . . . . . . . 2.4 Auxiliary Results for Theorems 2.4, 2.5 and 2.7 . . . . . 2.5 Properties of the Function U . . . . . . . . . . . . . . . . . . . 2.6 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Proof of Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Overtaking Optimal Programs . . . . . . . . . . . . . . . . . . 2.10 Applications to the Nonstationary RSS Model . . . . . . 2.11 Auxiliary Results for Theorem 2.23 . . . . . . . . . . . . . . 2.12 Proof of Theorem 2.23 . . . . . . . . . . . . . . . . . . . . . . .

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27 27 32 33 41 48 51 55 57 59 60 64 69

3 One-Dimensional Concave RSS Model 3.1 Preliminaries and Main Results . . 3.2 Auxiliary Results . . . . . . . . . . . . 3.3 Proof of Theorem 3.14 . . . . . . . . 3.4 Stability Results . . . . . . . . . . . . . 3.5 Proof of Theorem 3.26 . . . . . . . .

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71 71 78 84 89 91

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Turnpike Phenomenon . . . . . . . . . . . . . . . . . 1.2 Nonconcave (Nonconvex) Problems . . . . . . . . . . . 1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Stability of the Turnpike Phenomenon . . . . . . . . . 1.5 The RSS Model . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Overtaking Optimal Programs for the RSS Model 1.7 Turnpike Properties of the RSS Model . . . . . . . . . 1.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . .

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103 103 108 110 113 123 125 132 132 135 139 150 158 163

5 The Turnpike Phenomenon for Nonautonomous Problems . 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Turnpike Property . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 TP Implies (P1) and (P2) . . . . . . . . . . . . . . . . . . . . . . 5.5 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Completion of the Proof of Theorem 5.2 . . . . . . . . . . . 5.7 A Turnpike Result for Approximate Solutions . . . . . . . 5.8 An Auxiliary Result for Theorem 5.8 . . . . . . . . . . . . . . 5.9 Proof of Theorem 5.8 . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Stability of the Turnpike Phenomenon . . . . . . . . . . . . . 5.11 Proof of Theorem 5.11 . . . . . . . . . . . . . . . . . . . . . . . .

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165 165 169 171 176 185 196 199 200 203 210 212

6 Generic Turnpike Results for the One-Dimensional RSS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Auxiliary Results for Theorem 6.6 . . . . . . . . . . . . . . 6.5 Proof of Theorem 6.6 . . . . . . . . . . . . . . . . . . . . . . . 6.6 Proof of Theorem 6.10 . . . . . . . . . . . . . . . . . . . . . .

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225 225 231 235 254 270 271

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Model Description and Preliminaries . . . . . . . . . . . . . . . 7.2 Turnpike Results for a General Model . . . . . . . . . . . . . . . . . 7.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Proofs of Propositions 7.5–7.7 . . . . . . . . . . . . . . . . . . . . . . .

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273 273 283 286 290

4 Turnpike Properties for Autonomous Problems . . 4.1 The Model Description and Main Results . . . 4.2 A Controllability Lemma . . . . . . . . . . . . . . . 4.3 TP Implies ATP . . . . . . . . . . . . . . . . . . . . . . 4.4 Two Auxiliary Results . . . . . . . . . . . . . . . . . 4.5 ATP Implies TP . . . . . . . . . . . . . . . . . . . . . . 4.6 A Weak Turnpike Result . . . . . . . . . . . . . . . . 4.7 A Turnpike Result for Approximate Solutions 4.8 Auxiliary Results for Theorem 4.11 . . . . . . . . 4.9 Proof of Theorem 4.11 . . . . . . . . . . . . . . . . . 4.10 Stability of the Turnpike Phenomenon . . . . . . 4.11 A Subclass of Models . . . . . . . . . . . . . . . . . . 4.12 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . 4.13 Proof of Theorem 4.20 . . . . . . . . . . . . . . . . .

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8 Discrete Dispersive Dynamical Systems . . . . . . . . . . 8.1 Uniform Convergence to Global Attractors . . . . 8.2 Proof of Proposition 8.1 . . . . . . . . . . . . . . . . . 8.3 Proof of Theorem 8.2 . . . . . . . . . . . . . . . . . . . 8.4 An Auxiliary Result . . . . . . . . . . . . . . . . . . . . 8.5 Proofs of Theorems 8.3 and 8.4 . . . . . . . . . . . . 8.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Spaces of Set-Valued Mappings . . . . . . . . . . . 8.8 Attracting Sets . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Proof of Theorem 8.11 . . . . . . . . . . . . . . . . . . 8.10 Proof of Theorem 8.12 . . . . . . . . . . . . . . . . . . 8.11 Proof of Theorem 8.14 . . . . . . . . . . . . . . . . . . 8.12 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.13 Proof of Proposition 8.24 and Theorem 8.25 . . 8.14 Proof of Theorem 8.26 . . . . . . . . . . . . . . . . . . 8.15 Proof of Theorem 8.27 . . . . . . . . . . . . . . . . . . 8.16 Generic Results . . . . . . . . . . . . . . . . . . . . . . . 8.17 Dynamical Systems with a Lyapunov Function 8.18 Proof of Theorem 8.30 . . . . . . . . . . . . . . . . . . 8.19 Proofs of Propositions 8.31, 8.32 and 8.34 . . . . 8.20 Proof of Theorem 8.35 . . . . . . . . . . . . . . . . . . 8.21 Proof of Theorem 8.36 . . . . . . . . . . . . . . . . . . 8.22 Proof of Theorem 8.38 . . . . . . . . . . . . . . . . . .

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299 299 301 302 305 306 309 310 312 315 316 318 320 323 324 326 327 330 333 334 336 338 338

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Chapter 1

Introduction

Abstract The study of optimal control problems defined on infinite intervals and on sufficiently large intervals has been a rapidly growing area of research [2–4, 7– 13, 17, 19, 21, 23–26, 28, 30, 34, 35, 52, 53, 57, 59, 60, 65, 66, 68, 71, 80, 98, 100, 105, 107, 111, 113, 115, 116, 123, 124] which has various applications in engineering [1, 16, 55, 93], in models of economic growth [5, 14–16, 27, 31–33, 52, 54, 58, 64, 69, 79, 81, 89, 93, 104, 108, 114, 118–122, 126], in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals [6, 73, 91], in control with partial differential equations [29, 72, 88, 125] and in the theory of thermodynamical equilibrium for materials [18, 56, 61–63]. In this chapter we discuss turnpike properties and optimality criterions 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 an autonomous discrete-time control system with a compact metric space of states X . This 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 study the problems T −1 

T −1 v(xi , xi+1 ) → max, {(xi , xi+1 )}i=0 ⊂ Ω, x0 = z, x T = y,

(P1)

i=0

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. J. Zaslavski, Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model, Monographs in Mathematical Economics 4, https://doi.org/10.1007/978-981-16-2252-6_1

1

2

1 Introduction T −1 

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

(P2)

i=0

and

T −1 

T −1 v(xi , xi+1 ) → max, {(xi , xi+1 )}i=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 ⊂ X of the problem (P1) the inequality ρ(xi , x) ¯ ≤ T ≥ 2L and each solution {xi }i=0  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 car in an optimal way, then one should turn to a turnpike, spend most of time on it and then leave the turnpike to reach the required point. In the classical turnpike theory [27, 64, 79, 89] 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 T −1  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 assumptions. (See, for example, [93] 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 [93], 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.

1.1 The Turnpike Phenomenon

3

T −1 We study the problems (P1)–(P3) with the constraint {(xi , xi+1 )}i=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 [93]. They are also more realistic from the point of view of mathematical economics. As we have previously mentioned, 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 [96, 109, 121, 122]. 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 [m 1 , m 2 ] and the new values z, y ∈ X at the end points m 1 and m 2 . Namely, instead of solving this new problem on the “large” interval [m 1 , m 2 ] we can find an “approximate” solution of the problem (P1) on the “small” interval [m 1 , m 1 + L] with the values z, x¯ at the end points and an “approximate” solution of ¯ y at the end the problem (P1) on the “small” interval [m 2 − L , m 2 ] with the values x, ¯ points. Then the concatenation of the first solution, the constant sequence xi = x, i = m 1 + L , . . . , m 2 − L and the second solution is an “approximate” solution of the problem (P1) on the interval [m 1 , m 2 ] with the values z, y at the end points. Sometimes as an “approximate” solution of the problem (P1) we can choose any m2 satisfying admissible sequence {xi }i=m 1

xm 1 = z, xm 2 = y and xi = x¯ for all i = m 1 + L , . . . , m 2 − L .

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 T , is understood) if (xt , xt+1 ) ∈ Ω for all nonnegative integers t. A sequence {xt }t=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

4

1 Introduction T −1 

T −1 v(xi , xi+1 ) → max, {(xi , xi+1 )}i=0 ⊂ Ω, x0 = y, x T = z,

i=0 T −1 

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

i=0

and

T −1 

T −1 v(xi , xi+1 ) → max, {(xi , xi+1 )}i=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 v(xt , xt+1 ) ≤ T v(x, ¯ x) ¯ + c¯ for any natural number T and any pro(A2) t=0 T gram {xt }t=0 . The property (A2) implies that for each program {xt }∞ t=0 either the sequence T −1 

∞ v(xt , xt+1 ) − T v(x, ¯ x) ¯ T =1

t=0

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

∞ v(xt , xt+1 ) − T v(x, ¯ x) ¯ T =1

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

v(xt , xt+1 ) − T v(x, ¯ x) ¯ ≥ −M.

1.2 Nonconcave (Nonconvex) Problems

5

 It is not difficult to see that {X M : M ∈ (0, ∞)} is the set of all points x ∈ X for which there exists a (v)-good program {xt }∞ t=0 satisfying x 0 = x. T ⊂ X is called (Δ)-optimal Let T ≥ 1 be an integer and Δ ≥ 0. A program {xi }i=0  T  if for any program {xi }i=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 [121]. In particular, in Chap. 2 of [121] 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 T which satisfies x0 ∈ X M there exist nonnegative integers τ1 , τ2 ≤ L program {xt }t=0 ¯ ≤  for all t = τ1 , . . . , T − τ2 and if ρ(x0 , x) ¯ ≤ δ, then τ1 = 0. such that ρ(xt , x) An analogous turnpike result for approximate solutions of our first optimization problem is also proved in Chap. 2 of [121]. ∞ 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 →∞

T −1 

[v(yt , yt+1 ) − v(xt , xt+1 )] ≤ 0

t=0

holds. In Chap. 2 of [121] we prove the following result which establishes the existence of an overtaking optimal program. 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.

6

1 Introduction

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 [67, 76] 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.

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

T −1 

 u(xi , xi+1 ) :

T {xi }i=0

u(xi , xi+1 ) :

T {xi }i=0

is an (Ω) − program and x0 = x ,

i=0

σ(u, T, x, y) = sup

T −1  i=0

σ(u, T ) = sup

T −1 

 is an (Ω) − program and x0 = x, x T = y ,  T u(xi , xi+1 ) : {xi }i=0 is an (Ω) − program .

i=0

(Here we use the convention that the supremum of an empty set is −∞.)

1.4 Stability of the Turnpike Phenomenon

7

For every pair of points x, y ∈ X , every pair of nonnegative integers T1 , T2 which T2 −1 ⊂ M(Ω) define satisfies T1 < T2 and every finite sequence of functions {u t }t=T 1 T2 −1 σ({u t }t=T , T1 , T2 , x) 1

= sup

T2 −1 σ({u t }t=T , T1 , T2 , x, y) = sup 1

T2 −1 , T1 , T2 ) σ({u t }t=T 1

= sup

⎧ 2 −1 ⎨T ⎩ t=T1 ⎧ 2 −1 ⎨T ⎩ t=T1 ⎧ 2 −1 ⎨T ⎩

u t (xt , xt+1 ) :

T2 {xt }t=T 1

is an (Ω) − program and x T1 = x

⎫ ⎬ ⎭

,

T2 u t (xt , xt+1 ) : {xt }t=T is an (Ω) − program and x T1 = x, x T2 = y 1

u t (xt , xt+1 ) :

T2 {xt }t=T 1

t=T1

⎫ ⎬ ⎭

,

⎫ ⎬ is an (Ω) − program . ⎭

Let v ∈ M(Ω) be an upper semicontinuous function. Set v(x, y) = − v − 1 for all (x, y) ∈ (X × X ) \ Ω. Suppose that there exist a point x¯ ∈ X and real positive constants c¯ such that assumptions (A1)–(A3) of Sect. 1.2 hold. Let T ≥ 1 be an integer. Denote by YT the set of all x ∈ X for which there exists T such that x0 = x, ¯ x T = x. Denote by Y¯T the set of all x ∈ X for a program {xt }t=0 T ¯ which there exists a program {xt }t=0 such that x0 = x, x T = x. The following two theorems stated below are proved in [121]. They show that the turnpike phenomenon holds for approximate solutions of the optimal control problems of the types (P1) and (P2) with objective functions u t , t = 0, . . . , T − 1 belonging to a small neighborhood of v. Theorem 1.5 Let  > 0, L 0 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 every T −1 ⊂ M(Ω) for natural number T > 2L, every finite sequence of functions {u t }t=0 which u t − v ≤ δ, t = 0 . . . T − 1, T for which and every (Ω)-program {xt }t=0

x0 ∈ Y¯ L 0 , x T ∈ Y L 0 , T −1 

T −1 u t (xt , xt+1 ) ≥ σ({u t }t=0 , 0, T, x0 , x T ) − M0

t=0

and

τ +L−1 

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

t=τ

for every nonnegative integer τ ≤ T − L there exist integers τ1 ∈ [0, L], τ2 ∈ [T − L , T ] such that

8

1 Introduction

ρ(xt , x) ¯ ≤ , t = τ1 , . . . , τ2 . ¯ ≤ δ, then τ1 = 0 and if ρ(x T , x) ¯ ≤ δ, then τ2 = T . Moreover, if ρ(x0 , x) Theorem 1.6 Let  > 0, L 0 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 every T −1 ⊂ M(Ω) for which natural number T > 2L, every finite sequence {u t }t=0 u t − v ≤ δ, t = 0 . . . , T − 1 T for which and every (Ω)-program {xt }t=0

x0 ∈ Y¯ L 0 ,

T −1 

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

t=0

and

τ +L−1 

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

t=τ

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

1.5 The RSS Model In this section we discuss the Robinson–Solow–Srinivasan model (or the RSS model for short) which was introduced in the 1960s by Robinson, Solow and Srinivasan [74, 83, 84] and was studied by Robinson, Okishio and Stiglitz [70, 75, 85–87]. Recently, the RSS model was studied by Khan and Mitra [36–44], Khan and Piazza [45–47], Khan and Zaslavski [48–51] and Zaslavski [92, 94, 101–104, 106–109, 111–114, 117–119]. Many results on the RSS model are collected in [126]. 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

1.5 The RSS Model

9

xy =

n 

xi yi

i=1

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 all Let e(i), i = 1, . . . , n, be the ith unit vector in R n , and e be an element of R+ n 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 a 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) input-coefficients 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) = (z 1 (t + 1), . . . , z n (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)) + d x(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)

10

1 Introduction

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 sectors is described by az(t + 1) + ey(t) ≤ 1. Note that the flows 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 × R+ , x(t + 1) ≥ (1 − d)x(t), 0 ≤ y(t) ≤ x(t), (x(t), y(t)) ∈ R+ 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 T2 T2 −1 , {y(t)}t=T ) ({x(t)}t=T 1 1 n and for each integer t satisfying T1 ≤ t < T2 is called a program if x(T2 ) ∈ R+ relations (1.2) hold. We associate with every program {x(t), y(t)}∞ t=0 its gross investment sequence {z(t + 1)}∞ t=0 such that

z(t + 1) = x(t + 1) − (1 − d)x(t), t = 0, 1, . . . and a consumption sequence {by(t)}∞ t=0 . The results presented in this section were obtained in [36]. Proposition 1.7 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 actually be 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. A program {x ∗ (t), y ∗ (t)}∞ t=0 is weakly optimal if T  lim inf [w(by(t)) − w(by ∗ (t))] ≤ 0 T →∞

t=0

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

1.5 The RSS Model

11

n Λ(x, x  ) = {y ∈ R+ : 0 ≤ y ≤ x and ey ≤ 1 − a(x  − (1 − d)x)}, (x, x  ) ∈ Ω.

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

pi = w  (bσ (1 + daσ )−1 ) qi , y = (1 + daσ )−1 e(σ).

qi = ai bi /(1 + dai ), The following useful lemma plays an important role in the study of the RSS model. Lemma 1.8 w(b y) ≥ w(by) + px  − p x for any (x, x  ) ∈ Ω and for any y ∈  Λ(x, x ). Theorem 1.9 There exists a unique golden-rule stock x = (1 + daσ )−1 e(σ). We use the following notion of good programs introduced by Gale [27] and used in optimal control [16, 93, 121]. 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

n Proposition 1.10 Let x0 ∈ R+ . Then there exists a good program

{x(t), y(t)}∞ t=0 which satisfies x(0) = x0 . Proposition 1.11 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 1.11 easily implies the following result.

12

1 Introduction

Proposition 1.12 Every program which is not good is bad. For any (x, x  ) ∈ Ω and any y ∈ Λ(x, x  ) set p (x − x  ) − (w(by) − w(b y)). δ(x, y, x  ) =

(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 1.13 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 1.8 and (1.3). Proposition 1.14 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 1.14 implies the following result. Proposition 1.15 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)) : {x(t),

y(t)}∞ t=0

t=0

n Proposition 1.16 Let x0 ∈ R+ . Then

0 ≤ Δ(x0 ) < ∞

is a program such that x(0) = x0 .

1.5 The RSS Model

13

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 1.17 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 Theorem 1.18 Let x0 ∈ R+ . Then there exists a weakly optimal program {x(t), ∞ x , then the program {x(t), y(t)}∞ y(t)}t=0 satisfying x(0) = x0 . If x0 = 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 1.19 Let

ξσ = 1 − d − aσ−1 .

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.

1.6 Overtaking Optimal Programs for the RSS Model In this section we continue to use the assumptions introduced in Sect. 1.5. The following three theorems were obtained in [92].

14

1 Introduction

Theorem 1.20 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 1.21 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  [w(by(t)) − w(by ∗ (t))] ≤ 0 t=0

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

Theorem 1.22 Assume that for every good program {x(t), y(t)}∞ t=0 , x, x ). lim (x(t), y(t)) = (

t→∞

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

Corollary 1.23 Assume that the function w is strictly concave. Then for every point n x 0 ∈ R+ there exists an overtaking optimal program {x(t), y(t)}∞ t=0 satisfying x(0) = x0 . n Corollary 1.24 Assume that ξσ = −1. Then for every point x0 ∈ R+ there is an ∞ overtaking optimal program {x(t), y(t)}t=0 such that x(0) = x0 .

The following three theorems were obtained in [48]. Theorem 1.25 Assume that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

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.

1.6 Overtaking Optimal Programs for the RSS Model

15

Theorem 1.26 Assume that at least one of the following conditions holds: (a) w is strictly concave. (b) ξσ = −1. 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 1.27 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 ≤ δ the following inequality holds: {x(t), y(t)}∞ t=0 satisfying x(0) − x(t) − x , y(t) − x ≤  for all integers t ≥ 0. In [101] we studied the structure of good programs of the RSS model and proved the following three results. Theorem 1.28 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 1.29 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 1.30 Let w ∈ C 2 , w  (b x ) = 0 and let for every good program {u(t), v(t)}∞ t=0 ,

16

1 Introduction

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}, ∞  t=0 ∞  t=0 ∞ 

xi (t) < ∞, (xσ (t) − yσ (t)) < ∞, (yσ (t) − xσ )2 < ∞,

t=0

and the sequence {

T −1 t=0

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

1.7 Turnpike Properties of the RSS Model In this section we discuss the turnpike properties for the RSS 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) of Sect. 1.2 does not hold for the RSS model. We continue to use the assumptions introduced in Sect. 1.5. n and T ≥ 1 be a natural number. Set Let z ∈ R+ U (z, T ) = sup

T −1 

T −1 T w(by(t)) : ({x(t)}t=0 , {y(t)}t=0 )

t=0

 is a program such that x(0) = z . n Clearly, U (z, T ) is a finite number. Let x0 , x1 ∈ R+ , T1 , T2 be integers, 0 ≤ T1 < T2 . Define T −1 2  T2 T2 −1 w(by(t)) : ({x(t)}t=T , {y(t)}t=T ) U (x0 , x1 , T1 , T2 ) = sup 1 1 t=T1

 is a program such that x(T1 ) = x0 , x(T2 ) ≥ x1 .

1.7 Turnpike Properties of the RSS Model

17

(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 [50]. Theorem 1.31 Let M,  be positive numbers and  ∈ (0, 1). Then there exists a n natural number L such that for each integer T > L, each z 0 , z 1 ∈ R+ satisfying T −1 T −1 z 0 ≤ Me and az 1 ≤ d and each program ({x(t)}t=0 , {y(t)}t=0 ) which satisfies x(0) = z 0 , x(T ) ≥ z 1 ,

T −1 

w(by(t)) ≥ U (z 0 , z 1 , 0, T ) − M,

t=0

the following inequality holds: Card{i ∈ {0, . . . , T − 1} : max{ x(t) − x , y(t) − x } > } ≤ L . Theorem 1.32 Let M,  be positive numbers and  ∈ (0, 1). Then there exist a natural number L and a positive number γ such that for each integer T > n satisfying z 0 ≤ Me and az 1 ≤ d −1 and each program 2L, each z 0 , z 1 ∈ R+ T −1 T ({x(t)}t=0 , {y(t)}t=0 ) which satisfies x(0) = z 0 , x(T ) ≥ z 1 ,

T −1 

w(by(t)) ≥ U (z 0 , z 1 , 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. x ≤ γ then τ2 = T . Moreover, if x(0) − x ≤ γ then τ1 = 0 and if x(T ) − In [112] we continued to study the turnpike phenomenon for the RSS model and proved the following three turnpike results which are extensions of Theorem 1.32.

18

1 Introduction

Theorem 1.33 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 and  ∈ (0, 1). Then there exist a natural number n L and a positive number γ such that for each integer T > 2L, each z 0 , z 1 ∈ R+ T −1 T −1 satisfying z 0 ≤ Me and az 1 ≤ d and each program ({x(t)}t=0 , {y(t)}t=0 ) which satisfies x(0) = z 0 , x(T ) ≥ z 1 , 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), z 1 , 0, 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 1.34 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 number γ such that for each integer T > 2L, each z 0 ∈ R+ satisfying z 0 ≤ Me and T −1 T each program ({x(t)}t=0 , {y(t)}t=0 ) which satisfies x(0) = z 0 , 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

1.7 Turnpike Properties of the RSS Model

19

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 1.35 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 number γ such that for each integer T > 2L, each z 0 ∈ R+ satisfying z 0 ≤ Me and T −1 T each program ({x(t)}t=0 , {y(t)}t=0 ) which satisfies x(0) = z 0 , T −1 

w(by(t)) ≥ U (z 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. x ≤ γ then τ2 = T . Moreover, if x(0) − x ≤ γ then τ1 = 0 and x(T ) − n For every positive number M and every function φ : R+ → R 1 define

φ M = 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 z 0 , z 1 ∈ R+ define

T2 −1 U ({wt }t=T , z0 , z1 ) 1

= sup

⎧ 2 −1 ⎨T ⎩

wt (y(t)) :

t=T1

 T2 T2 −1 , {y(t)} ) is a program such that x(T ) = z , x(T ) ≥ z ({x(t)}t=T , 1 0 2 1 t=T 1 1 T2 −1 , z0 ) U ({wt }t=T 1

= sup

⎧ 2 −1 ⎨T ⎩

t=T1

wt (y(t)) :

 T2 T2 −1 , {y(t)} ) is a program such that x(T ) = z ({x(t)}t=T . 1 0 t=T1 1

20

1 Introduction

(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 1.36 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. T2 T2 −1 n there exists a program ({x(t)}t=T , {y(t)}t=T ) such that 1. For every point z 0 ∈ R+ 1 1

x(T1 ) = z 0 ,

T 2 −1

T2 −1 wt (y(t)) = U ({wt }t=T , z 0 ). 1

t=T1 T2 −1 n such that U ({wt }t=T , z 0 , z 1 ) is finite there 2. For every pair of points z 0 , z 1 ∈ R+ 1 T2 T2 −1 exists a program ({x(t)}t=T1 , {y(t)}t=T1 ) such that x(0) = z 0 , x(T2 ) ≥ z 1 and T 2 −1

T2 −1 wt (y(t)) = U ({wt }t=T , z 0 , z 1 ). 1

t=T1

The following stability results were obtained in [112]. They show that the turnpike phenomenon is stable under small perturbations of the utility functions. 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 > max{(ai d)−1 : i = 1, . . . , n},  > 0 and  ∈ (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 0 ≤ Me and az 1 ≤ d −1 , each finite sequence of functions z 0 , z 1 ∈ R+ n n and wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ such that wi − w(b(·)) M ≤ γ˜ T −1 T , {y(t)}t=0 ) such for every integer i ∈ {0, . . . , T − 1} and every program ({x(t)}t=0 that

x(0) = z 0 , x(T ) ≥ z 1 , τ +L−1 

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

t=τ

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

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

1.7 Turnpike Properties of the RSS Model

21

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

t→∞

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

x(0) = z 0 , τ +L−1 

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

t=τ

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

T −1 wt (y(t)) ≥ U ({wt }t=T ˜ −L , x(T − 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 1.39 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

Let M > max{(ai d)−1 : i = 1, . . . , n},  > 0 and  ∈ (0, 1). Then there exist a natural number L, a positive number γ and λ > 1 such that for each integer T > 2L,

22

1 Introduction

n each z 0 , z 1 ∈ R+ satisfying z 0 ≤ Me and az 1 ≤ d −1 , each finite sequence of funcn tions wi : R+ → R 1 , i = 0, . . . , T − 1 which are bounded on bounded subsets of n and such that wi − w(b(·)) M ≤ γ for each i ∈ {0, . . . , T − 1}, each sequence R+ T −1 {αi }i=0 ⊂ (0, 1] such that for each i, j ∈ {0, . . . , T − 1} satisfying | j − i| ≤ L the T −1 T inequality αi α−1 j ≤ λ holds and each program ({x(t)}t=0 , {y(t)}t=0 ) such that

x(0) = z 0 , x(T ) ≥ z 1 , τ +L−1 

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

t=τ

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

T −1 αt wt (y(t)) ≥ U ({αt wt }t=T −L , z 1 ) − γαT −L

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

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, n n satisfying z 0 ≤ Me, each finite sequence of functions wi : R+ → R1, each z 0 ∈ R+ n i = 0, . . . , T − 1 which are bounded on bounded subsets of R+ and such that wi − w(b(·)) M ≤ γ T −1 ⊂ (0, 1] such that for each for each i ∈ {0, . . . , T − 1}, each sequence {αi }i=0

i, j ∈ {0, . . . , T − 1} T satisfying | j − i| ≤ L the inequality αi α−1 j ≤ λ holds and each program ({x(t)}t=0 , T −1 {y(t)}t=0 ) such that

1.7 Turnpike Properties of the RSS Model

23

x(0) = z 0 , τ +L−1 

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

t=τ

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

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

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 ) − The following results were obtained in [117]. Theorem 1.41 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

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

x(0) = z 0 , x(T ) ≥ z 1 , T −1 

T −1 wt (y(t)) ≥ U ({wt }t=0 , z 0 , z 1 ) − M0

t=0

the following inequality holds: Card({t ∈ {0, . . . , T − 1} : max{ x(t) − x , y(t) − x } > }) ≤ L . Theorem 1.42 Suppose that for each good program {u(t), v(t)}∞ t=0 , x, x ). lim (u(t), v(t)) = (

t→∞

24

1 Introduction

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 satisfying z 0 ≤ Me, each finite sequence of functions wi : R+ → R1, i = z 0 ∈ R+ n 0, . . . , T − 1 which are bounded on bounded subsets of R+ and such that wi − w(b(·)) M ≤ γ˜ T −1 T , {y(t)}t=0 ) which satisfies for each i ∈ {0, . . . , T − 1} and each program ({x(t)}t=0

x(0) = z 0 , T −1 

T −1 wt (y(t)) ≥ U ({wt }t=0 , z 0 ) − M0

t=0

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

1.8 Concluding Remarks In this chapter we discussed turnpike properties and optimality criterions 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 of the second class are related to the RSS 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) of 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 : (x, x  ) ∈ Ω}. a(x) = {x  ∈ R+ 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 also possesses the following monotonicity property: n which satisfies x˜ ≥ x there exists y˜ ∈ a(x) ˜ for every (x, y) ∈ Ω and every x˜ ∈ R+ such that y˜ ≥ y, u(x, ˜ y˜ ) ≥ u(x, y).

1.8 Concluding Remarks

25

These two properties play a crucial role in the analysis of the turnpike phenomenon for the RSS model. This understanding leads us to the idea of developing a turnpike theory for large classes of optimal control problems possessing the strict contraction property and the monotonicity property described above which includes the RSS model as a particular case. This theory is developed in this book.

Chapter 2

Infinite Horizon Nonautonomous Optimization Problems

Abstract In this chapter we study infinite horizon optimal control problems with nonautonomous optimality criterions. The utility functions, which determine the optimality criterion, are nonconcave. The class of models contains, as a particular case, the Robinson–Solow–Srinivasan model. We establish the existence of good programs and optimal programs.

2.1 The Model Description and Main Results 1 Let R 1 (R+ ) be the set of real (non-negative) numbers and let R n be the n-dimensional Euclidean space with 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 n  xi yi xy = i=1

and let x , x > y, x ≥ y have their usual meaning. 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 x2 its Euclidean norm in R n . We assume that  ·  is a norm in R n . For every mapping a : X → 2Y \ {∅}, where X, Y are nonempty sets, set graph(a) = {(x, y) ∈ X × Y : y ∈ a(x)}.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. J. Zaslavski, Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model, Monographs in Mathematical Economics 4, https://doi.org/10.1007/978-981-16-2252-6_2

27

28

2 Infinite Horizon Nonautonomous Optimization Problems

Let K be a nonempty compact subset of R n . Denote by P(K ) the set of all nonempty closed subsets of K . For every pair of nonempty sets A, B ⊂ R n define H (A, B) = sup{sup inf x − y, sup inf x − y}. x∈A y∈B

y∈B x∈A

(2.1)

For every nonnegative integer t, let at : K → P(K ) be such that graph(at ) is a closed subset of R n × R n . Assume that there exists a number κ ∈ (0, 1) such that for every pair of points x, y ∈ K and every nonnegative integer t, H (at (x), at (y)) ≤ κx − y

(2.2)

and that for every nonnegative integer t the upper semicontinuous function u t : {(x, x ) ∈ K × K , x ∈ at (x)} → [0, ∞) satisfies sup{sup{u t (x, x ) : (x, x ) ∈ graph(at )} : t = 0, 1, . . . } < ∞.

(2.3)

A sequence {x(t)}∞ t=0 ⊂ K is called a program if x(t + 1) ∈ at (x(t)) for every nonnegative integer t. T2 ⊂ K is called a Let T1 , T2 be integers such that T1 < T2 . A sequence {x(t)}t=T 1 program if x(t + 1) ∈ at (x(t)) for every integer t satisfying T1 ≤ t < T2 . We suppose that the following assumptions hold: (A1) for every positive number δ there exists a positive number λ such that if an integer t ≥ 0 and if (x, x ) ∈ graph(at ) satisfies u t (x, x ) ≥ δ, then there exists z ∈ at (x) satisfying z ≥ x + λe;  x (t), (A2) there exist a program { x (t)}∞ t=0 and a positive number Δ such that u t (   x (t + 1)) ≥ Δ for every nonnegative integer t; (A3) for every nonnegative integer t, every (x, y) ∈ graph(at ) and every x˜ ∈ K ˜ for which which satisfies x˜ ≥ x there exists y˜ ∈ at (x) ˜ y˜ ) ≥ u t (x, y). y˜ ≥ y, u t (x, In the sequel we assume that the supremum of an empty set is −∞. For every point x0 ∈ K and every natural number T define U (x0 , T ) = sup

T −1 

u t (x(t), x(t + 1)) :

t=0

 T −1 is a program and x(0) = x0 . {x(t)}t=0

(2.4)

2.1 The Model Description and Main Results

29

Let x0 , x˜0 ∈ K and let T ≥ 1 be an integer. Define U (x0 , x˜0 , T ) = sup

T −1 

T −1 u t (x(t), x(t + 1)) : {x(t)}t=0 is a program

t=0

 such that x(0) = x0 , x(T ) ≥ x˜0 .

(2.5)

Let T ≥ 1 be an integer. Define (T ) = sup U

T −1 

 u t (x(t), x(t + 1)) :

T −1 {x(t)}t=0

is a program .

(2.6)

t=0

The results presented in this section were obtained in [102]. They are also discussed in Chap. 6 of [126]. Upper semicontinuity of u t , t = 0, 1, . . . implies the following two propositions. Proposition 2.1 For every x0 ∈ K and every integer T ≥ 1 there exists a program T which satisfies x(0) = x0 and {x(t)}t=0 T −1 

u t (x(t), x(t + 1)) = U (x0 , T ).

t=0 T Proposition 2.2 For every integer T ≥ 1 there exists a program {x(t)}t=0 satisfying T −1  u (x(t), x(t + 1)) = U (T ). t=0 t

For every x0 ∈ K and every pair of integers T1 < T2 define U (x0 , T1 , T2 ) = sup

T −1 2 

u t (x(t), x(t + 1)) :

t=T1

 T2 −1 {x(t)}t=T is a program and x(T ) = x 1 0 . 1

(2.7)

Upper semicontinuity of u t , t = 0, 1, . . . implies the following result. Proposition 2.3 For every x0 ∈ K and every pair of integers T1 < T2 there exists a T2 such that x(T1 ) = x0 and program {x(t)}t=T 1 T 2 −1

u t (x(t), x(t + 1)) = U (x0 , T1 , T2 ).

t=T1

Let x0 , x˜0 ∈ K and let T1 < T2 be integers. Define

30

2 Infinite Horizon Nonautonomous Optimization Problems

U (x0 , x˜0 , T1 , T2 ) = sup

T −1 2 

T2 u t (x(t), x(t + 1)) : {x(t)}t=T is a program and 1

t=T1

 x(T1 ) = x0 , x(T2 ) ≥ x˜0 .

(2.8)

Let T1 , T2 be integers such that T1 < T2 . Define (T1 , T2 ) = sup U

T −1 2 

 u t (x(t), x(t + 1)) :

T2 {x(t)}t=T 1

is a program .

(2.9)

t=T1

We will prove the following theorem, which is the main result of this section. Theorem 2.4 There exists a positive number M such that for every x0 ∈ K there ∞ ¯ = x0 and that for every pair of nonnegative exists a program {x(t)} ¯ t=0 such that x(0) integers T1 , T2 satisfying T1 < T2 , the inequality T −1  2    (T1 , T2 ) ≤ M u t (x(t), ¯ x(t ¯ + 1)) − U    t=T1

holds. Moreover, for every natural number T , T −1 

u t (x(t), ¯ x(t ¯ + 1)) = U˜ (x(0), ¯ x(T ¯ ), 0, T )

t=0

if the following properties hold: for every nonnegative integer t and every (z, z ) ∈ graph(at ) satisfying u t (z, z ) > 0 the function u t is continuous at the point (z, z ); for every nonnegative integer t and each z, z 1 , z 2 , z 3 ∈ K which satisfy z 1 ≤ z 2 ≤ z 3 and z i ∈ at (z), i = 1, 3 the inclusion z 2 ∈ at (z) is valid. ∞ The program {x(t)} ¯ t=0 whose existence is guaranteed by Theorem 2.4 in infinite horizon optimal control is considered as an (approximately) optimal program.

Theorem 2.5 Assume that {x(t)}∞ t=0 is a program, there exists a positive number M0 such that for every natural number T , T −1 

u t (x(t), x(t + 1)) ≥ U (0, T, x(0), x(T )) − M0

t=0

and that lim sup u t (x(t), x(t + 1)) > 0. t→∞

2.1 The Model Description and Main Results

31

Then there exists a positive number M1 such that for every pair of integers T1 ≥ 0 satisfying T2 > T1 the inequality T −1  2    (T1 , T2 ) ≤ M1 u t (x(t), x(t + 1)) − U    t=T1

holds. Theorem 2.4 is proved in Sect. 2.6, while Theorem 2.5 is proved in Sect. 2.7. Let M > 0 be as guaranteed by Theorem 2.4. ∞ Proposition 2.6 Let x0 ∈ K and let a program {x(t)} ¯ t=0 be as guaranteed by is a program. Then either the sequence Theorem 2.4. Assume that {x(t)}∞ t=0

T −1 

u t (x(t), x(t + 1)) −

t=0

T −1 

∞ u t (x(t), ¯ x(t ¯ + 1)) T =1

t=0

is bounded or T −1 

u t (x(t), x(t + 1)) −

t=0

T −1 

u t (x(t), ¯ x(t ¯ + 1)) → −∞ as T → ∞.

(2.10)

t=0

Proof Assume that the sequence T −1 

u t (x(t), x(t + 1)) −

t=0

T −1 

∞ u t (x(t), ¯ x(t ¯ + 1)) T =1

t=0

is not bounded. Then by Theorem 2.4, lim inf T →∞

T −1 

u t (x(t), x(t + 1)) −

t=0

T −1 

u t (x(t), ¯ x(t ¯ + 1)) = −∞.

t=0

Let Q > 0. Then there exists a natural number T0 which satisfies T 0 −1 t=0

u t (x(t), x(t + 1)) −

T 0 −1

u t (x(t), ¯ x(t ¯ + 1)) < −Q − M.

(2.11)

t=0

∞ In view of (2.11), the choice of {x(t)} ¯ t=0 and Theorem 2.4, for every natural number T > T0 , we have

32

2 Infinite Horizon Nonautonomous Optimization Problems T −1 

u t (x(t), x(t + 1)) −

t=0

=

T 0 −1

T −1 

u t (x(t), x(t + 1)) −

t=0

−

u t (x(t), ¯ x(t ¯ + 1))

t=0

T −1 

T 0 −1

u t (x(t), ¯ x(t ¯ + 1)) +

t=0

T −1 

u t (x(t), x(t + 1))

t=T0

u t (x(t), ¯ x(t ¯ + 1))

t=T0

(T0 , T ) − < −Q − M + U

T −1 

u t (x(t), ¯ x(t ¯ + 1)) < −Q.

t=T0

Since Q is an arbitrary positive number we conclude that (2.10) holds. This completes the proof of Proposition 2.6. Now assume that u t = u 0 and at = a0 , t = 0, 1, . . . . Let a positive number M be as guaranteed by Theorem 2.4 and set u = u 0 , a = a0 . The following result will be proved in Sect. 2.8. (0, p)/ p and Theorem 2.7 There exists μ = lim p→∞ U (0, p) − μ| ≤ 2M/ p for all natural numbers p. | p −1 U

2.2 Upper Semicontinuity of Cost Functions For every nonnegative integer t, let at : K → P(K ) be such that graph(at ) is a closed set and assume that for every nonnegative integer t an upper semicontinuous function n → [0, ∞) satisfies φt : R + n } : t = 0, 1, . . . } < ∞. sup{sup{φt (z) : z ∈ (K − K ) ∩ R+

(2.12)

For every nonnegative integer t and every point (x, x ) ∈ graph(at ) set n , x + z ∈ a(x)}. u t (x, x ) = sup{φt (z) : z ∈ R+

(2.13)

By (2.12) and (2.13), u t , t = 0, 1, . . . satisfy (2.3). Note that in many models of economic dynamics, cost functions u t , t = 0, 1, . . . are defined by (2.13). Lemma 2.8 For every nonnegative integer t the function u t : graph(at ) → [0, ∞) is upper semicontinuous. Proof Let t be a nonnegative integer and let {(x ( j) , y ( j) )}∞ j=1 ⊂ graph(at ) satisfy lim (x ( j) , y ( j) ) = (x, y).

j→∞

(2.14)

2.2 Upper Semicontinuity of Cost Functions

We claim that

33

u t (x, y) ≥ lim sup u t (x ( j) , y ( j) ). j→∞

Extracting a subsequence and re-indexing if necessary we may assume without loss of generality that there exists lim u t (x ( j) , y ( j) ).

j→∞

n satisfying In view of (2.13), for every natural number j there exists z ( j) ∈ R+

y ( j) + z ( j) ∈ at (x ( j) ), φt (z ( j) ) ≥ u t (x ( j) , y ( j) ) − 1/j.

(2.15)

Evidently, the sequence {z ( j) }∞ j=1 is bounded. Extracting a subsequence and reindexing, if necessary, we may assume without loss of generality that there exists z = lim z ( j) . j→∞

(2.16)

In view of (2.14)–(2.16), we have z≥0 and

(x, y + z) = lim (x ( j) , y ( j) + z ( j) ) ∈ graph(at ). j→∞

Together with (2.13)–(2.16) the relation above implies that u t (x, y) ≥ φt (z) ≥ lim sup φt (z ( j) ) ≥ lim sup[u t (x ( j) , y ( j) ) − 1/j] j→∞

= lim u t (x j→∞

( j)

,y

j→∞ ( j)

).

This completes the proof of Lemma 2.8.

2.3 The Nonstationary RSS Model In this section we consider a subclass of the class of infinite horizon optimal control problems considered in Sect. 2.1. Infinite horizon problems of this subclass correspond to the nonstationary RSS model.

34

2 Infinite Horizon Nonautonomous Optimization Problems

For every nonnegative integer t, let α(t) = (α1(t) , . . . , αn(t) )  0, b(t) = (b1(t) , . . . , bn(t) )  0, d

(t)

=

(d1(t) , . . . , dn(t) )

∈ ((0, 1])

(2.17)

n

and for every nonnegative integer t, let wt : [0, ∞) → [0, ∞) be a strictly increasing continuous function such that wt (0) = 0 for all integers t ≥ 0, inf{wt (z) : t = 0, 1, . . . } > 0 for all z > 0

(2.18)

and such that the following assumption holds: (A4) for every positive number  there exists a positive number δ such that for every nonnegative integer t and every z ∈ [0, δ] the inequality wt (z) ≤  is valid. n define Let t be a nonnegative integer. For every x ∈ R+ 

n at (x) = y ∈ R+ : yi ≥ (1 − di(t) )xi , i = 1, . . . , n, n 

αi(t) (yi

− (1 −

di(t) )xi )

 ≤1 .

(2.19)

i=1 n , at (x) is a nonempty closed bounded subset It is easy to see that for every x ∈ R+ n n n × R+ . Assume that of R+ and graph(at ) is a closed subset of R+

inf{di(t) : i = 1, . . . , n, t = 0, 1, . . . } > 0,

(2.20)

inf{eb(t) : t = 0, 1, . . . } > 0,

(2.21)

inf{αi(t) : sup{bi(t) : sup{αi(t) :

i = 1, . . . , n, t = 0, 1, . . . } > 0,

(2.22)

i = 1, . . . , n, t = 0, 1, . . . } < ∞,

(2.23)

i = 1, . . . , n, t = 0, 1, . . . } < ∞

(2.24)

and that for every positive number M, we have sup{wt (M) : t = 0, 1, . . . } < ∞, inf{wt (M) : t = 0, 1, . . . } > 0.

(2.25)

The constraint mappings at , t = 0, 1, . . . have already been defined. Let us now define the cost functions u t , t = 0, 1, . . . .

2.3 The Nonstationary RSS Model

35

For every nonnegative integer t and every (x, x ) ∈ graph(at ) put  u t (x, x ) = sup wt (b(t) y) : 0 ≤ y ≤ x,

ey +

n 

αi(t) (xi

− (1 −

di(t) )xi )

 ≤1 .

(2.26)

i=1

Fix numbers α∗ , α∗ > 0, d∗ > 0 which satisfy α∗ < αi(t) < α∗ , d∗ < di(t) , i = 1, . . . , n, t = 0, 1, . . . .

(2.27)

Lemma 2.9 Let a number M0 > (α∗ d∗ )−1 , an integer t ≥ 0 and let (x, x ) ∈ graph(at ) satisfy x ≤ M0 e. Then x ≤ M0 e. Proof In view of (2.19), n 

αi(t) (xi − (1 − di(t) )xi ) ≤ 1

i=1

and by (2.27), for every i = 1, . . . , n, we have xi ≤ (αi(t) )−1 + (1 − di(t) )xi ≤ α∗−1 + (1 − d∗ )xi ≤ α∗−1 + (1 − d∗ )M0 ≤ d∗ (α∗ d∗ )−1 + (1 − d∗ )M0 ≤ d∗ M0 + (1 − d∗ )M0 = M0 . This completes the proof of Lemma 2.9. Lemma 2.10 Let t be a nonnegative integer. Then the function u t : graph(at ) → [0, ∞) is upper semicontinuous. Moreover, if (x, y) ∈ graph(at ) and u t (x, y) > 0, then u t is continuous at (x, y). Proof Let ( j) ( j) (x, y) ∈ graph(at ), {(x ( j) , y ( j) )}∞ j=1 ⊂ graph(at ), lim (x , y ) = (x, y). j→∞

(2.28) We claim that

u t (x, y) ≥ lim sup u t (x ( j) , y ( j) ). j→∞

Extracting a subsequence and re-indexing we may assume that there exists lim j→∞ u t (x ( j) , y ( j) ). In view of (2.25) and (2.26), for every natural numn which satisfies ber j there exists z ( j) ∈ R+

36

2 Infinite Horizon Nonautonomous Optimization Problems

z ( j) ≤ x ( j) , ez ( j) + wt (b(t) z ( j) ) ≥ u t (x

n 

( j)

αi(t) (yi

i=1 ( j) ( j)

,y

( j)

− (1 − di(t) )xi ) ≤ 1,

) − 1/j.

(2.29) (2.30)

Extracting a subsequence and re-indexing we may assume without loss of generality that there exists (2.31) z = lim z ( j) . j→∞

By (2.28) and (2.31), 0 ≤ z ≤ x.

(2.32)

It follows from (2.28), (2.29) and (2.31) that ez +

n 

αi(t) (yi − (1 − di(t) )xi )

i=1

= lim [ez ( j) + j→∞

n 

( j)

αi(t) (yi

( j)

− (1 − di(t) )xi )] ≤ 1.

i=1

Combined with (2.26) and (2.30)–(2.32) the equation above implies that u t (x, y) ≥ wt (b(t) z) = lim wt (b(t) z ( j) ) = lim u t (x ( j) , y ( j) ). j→∞

j→∞

Hence u t is upper semicontinuous. Assume now that (x, y) ∈ graph(at ) satisfies u t (x, y) > 0

(2.33)

and show that u t is continuous at (x, y). Evidently, it is sufficient to show that u t is lower semicontinuous at (x, y). Assume that (x ( j) , y ( j) ) ∈ graph(at ) for all natural numbers j, lim (x ( j) , y ( j) ) = (x, y).

j→∞

Let  be a positive number. It is sufficient to show that lim inf u t (x ( j) , y ( j) ) ≥ u t (x, y) − . j→∞

n for which It follows from (2.26) and (2.33) that there exists z ∈ R+

(2.34)

2.3 The Nonstationary RSS Model

z ≤ x, ez +

37 n 

αi(t) (yi − (1 − di(t) )xi ) ≤ 1,

(2.35)

i=1

wt (b(t) z) > 0, wt (b(t) z) > u t (x, y) − /4.

(2.36)

By (2.18) and (2.36), there exists q ∈ {1, . . . , n} satisfying bq(t) z q > 0.

(2.37)

Relations (2.18) and (2.37) imply that there exists γ ∈ (0, 1) which satisfies wt (b(t) γz) ≥ wt (b(t) z) − /4.

(2.38)

In view of (2.34), (2.35) and (2.37), there exists an integer j0 ≥ 1 such that for every natural number j ≥ j0 , we have γz ≤ x ( j) , e(γz) +

n 

( j)

αi(t) (yi

( j)

− (1 − di(t) )xi ) ≤ 1.

(2.39)

i=1

It follows from (2.26), (2.36), (2.38) and (2.39) that for all natural numbers j ≥ j0 , we have u t (x ( j) , y ( j) ) ≥ wt (b(t) γz) ≥ wt (b(t) z) − /4 > u t (x, y) − /2. This implies that u t is lower semicontinuous at (x, y) and completes the proof of Lemma 2.10. For every vector x = (x1 , . . . , xn ) ∈ R n put x1 =

n 

|xi |, x∞ = max{|xi | : i = 1, . . . , n}.

(2.40)

i=1

In view of (2.19) and (2.27), for every nonnegative integer t, every pair of points x, y ∈ K and for  ·  =  ·  p , where p = 1, 2, ∞, we have H (at (x), at (y)) n n − ((1 − di(t) )yi )i=1  ≤ (1 − d∗ )x − y ≤ ((1 − di(t) )xi )i=1

(2.41)

(see (2.2)). Proposition 2.11 Let δ be a positive number. Then there exists a positive number λ such that for every nonnegative integer t and every (x, y) ∈ graph(at ) satisfying u t (x, y) ≥ δ the inclusion y + λe ∈ at (x) is valid.

38

2 Infinite Horizon Nonautonomous Optimization Problems

Proof Assumption (A4) implies that there exists a positive number δ0 such that for 1 such that wt (ξ) ≥ δ/2 we have every nonnegative integer t and every ξ ∈ R+

Put

ξ ≥ δ0 .

(2.42)

b∗ = sup{bi(t) : t = 0, 1, . . . , i = 1, . . . , n}

(2.43)

(see (2.23)). Fix a positive number λ for which λnα∗ < 2−1 b∗−1 δ0 .

(2.44)

(x, y) ∈ graph(at ), u t (x, y) ≥ δ.

(2.45)

Assume that an integer t ≥ 0,

n which satisfies In view of (2.26) and (2.45), there exists z ∈ R+

0 ≤ z ≤ x, ez +

n 

αi(t) (yi − (1 − di(t) )xi ) ≤ 1, wt (b(t) z) ≥ δ/2.

(2.46)

i=1

By (2.46) and the choice of δ0 , we have b(t) z ≥ δ0 .

(2.47)

Relations (2.43) and (2.47) imply that ez =

n  i=1

zi =

n 

(bi(t) )−1 bi(t) z i ≥ b∗−1 b(t) z ≥ b∗−1 δ0 .

(2.48)

i=1

We claim that y + λe ∈ at (x). Evidently (see (2.19) and (2.45)), for any i = 1, . . . , n, yi + λ ≥ yi ≥ (1 − di(t) )xi .

(2.49)

In view of (2.27), (2.44), (2.46) and (2.48), n 

αi(t) ((y + λe)i − (1 − di(t) )xi ) =

i=1

≤ 1 − ez + λ

n 

αi(t) (yi − (1 − di(t) )xi ) + λ

i=1 n  i=1

αi(t) ≤ 1 − b∗−1 δ0 + λnα∗ < 1

n  i=1

αi(t)

2.3 The Nonstationary RSS Model

39

and combined with (2.49) this implies that y + λe ∈ at (x). This completes the proof of Proposition 2.11.  Proposition 2.12 There exist a program { x (t)}∞ t=0 and Δ > 0 such that  for all integers t ≥ 0. x (t),  x (t + 1)) ≥ Δ u t ( Proof Fix numbers λ0 > 0, λ1 > 0 such that λ0 nα∗ < 1/2, λ1 < λ0 , λ1 n < 1/4.

(2.50)

In view of (2.21), there exists a positive number 0 satisfying

Set

eb(t) ≥ 0 , t = 0, 1, . . . .

(2.51)

 = inf{wt (λ1 0 ) : t = 0, 1, . . . }. Δ

(2.52)

 > 0. Define Relation (2.25) implies that Δ y(t) = λ1 e, t = 0, 1, . . .  x (t) = λ0 e, t = 0, 1, . . . , 

(2.53)

It follows from (2.27), (2.50) and (2.53) that for i = 1, . . . , n and t = 0, 1, . . . , we have xi (t) = λ0 di(t) > 0,  xi (t + 1) − (1 − di(t) ) n 

αi(t) [ xi (t + 1) − (1 − di(t) ) xi (t)]

i=1

=

(2.54)

n 

αi(t) di(t)

i=1

λ0 ≤ λ0

n 

αi(t) ≤ λ0 nα∗ < 1/2

(2.55)

i=1

and for all t = 0, 1, . . . , we have e y(t) +

n 

αi(t) [ xi (t + 1) − (1 − di(t) ) xi (t)] ≤ λ1 n + 1/2 < 1.

(2.56)

i=1

Thus { x (t)}∞ t=0 is a program. In view of (2.26), (2.50)–(2.53) and (2.56), for every nonnegative integer t, we have  x (t),  x (t + 1)) ≥ wt (b(t) y(t)) ≥ wt (λ1 eb(t) ) ≥ wt (λ1 0 ) ≥ Δ. u t ( This completes the proof of Proposition 2.12.

40

2 Infinite Horizon Nonautonomous Optimization Problems

Proposition 2.13 Let t be a nonnegative integer, (x, y) ∈ graph(at ) and let a point n satisfy x˜ ≥ x. Then there exists y˜ ∈ at (x) ˜ which satisfies that y˜ ≥ y and x˜ ∈ R+ ˜ y˜ ) ≥ u t (x, y). u t (x, n which satisfies Proof In view of (2.26), there exists z ∈ R+

0 ≤ z ≤ x, ez +

n 

αi(t) (yi − (1 − di(t) )xi ) ≤ 1, wt (b(t) z) = u t (x, y).

(2.57)

i=1

For all integers i = 1, . . . , n define y˜i = x˜i (1 − di(t) ) + yi − (1 − di(t) )xi .

(2.58)

It follows from (2.19), (2.57) and (2.58) that for i = 1, . . . , n, we have y˜i ≥ (1 − di(t) )x˜i , n 

αi(t) ( y˜i − (1 − di(t) )x˜i ) =

i=1

n 

αi(t) (yi − (1 − di(t) )xi ) ≤ 1 − ez.

i=1

˜ By the inequality x˜ ≥ x and (2.58), y˜ ≥ y. It is not difficult to see Thus y˜ ∈ at (x). that ˜ y˜ ) ≥ wt (b(t) z) = u t (x, y). u t (x, Proposition 2.13 is proved. It is not difficult to see that the following result holds. n , xi ∈ at (x), i = 1, 3, Proposition 2.14 Let an integer t ≥ 0, x, x1 , x2 , x3 ∈ R+ x1 ≤ x2 ≤ x3 . Then x2 ∈ a(xt ).

Thus we have defined the mappings at and the cost functions u t , t = 0, 1, . . . . The control system considered in this section is a special case of the control system studied in Sect. 2.1. As we have already mentioned, this control system corresponds to the nonstationary RSS model. Note that this control system satisfies the assumptions posed in Sect. 2.1 and therefore all the results stated there hold for this system. Indeed, choose M0 > (α∗ d∗ )−1 and put n : z ≤ M0 e}. K = {z ∈ R+

Lemma 2.9 implies that at (K ) ⊂ K , t = 0, 1, . . . . Relation (2.2) follows from (2.41). It is clear that (2.3) holds. By Lemma 2.10, u t is upper semicontinuous for every nonnegative integer t. Proposition 2.11 implies (A1). Assumption (A2) follows from Proposition 2.12 and assumption (A3) follows from Proposition 2.13.

2.4 Auxiliary Results for Theorems 2.4, 2.5 and 2.7

41

2.4 Auxiliary Results for Theorems 2.4, 2.5 and 2.7 Lemma 2.15 Let δ be a positive number. Then there exists an integer T0 ≥ 4 such that for every natural number τ1 ≥ 0, every natural number τ2 ≥ T0 + τ1 , every 2 satisfying program {x(t)}τt=τ 1 u τ2 −1 (x(τ2 − 1), x(τ2 )) ≥ δ

(2.59)

τ2 ˜ and every x˜0 ∈ K there exists a program {x(t)} t=τ1 which satisfies

˜ 2 ) ≥ x(τ2 ). x(τ ˜ 1 ) = x˜0 , x(τ Proof In view of assumption (A1), there exists a number λ ∈ (0, 1) such that the following property holds: (P1) For every nonnegative integer t and every (x, x ) ∈ graph(at ) which satisfies u t (x, x ) ≥ δ there exists z ∈ at (x) for which z ≥ x + λe. Fix a positive number D0 such that z ≤ D0 for all z ∈ K .

(2.60)

There exists a positive number c0 such that z2 ≤ c0 z for all z ∈ K .

(2.61)

Fix an integer T0 ≥ 4 satisfying 2D0 c0 κT0 < λ

(2.62)

(see (2.2)). 2 satisfies (2.59) Assume that integers τ1 ≥ 0, τ2 ≥ T0 + τ1 , a program {x(t)}τt=τ 1 n and that x˜0 ∈ K . It follows from (2.59) and property (P1) that there exists z ∈ R+ such that (2.63) z ∈ aτ2 −1 (x(τ2 − 1)), z ≥ x(τ2 ) + λe. τ2 −1 In view of (2.2), there exists a program {x(t)} ˜ t=τ1 such that

x(τ ˜ 1 ) = x˜0 , x(t ˜ + 1) − x(t + 1) ≤ κx(t) ˜ − x(t), t = τ1 , . . . , τ2 − 2. By (2.2) and (2.63), there exists ˜ 2 − 1)) x(τ ˜ 2 ) ∈ aτ2 −1 (x(τ

(2.64)

42

2 Infinite Horizon Nonautonomous Optimization Problems

which satisfies ˜ 2 − 1). x(τ ˜ 2 ) − z ≤ κx(τ2 − 1) − x(τ

(2.65)

τ2 Evidently, {x(t)} ˜ t=τ1 is a program. It follows from (2.60), (2.64) and (2.65) that

˜ 1 ) − x(τ1 ) ≤ κτ2 −τ1 (2D0 ) ≤ κT0 (2D0 ) x(τ ˜ 2 ) − z ≤ κτ2 −τ1 x(τ and by (2.61), ||x(τ ˜ 2 ) − z||2 ≤ 2D0 c0 κT0 . The inequality above implies that for all natural numbers i = 1, . . . , n, we have |x˜i (τ2 ) − z i | ≤ 2D0 c0 κT0 and in view of (2.62) and (2.63), we have x(τ ˜ 2 ) ≥ z − 2D0 c0 κT0 e ≥ x(τ2 ) + [λ − 2D0 c0 κT0 ]e ≥ x(τ2 ). This completes the proof of Lemma 2.15. Fix a number γ > 0 satisfying  γ < 1/2 and γ < 4−1 Δ.

(2.66)

Lemma 2.16 Let M1 be a positive number. Then there exist integers L 1 , L 2 ≥ 4 such T2 −1 that for every pair of integers T1 ≥ 0, T2 ≥ L 1 + L 2 + T1 , every program {x(t)}t=T 1 satisfying T 2 −1 u t (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M1 (2.67) t=T1

and every integer τ ∈ [T1 + L 1 , T2 − L 2 ] the following inequality holds: max{u t (x(t), x(t + 1)) : t = τ , . . . , τ + L 2 − 1} ≥ γ.

(2.68)

Proof Lemma 2.15 implies that there exists an integer L 1 ≥ 4 such that the following property holds: S2 satisfies (P2) If integers S1 ≥ 0, S2 ≥ S1 + L 1 , if a program {v(t)}t=S 1 u S2 −1 (v(S2 − 1), v(S2 )) ≥ γ S2 ˜ ˜ 1 ) = v˜0 , v(S ˜ 2) ≥ and if v˜0 ∈ K , then there exists a program {v(t)} t=S1 such that v(S v(S2 ). Fix a number M2 for which

M2 > u t (z, z ) for each integer t ≥ 0 and each (z, z ) ∈ graph(at )

(2.69)

2.4 Auxiliary Results for Theorems 2.4, 2.5 and 2.7

43

and an integer L 2 ≥ 1 which satisfies −1 (M1 + L 1 γ + 1) + 16Δ −1 (M1 + M2 + L 1 + 2). L 2 > 4(L 1 + 1) + 16Δ (2.70) T2 −1 satisfies Assume that integers T1 ≥ 0, T2 ≥ L 1 + L 2 + T1 , a program {x(t)}t=T 1 (2.67) and that an integer τ satisfies T1 + L 1 ≤ τ ≤ T2 − L 2 .

(2.71)

We claim that (2.68) is valid. Assume the contrary. Then u t (x(t), x(t + 1)) < γ, t = τ , . . . , τ + L 2 − 1.

(2.72)

There are two cases: u t (x(t), x(t + 1)) < γ, t = τ , . . . , T2 − 1;

(2.73)

max{u t (x(t), x(t + 1)) : t = τ , . . . , T2 − 1} ≥ γ.

(2.74)

Now we define a natural number τ0 as follows. If (2.73) holds, then we put τ0 = T2 . If (2.74) is valid, then in view of (2.72), there exists an integer τ0 ≥ 1 such that τ + L 2 ≤ τ0 ≤ T2 − 1, u τ0 (x(τ0 ), x(τ0 + 1)) ≥ γ,

(2.75) (2.76)

u t (x(t), x(t + 1)) < γ, t = τ , . . . , τ0 − 1.

(2.77)

It is not difficult to see that in both cases (2.77) is valid and that in both cases τ0 − τ ≥ L 2 .

(2.78)

Assume that (2.73) holds. Assumption (A2), the choice of L 1 , property (P2), (2.66), τ +L 1 (2.70) and (2.71) imply that there exists a program {x(t)} ˜ t=τ which satisfies x (τ + L 1 ). x(τ ˜ ) = x(τ ), x(τ ˜ + L 1) ≥ 

(2.79)

x(t) ˜ = x(t), t = T1 , . . . , τ .

(2.80)

Define In view of (2.79), (2.80), (A3) and (A2), there exists x(t) ˜ ∈ K , t = τ + L1 + T2 ˜ is a program, 1, . . . , T2 such that {x(t)} t=T1 x(t) ˜ ≥ x (t) for all integers t = τ + L 1 , . . . , T2 ,

(2.81)

u t (x(t), ˜ x(t ˜ + 1)) ≥ u t ( x (t),  x (t + 1)), t = τ + L 1 , . . . , T2 − 1.

(2.82)

44

2 Infinite Horizon Nonautonomous Optimization Problems

Assumption (A2), (2.66), (2.67), (2.70), (2.71), (2.73), (2.80) and (2.82) imply that M1 ≥ U (x(T1 ), T1 , T2 ) −

T 2 −1

u t (x(t), x(t + 1))

t=T1

≥

T 2 −1

u t (x(t), ˜ x(t ˜ + 1)) −

t=T1

=

T 2 −1

u t (x(t), ˜ x(t ˜ + 1)) −

τ +L 1 −1 

T 2 −1

T 2 −1

u t (x(t), x(t + 1))

t=τ

u t (x(t), ˜ x(t ˜ + 1)) +

T 2 −1

u t ( x (t),  x (t + 1)) −

T 2 −1

t=τ +L 1

t=τ

≥

u t (x(t), x(t + 1))

t=T1

t=τ

≥

T 2 −1

u t ( x (t),  x (t + 1)) −

t=τ +L 1

T 2 −1

u t (x(t), x(t + 1))

t=τ

u t (x(t), x(t + 1))

t=τ

 − (T2 − τ )γ = (T2 − τ − L 1 )(Δ  − γ) ≥ (T2 − τ − L 1 )Δ −1  (T2 − τ − L 1 ) − L 1 γ − L 1 γ ≥ Δ2  2 − L 1) − L 1γ ≥ 2−1 Δ(L  2 − L 1γ ≥ 4−1 ΔL and

−1 (M1 + L 1 γ). L 2 ≤ 8Δ

The inequality above contradicts (2.70). The contradiction we have reached proves that (2.73) is not true. Thus (2.74) holds and there exists an integer τ0 ≥ 1 satisfying (2.75)–(2.77). Assumption (A2), the choice of L 1 , property (P2), and (2.66) imply τ +L 1 that there exists a program {x(t)} ˜ t=τ such that x (τ + L 1 ). x(τ ˜ ) = x(τ ), x(τ ˜ + L 1) ≥ 

(2.83)

x(t) ˜ = x(t), t = T1 , . . . , τ .

(2.84)

Put By (A2), (A3), (2.75), (2.76) and (2.83), there exist x(t) ˜ ∈ K , t = τ + 1 + L 1, . . . , τ0 −L 1 ˜ is a program, τ0 − L 1 such that {x(t)} t=τ +L 1 x(t) ˜ ≥ x (t), t = τ + L 1 , . . . , τ0 − L 1 ,

(2.85)

u t (x(t), ˜ x(t ˜ + 1)) ≥ u t ( x (t),  x (t + 1)), t = τ + L 1 , . . . , τ0 − L 1 − 1.

(2.86)

τ0 −L 1 Evidently, {x(t)} ˜ t=T1 is a program. In view of the choice of L 1 , property (P2) and τ0 +1 (6.76) there exist x(t) ˜ ∈ K , t = τ0 − L 1 + 1, . . . , τ0 + 1 such that {x(t)} ˜ t=τ0 −L 1 is

2.4 Auxiliary Results for Theorems 2.4, 2.5 and 2.7

45

a program, x(τ ˜ 0 + 1) ≥ x(τ0 + 1).

(2.87)

τ0 +1 It is clear that {x(t)} ˜ t=T1 is a program. If T2 > τ0 + 1, then relation (2.87) and assumpT2 ˜ tion (A3) imply that there exist x(t) ˜ ∈ K , t = τ0 + 2, . . . , T2 such that {x(t)} t=τ0 +1 is a program,

x(t) ˜ ≥ x(t), t = τ0 + 1, . . . , T2 ,

(2.88)

u t (x(t), ˜ x(t ˜ + 1)) ≥ u t (x(t), x(t + 1)), t = τ0 + 1, . . . , T2 − 1.

(2.89)

It follows from (2.66), (2.67), (2.69)–(2.71), (2.74), (2.77), (2.78), (2.84), (2.86), (2.89) and assumption (A2) that M1 ≥ U (x(T1 ), T1 , T2 ) −

T 2 −1

u t (x(t), x(t + 1))

t=T1

≥

T 2 −1

u t (x(t), ˜ x(t ˜ + 1)) −

t=T1

= ≥

u t (x(t), ˜ x(t ˜ + 1)) −

u t (x(t), ˜ x(t ˜ + 1)) −

t=τ

≥

u t (x(t), x(t + 1))

t=T1

T 2 −1 t=τ τ0 

T 2 −1

τ0 −L 1 −1

T 2 −1 t=τ τ0 

u t (x(t), x(t + 1))

u t (x(t), x(t + 1))

t=τ

u t (x(t), ˜ x(t ˜ + 1)) − (τ0 − τ )γ − u τ0 (x(τ0 ), x(τ0 + 1))

t=τ +L 1

≥

τ0 −L 1 −1

u t ( x (t),  x (t + 1)) − (τ0 − τ )γ − u τ0 (x(τ0 ), x(τ0 + 1))

t=τ +L 1

 − γ)(τ0 − τ − 2L 1 ) − 2L 1 γ − M2  0 − τ − 2L 1 ) − (τ0 − τ )γ − M2 = (Δ ≥ Δ(τ  ≥ (Δ/2)(τ 0 − τ − 2L 1 ) − 2L 1 − M2 −1  − 2L 1 − M2  ≥ (Δ/2)(L 2 − 2L 1 ) − 2L 1 − M2 ≥ 4 L 2 Δ

and

 −1 (M1 + M2 + 2L 1 ). L 2 ≤ 4(Δ)

This inequality contradicts (2.70). The contradiction we have reached proves (2.68) and Lemma 2.16 itself. Lemma 2.17 Let M1 be a positive number. Then there exist integers L¯ 1 , L¯ 2 ≥ 1 and a positive number M2 such that for every pair of integers τ1 ≥ 0, τ2 ≥ L¯ 1 + L¯ 2 + τ1 2 and every program {x(t)}τt=τ satisfying 1

46

2 Infinite Horizon Nonautonomous Optimization Problems τ 2 −1

u t (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M1

(2.90)

t=τ1

the following assertion holds. If integers T1 , T2 ∈ [τ1 , τ2 − L¯ 2 ] satisfy L¯ 1 ≤ T2 − T1 , then T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M2 .

(2.91)

t=T1

Proof Let integers L 1 , L 2 ≥ 4 be as guaranteed by Lemma 2.16. Lemma 2.15 implies that there exists an integer L 3 ≥ 4 such that the following property holds: S2 satisfies (P3) If integers S1 ≥ 0, S2 ≥ L 3 + S1 , if a program {v(t)}t=S 1 u S2 −1 (v(S2 − 1), v(S2 )) ≥ γ S2 ˜ ˜ 1 ) = v˜0 , v(S ˜ 2) ≥ and if v˜0 ∈ K , then there exists a program {v(t)} t=S1 such that v(S v(S2 ). Fix a positive a number M0 for which

M0 > u t (z, z ) for every nonnegative integer t and every (z, z ) ∈ graph(at ), (2.92) integers L¯ 1 , L¯ 2 ≥ 1 and a number M2 > 0 satisfying L¯ 1 ≥ L 1 , L¯ 2 > 2(L 1 + L 2 + L 3 + 1),

(2.93)

M2 > M1 + M0 (L 3 + L 2 ).

(2.94)

2 satisfies Assume that integers τ1 ≥ 0, τ2 ≥ L¯ 1 + L¯ 2 + τ1 , a program {x(t)}τt=τ 1 (2.90) and integers T1 , T2 satisfy

T1 , T2 ∈ [τ1 , τ2 − L¯ 2 ], L¯ 1 ≤ T2 − T1 .

(2.95)

We claim that (2.91) is true. Proposition 2.3 implies that there exists a program T2 which satisfies {x (1) (t)}t=T 1 x (1) (T1 ) = x(T1 ),

T 2 −1

u t (x (1) (t), x (1) (t + 1)) = U (x(T1 ), T1 , T2 ).

(2.96)

t=T1

It follows from (2.93) and (2.95) that T1 + L 1 ≤ T1 + L¯ 1 + L 3 ≤ T2 + L 3 ≤ τ2 − L¯ 2 + L 3 ≤ τ2 − 2L 2 − L 3 . (2.97) By the choice of the integers L 1 , L 2 , Lemma 2.16, (2.90), (2.93) and (2.97), we have

2.4 Auxiliary Results for Theorems 2.4, 2.5 and 2.7

47

max{u t (x(t), x(t + 1)) : t = T2 + L 3 , . . . , T2 + L 2 + L 3 − 1} ≥ γ. Hence there exists an integer τ ∈ [T2 + L 3 , . . . , T2 + L 3 + L 2 − 1] for which u τ (x(τ ), x(τ + 1)) ≥ γ.

(2.98)

+1 such that Property (P3) and (2.98) imply that there exists a program {x (2) (t)}τt=T 2

x (2) (T2 ) = x (1) (T2 ), x (2) (τ + 1) ≥ x(τ + 1).

(2.99)

Define ˜ = x (1) (t), t = T1 + 1, . . . , T2 , x(t) ˜ = x(t), t = τ1 , . . . , T1 , x(t) x(t) ˜ = x (2) (t), t = T2 + 1, . . . , τ + 1.

(2.100)

τ +1 It is easy to see that {x(t)} ˜ t=τ1 is a program. By (2.99) and (2.100), we have

x(τ ˜ + 1) ≥ x(τ + 1).

(2.101)

Assumption (A3) and (2.101) imply that there exist x(t) ˜ ∈ K , t = τ + 2, . . . , τ2 τ2 such that (x(t)} ˜ t=τ1 is a program, x(t) ˜ ≥ x(t), t = τ + 1, . . . , τ2 , u t (x(t), ˜ x(t ˜ + 1)) ≥ u t (x(t), x(t + 1)), t = τ + 1, . . . , τ2 − 1.

(2.102) (2.103)

¯ By (2.90), (2.92), (2.94), (2.96), (2.100), (2.103) and the choice of L, M1 ≥ U (x(τ1 ), τ1 , τ2 ) −

τ 2 −1

u t (x(t), x(t + 1))

t=τ1

≥

τ 2 −1

u t (x(t), ˜ x(t ˜ + 1)) −

t=τ1

= ≥

τ 2 −1 t=T1 τ 

T 2 −1 t=T1

u t (x(t), x(t + 1))

t=τ1

u t (x(t), ˜ x(t ˜ + 1)) − u t (x(t), ˜ x(t ˜ + 1)) −

t=T1

≥

τ 2 −1

τ 2 −1 t=T1 τ 

u t (x(t), x(t + 1)) u t (x(t), x(t + 1))

t=T1

u t (x(t), ˜ x(t ˜ + 1)) −

T 2 −1 t=T1

u t (x(t), x(t + 1)) −

τ  t=T2

u t (x(t), x(t + 1))

48

2 Infinite Horizon Nonautonomous Optimization Problems

≥ U (x(T1 ), T1 , T2 ) −

T 2 −1

u t (x(t), x(t + 1)) − (τ − T2 + 1)M0

t=T1

and T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M1 − M0 (L 3 + L 2 )

t=T1

> U (x(T1 ), T1 , T2 ) − M2 . This completes the proof of Lemma 2.17.

2.5 Properties of the Function U It is not difficult to see that the following auxiliary result holds. Proposition 2.18 Let τ1 ≥ 0, τ1 > τ1 be integers, Δ ≥ 0, T1 , T2 be integers such 2 be a program satisfying that τ1 ≤ T1 < T2 ≤ τ2 and let {x(t)}τt=τ 1 τ 2 −1

u t (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − Δ.

t=τ1

Then

T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − Δ.

t=T1

Lemma 2.19 There exist an integer L ≥ 1 and a positive number M1 such that for every pair of points x0 , x˜0 ∈ K and every pair of integers T1 ≥ 0, T2 ≥ T1 + L the inequality |U (x0 , T1 , T2 ) − U (x˜0 , T1 , T2 )| ≤ M1 is valid. Proof Let integers L 1 , L 2 ≥ 4 be as guaranteed by Lemma 2.16 with M1 = 1. Lemma 2.15 implies that there exists a natural number L 3 ≥ 4 such that the following property holds: S2 satisfies (P4) If integers S1 ≥ 0, S2 ≥ S1 + L 3 , a program {v(t)}t=S 1 u S2 −1 (v(S2 − 1), v(S2 )) ≥ γ S2 ˜ ˜ 1 ) = v˜0 , v(S ˜ 2) ≥ and if v˜0 ∈ K , then there exists a program {v(t)} t=S1 such that v(S v(S2 ).

2.5 Properties of the Function U

49

Fix an integer

a number

L > 2(L 1 + L 2 + L 3 + 1),

(2.104)

M0 > u t (z, z ), t = 0, 1, . . . , (z, z ) ∈ graph(at )

(2.105)

M1 = M0 (L 1 + L 2 + L 3 ).

(2.106)

and set Assume that x0 , x˜0 ∈ K and that integers T1 ≥ 0, T2 ≥ T1 + L. Proposition 2.3 T2 which satisfies implies that there exists a program {x(t)}t=T 1 x(T1 ) = x0 ,

T 2 −1

u t (x(t), x(t + 1)) = U (x0 , T1 , T2 ).

(2.107)

t=T1

By (2.104), we have T1 + L 1 + L 3 < T1 + L − L 2 ≤ T2 − L 2 .

(2.108)

By the choice of L 1 , L 2 , Lemma 2.16, (2.104) and (2.107), max{u t (x(t), x(t + 1)) : t = L 3 + L 1 + T1 , . . . , L 3 + L 1 + L 2 + T1 − 1} ≥ γ. Thus there exists an integer τ ∈ {T1 + L 1 + L 3 , . . . , T1 + L 3 + L 1 + L 2 − 1}

(2.109)

u τ (x(τ ), x(τ + 1)) ≥ γ.

(2.110)

such that

Property (P4), the choice of L 3 , (2.109) and (2.110) imply that there exists a program τ +1 {x(t)} ˜ t=T1 satisfying ˜ + 1) ≥ x(τ + 1). x(T ˜ 1 ) = x˜0 , x(τ

(2.111)

Assumption (A3) and (2.111) imply that there exist x(t) ˜ ∈ K , t = τ + 2, . . . , T2 T2 such that {x(t)} ˜ is a program, t=τ +1 x(t) ˜ ≥ x(t), t = τ + 1, . . . , T2 ,

(2.112)

u t (x(t), ˜ x(t ˜ + 1)) ≥ u t (x(t), x(t + 1)), t = τ + 1, . . . , T2 − 1.

(2.113)

T2 Evidently, {x(t)} ˜ t=T1 is a program. In view of (2.105)–(2.107), (2.109), (2.111) and (2.113), we have

50

2 Infinite Horizon Nonautonomous Optimization Problems U (x˜0 , T1 , T2 ) ≥

T 2 −1

u t (x(t), ˜ x(t ˜ + 1)) =

t=T1

−

T 2 −1

u t (x(t), x(t + 1))

t=T1

T 2 −1

u t (x(t), x(t + 1)) +

t=T1

⎡

≥ U (x0 , T1 , T2 ) − ⎣

u t (x(t), ˜ x(t ˜ + 1))

t=T1 τ  t=T1

≥ U (x0 , T1 , T2 ) −

T 2 −1

τ 

u t (x(t), x(t + 1)) −

τ 

⎤ u t (x(t), ˜ x(t ˜ + 1))⎦

t=T1

u t (x(t), x(t + 1)) ≥ U (x0 , T1 , T2 ) − (τ − T1 )M0

t=T1

≥ U (x0 , T1 , T2 ) − (L 1 + L 2 + L 3 )M0 = U (x0 , T1 , T2 ) − M1 .

Therefore we have shown that for each x0 , x˜0 ∈ K and each pair of integers T1 ≥ 0, T2 ≥ T1 + L, U (x˜0 , T1 , T2 ) ≥ U (x0 , T1 , T2 ) − M1 . Lemma 2.19 is proved. Corollary 2.20 There exists a positive number M1 and an integer L ≥ 1 such that for every pair of integers T1 ≥ 0, T2 ≥ T1 + L and every x0 ∈ K the inequality (T1 , T2 )| ≤ M1 |U (x0 , T1 , T2 ) − U holds. Lemmas 2.16 and 2.17 and Corollary 2.20 imply the following auxiliary result. Lemma 2.21 Let M1 > 0. Then there exist natural numbers L¯ 1 , L¯ 2 and M2 > 0 such that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L¯ 1 + L¯ 2 and each program 2 which satisfies {x(t)}τt=τ 1 τ 2 −1

u t (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M1

t=τ1

the following assertion holds: If integers T1 , T2 ∈ [τ1 , τ2 − L¯ 2 ] satisfy L¯ 1 ≤ T2 − T1 , then T 2 −1 t=T1

(T1 , T2 ) − M2 . u t (x(t), x(t + 1)) ≥ U

2.6 Proof of Theorem 2.4

51

2.6 Proof of Theorem 2.4 Let M1 = 1 and let integers L¯ 1 , L¯ 2 ≥ 1 and a positive number M2 be as guaranteed by Lemma 2.21. Let x0 ∈ K be given. Proposition 2.3 implies that for every integer k ≥ 1, there exists a program {x (k) (t)}kt=0 satisfying x (k) (0) = x0 ,

k−1 

u t (x (k) (t), x (k) (t + 1)) = U (x0 , 0, k).

(2.114)

t=0

By the choice of L¯ 1 , L¯ 2 , M2 and Lemma 2.21 the following property holds: (i) For every natural number k ≥ L¯ 1 + L¯ 2 and every pair of integers T1 , T2 ∈ [0, k − L¯ 2 ] satisfying L¯ 1 ≤ T2 − T1 , T 2 −1

(T1 , T2 ) − M2 . u t (x (k) (t), x (k) (t + 1)) ≥ U

t=T1

Evidently, there exists a strictly increasing sequence of natural numbers {k j }∞ j=1 such that for every nonnegative integer t there exists x(t) ¯ = lim x (k j ) (t). j→∞

(2.115)

∞ It is clear that {x(t)} ¯ t=0 is a program. By (2.114) and (2.115), we have

x(0) ¯ = x0 .

(2.116)

In view of (2.115), property (i) and upper semicontinuity of the functions u t , t = 0, 1, . . . , the following property holds: (ii) For each pair of nonnegative integers T1 , T2 which satisfy T2 − T1 ≥ L¯ 1 , T −1  2    (T1 , T2 ) ≤ M2 . u t (x(t), ¯ x(t ¯ + 1)) − U   

(2.117)

t=T1

Fix a number M0 > 0 such that M0 > u t (z, z ) for every integer t ≥ 0 and every (z, z ) ∈ graph(at ). Put

M = M2 + M0 L¯ 1 .

(2.118)

(2.119)

Assume that nonnegative integers T1 , T2 satisfy T1 < T2 . If T2 − T1 ≥ L¯ 1 , then property (ii), (2.117) and (2.118) imply

52

2 Infinite Horizon Nonautonomous Optimization Problems

T −1  2    (T1 , T2 ) ≤ M2 ≤ M. u t (x(t), ¯ x(t ¯ + 1)) − U    t=T1

If T2 − T1 ≤ L¯ 1 , then in view of (2.118) and (2.119), we have T −1  2    (T1 , T2 ) ≤ (T2 − T1 )M0 ≤ M0 L¯ 1 < M. u t (x(t), ¯ x(t ¯ + 1)) − U    t=T1

Thus in both cases T −1  2    (T1 , T2 ) ≤ M. u t (x(t), ¯ x(t ¯ + 1)) − U   

(2.120)

t=T1

Assume now that the following properties hold: (iii) For every nonnegative integer t and every (z, z ) ∈ graph(at ) satisfying u t (z, z ) > 0 the function u t is continuous at (z, z ). (iv) If an integer t ≥ 0 and z, z 1 , z 2 , z 3 ∈ K satisfy z i ∈ at (z), i = 1, 3 and z 1 ≤ z 2 ≤ z 3 , then z 2 ∈ at (z). In order to complete the proof of the theorem it is sufficient to show that for every natural number T , we have T −1 

u t (x(t), ¯ x(t ¯ + 1)) = U (x(0), x(T ), 0, T ).

(2.121)

t=0

Denote by E the set of all natural numbers τ such that ¯ − 1), x(τ ¯ )) > 0. u τ −1 (x(τ

(2.122)

Assumption (A2) and (2.120) imply that the set E is infinite. By Proposition 2.18, it is sufficient to show that (2.121) is valid for all T = τ − 1, where τ ∈ E. Let τ ∈ E and T = τ − 1. We claim that (2.121) holds. Assume the contrary. T and a number Δ > 0 such that Then there exist a program {x(t)}t=0 x(0) = x(0), ¯ x(T ) ≥ x(T ¯ ), T −1  t=0

u t (x(t), x(t + 1)) ≥

T −1 

(2.123) u t (x(t), ¯ x(t ¯ + 1)) + 2Δ.

(2.124)

t=0

In view of the inclusion τ ∈ E and the definition of E, ¯ ), x(T ¯ + 1)) = u τ −1 (x(τ ¯ − 1), x(τ ¯ )) > 0. u T (x(T

(2.125)

2.6 Proof of Theorem 2.4

53

It follows from (2.125) and (A1) that there exists a number λ0 ∈ (0, 1) and ¯ − 1)) = aT (x(T ¯ )) z 0 ∈ aτ −1 (x(τ

(2.126)

¯ ) + λ0 e = x(T ¯ + 1) + λ0 e. z 0 ≥ x(τ

(2.127)

such that There exists a number c0 > 1 such that y ≤ c0 y2 ≤ c02 y for all y ∈ R n .

(2.128)

In view of (2.125), (2.127) and properties (iii) and (iv) we may assume without loss of generality that ¯ − 1), z 0 ) − u τ −1 (x(τ ¯ − 1), x(τ ¯ ))| ≤ Δ/4. |u τ −1 (x(τ

(2.129)

Assumption (A3), (2.123) and (2.126) imply that there exists z 1 ∈ aT (x(T )) such that ¯ ), z 0 ). (2.130) z 1 ≥ z 0 , u T (x(T ), z 1 ) ≥ u T (x(T Fix a positive number

δ < min{1, λ0 , Δτ −1 }.

(2.131)

∞ By the construction of the program {x(t)} ¯ t=0 (see (2.115)) and upper semicontinuity of u t , t = 0, 1, . . . there is a natural number k > τ + 4 such that

¯ x (k) (t) − x(t) 2 ≤ δ, t = 0, . . . , τ + 2, u t (x

(k)

(t), x

(k)

(t + 1)) ≤ u t (x(t), ¯ x(t ¯ + 1)) + δ, t = 0, . . . , τ + 2.

(2.132) (2.133)

Define x(t) ˜ = x(t), t = 0, . . . , τ − 1.

(2.134)

We claim that z 1 ≥ x (k) (τ ). In view of (2.132), ¯ )2 ≤ δ. x (k) (τ ) − x(τ

(2.135)

By (2.127), (2.130), (2.131) and (2.135), we have ¯ ) + δe ≤ x(τ ¯ ) + λ0 e ≤ z 0 ≤ z 1 . x (k) (τ ) ≤ x(τ

(2.136)

x(τ ˜ ) = z1.

(2.137)

Put

Since

54

2 Infinite Horizon Nonautonomous Optimization Problems

z 1 ∈ aT (x T ) = aτ −1 (x˜τ −1 ), τ {x(t)} ˜ t=0 is a program. In view of (2.116), (2.123), (2.134), (2.136) and (2.137),

˜ ) ≥ x (k) (τ ). x(0) ˜ = x(0) ¯ = x0 , x(τ

(2.138)

By (2.124), (2.129), (2.130), (2.134), (2.137) and the equality T = τ − 1, τ −1 

u t (x(t), ˜ x(t ˜ + 1)) −

t=0

≥

τ −1 

u t (x(t), ¯ x(t ¯ + 1))

t=0

τ −2 

u t (x(t), x(t + 1)) + u τ −1 (x(τ ˜ − 1), x(τ ˜ )) −

t=0

≥

τ −1 

u t (x(t), ¯ x(t ¯ + 1))

t=0

τ −2 

u t (x(t), ¯ x(t ¯ + 1)) + 2Δ + u τ −1 (x(τ − 1), z 1 ) −

t=0

τ −1 

u t (x(t), ¯ x(t ¯ + 1))

t=0

≥ 2Δ +

τ −2 

u t (x(t), ¯ x(t ¯ + 1)) + u τ −1 (x(τ ¯ − 1), z 0 ) −

t=0

τ −1 

u t (x(t), ¯ x(t ¯ + 1))

t=0

≥ 2Δ + u τ −1 (x(τ ¯ − 1), z 0 ) − u τ −1 (x(τ ¯ − 1), x(τ ¯ )) ≥ (3/2)Δ.

(2.139)

It follows from (2.131), (2.133) and (2.139) that τ −1 

u t (x(t), ˜ x(t ˜ + 1)) −

t=0

=

τ −1 

u t (x (k) (t), x (k) (t + 1))

t=0

u t (x(t), ˜ x(t ˜ + 1)) −

t=0

−

τ −1 

τ −1 

τ −1 

u t (x(t), ¯ x(t ¯ + 1)) +

t=0

τ −1 

u t (x(t), ¯ x(t ¯ + 1))

t=0

u t (x (k) (t), x (k) (t + 1)) ≥ (3/2)Δ − δτ ≥ Δ/2.

(2.140)

t=0

In view of (2.138) and (2.140), we have U (x0 , x (k) (τ ), 0, τ ) ≥

τ −1 

u t (x (k) (t), x (k) (t + 1)) + Δ/2.

t=0

This inequality contradicts (2.114). The contradiction we have reached proves that (2.121) holds for all T = τ − 1, where τ ∈ E. Theorem 2.4 is proved.

2.7 Proof of Theorem 2.5

55

2.7 Proof of Theorem 2.5 In the sequel we assume that the sum over an empty set is zero. There exist a positive ∞ such that number Δ and a strictly increasing sequence of natural numbers {τi }i=1 τ1 ≥ 4 and that u τi −1 (x(τi−1 ), x(τi )) ≥ Δ for all natural numbers i.

(2.141)

Let a positive number M be as guaranteed by Theorem 2.4. Lemma 2.15 implies that there exists an integer L 0 ≥ 4 such that the following property holds: (P5) For every nonnegative integer S1 , every integer S2 ≥ S1 + L 0 , every program S2 satisfying {v(t)}t=S 1 u S2 −1 (v(S2 − 1), v(S2 )) ≥ Δ S2 ˜ ˜ 1 ) = v˜0 , v(S ˜ 2) ≥ and every v˜0 ∈ K , there exists a program {v(t)} t=S1 such that v(S v(S2 ). Corollary 2.20 and (2.3) imply that there exists a positive number M∗ such that

(T1 , T2 )| ≤ M∗ |U (v0 , T1 , T2 ) − U for every v0 ∈ K and every pair of integers T1 < T2 , (2.142) u t (z, z ) ≤ M∗ for every nonnegative integer t, and every (z, z ) ∈ graph(at ). (2.143) Fix a number M1 > L 0 M∗ + M0 + 3M.

(2.144)

∞ Theorem 2.4 implies that there exists a program {x(t)} ¯ t=0 such that

x(0) ¯ = x(0)

(2.145)

and that for each pair of integers S1 , S2 satisfying S1 < S2 ,  S −1  2    (S1 , S2 ) ≤ M. u t (x(t), ¯ x(t ¯ + 1)) − U   

(2.146)

t=S1

Assume that T1 , T2 are integers such that 0 ≤ T1 < T2 . We claim that T −1  2    (T1 , T2 ) ≤ M1 . u t (x(t), x(t + 1)) − U    t=T1

If T2 ≤ T1 + L 0 , then this inequality follows from (2.143) and (2.144).

(2.147)

56

2 Infinite Horizon Nonautonomous Optimization Problems

Assume that T2 > T1 + L 0 . There exists an integer i ≥ 1 such that τi > T2 + 2L 0 .

(2.148)

τi By (6.141), (6.148) and (P5), there exists a program {x(t)} ˜ t=τi −L 0 such that

¯ i − L 0 ), x(τ ˜ i ) ≥ x(τi ). x(τ ˜ i − L 0 ) = x(τ

(2.149)

x(t) ˜ = x(t), ¯ t = 0, . . . , τi − L 0 − 1.

(2.150)

Define τi Evidently, {x(t)} ˜ t=0 is a program and by (2.149), (2.150) and (2.145), τ i −1

u t (x(t), x(t + 1)) ≥

t=0

τ i −1

u t (x(t), ˜ x(t ˜ + 1)) − M0 .

(2.151)

t=0

By (2.143) and (2.151), τ i −1

u t (x(t), x(t + 1)) ≥

t=0

τi −L 0 −1 

u t (x(t), ¯ x(t ¯ + 1)) − M0

t=0

≥

τ i −1

u t (x(t), ¯ x(t ¯ + 1)) − M0 − L 0 M∗ .

t=0

Together with (2.146) this implies that − (M0 + L 0 M∗ ) ≤

τ i −1

u t (x(t), x(t + 1)) −

t=0

≤

u t (x(t), ¯ x(t ¯ + 1))

t=0

  {u t (x(t), x(t + 1)) : 0 ≤ t < T1 } − {u t (x(t), ¯ x(t ¯ + 1)) : 0 ≤ t < T1 }

+

T 2 −1

u t (x(t), x(t + 1)) −

t=T1

+

τ i −1

≤M+

T 2 −1

u t (x(t), ¯ x(t ¯ + 1))

t=T1

u t (x(t), x(t + 1)) −

t=T2 T 2 −1

τ i −1

u t (x(t), ¯ x(t ¯ + 1))

t=T2

(T1 , T2 ) − M) + U (T2 , τi ) − u t (x(t), x(t + 1)) − (U

t=T1

≤

τ i −1

T 2 −1

(T1 , T2 ) + 3M u t (x(t), x(t + 1)) − U

t=T1

and combined with (2.144) this implies that

τi  t=T2

u t (x(t), ¯ x(t ¯ + 1))

2.7 Proof of Theorem 2.5 T 2 −1

57

(T1 , T2 ) ≥ −3M − (M0 + L 0 M∗ ) > −M1 . u t (x(t), x(t + 1)) − U

t=T1

This completes the proof of Theorem 2.5.

2.8 Proof of Theorem 2.7 ∞ Let x0 ∈ K , M > 0 and let {x(t)} ¯ t=0 be as guaranteed by Theorem 2.4. Then for every pair of integers T1 , T2 ≥ 0 satisfying T1 < T2 ,

T −1  2    (T1 , T2 ) ≤ M. u t (x(t), ¯ x(t ¯ + 1)) − U   

(2.152)

t=T1

Fix a positive number Δ such that Δ > u(z, z ) for each (z, z ) ∈ graph(a).

(2.153)

Let p be a natural number. We show that for all sufficiently large natural numbers T,

  T −1     −1   u(x(t), ¯ x(t ¯ + 1)) ≤ 2M/ p.  p U (0, p) − T −1  

(2.154)

t=0

Assume that T ≥ p is a natural number. Then there exist integers q, s such that q ≥ 1, 0 ≤ s < p, T = pq + s.

(2.155)

By (2.155) we have T −1

T −1

⎛ (0, p) = T −1 ⎝ u(x(t), ¯ x(t ¯ + 1)) − p −1 U

t=0

+

pq−1 

u(x(t), ¯ x(t ¯ + 1))

t=0

  (0, p) {u(x(t), ¯ x(t ¯ + 1)) : t is an integer such that pq ≤ t ≤ T − 1} − p −1 U

= T −1

 {u(x(t), ¯ x(t ¯ + 1)) : t is an integer such that pq ≤ t ≤ T − 1}

+ (T −1 pq)( pq)−1

q−1  (i+1) p−1

⎡

i=0

(0, p) u(x(t), ¯ x(t ¯ + 1)) − p −1 U

t=i p

⎤

q−1  (i+1) p−1 (0, p)) + q U (0, p)⎦ − p −1 U (0, p) = (T −1 pq)( pq)−1 ⎣ ( u(x(t), ¯ x(t ¯ + 1)) − U

+ T −1



i=0

t=i p

 u(x(t), ¯ x(t ¯ + 1)) : t is an integer such that pq ≤ t ≤ T − 1 .

(2.156)

58

2 Infinite Horizon Nonautonomous Optimization Problems

In view of (2.152), (2.153), (2.155) and (2.156), |T −1

T −1 

(0, p)| u(x(t), ¯ x(t ¯ + 1)) − p −1 U

t=0

(0, p)|q/T − 1/ p| pΔ + ( pq)−1 q M + U (0, p)s( pT )−1 → M/ p as T → ∞. ≤ T −1 pΔ + M/ p + U

≤T

−1

Since p is an arbitrary natural number we conclude that  T

−1

T −1 

∞ u(x(t), ¯ x(t ¯ + 1)) T =1

t=0

is a Cauchy sequence. Evidently, there exists lim T −1

T →∞

T −1 

u(x(t), ¯ x(t ¯ + 1))

t=0

and for every integer p ≥ 1, we have   T −1     −1   u(x(t), ¯ x(t ¯ + 1)) ≤ 2M/ p.  p U (0, p) − lim T −1 T →∞  

(2.157)

t=0

Since (2.157) is true for every integer p ≥ 1 we conclude that lim T −1

T →∞

Define

T −1 

(0, p)/ p. u(x(t), ¯ x(t ¯ + 1)) = lim U

(2.158)

(0, p)/ p. μ = lim U

(2.159)

p→∞

t=0

p→∞

It follows from (2.157)–(2.159) that, for every integer p ≥ 1, we have (0, p) − μ| ≤ 2M/ p. | p −1 U This completes the proof of Theorem 2.7.

2.9 Overtaking Optimal Programs

59

2.9 Overtaking Optimal Programs In this section we study the existence of overtaking optimal solutions for a large class of infinite horizon discrete-time optimal control problems. This class contains optimal control problems arising in economic dynamics which describe a model proposed by Robinson, Solow and Srinivasan with nonconcave utility functions representing the preferences of the planner. The results of this section were obtained in [107]. We continue to use the notation and definitions introduced in Sect. 2.1. In particular, we assume that for every nonnegative integer t, at : K → P(K ) is such that graph(at ) is a closed subset of R n × R n . We also suppose that there exists κ ∈ (0, 1) such that for every pair of points x, y ∈ K and every nonnegative integer t, we have H (at (x), at (y)) ≤ κx − y

(2.160)

and that for every nonnegative integer t, the upper semicontinuous function u t : graph(at ) → [0, ∞) satisfies lim sup{u t (x, x ) : (x, x ) ∈ graph(at )} = 0.

t→∞

(2.161)

In Sect. 2.1 we introduced assumptions (A1)–(A3). Here we assume that (A3) holds, but we do not assume (A1) and (A2). Namely, we assume: (A3) for every nonnegative integer t, every (x, y) ∈ graph(at ) and every x˜ ∈ K ˜ for which which satisfies x˜ ≥ x there exists y˜ ∈ at (x) ˜ y˜ ) ≥ u t (x, y). y˜ ≥ y, u t (x, We also suppose that the following assumptions hold: (A5) there exist a positive γ and a sequence of positive numbers {Δt }∞ t=0 such that: (i) for every nonnegative integer s and every point z 0 ∈ K there exists a sequence {x(t)}∞ t=s ⊂ K such that x(s) = z 0 , x(t + 1) ∈ at (x(t)) for all integers t ≥ s and that u t (x(t), x(t + 1)) ≥ Δt for every natural number t ≥ s + 1; (ii) for every nonnegative integer t, if (x, x ) ∈ graph(at ) satisfies u t (x, x ) ≥ Δt , then there exists z ∈ at (x) such that z ≥ x + γe; For every point x0 ∈ K and every natural number T define U (x0 , T ) = sup

T −1  t=0

T −1 {x(t)}t=0

u t (x(t), x(t + 1)) :

 is a program and x(0) = x0 .

(2.162)

60

2 Infinite Horizon Nonautonomous Optimization Problems

Upper semicontinuity of u t , t = 0, 1, . . . implies the following result. Proposition 2.22 For every x0 ∈ K and every integer T ≥ 1 there exists a program T such that x(0) = x0 and {x(t)}t=0 T −1 

u t (x(t), x(t + 1)) = U (x0 , T ).

t=0

We prove the following theorem. Theorem 2.23 For every z ∈ K there exists a program {x z (t)}∞ t=0 such that x z (0) = z and the following assertion holds: For every positive number δ there exists an integer L (δ) ≥ 1 such that for every natural number S ≥ L (δ) and every z ∈ K , S−1 

u t (x z (t), x z (t + 1)) ≥ U (z, S) − δ.

t=0

Theorem 2.23 easily implies the following corollary. Corollary 2.24 Let z ∈ K and let a program {x z (t)}∞ t=0 be as guaranteed by Theorem satisfying x(0) = z, 2.23. Then for every program {x(t)}∞ t=0 lim sup T →∞

T −1 

u t (x(t), x(t + 1)) −

t=0

T −1 

u t (x z (t), x z (t + 1)) ≤ 0.

t=0

2.10 Applications to the Nonstationary RSS Model In this section we consider a subclass of the class of infinite horizon optimal control problems considered in Sect. 2.9. Infinite horizon problems of this subclass correspond to the nonstationary RSS model. For every nonnegative integer t let α(t) = (α1(t) , . . . , αn(t) )  0, b(t) = (b1(t) , . . . , bn(t) )  0, d

(t)

=

(d1(t) , . . . , dn(t) )

∈ ((0, 1])

(2.163)

n

and for every nonnegative integer t let wt : [0, ∞) → [0, ∞) be a strictly increasing continuous function such that wt (0) = 0, lim wt (z) = 0 for all z > 0. t→∞

n Let t ≥ 0 be an integer. For every x ∈ R+ define

(2.164)

2.10 Applications to the Nonstationary RSS Model

61



n at (x) = y ∈ R+ : yi ≥ (1 − di(t) )xi , i = 1, . . . , n,  n  (t) (t) αi (yi − (1 − di )xi ) ≤ 1 .

(2.165)

i=1

It is not difficult to see that for every x ∈ R n , at (x) is a nonempty closed bounded n n n and graph(at ) is a closed subset of R+ × R+ . Suppose that subset of R+ inf{di(t) : i = 1, . . . , n, t = 0, 1, . . . } > 0, inf{eb

(t)

: t = 0, 1, . . . } > 0,

inf{αi(t) : sup{bi(t) : sup{αi(t) :

(2.166) (2.167)

i = 1, . . . , n, t = 0, 1, . . . } > 0,

(2.168)

i = 1, . . . , n, t = 0, 1, . . . } < ∞,

(2.169)

i = 1, . . . , n, t = 0, 1, . . . } < ∞.

(2.170)

The constraint mappings at , t = 0, 1, . . . have already been defined. Let us now define the cost functions u t , t = 0, 1, . . . . For every nonnegative integer t and every (x, x ) ∈ graph(at ) define  u t (x, x ) = sup wt (b(t) y) : 0 ≤ y ≤ x,

 ≤1 .

(2.171)

α∗ < αi(t) < α∗ , d∗ < di(t) , i = 1, . . . , n, t = 0, 1, . . . .

(2.172)

ey +

n 

αi(t) (xi

− (1 −

di(t) )xi )

i=1

Fix α∗ , α∗ > 0, d∗ > 0 such that

Lemma 2.25 Let a number M0 > (α∗ d∗ )−1 , an integer t ≥ 0 and let (x, x ) ∈ graph(at ) satisfy x ≤ M0 e. Then x ≤ M0 e. For the proof see Lemma 2.9. Lemma 2.26 Let t ≥ 0 be an integer. Then the function u t : graph(at ) → [0, ∞) is upper semicontinuous. For the proof see Lemma 2.10. n Proposition 2.27 Let t ≥ 0 be an integer, (x, y) ∈ graph(at ) and let x˜ ∈ R+ satisfy ˜ such that y˜ ≥ y and u t (x, ˜ y˜ ) ≥ u t (x, y). x˜ ≥ x. Then there is y˜ ∈ at (x)

For the proof see Proposition 2.13. Set

62

2 Infinite Horizon Nonautonomous Optimization Problems

β = inf{b(t) e : t = 0, 1, . . . }, b∗ = sup{bi(t) : i = 1, . . . , n, t = 0, 1, . . . }. (2.173) In view of (2.167) and (2.169), β > 0 and b∗ < ∞. Proposition 2.28 There exist γ > 0 and a sequence of positive numbers {Δt }∞ t=0 such that: n there exists a sequence (i) for every nonnegative integer s and every x0 ∈ R+ ∞ n {x(t)}t=s ⊂ R+ such that x(s) = x0 , x(t + 1) ∈ at (x(t)) for every integer t ≥ s and u t (x(t), x(t + 1)) ≥ Δt for every integer t ≥ s + 1; (ii) for every nonnegative integer t if (x, x ) ∈ graph(at ) satisfies u t (x, x ) ≥ Δt , then x + γe ∈ at (x). Proof Choose numbers λ0 , λ1 ∈ (0, 1) for which λ0 n(1 + α∗ ) < 1 and λ1 < λ0

(2.174)

and set γ = (b∗ )−1 βλ1 n −1 α∗−1 , Δt = wt (βλ1 ), t = 0, 1, . . . .

(2.175)

n . Define Let an integer s ≥ 0 and x0 ∈ R+

x(s) = x0 , y(s) = 0, xi (s + 1) = (1 − di(s) )xi (s) + λ0 for i = 1, . . . , n

(2.176)

and for every integer t ≥ s + 1 define xi (t + 1) = (1 − di(t) )xi (t) + λ0 , i = 1, . . . , n, y(t) = λ1 e.

(2.177)

It follows from (2.165), (2.172), (2.174), (2.176) and (2.177) that x(t + 1) ∈ at (x(t)) for all integers t ≥ s. By (2.172), (2.174) and (2.177), for all integers t > s, ey(t) +

n 

αi(t) [xi (t + 1) − (1 − di(t) )xi (t)]

i=1

=

n 

αi(t) λ0 + λ1 n ≤ α∗ λ0 n + λ1 n ≤ λ0 n(1 + α∗ ) < 1.

i=1

Combined with (2.171), (2.173), (2.175) and (2.177) this implies that for all integers t ≥ s + 1, u t (x(t), x(t + 1)) ≥ wt (b(t) λ1 e) ≥ wt (λ1 β) = Δt . Therefore (i) is true.

2.10 Applications to the Nonstationary RSS Model

63

Let us now prove that the property (ii) is true. Assume that an integer t ≥ 0 and that (2.178) (x, x ) ∈ graph(at ), u t (x, x ) ≥ Δt . In follows from (2.165), (2.171) and (2.178) that xi ≥ (1 − di(t) )xi , i = 1, . . . , n

(2.179)

n satisfying and there exists y ∈ R+

y ≤ x, ey +

n 

αi(t) (xi − (1 − di(t) )xi ) ≤ 1,

i=1 (t)

wt (b y) = u t (x, x ).

(2.180)

In view of (2.175), (2.178) and (2.180), we have wt (b(t) y) ≥ Δt ≥ wt (βλ1 ) and since the function wt is strictly increasing, relation (2.173) implies that βλ1 ≤ b(t) y ≤ b∗ ey and

ey ≥ (b∗ )−1 βλ1 .

(2.181)

We claim that x + γe ∈ at (x). In view of (2.179), for i = 1, . . . , n, xi + γ ≥ xi ≥ (1 − di(t) )xi .

(2.182)

It follows from (2.172), (2.175), (2.180) and (2.181) that for integers i = 1, . . . , n, we have n 

αi(t) (xi + γ − (1 − di(t) )xi ) =

i=1

n  i=1

≤

n 

αi(t) (xi − (1 − di(t) )xi ) + γ

n 

αi(t)

i=1

αi(t) (xi − (1 − di(t) )xi ) + γnα∗

i=1

=

n 

αi(t) (xi − (1 − di(t) )xi ) + (b∗ )−1 βλ1

i=1

≤

n  i=1

αi(t) (xi − (1 − di(t) )xi ) + ey ≤ 1.

64

2 Infinite Horizon Nonautonomous Optimization Problems

Combined with (2.182) and (2.165) the relation above implies that x + γe ∈ at (x). Therefore (ii) holds and Proposition 2.28 holds too. For every x = (x1 , . . . , xn ) ∈ R n put x1 =

n 

|xi |, x∞ = max{|xi | : i = 1, . . . , n}.

i=1 n It is easy to see that for every nonnegative integer t, every pair of vectors x, y ∈ R+ and for  ·  =  ·  p , where p = 1, 2, ∞,

H (at (x), at (y)) n n − ((1 − di(t) )yi )i=1  ≤ (1 − d∗ )x − y. ≤ ((1 − di(t) )xi )i=1

(2.183)

Thus we have defined the mappings at and the cost functions u t , t = 0, 1, . . . . The control system considered in this section is a special case of the control system studied in Sect. 2.9. As we have already mentioned, this control system corresponds to the nonstationary RSS model. Note that this control system satisfies the assumptions posed in Sect. 2.9 and therefore all the results stated there hold for this system. Indeed, fix M0 > (α∗ d∗ )−1 and set n : z ≤ M0 e}. K = {z ∈ R+

In view of Lemma 6.25, at (K ) ⊂ K , t = 0, 1, . . . . Relation (2.160) follows from (2.183). Evidently, (2.161) is true by (2.164). Lemma 2.26 implies that u t is upper semicontinuous for every nonnegative integer t. By Proposition 2.28, assumption (A5) holds. (A3) follows from Proposition 2.27.

2.11 Auxiliary Results for Theorem 2.23 Lemma 2.29 Let δ be a positive number. Then there exists an integer T0 ≥ 4 such 2 that for every integer τ1 ≥ 0, every integer τ2 ≥ T0 + τ1 , every program {x(t)}τt=τ 1 n for which there exists z ∈ R satisfying z ∈ aτ2 −1 (x(τ2 − 1)) and z ≥ x(τ2 ) + δe τ2 ˜ and every x˜0 ∈ K there exists a program {x(t)} t=τ1 such that

˜ 2 ) ≥ x(τ2 ). x(τ ˜ 1 ) = x˜0 , x(τ

(2.184)

2.11 Auxiliary Results for Theorem 2.23

65

Proof Fix a positive number D0 for which z ≤ D0 for all z ∈ K .

(2.185)

There exists a positive number c0 such that z2 ≤ c0 z for all z ∈ K .

(2.186)

Fix an integer T0 ≥ 4 such that 8D0 c0 κT0 < δ.

(2.187)

2 n and z ∈ R+ satAssume that integers τ1 ≥ 0, τ2 ≥ T0 + τ1 , a program {x(t)}τt=τ 1 τ2 −1 isfy (2.184) and that x˜0 ∈ K . In view of (2.160) there exists a program {x(t)} ˜ t=τ1 such that

x(τ ˜ 1 ) = x˜0 , x(t ˜ + 1) − x(t + 1) ≤ κx(t) ˜ − x(t), t = τ1 , . . . , τ2 − 2.

(2.188)

By (2.160) and (2.184), there exists ˜ 2 − 1)) x(τ ˜ 2 ) ∈ aτ2 −1 (x(τ

(2.189)

˜ 2 − 1). x(τ ˜ 2 ) − z ≤ κx(τ2 − 1) − x(τ

(2.190)

such that τ2 Evidently, {x(t)} ˜ t=τ1 is a program. It follows from (2.185), (2.189) and (2.190) that

˜ 1 ) − x(τ1 ) ≤ κτ2 −τ1 (2D0 ) ≤ κT0 (2D0 ) x(τ ˜ 2 ) − z ≤ κτ2 −τ1 x(τ and by (2.186) x(τ ˜ 2 ) − z2 ≤ 2D0 c0 κT0 . This implies that for all integers i = 1, . . . , n, |x˜i (τ2 ) − z i | ≤ 2D0 c0 κT0 . Combined with (2.184) and (2.187), this implies that x(τ ˜ 2 ) ≥ z − 2D0 c0 κT0 e ≥ x(τ2 ). This completes the proof of Lemma 2.29.

66

2 Infinite Horizon Nonautonomous Optimization Problems

Lemma 2.30 Let δ be a positive number. Then there exists an integer L¯ ≥ 1 such that for every integer L ≥ L¯ there exists an integer τ ≥ L for which the following assertion holds: T satisfying For every integer T ≥ τ and every program {x(t)}t=0 T −1 

u t (x(t), x(t + 1)) = U (x(0), T )

(2.191)

t=0

the inequality L−1 

u t (x(t), x(t + 1)) ≥ U (x(0), L) − δ

t=0

is valid. Proof Lemma 2.29 implies that there exists an integer L 0 ≥ 4 such that the following property holds: 2 there exists (P6) If integers τ1 ≥ 0 and τ2 ≥ L 0 + τ1 , if for a program ({x(t)}τt=τ 1 n z ∈ R+ which satisfies z ∈ aτ2 −1 (x(τ2 − 1)) and z ≥ x(τ2 ) + γe τ2 ˜ and if x˜0 ∈ K , then there exists a program {x(t)} t=τ1 such that

˜ 2 ) ≥ x(τ2 ). x(τ ˜ 1 ) = x˜0 , x(τ In view of (2.161), there exists an integer L¯ ≥ 1 such that for every natural number ¯ we have t ≥ L, sup{u t (x, x ) : (x, x ) ∈ graph(at )} ≤ δ(32L 0 )−1 .

(2.192)

Assume that an integer L ≥ L¯ and fix a natural number τ ≥ L + L 0 + 2.

(2.193)

T satisfies (2.191). We Assume that an integer T ≥ τ and that a program {x(t)}t=0 claim that L−1  u t (x(t), x(t + 1)) ≥ U (x(0), L) − δ. t=0 L Proposition 2.22 implies that there exists a program {x(t)} ˜ t=0 satisfying

x(0) ˜ = x(0),

L−1  t=0

u t (x(t), ˜ x(t ˜ + 1)) = U (x(0), L).

(2.194)

2.11 Auxiliary Results for Theorem 2.23

67

There are two cases: u t (x(t), x(t + 1)) < Δt , t = L + L 0 , . . . , T − 1; max{u t (x(t), x(t + 1)) − Δt : t = L + L 0 , . . . , T − 1} ≥ 0.

(2.195) (2.196)

Assume that (2.195) is valid. Assumption (A5) implies that there exists a program T which satisfies {x (1) (t)}t=0 ˜ t = 0, . . . , L , x (1) (t) = x(t), u t (x (1) (t), x (1) (t + 1)) ≥ Δt , t = L + 1, . . . , T − 1. It follows from (2.191), (2.192), (2.194), (2.195) and (2.197) that 0≥

T −1 

u t (x (1) (t), x (1) (t + 1)) −

T −1 

t=0

=

L−1 

u t (x(t), ˜ x(t ˜ + 1)) +

t=0

−

u t (x(t), x(t + 1))

t=0

T −1 

T −1 

u t (x (1) (t), x (1) (t + 1))

t=L

u t (x(t), x(t + 1))

t=0

≥ U (x(0), L) +

T −1  t=L+1

≥ U (x(0), L) −

L−1 

Δt −

T −1 

u t (x(t), x(t + 1))

t=0

u t (x(t), x(t + 1))

t=0

+

T −1 

Δt −

L+L 0 −1 

t=L+1

≥ U (x(0), L) −

u t (x(t), x(t + 1)) −

t=L L−1 

T −1 

Δt

t=L+L 0

u t (x(t), x(t + 1)) − δ

t=0

and

L−1 

u t (x(t), x(t + 1)) ≥ U (x(0), L) − δ.

t=0

Assume that (2.196) is true. Then there exists an integer S0 for which

(2.197)

68

2 Infinite Horizon Nonautonomous Optimization Problems

L + L 0 ≤ S0 − 1 ≤ T − 1,

(2.198)

u S0 −1 (x(S0 − 1), x(S0 )) ≥ Δ S0 −1 , u t (x(t), x(t + 1)) < Δt for every integer t satisfying L 0 + L ≤ t < S0 − 1. (2.199) S0 −L 0 Assumption (A5) implies that there exists a program {x (2) (t)}t=0 which satisfies

˜ t = 0, . . . , L , x (2) (t) = x(t),

(2.200)

u t (x (2) (t), x (2) (t + 1)) ≥ Δt for all integers t satisfying L + 1 ≤ t ≤ S0 − L 0 − 1. (2.201) It follows from (2.198) and (A5) that there exists y ∈ R n such that y ∈ a S0 −1 (x(S0 − 1)) and y ≥ x(S0 ) + γe.

(2.202)

S0 such that Property (P6) and (2.202) imply that there exists a program {x (2) (t)}t=S 0 −L 0

x (2) (S0 ) ≥ x(S0 ).

(2.203)

S0 is a program. In view of (2.203) and (A3), there exists a Evidently, {x (2) (t)}t=0 T such that program {x (2) (t)}t=0

u t (x (2) (t), x (2) (t + 1)) ≥ u t (x(t), x(t + 1)), t = S0 , . . . , T − 1.

(2.204)

It follows from (2.104), (2.191), (2.192), (2.194), (2.199)–(2.201) that 0≥

T −1 

u t (x (2) (t), x (2) (t + 1)) − U (x(0), T )

t=0

=

T −1 

u t (x (2) (t), x (2) (t + 1)) −

T −1 

t=0

≥

L−1 

u t (x(t), ˜ x(t ˜ + 1)) −

t=0

+

u t (x(t), x(t + 1))

t=0 L−1 

u t (x(t), x(t + 1))

t=0

S 0 −1

u t (x (2) (t), x (2) (t + 1)) −

t=L

S 0 −1

u t (x(t), x(t + 1))

t=L

≥ U (x(0), L) −

L−1 

u t (x(t), x(t + 1)) +

t=0

−

S0 −L 0 −1 t=L

u t (x(t), x(t + 1)) −

S0 −L 0 −1

u t (x (2) (t), x (2) (t + 1))

t=L S 0 −1 t=S0 −L 0

u t (x(t), x(t + 1))

2.11 Auxiliary Results for Theorem 2.23

≥ U (x(0), L) −

L−1 

69

u t (x(t), x(t + 1))

t=0

−

L 0 +L−1

u t (x(t), x(t + 1)) −

S 0 −1

u t (x(t), x(t + 1))

t=S0 −L 0

t=L

≥ U (x(0), L) −

L−1 

u t (x(t), x(t + 1)) − δ,

t=0 L−1 

u t (x(t), x(t + 1)) ≥ U (x(0), L) − δ.

t=0

Thus in both cases the inequality above holds. This completes the proof of Lemma 2.30.

2.12 Proof of Theorem 2.23 Proposition 2.22 implies that for every z ∈ K and every integer T ≥ 1 there exists a T such that program {x z,T (t)}t=0 x z,T (0) = z,

T −1 

u t (x z,T (t), x z,T (t + 1)) = U (z, T ).

(2.205)

t=0

Let δ be a positive number. Lemma 2.30 implies that there exists an integer L δ ≥ 1 such that the following property holds: (P7) For every integer L ≥ L δ there exists an integer τ L ≥ L such that for every natural number T ≥ τ L and every z ∈ K L−1 

u t (x z,T (t), x z,T (t + 1)) ≥ U (z, L) − δ/4.

t=0

Let z ∈ K . There exists a strictly increasing sequence of natural numbers {Tk }∞ k=1 and a program {x z (t)}∞ t=0 such that for each integer t ≥ 0, we have x z,Tk (t) → x z (t) as k → ∞.

(2.206)

x z (0) = z.

(2.207)

Evidently,

70

2 Infinite Horizon Nonautonomous Optimization Problems

Let an integer L satisfy L ≥ L δ and let an integer τ L ≥ L be as guaranteed by the property (P7). By (2.206) and upper semicontinuity of the functions u t , t = 0, 1, . . . there exists an integer k ≥ 1 such that Tk ≥ τ L ,

L−1 

u t (x z (t), x z (t + 1)) ≥

t=0

L−1 

u t (x z,Tk (t), x z,Tk (t + 1)) − δ/4. (2.208)

t=0

In view of (P7), (2.208) and the choice of τ L , L−1 

u t (x z,Tk (t), x z,Tk (t + 1)) ≥ U (z, L) − δ/4.

t=0

Combined with (2.208) this implies that L−1  t=0

Theorem 2.23 is proved.

u t (x z (t), x z (t + 1)) ≥ U (z, L) − δ.

Chapter 3

One-Dimensional Concave RSS Model

Abstract In this chapter we show the stability of the turnpike phenomenon under small perturbations of objective functions, for a class of discrete-time concave optimal control problems. These control problems arise in economic dynamics and describe the nonstationary Robinson–Solow–Srinivasan model. We study the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

3.1 Preliminaries and Main Results 1 Denote by Card(E) the cardinality of a set E. Let R 1 (R+ ) be the set of real (nonY negative) numbers. For each mapping a : X → 2 \ {∅}, where X, Y are nonempty sets, put graph(a) = {(x, y) ∈ X × Y : y ∈ a(x)}. For each integer t ≥ 0 let

αt > 0, dt ∈ (0, 1]

(3.1)

and for each integer t ≥ 0 let wt : [0, ∞) → [0, ∞) be a strictly increasing continuous function such that wt (0) = 0 and inf{wt (z) : an integer t ≥ 0} > 0 for all z > 0.

(3.2)

We suppose that the following assumption holds: (A1) for each  > 0 there exists δ > 0 such that wt (δ) ≤  for each integer t ≥ 0. We now give a formal description of the model. 1 set Let t ≥ 0 be an integer. For each x ∈ R+ 1 : y ≥ (1 − dt )x and αt (y − (1 − dt )x) ≤ 1}. at (x) = {y ∈ R+

(3.3)

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. J. Zaslavski, Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model, Monographs in Mathematical Economics 4, https://doi.org/10.1007/978-981-16-2252-6_3

71

72

3 One-Dimensional Concave RSS Model

1 It is clear that for each x ∈ R+ ,

at (x) = [(1 − dt )x, αt−1 + (1 − dt )x] 1 1 × R+ . Suppose that and that graph(at ) is a closed subset of R+

inf{dt : t = 0, 1, . . . } > 0, inf{αt : t = 0, 1, . . . } > 0,

(3.4) (3.5)

sup{αt : t = 0, 1, . . . } < ∞, sup{wt (M) : t = 0, 1, . . . } < ∞ for each M > 0.

(3.6) (3.7)

The constraint mappings at , t = 0, 1, . . . have already been defined. Now we define the cost functions u t , t = 0, 1, . . . . For each integer t ≥ 0 and each (x, x  ) ∈ graph(at ) set u t (x, x  ) = sup{wt (y) : 0 ≤ y ≤ x and y + αt (x  − (1 − dt )x) ≤ 1}.

(3.8)

Clearly, for each integer t ≥ 0 and each (x, x  ) ∈ graph(at ), u t (x, x  ) = wt (min{x, 1 − αt (x  − (1 − dt )x)}). Choose α∗ , α∗ , d∗ > 0 such that α∗ < αt < α∗ , d∗ < dt for all integers t ≥ 0.

(3.9)

Clearly, for each integer t ≥ 0 the function u t : graph(at ) → [0, ∞) is upper semicontinuous. 1 A sequence {x(t)}∞ t=0 ⊂ R+ is called a program if x(t + 1) ∈ at (x(t)) for all T2 1 integers t ≥ 0. Let T1 , T2 be integers such that T1 < T2 . A sequence {x(t)}t=T ⊂ R+ 1 is called a program if x(t + 1) ∈ at (x(t)) for all integers t satisfying T1 ≤ t < T2 . In the sequel we assume that the supremum over an empty set is −∞. 1 and each pair of integers T1 < T2 set For each x0 ∈ R+ T 2 −1 T2 U (x0 , T1 , T2 ) = sup u t (x(t), x(t + 1)) : {x(t)}t=T 1 t=T1

 is a program and x(T1 ) = x0 . 1 Let x0 , x˜0 ∈ R+ and let T1 < T2 be integers. Set

(3.10)

3.1 Preliminaries and Main Results

U (x0 , x˜0 , T1 , T2 ) = sup

73

T 2 −1

T2 u t (x(t), x(t + 1)) : {x(t)}t=T 1

t=T1

 is a program such that x(T1 ) = x0 , x(T2 ) ≥ x˜0 .

(3.11)

Let T1 , T2 be integers such that T1 < T2 . Set M (T1 , T2 ) = sup U

T 2 −1 t=T1

T2 {x(t)}t=T 1

u t (x(t), x(t + 1)) :  is a program and x(T1 ) ≤ M .

(3.12)

Upper semicontinuity of u t , t = 0, 1, . . . , compactness of sets of admissible programs and the optimization theorem of Weierstrass imply the following results. 1 Proposition 3.1 For each x0 ∈ R+ and each pair of integers T1 < T2 there exists a T2 program {x(t)}t=T1 such that x(T1 ) = x0 and T 2 −1

u t (x(t), x(t + 1)) = U (x0 , T1 , T2 ).

t=T1

Proposition 3.2 For each natural number T and each M > 0 there exists a program T such that {x(t)}t=0 T −1  M (0, T ) u t (x(t), x(t + 1)) = U t=0

and x(0) ≤ M. Fix

M∗ > (α∗ d∗ )−1 + 1.

(3.13)

It is clear that the model considered here is a particular case of the model discussed in Chap. 2 with n = 1 (see Sect. 2.3). Therefore all the results of Chap. 2 can be applied. Theorem 2.4 and Lemma 2.9 imply the following result. Theorem 3.3 There exists M¯ > 0 such that for each x0 ∈ [0, M∗ ] there exists a ∞ ¯ = x0 , for each pair of integers T1 , T2 ≥ 0 satisfying program {x(t)} ¯ t=0 such that x(0) T1 < T2 , T −1  2    M∗ (T1 , T2 ) ≤ M¯ u t (x(t), ¯ x(t ¯ + 1)) − U    t=T1

74

3 One-Dimensional Concave RSS Model

and that for each integer T > 0, T −1 

u t (x(t), ¯ x(t ¯ + 1)) = U (x(0), ¯ x(T ¯ ), 0, T ).

t=0

Lemma 2.9 and Proposition 2.6 imply the following result. ∞ Theorem 3.4 Let x0 ∈ [0, M∗ ] and let a program {x(t)} ¯ t=0 be as guaranteed by ∞ Theorem 3.3. Assume that {x(t)}t=0 is a program. Then either the sequence

T −1 

u t (x(t), x(t + 1)) −

t=0

T −1 

∞ u t (x(t), ¯ x(t ¯ + 1))

t=0

T =1

is bounded or T −1 

u t (x(t), x(t + 1)) −

t=0

T −1 

u t (x(t), ¯ x(t ¯ + 1)) → −∞ as T → ∞.

t=0

Let M¯ > 0 be as guaranteed by Theorem 3.3. Fix x∗0 ∈ [0, M∗ ] and let a program {x (t)}∞ t=0 be as guaranteed by Theorem 3.3. Namely, ∗

x ∗ (0) = x∗0 , T −1 

u t (x ∗ (t), x ∗ (t + 1)) = U (x ∗ (0), x ∗ (T ), 0, T )

(3.14)

t=0

for each integer T > 0 and T −1  2     ∗ ∗  u t (x (t), x (t + 1)) − U M∗ (T1 , T2 ) ≤ M¯   

(3.15)

t=T1

for each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 . For each integer t ≥ 0 set y ∗ (t) = min{x ∗ (t), 1 − αt (x ∗ (t + 1) − (1 − dt )x ∗ (t))}.

(3.16)

We show that the program {x ∗ (t)}∞ t=0 is the turnpike for the model. A function w : [0, ∞) → R is called strictly concave if for each x, y ∈ [0, ∞) satisfying x = y and each α ∈ (0, 1), w(αx + (1 − α)y) > αw(x) + (1 − α)w(y). The following two results are consequences of the optimization theorem of Weierstrass.

3.1 Preliminaries and Main Results

75

Proposition 3.5 Assume that w : [0, ∞) → [0, ∞) is continuous strictly concave function. Let , M > 0. Then there exists δ0 > 0 such that for each x, y ∈ [0, M] satisfying |x − y| ≥ , w(2−1 (x + y)) − 2−1 w(x) − 2−1 w(y) ≥ δ0 . Proposition 3.6 Assume that w : [0, ∞) → [0, ∞) is a strictly increasing continuous function, M > 0 and  ∈ (0, M). Then inf{w(x) − w(y) : x, y ∈ [0, M] and x ≥ y + } > 0. We suppose that the following assumptions hold. (A2) For each , M > 0 there exists 0 > 0 such that for each x, y ∈ (0, M] satisfying |x − y| ≥  and each integer t ≥ 0, wt (2−1 (x + y)) − 2−1 wt (x) − 2−1 wt (y) ≥ 0 . (A3) For each M > 0 and each  ∈ (0, M] there is 1 > 0 such that for each integer t ≥ 0 and each x, y ∈ [0, M] satisfying x ≥ y + , wt (x) − wt (y) ≥ 1 . (A4) For each M > 0 and each  > 0 there exists δ > 0 such that for each integer t ≥ 0 and each x, y ∈ [0, M] satisfying |x − y| ≤ δ the inequality |wt (x) − wt (y)| ≤  holds. Note that (A2) is an assumption of uniform concavity of the functions wt , t = 0, 1, . . . , (A3) is an assumption of uniform strict monotonicity of the functions wt , t = 0, 1, . . . and (A4) is an assumption of uniform equicontinuity of the functions wt , t = 0, 1, . . . . It is easy to see that (A1) follows from (A4) and (3.2). We assume that (3.17) d ∗ := sup{dt : t = 0, 1, . . . } < 1. In [103] we studied the structure of optimal programs of the model and obtained the following turnpike theorems which are also presented in [126]. Theorem 3.7 Let M > 0 and  > 0. Then there exists a natural number Q such T2 that for each pair of integers T1 ≥ 0 and T2 ≥ Q + T1 and each program {x(t)}t=T 1 which satisfies x(T1 ) ≤ M∗ and T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M

t=T1

the following inequality holds:

76

3 One-Dimensional Concave RSS Model

Card({t ∈ {T1 , . . . , T2 } : |x(t) − x ∗ (t)| > }) ≤ Q. Theorem 3.8 Let M,  > 0. Then there exist a natural number p and δ > 0 such T2 that for each pair of integers T1 ≥ 0, T2 ≥ 2 p + T1 and each program {x(t)}t=T 1 satisfying x(T1 ) ≤ M∗ , T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ

t=T1

and U (x(T1 ), x(T2 ), T1 , T2 ) ≥ U (x(T1 ), T1 , T2 ) − M the inequality |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [T1 + p, T2 − p]. Theorem 3.9 Let M > 0 and  > 0. Then there exist a natural number p and δ > 0 T2 such that for each pair of integers T1 ≥ 0, T2 ≥ p + T1 and each program {x(t)}t=T 1 ∗ which satisfies x(T1 ) ≤ M∗ , |x(T1 ) − x (T1 )| ≤ δ, T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ

t=T1

and U (x(T1 ), x(T2 ), T1 , T2 ) ≥ U (x(T1 ), T1 , T2 ) − M the inequality |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [T1 , T2 − p]. Theorem 3.10 Let  > 0. Then there exist a natural number p and δ > 0 such that T2 for each pair of integers T1 ≥ 0, T2 ≥ 2 p + T1 and each program {x(t)}t=T which 1 satisfies x(T1 ) ≤ M∗ and T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − δ

t=T1

the inequality |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [T1 + p, T2 − p]. Theorem 3.11 Let  > 0. Then there exist a natural number p and δ > 0 such that T2 for each pair of integers T1 ≥ 0, T2 ≥ p + T1 and each program {x(t)}t=T which 1 ∗ satisfies x(T1 ) ≤ M∗ , |x(T1 ) − x (T1 )| ≤ δ and T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − δ

t=T1

the inequality |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [T1 , T2 − p].

3.1 Preliminaries and Main Results

77

Theorems 3.10 and 3.11 easily follow from Theorems 3.8 and 3.9 respectively. A program {x(t)}∞ t=0 is called good if the sequence T −1 

u t (x(t), x(t + 1)) −

t=0

T −1 

∞ ∗

∗

u t (x (t), x (t + 1)) T =1

t=0

is bounded. In view of Theorem 3.4, if the sequence {x(t)}∞ t=0 is not good, then lim

T −1 

T →∞

u t (x(t), x(t + 1)) −

t=0

T −1 

u t (x ∗ (t), x ∗ (t + 1)) = −∞.

t=0

In [103] we proved the following result. ∗ Theorem 3.12 Assume that a program {x(t)}∞ t=0 is good. Then x(t) − x (t) → 0 as t → ∞. ∞  A program {x(t)}∞ t=0 is called overtaking optimal if for each program {x (t)}t=0  satisfying x (0) = x(0),

lim sup T →∞

T −1 

u t (x  (t), x  (t + 1)) −

t=0

T −1 

u t (x(t), x(t + 1)) ≤ 0.

t=0

In [103] we proved the following result. ∞ Theorem 3.13 Let x0 ∈ [0, M∗ ] and let a program {x(t)} ¯ t=0 be as guaranteed by ∞ is a unique overtaking optimal program with the initial Theorem 3.3. Then {x(t)} ¯ t=0 state x0 .

In this chapter we are interested in the stability of the turnpike phenomenon under small perturbations of objective function. We prove the following turnpike result for approximate solutions, which is new. Theorem 3.14 Let M,  > 0. Then there exist a natural number p and δ > 0 such T2 that for each pair of integers T1 ≥ 0, T2 ≥ 2 p + T1 and each program {x(t)}t=T 1 satisfying x(T1 ) ≤ M∗ , τ + p−1



u t (x(t), x(t + 1)) ≥ U (x(τ ), x(τ + p), τ , τ + p) − δ

t=τ

for all τ ∈ {T1 , . . . , T2 − p}, T 2 −1 T2 − p

u t (x(t), x(t + 1)) ≥ U (x(T2 − p), T2 − p, T2 ) − M.

78

3 One-Dimensional Concave RSS Model

Then there exist integers τ1 ∈ [T1 , T1 + p] and τ2 ∈ [T2 − p, T2 ] such that |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [τ1 , τ2 ]. Moreover, if

|x(T1 ) − x ( T1 )| ≤ δ,

then τ1 = T1 .

3.2 Auxiliary Results We begin with the result which follows from Lemma 2.9 and Corollary 2.20. Proposition 3.15 There exist M1 > 0 and a natural number L such that for each pair of integers T1 ≥ 0, T2 ≥ T1 + L and each x0 ∈ [0, M∗ ] the inequality M∗ (T1 , T2 )| ≤ M1 holds. |U (x0 , T1 , T2 ) − U The following lemma shows the uniform equicontinuity of the functions u t , t = 0, 1, . . . . Lemma 3.16 Let  > 0 and M ≥ M∗ . Then there exists δ > 0 such that for each integer t ≥ 0 and each (x, x  ), (y, y  ) ∈ graph(at ) satisfying

x, y ≤ M, |x − y|, |x  − y  | ≤ δ

the inequality |u t (x, x  ) − u t (y, y  )| ≤  holds. The next result easily follows from (3.8) and the strict monotonicity of wt , t = 0, 1, . . . . Lemma 3.17 Let t ≥ 0 be an integer, (x, x  ) ∈ graph(at ) and let y ∈ [0, x] satisfy y + αt (x  − (1 − dt )x) ≤ 1. Then wt (y) = u t (x, x  ) if and only if y = min{x, 1 − αt (x  − (1 − dt )x)}.  > 0 and a program { By Proposition 2.12, there exist Δ x (t)}∞ t=0 such that  x (0) < 1 and that for all integers t ≥ 0.

 x (t),  x (t + 1)) ≥ Δ u t (

3.2 Auxiliary Results

79

Let a positive number γ satisfy  γ < 1/2 and γ < 4−1 Δ.

(3.18)

Lemmas 2.9 and 2.16 imply the following result. Lemma 3.18 Let M1 > 0. Then there exist integers L 1 , L 2 ≥ 4 such that for each T2 satisfying pair of integers T1 ≥ 0, T2 ≥ L 1 + L 2 + T1 , each program {x(t)}t=T 1 x(T1 ) ≤ M∗ ,

T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), T1 , T2 ) − M1

t=T1

and each integer τ ∈ [T1 + L 1 , T2 − L 2 ] the following inequality holds: max{u t (x(t), x(t + 1)) : t = τ , . . . , τ + L 2 − 1} ≥ γ. Lemma 3.19 Let t ≥ 0 be an integer, x ∈ [0, M∗ ], y ∈ at (x). Then y ≤ M∗ . Proof By (3.3), (3.9) and (3.13), y ≤ (1 − dt )x + αt−1 ≤ (1 − dt )M∗ + dt−1 ≤ (1 − d∗ )M∗ + α∗−1 ≤ M∗ − d∗ M∗ + α∗−1 ≤ M∗ − d∗ . Lemma 3.19 is proved. Recall that M¯ is as guaranteed by Theorem 3.3. By (3.15), for each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 , we have T −1  2    ¯ M∗ (T1 , T2 ) ≤ M. u t (x ∗ (t), x ∗ (t + 1)) − U    t=T1

Lemma 3.19 and (3.14) imply that x ∗ (t) ≤ M∗ , t = 0, 1, . . . .

(3.19)

Lemmas 3.18, (3.15) and (3.19) imply the following result. Lemma 3.20 There exist integers L 1 , L 2 ≥ L 1 such that for each integer τ ≥ L 1 , max{u t (x ∗ (t), x ∗ (t + 1)) : t = τ , . . . , τ + L 2 − 1} ≥ γ. The following results were proved in [103].

80

3 One-Dimensional Concave RSS Model

Lemma 3.21 Let  > 0. Then there exist a natural number L 0 ≥ 4 and δ > 0 such that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L 0 and each z 1 , z 2 ∈ [0, M∗ ] satisfying |z i − x ∗ (τi )| ≤ δ, i = 1, 2, the inequality |U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) − U (z 1 , z 2 , τ1 , τ2 )| ≤  holds. Lemma 3.22 Let  > 0. Then there exist a natural number L 0 and δ > 0 such that 2 satisfying for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L 0 and each program {x(t)}τt=τ 1 x(τ1 ) ≤ M∗ , |x(τi ) − x ∗ (τi )| ≤ δ, i = 1, 2, and

τ 2 −1

u t (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ

t=τ1

the following inequality holds for all t = τ1 , . . . , τ2 − L 0 : |x ∗ (t) − x(t)| ≤ . Lemma 3.23 There exist an integer L ≥ 4, γ0 ∈ (0, γ) and M1 > 0 such that the following assertion holds. Let S1 ≥ 0, S2 ≥ S1 + L be a pair of integers and z 0 , z 1 ∈ [0, M∗ ] satisfy

|z 1 − x ∗ (S2 )| ≤ γ0 .

S2 Then there exists a program {x(t)} ˜ t=S1 such that

x(S ˜ 1) = z0 , x(S ˜ 2 ) ≥ z1 and

S 2 −1 t=S1

u t (x(t), ˜ x(t ˜ + 1)) ≥

S 2 −1 t=S1

u t (x ∗ (t), x ∗ (t + 1)) − M1 .

3.2 Auxiliary Results

81

Proof By Lemma 3.20, there exist integers L 1 , L 2 ≥ 4 such that for each integer τ ≥ L 1, (3.20) max{u t (x ∗ (t), x ∗ (t + 1)) : t = τ , . . . , τ + L 2 − 1} ≥ γ. By Lemmas 3.19 and 2.15, there exists an integer L 3 ≥ 4 such that the following property holds: (P1) For every integer τ1 ≥ 0, every natural number τ2 ≥ L 3 + τ1 , every program 2 satisfying {x(t)}τt=τ 1 u τ2 −1 (x(τ2 − 1), x(τ2 )) ≥ γ and every x˜0 ∈ [0, M∗ ] τ2 there exists a program {x(t)} ˜ t=τ1 which satisfies

˜ 2 ) ≥ x(τ2 ). x(τ ˜ 1 ) = x˜0 , x(τ By Lemma 3.21, there exist a natural number L 4 ≥ 4 and γ0 ∈ (0, γ) such that the following property holds: (P2) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L 4 and each z 1 , z 2 ∈ [0, M∗ ] satisfying

|z i − x ∗ (τi )| ≤ γ0 , i = 1, 2,

the inequality |U (x ∗ (τ1 ), x ∗ (τ2 ), τ1 , τ2 ) − U (z 1 , z 2 , τ1 , τ2 )| < γ/4 holds. Fix a natural number L > 4(L 1 + L 2 + L 3 + L 4 + 1)

(3.21)

M1 = γ + (L 1 + L 2 + L 3 ) sup{wt (M∗ ) : t = 0, 1, . . . }.

(3.22)

S2 ≥ S1 + L

(3.23)

and set Let S1 ≥ 0,

82

3 One-Dimensional Concave RSS Model

be a pair of integers and

satisfy

z 0 , z 1 ∈ [0, M∗ ]

(3.24)

|z 1 − x ∗ (S2 )| ≤ γ0 .

(3.25)

It follows from (3.20) and the choice of L 1 , L 2 that there exists an integer

such that

τ ∈ [S1 + L 1 + L 3 , S1 + L 1 + L 2 + L 3 − 1]

(3.26)

u τ (x ∗ (τ ), x ∗ (τ + 1)) ≥ γ.

(3.27)

τ Property (P1), (3.24) and (3.27) imply that there exists a program {x(t)} ˜ t=S1 which satisfies ˜ ) ≥ x ∗ (τ ). (3.28) x(S ˜ 1 ) = z 0 , x(τ

In view of (3.21), (3.23) and (3.26), S2 − τ ≥ S1 + L − (S1 + L 1 + L 2 + L 3 ) > 4L 4 + 2L 3 .

(3.29)

Property (P2), (3.25), (3.26) and (3.29) imply that |U (x ∗ (τ ), z 1 , τ , S2 ) − U (x ∗ (τ ), x ∗ (S2 ), τ , S2 )| < γ/4.

(3.30)

S2 which satisfies In view of (3.30), there exists a program {y(t)}t=τ

y(τ ) = x ∗ (τ ), y(S2 ) ≥ z 1 ,  S −1  2     ∗ ∗ u t (y(t), y(t + 1)) − U (x (τ ), x (S2 ), τ , S2 ) < γ/4.   

(3.31) (3.32)

t=τ

Equation (3.28) implies that there exist x(t), ˜ t = τ + 1, . . . , S2 S2 such that {x(t)} ˜ t=S1 is a program,

x(t) ˜ ≥ y(t), t = τ + 1, . . . , S2 ,

(3.33)

u t (x(t), ˜ x(t ˜ + 1)) ≥ u t (y(t), y(t + 1)), t = τ , . . . , S2 − 1.

(3.34)

By (3.28), (3.31) and (3.33), ˜ 2 ) ≥ y(S2 ) ≥ z 1 . x(S ˜ 1 ) = z 0 , x(S

(3.35)

3.2 Auxiliary Results

83

It follows from (3.14), (3.22), (3.24), (3.26), (3.32), (3.34) and (3.35) that S 2 −1

u t (x(t), ˜ x(t ˜ + 1))

t=S1

=

τ −1 

u t (x(t), ˜ x(t ˜ + 1)) +

S 2 −1

u t (x(t), ˜ x(t ˜ + 1))

t=τ

t=S1

≥

S 2 −1

u t (y(t), y(t + 1))

t=τ

≥ U (x ∗ (τ ), x ∗ (S2 ), τ , S2 ) − γ/4 =

S 2 −1

u t (x ∗ (t), x ∗ (t + 1)) − γ/4

t=τ

≥

S 2 −1

u t (x ∗ (t), x ∗ (t + 1))

t=S1

− γ/4 − sup{wt (M∗ ) : t = 0, 1, . . . }(τ − S1 ) ≥

S 2 −1

u t (x ∗ (t), x ∗ (t + 1))

t=S1

− γ/4 − sup{wt (M∗ ) : t = 0, 1, . . . }(L 1 + L 2 + L 3 ) ≥

S 2 −1

u t (x ∗ (t), x ∗ (t + 1)) − M1 .

t=S1

Lemma 3.23 is proved. Corollary 3.24 There exist an integer L ≥ 4, γ0 ∈ (0, γ) and M1 > 0 such that for each pair of integers S1 ≥ 0, S2 ≥ S1 + L and each z 0 , z 1 ∈ [0, M∗ ] satisfying

the inequality

holds.

|z 1 − x ∗ (S2 )| ≤ γ0 . M∗ (S1 , S2 ) − M1 U (z 0 , z 1 , S1 , S2 ) ≥ U

84

3 One-Dimensional Concave RSS Model

3.3 Proof of Theorem 3.14 We may assume without loss of generality that  < 1. By Corollary 3.24, there exist an integer L 0 ≥ 4, δ0 ∈ (0, γ) and M1 > 0 such that the following property holds: (P3) For each pair of integers S1 ≥ 0, S2 ≥ S1 + L 0 and each z 0 , z 1 ∈ [0, M∗ ] satisfying

|z 1 − x ∗ (S2 )| ≤ δ0

the inequality

M∗ (S1 , S2 ) − M1 U (z 0 , z 1 , S1 , S2 ) ≥ U

is true. Lemma 3.22 implies that there exist a natural number L 1 and δ1 ∈ (0, ) such that the following property holds: 2 (P4) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L 1 and each program {x(t)}τt=τ 1 satisfying x(τ1 ) ≤ M∗ , |x(τi ) − x ∗ (τi )| ≤ δ1 , i = 1, 2, and

τ 2 −1

u t (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ1

t=τ1

the inequality

|x ∗ (t) − x(t)| ≤ 

is true for all t = τ1 , . . . , τ2 − L 1 . Choose a positive number δ < min{δ0 , δ1 }/4.

(3.36)

By Theorem 3.7, there exists a natural number L 2 such that the following property holds: (P5) For each pair of integers τ1 ≥ 0 and τ2 ≥ τ1 + L 2 and each program T2 which satisfies {x(t)}t=T 1 x(τ1 ) ≤ M∗ , τ 2 −1 t=τ1

u t (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M1 − M − 2

3.3 Proof of Theorem 3.14

85

the following inequality holds: Card({t ∈ {τ1 , . . . , τ2 } : |x(t) − x ∗ (t)| > δ}) ≤ L 2 . Fix a natural number p > 8L 2 + 8L 1 + 8L 0 + 8.

(3.37)

T1 ≥ 0, T2 ≥ 2 p + T1

(3.38)

Assume that T2 satisfies are integers and that a program {x(t)}t=T 1

x(T1 ) ≤ M∗ ,

(3.39)

τ + p−1



u t (x(t), x(t + 1)) ≥ U (x(τ ), x(τ + p), τ , τ + p) − δ

(3.40)

t=τ

and for all τ ∈ {T1 , . . . , T2 − p}, T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T2 − p), T2 − p, T2 ) − M.

(3.41)

t=T2 − p

In view of (3.39) and Lemma 3.19, x(t) ≤ M∗ , t = T1 , . . . , T2 .

(3.42)

Equations (3.37) and (3.41) imply that T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T2 − L 2 ), T2 − L 2 , T2 ) − M.

(3.43)

t=T2 −L 2

Property (P5), (3.42) and (3.43) imply that there exists

such that

t0 ∈ {T2 − L 2 , . . . , T2 }

(3.44)

|x(t0 ) − x ∗ (t0 )| ≤ δ.

(3.45)

t0 − p ≥ T2 − 2 p ≥ T1 .

(3.46)

By (3.37), (3.38) and (3.44),

86

3 One-Dimensional Concave RSS Model

It follows from (3.40), (3.44) and (3.46) that t0 −1 

u t (x(t), x(t + 1)) ≥ U (x(t0 − p), x(t0 ), t0 − p, t0 ) − δ.

(3.47)

t=t0 − p

In view of (3.37) and (3.47), t0 −1 

u t (x(t), x(t + 1))

t=t0 −2L 2 −2L 0 −2L 1

≥ U (x(t0 − 2L 2 − 2L 0 − 2L 1 ), x(t0 ), t0 − 2L 2 − 2L 0 − 2L 1 , t0 ) − δ.

(3.48)

Property (P3), (3.39), (3.42), (3.45) and (3.48) imply that t0 −1 

u t (x(t), x(t + 1))

t=t0 −2L 2 −2L 0 −2L 1

M∗ (t0 − 2L 2 − 2L 0 − 2L 1 , t0 ) − M1 − 1. ≥U

(3.49)

Property (P5), (3.42) and (3.49) imply that there exists an integer t1 such that

and

t1 ∈ [t0 − L 0 − L 1 − L 2 , t0 − L 0 − L 1 ]

(3.50)

|x(t1 ) − x ∗ (t1 )| ≤ δ.

(3.51)

Assume that k ≥ 1 is an integer and that we have defined a strictly decsreasing sequence of nonnegative integers ti , i = 0, . . . , k such that t0 ∈ [T2 − L 2 , T2 ],

(3.52)

ti ∈ [T1 , T2 ], |x(ti ) − x ∗ (ti )| ≤ δ, i = 0, . . . , k

(3.53)

and that for all i = 0, . . . , k − 1, ti+1 ∈ [ti − L 0 − L 1 − L 2 , ti − L 0 − L 1 ].

(3.54)

(Note that in view of (3.44), (3.45), (3.50) and (3.51) our assumption holds for k = 1.) There are two cases: tk ≥ T1 + 2L 1 + 2L 0 + 2L 2 ;

(3.55)

tk < T1 + 2L 1 + 2L 2 + 2L 0 .

(3.56)

In the case of (3.56), the construction of the sequence is completed.

3.3 Proof of Theorem 3.14

87

Assume that (3.55) holds and define tk+1 . Since T2 − T1 ≥ 2 p it follows from (3.37) that there exists an integer S ∈ [T1 , T2 ] such that [tk − 2L 2 − 2L 1 − 2L 0 , tk ] ⊂ [S, S + p] ⊂ [T1 , T2 ].

(3.57)

In view of (3.40), 

S+ p−1

u t (x(t), x(t + 1)) ≥ U (x(S), x(S + p), S, S + p) − δ.

(3.58)

t=S

By (3.57) and (3.58), tk −1 

u t (x(t), x(t + 1))

t=tk −2L 2 −2L 0 −2L 1

≥ U (x(tk − 2L 2 − 2L 0 − 2L 1 ), x(tk ), tk − 2L 2 − 2L 0 − 2L 1 , tk ) − δ.

(3.59)

Property (P3), (3.36), (3.39), (3.53) and (3.59) imply that tk −1 

M∗ (tk − 2L 2 − 2L 0 − 2L 1 , tk ) − M1 − 1. u t (x(t), x(t + 1)) ≥ U

t=tk −2L 2 −2L 0 −2L 1

(3.60) Property (P5), (3.39) and (3.60) imply that there exists an integer tk+1 ∈ [tk − L 0 − L 1 − L 2 , tk − L 0 − L 1 ] such that

|x(tk+1 ) − x ∗ (tk+1 )| ≤ δ.

Thus the assumption made for k holds for k + 1. By induction we have constructed a strictly decreasing sequence of nonnegative integers ti ∈ [T1 , T2 ], i = 0, . . . , q such that (3.52) holds, (3.61) |x(ti ) − x ∗ (ti )| ≤ δ, i = 0, . . . , q, and that for all i = 0, . . . , q − 1, ti+1 ∈ [ti − L 0 − L 1 − L 2 , ti − L 0 − L 1 ], T1 ≤ tq < T1 + 2L 0 + 2L 1 + 2L 2 .

(3.62) (3.63)

By (3.37), (3.38), (3.52), (3.62) and (3.63), q ≥ 4.

(3.64)

88

3 One-Dimensional Concave RSS Model

If

|x(T1 ) − x ∗ (T1 )| ≤ δ

and tq > T1 , then we set tq+1 = T1 , Q = q + 1. Otherwise set Q = q. Let an integer i satisfy Q ≥ i > 1. In view of (3.61), |x(ti ) − x ∗ (ti )| ≤ δ, |x(ti−2 ) − x ∗ (ti−2 )| ≤ δ.

(3.65)

It follows from (3.37), (3.62) and (3.63) that 2−1 p > 2(2L 0 + 2L 1 + 2L 2 ) ≥ ti−2 − ti ≥ L 0 + L 1 .

(3.66)

By (3.66), there exists an integer S such that [ti , ti−2 ] ⊂ [S, S + p] ⊂ [T1 , T2 ].

(3.67)

It follows from (3.40) and (3.67) that 

S+ p−1

u t (x(t), x(t + 1)) ≥ U (x(S), x(S + p), S, S + p) − δ,

t=S ti−2 −1



u t (x(t), x(t + 1)) ≥ U (x(ti ), x(ti−2 ), ti , ti−2 ) − δ.

t=ti

In view of (3.65), (3.66) and (3.68), |x(t) − x ∗ (t)| ≤ , t ∈ {ti , . . . , ti−2 − L 1 }. The relation above, (3.37), (3.52) and (3.62) imply that |x(t) − x ∗ (t)| ≤  for all  {{ti , . . . , ti−2 − L 1 } : i = 2, . . . , Q}  ⊃ {{ti , . . . , ti−1 } : i = 2, . . . , Q}

t∈

⊃ [t Q , t1 ] ⊃ [t Q , T2 − p]. Theorem 3.14 is proved.

(3.68)

3.4 Stability Results

89

3.4 Stability Results Let t ≥ 0 be an integer, ψ : graph(at ) → R 1 and let M > 0. Set

ψ M = sup{|ψ(z, z  )| : (z, z  ) ∈ graph(at ), z ≤ M}.

(3.69)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 and let vt : graph(at ) → R 1 , t = T1 , . . . , T2 − 1 be bounded on bounded sets functions. 1 define For each pair of points z 0 , z 1 ∈ R+ T2 −1 U ({vt }t=T , z0 , z1) 1

T 2 −1 T2 = sup vt (x(t), x(t + 1)) : {x(t)}t=T 1 t=T1

 is a program such that x(T1 ) = z 0 , x(T2 ) ≥ z 1 ,

(3.70)

T 2 −1 T2 −1 T2 U ({vt }t=T , z ) = sup vt (x(t), x(t + 1)) : {x(t)}t=T 0 1 1 t=T1

 is a program such that x(T1 ) = z 0 .

(3.71)

Recall that the supremum over an empty set is −∞. The results of this section show that the turnpike phenomenon is stable under small perturbations of the objective functions. Theorem 3.25 Let M0 ,  > 0. Then there exist a natural number p and δ˜ > 0 such that for each pair of integers T1 ≥ 0, T2 ≥ 2 p + T1 , each finite sequence of functions vt : graph(at ) → R 1 , t = T1 , . . . , T2 − 1 which are bounded on bounded sets and satisfy ˜

u t − vt M∗ ≤ δ, T2 satisfying for each t ∈ {T1 , . . . , T2 − 1} and every program {x(t)}t=T 1

x(T1 ) ≤ M∗ , τ + p−1



τ + p−1

vt (x(t), x(t + 1)) ≥ U ({vt }t=τ

, x(τ ), x(τ + p)) − δ˜

t=τ

for all τ ∈ {T1 , . . . , T2 − p}, T 2 −1 t=T2 − p

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T , T2 − p, T2 ) − M0 2− p

90

3 One-Dimensional Concave RSS Model

there exist integers τ1 ∈ [T1 , T1 + p] and τ2 ∈ [T2 − p, T2 ] such that |x(t) − x ∗ (t)| ≤  holds for all integers t ∈ [τ1 , τ2 ]. Moreover, if

˜ |x(T1 ) − x ∗ (T1 )| ≤ δ,

then τ1 = T1 . Proof Theorem 3.25 follows easily from Theorem 3.14. Namely, let M = M0 + 1, a natural number p and δ ∈ (0, 1) be as guaranteed by Theorem 3.14. Set δ˜ = δ(4 p + 4)−1 . Now it is easy to see that the assertion of Theorem 3.25 is true. For each integer t ≥ 0 denote by At the set of all functions vt : graph(at ) → R 1 which satisfy the following property: 1 satisfying x˜ ≥ x there exists (A5) for every (x, y) ∈ graph(at ) and every x˜ ∈ R+ ˜ for which y˜ ∈ at (x) y˜ ≥ y, v(x, ˜ y˜ ) ≥ v(x, y). Clearly, for each integer t ≥ 0, u t ∈ At . The following theorem is proved in the next section. Theorem 3.26 Let M > 0 and  > 0. Then there exists a natural number Q and δ > 0 such that for each pair of integers T1 ≥ 0 and T2 ≥ Q + T1 , each finite sequence of functions vt : graph(at ) → R 1 , t = T1 , . . . , T2 − 1 which are bounded on bounded sets and satisfy vt ∈ At , u t − vt M∗ ≤ δ T2 satisfying for each t ∈ {T1 , . . . , T2 − 1} and every program {x(t)}t=T 1

x(T1 ) ≤ M∗ , T 2 −1

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T , x(T1 )) − M 1

t=T1

the inequality Card({t ∈ {T1 , . . . , T2 } : |x(t) − x ∗ (t)| > }) ≤ Q holds. The results of this section are new.

3.5 Proof of Theorem 3.26

91

3.5 Proof of Theorem 3.26 We may assume without loss of generality that  < 1. Lemma 3.22 implies that there exist a natural number L 0 and δ0 ∈ (0, ) such that the following property holds: 2 (P6) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L 0 and each program {x(t)}τt=τ 1 satisfying x(τ1 ) ≤ M∗ , |x(τi ) − x ∗ (τi )| ≤ δ0 , i = 1, 2, and

τ 2 −1

u t (x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − 4δ0

t=τ1

the inequality

|x ∗ (t) − x(t)| ≤ 

is true for all t = τ1 , . . . , τ2 − L 0 . By Corollary 3.24, there exist an integer L 1 ≥ 4, δ1 ∈ (0, γ) and M1 > 0 such that the following property holds: (P7) For each pair of integers S1 ≥ 0, S2 ≥ S1 + L 1 and each z 0 , z 1 ∈ [0, M∗ ] satisfying

|z 1 − x ∗ (S2 )| ≤ δ1

the inequality

M∗ (S1 , S2 ) − M1 U (z 0 , z 1 , S1 , S2 ) ≥ U

is true. Set δ2 = min{δ0 , δ1 }.

(3.72)

By Theorem 3.7, there exists a natural number L 2 such that the following property holds: (P8) For each pair of integers τ1 ≥ 0 and τ2 ≥ L 2 + τ1 and each program T2 which satisfies {x(t)}t=T 1 x(τ1 ) ≤ M∗ , τ 2 −1 t=τ1

u t (x(t), x(t + 1)) ≥ U (x(τ1 ), τ1 , τ2 ) − M − M1 − 2

92

3 One-Dimensional Concave RSS Model

the following inequality holds: Card({t ∈ {τ1 , . . . , τ2 } : |x(t) − x ∗ (t)| > δ2 }) ≤ L 2 . Choose a natural number Q > 80(L 0 + L 1 + L 2 + 1) + 12(L 0 + L 1 + L 2 )δ0−1 M and a positive number

(3.73)

δ ≤ Q −1 δ2 .

(3.74)

T1 ≥ 0, T2 ≥ Q + T1

(3.75)

Assume that

are integers, vt : graph(at ) → R 1 , t = T1 , . . . , T2 − 1 are bounded on bounded sets functions such that vt ∈ At , t = T1 , . . . , T2 − 1,

u t − vt M∗ ≤ δ, t ∈ {T1 , . . . , T2 − 1}

(3.76) (3.77)

T2 and that a program {x(t)}t=T satisfies 1

x(T1 ) ≤ M∗ and

T 2 −1

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T , x(T1 )) − M. 1

(3.78)

(3.79)

t=T1

Lemma 3.19 and (3.78) imply that x(t) ≤ M∗ , t = T1 , . . . , T2 .

(3.80)

In view of (3.79), T 2 −1

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T , x(T2 − L 2 )) − M. 2 −L 2

t=T2 −L 2

By (3.72)–(3.74), (3.77) and (3.81),

(3.81)

3.5 Proof of Theorem 3.26 T 2 −1

93

T2 −1 u t (x(t), x(t + 1)) ≥ U ({u t }t=T , x(T2 − L 2 )) − M − 2δL 2 2 −L 2

t=T2 −L 2

≥ U (x(T2 − L 2 ), T2 − L 2 , T2 ) − M − 2.

(3.82)

Property (P8), (3.80) and (3.82) imply that there exists an integer

such that

t0 ∈ {T2 − L 2 , . . . , T2 }

(3.83)

|x(t0 ) − x ∗ (t0 )| ≤ δ2 .

(3.84)

In view of (3.73), (3.75) and (3.83), t0 − 2L 2 − L 0 − L 1 > T1 .

(3.85)

By (A5), (3.76), (3.79), (3.83) and (3.85), t0 −1 

vt (x(t), x(t + 1))

t=t0 −2L 0 −2L 1 −2L 2 t0 −1 , x(t0 − 2L 0 − 2L 1 − 2L 2 ), x(t0 )) − M. ≥ U ({vt }t=t 0 −2L 0 −2L 1 −L 2

(3.86)

Equations (3.72)–(3.74), (3.77) and (3.86) imply that t0 −1 

u t (x(t), x(t + 1))

t=t0 −2L 0 −2L 1 −2L 2

≥ U (x(t0 − 2L 0 − 2L 1 − L 2 ), x(t0 ), t0 − 2L 0 − 2L 1 − 2L 2 , t0 ) − M − 2. (3.87) Property (P7), (3.72), (3.80) and (3.84) imply that t0 −1 

u t (x(t), x(t + 1))

t=t0 −2L 0 −2L 1 −2L 2

M∗ (t0 − 2L 0 − 2L 1 − L 2 , t0 ) − M1 − M − 2. ≥U

(3.88)

In view of (P8), (3.80) and (3.88), there exists an integer

such that

t1 ∈ [t0 − L 0 − L 1 − L 2 , t0 − L 0 − L 1 ]

(3.89)

|x(t1 ) − x ∗ (t1 )| ≤ δ2 .

(3.90)

94

3 One-Dimensional Concave RSS Model

Assume that k ≥ 1 is an integer and that we have defined a strictly decsreasing sequence of nonnegative integers ti ∈ [T1 , T2 ], i = 0, . . . , k such that t0 ∈ [T2 − L 2 , T2 ], ∗

|x(ti ) − x (ti )| ≤ δ2 , i = 0, . . . , k

(3.91) (3.92)

and that for all i = 0, . . . , k − 1, ti+1 ∈ [ti − L 0 − L 1 − L 2 , ti − L 0 − L 1 ].

(3.93)

Note that in view of (3.83)–(3.85), (3.89) and (3.90), our assumption holds for k = 1.) There are two cases: tk ≥ T1 + 2L 2 + 2L 0 + 2L 1 ; tk < T1 + 2L 2 + 2L 1 + 2L 0 .

(3.94) (3.95)

In the case of (3.95), the construction of the sequence ti , i = 0, . . . , k is completed. Assume that (3.94) holds and define tk+1 . It follows from (A5), (3.76) and (3.79) that tk −1 

vt (x(t), x(t + 1))

t=tk −2L 2 −2L 0 −2L 1 k −1 ≥ U ({vt }tt=t , x(tk − 2L 2 − 2L 0 − 2L 1 ), x(tk )) − M. k −2L 2 −2L 0 −2L 1

(3.96)

By (3.72)–(3.74) and (3.96), tk −1 

u t (x(t), x(t + 1))

t=tk −2L 2 −2L 0 −2L 1 k −1 ≥ U ({u t }tt=t , x(tk − 2L 2 − 2L 0 − 2L 1 ), x(tk )) k −2L 2 −2L 0 −2L 1

− M − 2δ(2L 0 + 2L 1 + 2L 2 ) ≥ U (x(tk − 2L 2 − 2L 0 − 2L 1 ), x(tk ), tk − 2L 2 − 2L 0 − 2L 1 , tk ) − M − 2. (3.97) By property (P7), (3.72), (3.80), (3.92) and (3.97), tk −1 

u t (x(t), x(t + 1))

t=tk −2L 2 −2L 0 −2L 1

M∗ (tk − 2L 2 − 2L 0 − 2L 1 , tk ) − M − 2 − M1 . ≥U

(3.98)

3.5 Proof of Theorem 3.26

95

Property (P8) and (3.98) imply that there exists an integer tk+1 ∈ [tk − L 0 − L 1 − L 2 , tk − L 0 − L 1 ] such that

|x(tk+1 ) − x ∗ (tk+1 )| ≤ δ2 .

Thus the assumption made for k holds for k + 1. By induction we have constructed a strictly decreasing sequence of nonnegative integers ti ∈ [T1 , T2 ], i = 0, . . . , q such that (3.91) holds, (3.99) |x(ti ) − x ∗ (ti )| ≤ δ2 , i = 0, . . . , q, and that for all i = 0, . . . , q − 1, ti+1 ∈ [ti − L 0 − L 1 − L 2 , ti − L 0 − L 1 ],

(3.100)

T1 ≤ tq < T1 + 2L 0 + 2L 1 + 2L 2 .

(3.101)

Now we define a finite strictly increasing sequence of elements of {t0 , . . . , tq }. Set S0 = tq . If

t0 −1 

(3.102)

t0 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=t , x(tq ), x(t0 )) − δ0 , q

t=tq

then we set S1 = t0 and the construction is completed. Otherwise there exists a integer j ∈ {0, . . . , q} \ {q} such that t j −1 

t −1

j vt (x(t), x(t + 1)) < U ({vt }t=t , x(tq ), x(t j )) − δ0 q

(3.103)

t=tq

and if an integer i satisfies q > i > j, then ti −1 

i −1 vt (x(t), x(t + 1)) ≥ U ({vt }tt=t , x(tq ), x(ti )) − δ0 q

t=tq

and set S1 = t j .

(3.104)

Assume that k ≥ 1 is an integer and that we have defined a strictly increasing finite k ⊂ {t0 , . . . , tq } such that sequence {Si }i=0

96

3 One-Dimensional Concave RSS Model

S0 = tq , for each j ∈ {0, . . . , k − 1}, S j+1 −1



S

−1

j+1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(S j ), x(S j+1 )) − δ0 j

(3.105)

t=S j

and if an integer i ∈ {0, . . . , q} satisfies S j < ti < S j+1 , then

ti −1 

i −1 vt (x(t), x(t + 1)) ≥ U ({vt }tt=S , x(S j ), x(ti )) − δ0 . j

(3.106)

t=S j

(Note that in view of (3.103) and (3.104), our assumption holds for k = 1.) If Sk = t0 , then our construction is completed. If t0 −1  t0 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=S , x(Sk ), x(t0 )) − δ0 , k t=Sk

then we set Sk+1 = t0 and the construction is completed. Assume that t0 −1 

t0 −1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(Sk ), x(t0 )) − δ0 . k

t=Sk

It is not difficult to see that there exists a unique integer Sk+1 > Sk such that Sk+1 ∈ {t0 , . . . , tq } and

Sk+1 −1



S

−1

k+1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(Sk ), x(Sk+1 )) − δ0 k

t=Sk

and if an integer S ∈ {t0 , . . . , tq } satisfies Sk < S < Sk+1 ,

3.5 Proof of Theorem 3.26

then

S−1 

97

S−1 vt (x(t), x(t + 1)) ≥ U ({vt }t=S , x(Sk ), x(S)) − δ0 . k

t=Sk

Thus the assumption made for k holds for k + 1. By induction we have constructed p a strictly increasing sequence of integers {Si }i=0 ⊂ {t0 , . . . , tq } such that S0 = tq , S p = t0 ,

(3.107)

for each j ∈ {0, . . . , p − 1} \ { p − 1}, S j+1 −1



S

−1

j+1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(S j ), x(S j+1 )) − δ0 j

(3.108)

t=S j

and for each j ∈ {0, . . . , p − 1} and each S ∈ {t0 , . . . , tq } satisfying S j < S < S j+1 , we have S−1 

S−1 vt (x(t), x(t + 1)) ≥ U ({vt }t=S , x(S j ), x(S)) − δ0 . j

(3.109)

t=S j

We show that

p ≤ 4 + δ0−1 M.

(3.110)

We may assume without loss of generality that p ≥ 3. T2 By induction, we define a program {x(t)} ˜ t=T1 . Set

x(t) ˜ = x(t), t = T1 , . . . , S0 .

(3.111)

In view of (3.108), S 1 −1

S1 −1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(S0 ), x(S1 )) − δ0 . 0

(3.112)

t=S0 S1 ˜ By (3.111) and (3.112), there exists x(t), ˜ t = S0 + 1, . . . , S1 such that {x(t)} t=T1 is a program,

98

3 One-Dimensional Concave RSS Model

x(S ˜ 1 ) ≥ x(S1 ), S 1 −1

(3.113)

vt (x(t), x(t + 1))
1 such that c∗−1 x ≤ x2 ≤ c∗ x, x ∈ R n .

4.1 The Model Description and Main Results

105

The model studied in this chapter is a particular case of the model considered in Chap. 2 with u t = u 0 , at = a0 , t = 0, 1, . . . and all the results of Chap. 2 can be applied for our model. Set diam(K ) = sup{z 1 − z 2  : z 1 , z 2 ∈ K }. For every point x0 ∈ K and every natural number T define  T −1 U (x0 , T ) = sup u(x(t), x(t + 1)) : t=0 T {x(t)}t=0

 is a program and x(0) = x0 .

Let x0 , x˜0 ∈ K and let T ≥ 1 be an integer. Define U (x0 , x˜0 , T ) = sup

 T −1 t=0

T {x(t)}t=0

u(x(t), x(t + 1)) :

 is a program such that x(0) = x0 , x(T ) ≥ x˜0 .

Let T ≥ 1 be an integer. Define (T ) = sup U

T −1 

 T u(x(t), x(t + 1)) : {x(t)}t=0 is a program .

t=0

For every x0 ∈ K and every pair of integers T1 < T2 define U (x0 , T1 , T2 ) = sup

T 2 −1

u(x(t), x(t + 1)) :

t=T1

 T2 {x(t)}t=T is a program and x(T ) = x 1 0 . 1

(4.3)

Let x0 , x˜0 ∈ K and let T1 < T2 be integers. Define U (x0 , x˜0 , T1 , T2 ) = sup

T 2 −1 t=T1

T2 {x(t)}t=T 1

u(x(t), x(t + 1)) :  is a program and x(T1 ) = x0 , x(T2 ) ≥ x˜0 .

(4.4)

106

4 Turnpike Properties for Autonomous Problems

Let T1 , T2 be integers such that T1 < T2 . Define (T1 , T2 ) = sup U

T −1 2 

 u(x(t), x(t + 1)) :

T2 {x(t)}t=T 1

is a program .

t=T1

The following result was proved in Chap. 2 (see Theorem 2.2). Theorem 4.1 There exists a positive number M¯ such that for every x0 ∈ K there ∞ ¯ = x0 and that for every pair of nonnegative exists a program {x(t)} ¯ t=0 such that x(0) integers T1 , T2 satisfying T1 < T2 , the inequality T −1  2    (T1 , T2 ) ≤ M¯ u(x(t), ¯ x(t ¯ + 1)) − U    t=T1

holds. Moreover, for every natural number T , T −1 

u(x(t), ¯ x(t ¯ + 1)) = U˜ (x(0), ¯ x(T ¯ ), 0, T )

t=0

if the following properties hold: for every (z, z ) ∈ graph(a) satisfying u(z, z ) > 0 the function u is continuous at the point (z, z ); for each z, z 1 , z 2 , z 3 ∈ K satisfying z 1 ≤ z 2 ≤ z 3 and z i ∈ a(z), i = 1, 3 the inclusion z 2 ∈ a(z) is valid. Let M¯ > 0 be as guaranteed by Theorem 4.1. ∞ Proposition 4.2 (Chap. 2, Proposition 2.6) Let x0 ∈ K and let a program {x(t)} ¯ t=0 ∞ be as guaranteed by Theorem 4.1. Assume that {x(t)}t=0 is a program. Then either the sequence

T −1 

u(x(t), x(t + 1)) −

t=0

T −1 

∞ u(x(t), ¯ x(t ¯ + 1))

t=0

T =1

is bounded or T −1 

u(x(t), x(t + 1)) −

t=0

T −1 

u(x(t), ¯ x(t ¯ + 1)) → −∞ as T → ∞.

t=0

By Theorem 2.7, there exists (0, p)/ p μ = lim U p→∞

(4.5)

4.1 The Model Description and Main Results

and

107

¯ p for all natural numbers p. (0, p) − μ| ≤ 2 M/ | p −1 U

(4.6)

Theorem 4.1, Proposition 4.2 and (4.6) imply the following result. Proposition 4.3 Assume that {x(t)}∞ t=0 is a program. Then either the sequence T −1 

∞ u(x(t), x(t + 1)) − T μ T =1

t=0

is bounded or T −1 

u(x(t), x(t + 1)) − T μ → −∞ as T → ∞.

t=0

In view of (A2), μ > 0.

(4.7)

A program {x(t)}∞ t=0 is good if the sequence T −1 

∞ u(x(t), x(t + 1)) − T μ

t=0

T =1

is bounded. If T −1 

u(x(t), x(t + 1)) − T μ → −∞ as T → ∞,

t=0

then it is called bad. Evidently, μ ≥ sup{u(x, x) : x ∈ K , x ∈ a(x)}.

(4.8)

μ = sup{u(x, x) : x ∈ K , x ∈ a(x)}.

(4.9)

Suppose that Clearly, there exists x ∗ ∈ K such that (x ∗ , x ∗ ) ∈ graph(a), μ = u(x ∗ , x ∗ ).

(4.10)

We suppose that the function u is continuous at (x ∗ , x ∗ ).

(4.11)

108

4 Turnpike Properties for Autonomous Problems

We say that the function u has the asymptotic turnpike property (or ATP for short) if for every good program {x(t)}∞ t=0 , lim x(t) = x ∗ .

t→∞

We say that the function u has the turnpike property (or TP for short) if for each , M > 0 there exist δ > 0 and a natural number L such that for each integer T > 2L T and each program {x(t)}t=0 which satisfies T −1 

u(x(t), x(t + 1)) ≥ U (x(0), x(T ), 0, T ) − δ

t=0

and

T −1 

u(x(t), x(t + 1)) ≥ T μ − M

t=0

we have

x(t) − x ∗  ≤ , t = L , . . . , T − L .

In this chapter we prove the following result. Theorem 4.4 The function u has ATP if and only if it has TP.

4.2 A Controllability Lemma Lemma 4.5 Let  > 0. Then there exists δ > 0 such that for each x, x ∈ K satisfying x − x ∗ , x − x ∗  ≤ δ there exists x¯ ∈ K such that x¯ ≥ x , x¯ − x ∗  ≤ , x¯ ∈ a(x). Proof By (4.7) and (4.10),

μ = u(x ∗ , x ∗ ) > 0.

(4.12)

In view of (4.11) and (4.12), there exists δ0 ∈ (0, ) such that for each (z, z ) ∈ graph(a) satisfying z − x ∗ , z − x ∗  ≤ δ0 we have

u(z, z ) ≥ μ/2.

(4.13)

4.2 A Controllability Lemma

109

Assumption (A1) implies that there exists λ > 0 such that the following property holds: (i) If (z, z ) ∈ graph(a) satisfies u(z, z ) ≥ μ/2, then there exists z¯ ∈ a(z) such that z¯ ≥ z + λe. Choose δ ∈ (0, δ0 ) such that

δ < (2c∗ )−1 λ, δ ≤ c∗−2 n −1 .

Let

satisfy

(4.14)

x, x ∈ K x − x ∗ , x − x ∗  ≤ δ.

(4.15)

H (a(x), a(x ∗ )) ≤ κx − x ∗  ≤ κδ.

(4.16)

By (4.1) and (4.15),

By (4.10) and (4.16), there exists

such that

z ∈ a(x)

(4.17)

z − x ∗  < δ.

(4.18)

In view of the choice of δ0 (see (4.13)), (4.15), (4.17) and (4.18), u(x, z) ≥ μ/2.

(4.19)

Property (i), (4.17) and (4.19) imply that there exists z¯ ∈ a(x)

(4.20)

z¯ ≥ z + λe.

(4.21)

such that

By (4.3) and (4.18),

z − x ∗  < δc∗ .

110

4 Turnpike Properties for Autonomous Problems

For all i = 1, . . . , n,

xi∗ − δc∗ ≤ z i ≤ xi∗ + δc∗ .

(4.22)

Equations (4.21) and (4.22) imply that for all i = 1, . . . , n, z¯ i ≥ z i + λ ≥ λ + xi∗ − c∗ δ ≥ xi∗ + δc∗ ≥ z i .

(4.23)

By (A4), (4.17), (4.20) and (4.23), x ∗ + δc∗ e ∈ a(x).

(4.24)

It follows from (4.3) and (4.15) that

Set

x ∗ + δc∗ e ≥ x .

(4.25)

x¯ = x ∗ + δc∗ e.

(4.26)

In view of (4.14) and (4.26), x¯ − x ∗  = δc∗ e ≤ δc∗2 n 1/2 ≤ . Lemma 4.5 is proved.

4.3 TP Implies ATP Proposition 4.6 Assume that TP holds. Then ATP holds. Proof Assume that {x(t)}∞ t=0 is a good program. Then there exists M1 > 0 such that T −1      u(x(t), x(t + 1)) − T μ ≤ M1   

(4.27)

t=0

for all natural numbers T . In view of (4.27), for each pair of integers T2 > T1 ≥ 0, T −1  2     u(x(t), x(t + 1)) − (T2 − T1 )μ ≤ 2M1 .   

(4.28)

t=T1

Let δ > 0. We show that there exists a natural number Tδ such that for each integer T > Tδ ,

4.3 TP Implies ATP T −1 

111

u(x(t), x(t + 1)) ≥ U (x(Tδ ), x(T ), Tδ ), T ) − δ.

t=Tδ ∞ Assume the contrary. Then there exist sequences of natural numbers {Ti }i=1 and ∞ {Si }i=1 such that for every integer i ≥ 1,

Ti < Si < Ti+1 and

S i −1

u(x(t), x(t + 1)) < U (x(Ti ), x(Si ), Ti , Si ) − δ.

(4.29)

t=Ti

Set x(t) ˜ = x(t), t = 0, . . . , T1 .

(4.30)

In view of (4.29), there exist x(t) ˜ ∈ K , t = T1 + 1, . . . , S1 S1 such that {x(t)} ˜ t=0 is a program,

x(S ˜ 1 ) ≥ x(S1 ) and

S 1 −1

u(x(t), x(t + 1))
Tδ , T −1  u(x(t), x(t + 1)) ≥ U (x(Tδ ), x(T ), Tδ , T ) − δ. t=Tδ

4.3 TP Implies ATP

113

Let  > 0. In view of TP, there exist δ0 > 0 and a natural number L 0 such that the following property holds: T which satisfies (ii) For each integer T > 2L 0 and each program {y(t)}t=0 T −1 

u(y(t), y(t + 1))

t=0

≥ max{T μ − 2M1 , U (y(0), y(T ), 0, T ) − δ0 } we have

x(t) − x ∗  ≤ , t = L 0 , . . . , T − L 0 .

Property (i) implies that there exists a natural number Tδ0 such that for each integer T > Tδ0 , T −1  u(x(t), x(t + 1)) ≥ U (x(Tδ0 ), x(T ), Tδ0 , T ) − δ0 . (4.40) t=Tδ0

Let an integer T > Tδ0 + 2L 0 . Then (4.40) is true. In view of (4.28), T −1 

u(x(t), x(t + 1)) ≥ (T − Tδ0 )μ − 2M1 .

(4.41)

t=Tδ0

Property (ii), (4.40) and (4.41) imply that x(t) − x ∗  ≤ , t = Tδ0 + L 0 , . . . , T − L 0 . This implies that

x(t) − x ∗  ≤ 

for all integers t ≥ Tδ0 + L 0 . Therefore ATP holds. This completes the proof of Proposition 4.6.

4.4 Two Auxiliary Results Recall that M¯ > 0 is as guaranteed by Theorem 4.1. Lemma 4.7 Assume that ATP holds and M0 ,  > 0. Then there exists a natural L which satisfies number L ≥ 2 such that for each program {x(t)}t=0 L−1  t=0

u(x(t), x(t + 1)) ≥ Lμ − M0

114

4 Turnpike Properties for Autonomous Problems

there exists an integer τ ∈ {1, . . . , L − 1} such that x(τ ) − x ∗ , x(τ + 1) − x ∗  ≤  and u(x(τ ), x(τ + 1)) ≥ μ/2. Proof Assume the contrary. Then for every natural number k ≥ 2 there exists a program {xk (t)}kt=0 such that k−1 

u(xk (t), xk (t + 1)) ≥ kμ − M0

(4.42)

t=0

and that {t ∈ {1, . . . , k − 1} : xk (t) − x ∗ , xk (t + 1) − x ∗  ≤ , u(xk (t), xk (t + 1)) ≥ μ/2} = ∅. Let k ≥ 2 be an integer and

(4.43)

T ∈ {1, . . . , k − 1}.

By (4.6) and (4.42), T −1 

u(xk (t), xk (t + 1))

t=0

=

k−1 

u(xk (t), xk (t + 1)) −



{u(xk (t), xk (t + 1)) : t ∈ {T, . . . , k} \ {k}}

t=0

¯ = T μ − M0 − 2 M. ¯ ≥ kμ − M0 − ((k − T )μ + 2 M) Thus for each natural number k ≥ 2 and each T ∈ {1, . . . , k}, T −1 

¯ u(xk (t), xk (t + 1)) ≥ T μ − M0 − 2 M.

(4.44)

t=0

Extracting subsequences and using the diagonalization process we obtain a program ∞ {x(t)}∞ t=0 and a strictly increasing sequence of natural numbers {k p } p=1 such that for every integer t ≥ 0, (4.45) x(t) = lim xk p (t). p→∞

Clearly, {x(t)}∞ t=0 is a program. Since the function u is upper semicontinuous it follows from (4.44) and (4.45) that for every integer T > 0,

4.4 Two Auxiliary Results T −1 

115

¯ u(x(t), x(t + 1)) ≥ T μ − M0 − 2 M.

(4.46)

t=0

Thus {x(t)}∞ t=0 is a good program. In view of ATP, lim x(t) = x ∗ .

t→∞

(4.47)

Equation (4.47) implies that there exists a natural number S0 such that x(t) − x ∗  ≤ /4 for all integers t ≥ S0 .

(4.48)

¯ −1 + 2. S1 > S0 + (2M0 + 9 M)μ

(4.49)

Choose an integer

By (4.45), there exists an integer p0 ≥ 8 such that k p0 > S1 + 8,

(4.50)

x(t) − xk p0 (t) ≤ /4, t = 0, . . . , S1 + 8.

(4.51)

It follows from (4.48)–(4.51) that for all integers t = S0 , . . . , S1 + 8, xk p0 (t) − x ∗  ≤ /2.

(4.52)

In view of (4.44), S 1 −1

¯ u(xk p0 (t), xk p0 (t + 1)) ≥ (S1 − S0 )μ − M0 − 4 M.

(4.53)

t=S0

By (4.43) and (4.52), u(xk p0 (t), xk p0 (t + 1)) < μ/2, t = S0 , . . . , S1 − 1.

(4.54)

It follows from (4.53) and (4.54) that (S1 − S0 )μ − M0 − 4 M¯ ≤ (S1 − S0 )μ/2 and

¯ (S1 − S0 )μ ≤ 2M0 + 8 M.

This contradicts (4.49). The contradiction we have reached completes the proof of Lemma 4.7. Lemma 4.8 Assume that ATP holds and 0 ∈ (0, 1). Then there exists δ > 0 such T that for each integer T ≥ 1 and each program {x(t)}t=0 which satisfies

116

4 Turnpike Properties for Autonomous Problems

x(0) − x ∗  ≤ δ, x(T ) − x ∗  ≤ δ and

T −1 

u(x(t), x(t + 1)) ≥ U (x(0), x(T ), 0, T ) − δ

t=0

the inequality

x(t) − x ∗  ≤ 0

holds for all integers t = 0, . . . , T . Proof Set

 = 0 (2 + 8nc∗ )−1 .

(4.55)

By Lemma 4.5, (4.10) and (4.11), for each integer k ≥ 1, there exists δk ∈ (0, 8−k )

(4.56)

such that the following property holds: (i) For each x, x ∈ K satisfying x − x ∗ , x − x ∗  ≤ δk there exists x¯ ∈ K such that x¯ ≥ x , x¯ − x ∗  ≤ 8−k , x¯ ∈ a(x) and

|u(x, x) ¯ − μ| ≤ 8−k .

Assume that the lemma does not hold. Then for every integer k ≥ 1 there exist a Tk such that natural number Tk and a program {xk (t)}t=0 xk (0) − x ∗  ≤ δk , xk (Tk ) − x ∗  ≤ δk , T k −1

u(xk (t), xk (t + 1)) ≥ U (xk (0), xk (T ), 0, Tk ) − δk

(4.57) (4.58)

t=0

and that

max{xk (t) − x ∗  : t = 0, . . . , Tk } > 0 .

(4.59)

We may assume without loss of generality that {δk }∞ k=1 is a decreasing sequence. Let k be a natural number. By (4.56), (4.57) and (4.59), Tk ≥ 2.

4.4 Two Auxiliary Results

117

Set ξ(0) = xk (0).

(4.60)

By property (i), (4.57) and (4.60), there exists ξ(1) ∈ K such that

and

ξ(1) ≥ x ∗ , ξ(1) ∈ a(ξ0 )

(4.61)

|u(ξ(0), ξ(1)) − μ| ≤ 8−k .

(4.62)

Property (i) and (4.57) imply that there exists ηk ∈ K such that

and

ηk ≥ xk (Tk ), ηk ∈ a(x ∗ )

(4.63)

|u(x ∗ , ηk ) − μ| ≤ 8−k .

(4.64)

Assumption (A3), (4.61) and (4.63) imply that there exist ξ(t) ∈ K , t = 2, . . . , Tk Tk such that {ξ(t)}t=0 is a program, for all integers t = 1, . . . , Tk − 1, ξ(t) ≥ x ∗ ,

(4.65)

if t ∈ {1, . . . , Tk − 1} \ {Tk − 1}, then u(ξ(t), ξ(t + 1)) ≥ u(x ∗ , x ∗ ) = μ, ξ(Tk ) ≥ ηk ,

(4.66) (4.67)

u(ξ(Tk − 1), ξ(Tk )) ≥ u(x ∗ , ηk ).

(4.68)

By (4.56), (4.58), (4.60), (4.62)–(4.64), (4.66)–(4.68), T k −1

u(xk (t), xk (t + 1)) ≥ U (xk (0), xk (T ), 0, Tk ) − 8−k 

t=0

≥

T k −1

u(ξ(t), ξ(t + 1)) − 8−k  ≥ Tk μ − 8−k  − 2 · 8−k .

(4.69)

t=0

Now we construct a program {x(t)}∞ t=0 . Set x(t) = x1 (t), t = 0, . . . , T1 .

(4.70)

118

4 Turnpike Properties for Autonomous Problems

Assume that k ≥ 1 is an integer and that we defined x(t) ∈ K , t = 0, . . . , 1) − 1 such that  k

{x(t)}t=0i=1

k

i=1 (Ti

+

(Ti +1)−1

is a program, (4.70) is true, for every p ∈ {1, . . . , k} \ {k}, p

p

  u x (Ti + 1) − 1 , x (Ti + 1) ≥ μ − 8− p  i=1

(4.71)

i=1

and for a every integer p ∈ {1, . . . , k} \ {1} and every t ∈ {0, . . . , T p }, x

p−1 

(Ti + 1) + t

≥ x p (t),

i=1

x

p−1 

(4.72)

((Ti + 1) + t) − x p (t) ≤ 4c∗

p−1 

i=1

−i

8

e,

(4.73)

i=1

for all t = 0, . . . .T p − 1,

p−1

p−1   (Ti + 1) + t , x (Ti + 1) + t + 1 u x i=1

i=1

≥ u(x p (t), x p (t + 1)).

(4.74)



k  h (Ti + 1) − 1 = xk (Tk ).

Set

(4.75)

i=1

k

Property (i), (4.57) and (4.75) imply that there exists h(

i=1 (Ti

+ 1)) ∈ K such that

k

 h (Ti + 1) ≥ xk+1 (0),

(4.76)

i=1

k

k

  h (Ti + 1) ∈ a h (Ti + 1) − 1 , i=1

(4.77)

i=1

  k

    ∗ (Ti + 1) − x  ≤ 8−k , h   i=1  k 

k

      (Ti + 1) − 1 ,h (Ti + 1) − μ ≤ 8−k . u h   i=1

i=1

(4.78)

(4.79)

4.4 Two Auxiliary Results

119

By (A3) and (4.76), there exist h

k 

(Ti + 1) + t

∈ K , t = 1, . . . , Tk+1

i=1

such that

Tk+1  k  (Ti + 1) + t h i=1

t=0

is a program and for all t = 0, . . . , Tk+1 − 1, k

k

  u h (Ti + 1) + t , h (Ti + 1) + t + 1 i=1

i=1

≥ u(xk+1 (t), xk+1 (t + 1)),

(4.80)

for all t = 1, . . . , Tk+1 , k

 h (Ti + 1) + t ≥ xk+1 (t),

(4.81)

i=1

k

k

  h (Ti + 1) + t − xk+1 (t) ≤ h (Ti + 1) − xk+1 (0). i=1

(4.82)

i=1

In view of (4.3),   k

     (Ti + 1) − xk+1 (0) ≤ c∗ (8−k + 8−k−1 ). h   i=1

2

Together with (4.82) this implies that for all integers t = 0, . . . , Tk+1 ,

k  h (Ti + 1) + t

− xk+1 (t) ≤ c∗ (8−k + 8−k−1 )e.

i=1

By (A3), (4.70), (4.72), (4.73), (4.75) and (4.79), there exist

k  x (Ti + 1) + t i=1

∈ K , t = 0, . . . , Tk+1

(4.83)

120

4 Turnpike Properties for Autonomous Problems

such that k+1

{x(t)}t=0i=1

(Ti +1)−1

is a program,

for all t = 0, . . . , Tk+1 + 1,

k

k   x (Ti + 1) + t − 1 ≥ h (Ti + 1) − 1 ,

i=1

i=1

i=1

i=1

(4.84)

k

k   x (Ti + 1) + t − 1 − h (Ti + 1) + t − 1 k

k

  ≤x (Ti + 1) − 1 − h (Ti + 1) − 1 i=1

i=1

k

k−1

  −i ≤x (Ti + 1) − 1 − xk (Tk ) ≤ 4c∗ 8 e, i=1

(4.85)

i=1

k

k   (Ti + 1) − 1 , x (Ti + 1) u x

i=1

≥ u xk (Tk ), h



i=1

k  (Ti + 1)



≥ μ − 8−k ,

(4.86)

i=1

for all t = 0, . . . , Tk+1 − 1, k

k

  u x (Ti + 1) + t , x (Ti + 1) + t + 1 i=1

i=1

k

k

  ≥u h (Ti + 1) + t , h (Ti + 1) + t + 1 . i=1

i=1

By (4.76), (4.81), (4.83)–(4.85), for all t = 0, . . . , Tk+1 ,

(4.87)

4.4 Two Auxiliary Results

121



k  x (Ti + 1) + t



i=1

k  x (Ti + 1) + t

≥ xk+1 (t),

i=1

=x

k 

− xk+1 (t)

(Ti + 1) + t

i=1



k  +h (Ti + 1) + t i=1

≤ 4

k−1 

(4.88)

−h

k 

(Ti + 1) + t

i=1

− xk+1 (t)

−i

8

−k

c∗ e + (8

−k−1

+8

)c∗ e ≤ 4

i=1

k 

−i

8

c∗ e.

(4.89)

i=1

In view of (4.80) and (4.87), for all t = 0, . . . , Tk+1 − 1, k

k

  u x (Ti + 1) + t , x (Ti + 1) + t + 1 i=1

i=1

≥ u(xk+1 (t), xk+1 (t + 1)).

(4.90)

It follows from (4.83) and (4.88)–(4.90) that the assumption made for k also holds for k + 1. Therefore by induction we constructed the program {x(t)}∞ t=0 such that (4.70) holds, for all integers p ≥ 1, (4.71) is valid, for all integers p ≥ 2 and all integers t = 0, . . . , T p (4.72) and (4.73) hold and for all integers t = 0, . . . , T p−1 , (4.74) is true. Let k ≥ 3 be an integer. Then 

 u(x(t), x(t + 1)) : t = 0, . . . ,

k 

 (Ti + 1) − 1

i=1

=

T 1 −1

u(x(t), x(t + 1))

t=0

+

T p+1 k−1  



p

p   u x (Ti + 1) + t − 1 , x (Ti + 1) + t .

p=1 t=0

i=1

(4.91)

i=1

In view of (4.69) and (4.70), T 1 −1 i=0

u(x(t), x(t + 1)) ≥ T1 μ − 3 · 8−1 .

(4.92)

122

4 Turnpike Properties for Autonomous Problems

By (4.69), (4.71) and (4.74), for all p = 1, . . . , k − 1, T p+1 

p

p

  u x (Ti + 1) + t − 1 , x (Ti + 1) + t

t=0

i=1

i=1

p

p

  ≥u x (Ti + 1) − 1 , x (Ti + 1) i=1

i=1

p

p

   + u x (Ti + 1) + t , x (Ti + 1) + t + 1 T p+1 −1

t=0

i=1

i=1

T p+1 −1

≥ μ − 8− p  +



u(x p+1 (t), x p+1 (t + 1))

t=0

≥ μ − 8− p  + T p+1 μ − 3 · 8− p−1 .

(4.93)

In view of (4.91)–(4.93), 



⎛ ≥ μ⎝

 k  (Ti + 1) − 1 u(x(t), x(t + 1)) : t = 0, . . . , i=1

⎞

k k   (T p + 1) − 1⎠ − 4 8− p . p=1

p=1

This implies that {x(t)}∞ t=0 is a good program. By ATP, lim x(t) = x ∗ .

(4.94)

t→∞

Equation (4.94) implies that there exists an integer S0 > 1 such that x(t) − x ∗  ≤ 4−1  for all integers t ≥ S0 .

(4.95)

Choose a natural number p ≥ 4 such that p−1  (Ti + 1) − 1 > S0 + 4.

(4.96)

i=1

It follows from (4.95) and (4.96) that  p−1  p   x(t) − x  ≤ , t ∈ (Ti + 1) − 1), . . . , (Ti + 1) + 1) . ∗

i=1

i=1

(4.97)

4.4 Two Auxiliary Results

123

By (4.72), (4.73) and (4.97), for each t = 0, . . . , T p ,  p−1

     −1  (Ti + 1) − 1 + t  c∗ x p (t) − x   i=1  p−1

      ≤ x p (t) − x (Ti + 1 − 1) + t  ≤ 4c∗ n   i=1

and

2

x p (t) − x ∗  ≤  + 4nc∗ < 0 .

This contradicts (4.59). The contradiction we have reached completes the proof of Lemma 4.8.

4.5 ATP Implies TP We prove the following result. Theorem 4.9 Assume that ATP holds,  ∈ (0, 1) and M > 0. Then there exist δ > 0 and a natural number L such that for each integer T > 2L and each program T satisfying {x(t)}t=0 T −1 

u(x(t), x(t + 1)) ≥ U (x(0), x(T ), 0, T ) − δ

t=0

and

T −1 

u(x(t), x(t + 1)) ≥ T μ − M

t=0

there exist integers τ1 ∈ [0, L] and τ2 ∈ [T − L , T ] such that x(t) − x ∗  ≤ , t = τ1 , . . . , τ2 . Moreover, if x(0) − x ∗  ≤ δ, then τ1 = 0 and if x(T ) − x ∗  ≤ δ, then τ2 = T . Proof By Lemma 4.8, there exists δ ∈ (0, ) such that the following property holds: T (i) For each integer T ≥ 1 and each program {x(t)}t=0 which satisfies x(0) − x ∗  ≤ δ, x(T ) − x ∗  ≤ δ

124

4 Turnpike Properties for Autonomous Problems

and

T −1 

u(x(t), x(t + 1)) ≥ U (x(0), x(T ), 0, T ) − δ

t=0

the inequality

x(t) − x ∗  ≤ 

holds for all integers t = 0, . . . , T . By Lemma 4.7, there exists a natural number L ≥ 2 such that the following property holds: L which satisfies (ii) For each program {x(t)}t=0 L−1 

u(x(t), x(t + 1)) ≥ Lμ − M − 4 M¯

t=0

there exists an integer τ ∈ {1, . . . , L − 1} such that x(τ ) − x ∗ , x(τ + 1) − x ∗  ≤ δ. T satisfies Assume that an integer T > 2L and a program {x(t)}t=0 T −1 

u(x(t), x(t + 1)) ≥ U (x(0), x(T ), 0, T ) − δ

(4.98)

t=0

and

T −1 

u(x(t), x(t + 1)) ≥ T μ − M.

(4.99)

t=0

Let 0 ≤ T1 < T2 ≤ T be integers. In view of (4.6) and (4.99), T 2 −1

u(x(t), x(t + 1))

t=T1

≥

T −1 

u(x(t), x(t + 1))

t=0

 {u(x(t), x(t + 1)) : t ∈ {0, . . . , T1 } \ {T1 }}  − {u(x(t), x(t + 1)) : t ∈ {T2 , . . . , T } \ {T }} −

¯ − ((T − T2 )μ + 2 M) ¯ = (T2 − T1 )μ − 4 M¯ − M. ≥ T μ − M − (T1 μ + 2 M) (4.100)

4.5 ATP Implies TP

125

Property (ii) and (4.100) imply that there exist integers τ1 ∈ [0, L], τ2 ∈ [T − L , T ] such that

x(τi ) − x ∗  ≤ δ, i = 1, 2.

If

(4.101)

x(0) − x ∗  ≤ δ,

then we may assume that τ1 = 0 and if x(T ) − x ∗  ≤ δ, then we may assume that τ2 = T. Assumption (A3) and (4.98) imply that τ 2 −1

u(x(t), x(t + 1)) ≥ U (x(τ1 ), x(τ2 ), τ1 , τ2 ) − δ.

(4.102)

t=τ1

By property (i), (4.101) and (4.102), x(t) − x ∗  ≤ , t = τ1 , . . . , τ2 . Theorem 4.9 is proved.

4.6 A Weak Turnpike Result Theorem 4.10 Let ATP hold, M > 0 and  > 0. Then there exists a natural number T satisfying Q such that for each integer T > 0 and each program {x(t)}t=0 T −1 

u(x(t), x(t + 1)) ≥ T μ − M

t=0

the inequality Card({t ∈ {0, . . . , T } : x(t) − x ∗ (t) > }) ≤ Q holds.

126

4 Turnpike Properties for Autonomous Problems

Proof By Lemma 4.8, there exists δ ∈ (0, /2) such that the following property holds: T which satisfies (i) For each integer T ≥ 1 and each program {x(t)}t=0 x(0) − x ∗  ≤ δ, x(T ) − x ∗  ≤ δ and

T −1 

u(x(t), x(t + 1)) ≥ U (x(0), x(T ), 0, T ) − δ

t=0

the inequality

x(t) − x ∗  ≤ 0

holds for all integers t = 0, . . . , T . By Lemma 4.7, there exists a natural number L ≥ 2 such that the following property holds: L which satisfies (ii) For each program {x(t)}t=0 L−1 

u(x(t), x(t + 1)) ≥ Lμ − M − 4 M¯

t=0

there exists an integer τ ∈ {1, . . . , L − 1} such that x(τ ) − x ∗ , x(τ + 1) − x ∗  ≤ δ. Choose an integer Q > 8L + 8 + 10L(4 + δ −1 (6 M¯ + M)).

(4.103)

T satisfies Assume that an integer T > Q and that a program {x(t)}t=0 T −1 

u(x(t), x(t + 1)) ≥ T μ − M.

t=0

Let 0 ≤ T1 < T2 ≤ T be integers. In view of (4.6) and (4.104),

(4.104)

4.6 A Weak Turnpike Result T 2 −1

127

u(x(t), x(t + 1))

t=T1

≥

T −1 

u(x(t), x(t + 1))

t=0

 {u(x(t), x(t + 1)) : t ∈ {0, . . . , T1 } \ {T1 }}  − {u(x(t), x(t + 1)) : t ∈ {T2 , . . . , T } \ {T }} −

¯ − ((T − T2 )μ + 2 M) ¯ = (T2 − T1 )μ − 4 M¯ − M. ≥ T μ − M − (T1 μ + 2 M) Thus we showed that the following property holds: (iii) For each pair of integers T1 , T2 satisfying 0 ≤ T1 < T2 ≤ T , T 2 −1

u(x(t), x(t + 1)) ≥ (T2 − T1 )μ − 4 M¯ − M.

t=T1

Properties (ii) and (iii) and (4.103) imply that there exists t0 ∈ {0, . . . , L} such that

x(t0 ) − x ∗  ≤ δ.

Using properties (ii) and (iii), we construct by induction a sequence of nonnegative q integers {ti }i=0 such that (4.105) t0 ∈ [0, L], for all integers i = 0, . . . , q − 1, ti+1 − ti ∈ [L , 2L],

(4.106)

T ≥ tq > T − 2L , x(ti ) − x ∗  ≤ δ, i = 0, . . . , q.

(4.107)

By induction we define a strictly increasing finite sequence {Si : i = 0, . . . , k} ⊂ {t0 , . . . , tq }. Set S0 = t0 .

(4.108)

128

4 Turnpike Properties for Autonomous Problems

In view of (4.103), (4.105) and (4.106), S0 < tq . Assume that k ≥ 1 is an integer and that we have defined a strictly increasing finite sequence k ⊂ {t0 , . . . , tq } {Si }i=0 such that (4.108) holds, if an integer p satisfies 0 ≤ p < k, then S p+1 −1



u(x(t), x(t + 1)) < U (x(S p ), x(S p+1 ), S p , S p+1 ) − δ

(4.109)

t=S p

and if an integer S ∈ {t0 , . . . , tq } satisfies S p < S < S p+1 , then

S−1 

u(x(t), x(t + 1)) ≥ U (x(S p ), x(S), S, S p ) − δ.

(4.110)

t=S p

(Note that our assumption holds for k = 0.) If Sk = tq , then our construction is completed. If tq −1  u(x(t), x(t + 1)) ≥ U (x(Sk ), x(tq ), Sk , tq ) − δ, t=Sk

then we set Sk+1 = tq and the construction is completed. Assume that tq −1



t=Sk

u(x(t), x(t + 1)) < U (x(Sk ), x(tq ), Sk , tq ) − δ.

4.6 A Weak Turnpike Result

129

It is not difficult to see that there exists an integer Sk+1 ∈ {t0 , . . . , tq } such that Sk < Sk+1 and

Sk+1 −1



u(x(t), x(t + 1)) < U (x(Sk ), x(Sk+1 , Sk , Sk+1 ) − δ

t=Sk

and if an integer S ∈ {t0 , . . . , tq } satisfies Sk < S < Sk+1 , then

S−1 

u(x(t), x(t + 1)) ≥ U (x(Sk ), x(S), Sk , S) − δ.

t=Sk

Thus the assumption made for k holds for k + 1. By induction we have constructed a strictly increasing sequence of integers j

{Si }i=0 ⊂ {t0 , . . . , tq } such that S0 = t0 , S j = tq , for each integer p ∈ {0, . . . , j − 1} \ { j − 1} (4.109) holds, and for each p ∈ {0, . . . , j − 1} and each S ∈ {t0 , . . . , tq } satisfying S p < S < S p+1 , (4.110) is true. We show that

j ≤ 4 + δ −1 (6 M¯ + M).

(4.111)

130

4 Turnpike Properties for Autonomous Problems

We may assume without loss of generality that j ≥ 4. Property (iii) and (4.6) imply that S j−1 −1



u(x(t), x(t + 1)) ≥ (S j−1 − S0 )μ − 4 M¯ − M

t=S0

(S0 , S j−1 ) − 6 M¯ − M. ≥U

(4.112)

In view of (4.109), for each integer p satisfying 0 ≤ p < j − 1, S

p+1 such that there exists a program {z p (t)}t=S p

z p (S p ) = x(S p ), z p (S p+1 ) ≥ x(S p+1 ), S p+1 −1



(4.113)

S p+1 −1

u(x(t), x(t + 1))
U (x(Si ), x(S), Si , S) − δ.

(4.121)

(4.122)

t=Si

In view of (4.107),

x(Si ) − x ∗  ≤ δ, x(S) − x ∗  ≤ δ.

Property (i), (4.122) and (4.123) imply that x(t) − x ∗  ≤ , t = Si , . . . , S. Together with (4.121) this implies that x(t) − x ∗  ≤ , t = Si , . . . , Si+1 − 2L . This implies that {t ∈ [0, T ] : x(t) − x ∗ (t) > } ⊂ {0, . . . , t0 } ∪ {tq , . . . , T } ∪ {{Si , . . . , Si+1 } : i ∈ {0, . . . , j − 1}, Si+1 − Si < 8L} ∪ {Si+1 − 2L + 1, . . . , Si+1 } : i ∈ E}. Combined with (4.103), (4.105), (4.106) and (4.111) this implies that

(4.123)

132

4 Turnpike Properties for Autonomous Problems

Card({t ∈ [0, T ] : x(t) − x ∗ (t) > }) ≤ 3L + 2 + 8L j + 2L j ≤ 3L + 2 + 10L j ≤ 3L + 2 + 10L(4 + δ −1 (6 M¯ + M)) ≤ Q. Theorem 4.10 is proved.

4.7 A Turnpike Result for Approximate Solutions In this chapter we prove the following turnpike result. Its proof is given in Sect. 4.9. Its based on two auxiliary results given in the next section. Theorem 4.11 Assume that ATP holds,  ∈ (0, 1) and M > 0. Then there exist δ > 0 and a natural number L such that for each integer T > 2L and each program T {x(t)}t=0 satisfying τ +L−1 

u(x(t), x(t + 1)) ≥ U (x(τ ), x(τ + L), τ , τ + L) − δ

t=τ

for all τ = 0, . . . , T − L and T −1 

u(x(t), x(t + 1)) ≥ Lμ − M

t=T −L

there exist integers τ1 ∈ [0, L] and τ2 ∈ [T − L , T ] such that x(t) − x ∗  ≤ , t = τ1 , . . . , τ2 . Moreover, if x(0) − x ∗  ≤ δ, then τ1 = 0 and if x(T ) − x ∗  ≤ δ, then τ2 = T .

4.8 Auxiliary Results for Theorem 4.11 Lemma 4.12 There exist γ0 ∈ (0, 1) and a natural number Q 0 such that for each Q0 x ∈ K and each y ∈ K satisfying y − x ∗  ≤ γ0 there exists a program {x(t)}t=0 such that x(0) = x, x(Q 0 ) ≥ y. Proof By Lemma 4.5, there exists γ0 ∈ (0, 1) such that the following property holds: (i) For each x, x ∈ K satisfying

4.8 Auxiliary Results for Theorem 4.11

133

x − x ∗ , x − x ∗  ≤ γ0 there exists x¯ ∈ K such that

x¯ ≥ x , x¯ ∈ a(x).

Choose a natural number Q 0 ≥ 3 such that

Let x, y ∈ K satisfy

κ Q 0 −1 c∗ (diam(K ) + 1) < γ0 .

(4.124)

y − x ∗  ≤ γ0 .

(4.125)

Property (i), (A3), (4.1) and (4.10) imply that there exists a program {ξ(t)}∞ t=0 such that ξ(0) = x (4.126) and that

ξ(t + 1) − x ∗  ≤ ξ(t) − x ∗  for all integers t ≥ 0.

(4.127)

By (4.4), (4.126) and (4.127), for all integers t ≥ 1, ξ(t) − x ∗  ≤ κt x − x ∗  ≤ diam(K )κt . Together with (4.124) this implies that ξ(Q 0 − 1) − x ∗  < γ0 .

(4.128)

Property (i), (4.125) and (4.128) imply that there exists ξ¯ ∈ K such that ξ¯ ≥ y, ξ¯ ∈ a(ξ(Q 0 − 1)). Set

(4.129)

¯ x(t) = ξ(t), t = 0, . . . , Q 0 − 1, x(Q 0 ) = ξ.

Q0 is a program satisfying In view of (4.126) and (4.129), {x(t)}t=0

x(0) = x, x(Q 0 ) ≥ y. Lemma 4.12 is proved. Lemma 4.13 There exist γ1 ∈ (0, 1), M1 > 0 and a natural number Q 1 such that for each x ∈ K , each y ∈ K satisfying y − x ∗  ≤ γ1

134

4 Turnpike Properties for Autonomous Problems

T and each integer T ≥ Q 1 there exists a program {x(t)}t=0 such that

x(0) = x, x(T ) ≥ y and

T −1 

u(x(t), x(t + 1)) ≥ T μ − M1 .

t=0

Proof By Lemma 4.12, there exist γ0 ∈ (0, 1) and a natural number Q 0 such that the following property holds: (i) For each x ∈ K and each y ∈ K satisfying y − x ∗  ≤ γ0 Q0 there exists a program {x(t)}t=0 such that

x(0) = x, x(Q 0 ) ≥ y. By Lemma 4.5, there exists γ1 ∈ (0, 1) such that the following property holds: (ii) For each x, y ∈ K satisfying x − x ∗ , y − x ∗  ≤ γ1 there exists x¯ ∈ a(x) such that

x¯ ≥ y, x¯ − x ∗  ≤ γ0 .

Let Q 1 = Q 0 + 4, M1 = μ(Q 0 + 2).

(4.130)

Assume that x ∈ K , y ∈ K satisfies y − x ∗  ≤ γ1

(4.131)

and an integer T ≥ Q 1 . Q0 By property (i), there exists a program {x(t)}t=0 such that

x(0) = x, x(Q 0 ) ≥ x ∗ .

(4.132)

Assumption (A3), (4.10) and (4.132) imply that there exist x(t) ∈ K , t = Q 0 + T −1 is a program, 1, . . . , T − 1 such that {xt }t=0

4.8 Auxiliary Results for Theorem 4.11

135

x(t) ≥ x ∗ , t = Q 0 + 1, . . . , T − 1,

(4.133)

u(x(t), x(t + 1)) ≥ μ, t = Q 0 , . . . , T − 2.

(4.134)

Property (ii) and (4.131) imply that there exists x¯ ∈ a(x ∗ ) such that x¯ ≥ y. Together with (A3) and (4.133) this implies that there exists x(T ) ∈ a(x(T − 1)) such that x(T ) ≥ x¯ ≥ y. T is a program. In view of (4.134), Clearly, {xt }t=0 T −1 

u(x(t), x(t + 1)) ≥ μ(T − Q 0 − 2) = T μ − μ(Q 0 + 2) = T μ − M1 .

t=0

Lemma 4.13 is proved.

4.9 Proof of Theorem 4.11 By Lemma 4.13, there exist γ0 ∈ (0, 1), M0 > 0 and a natural number L 0 such that the following property holds: (i) For each x ∈ K , each y ∈ K satisfying y − x ∗  ≤ γ0 T and each integer T ≥ L 0 there exists a program {x(t)}t=0 such that

x(0) = x, x(T ) ≥ y and

T −1  t=0

u(x(t), x(t + 1)) ≥ T μ − M0 .

136

4 Turnpike Properties for Autonomous Problems

By Lemma 4.8, there exists δ1 ∈ (0, min{, γ0 , 1} such that the following property holds: S which satisfies (ii) For each integer S ≥ 1 and each program {ξ(t)}t=0 ξ(0) − x ∗  ≤ δ1 , ξ(S) − x ∗  ≤ δ1 and

S−1 

u(ξ(t), ξ(t + 1)) ≥ U (ξ(0), ξ(S), 0, S) − δ1

t=0

the inequality

ξ(t) − x ∗  ≤ 

holds for all integers t = 0, . . . , S. Choose δ ∈ (0, 2−1 δ1 ), M1 > M¯ + M + M0 + 1.

(4.135) (4.136)

By Theorem 4.10, there exists a natural number L 1 such that the following property holds: T satisfying (iii) For each integer T ≥ L 1 and each program {ξ(t)}t=0 T −1 

u(ξ(t), ξ(t + 1)) ≥ T μ − M1

t=0

we have

Card({t ∈ {0, . . . , T } : x(t) − x ∗  > δ}) ≤ L 1 .

Choose a natural number L > 8L 1 + 8L 0 + 8.

(4.137)

T satisfies Assume that an integer T > 2L and a program {x(t)}t=0 τ +L−1  t=τ

u(x(t), x(t + 1)) ≥ U (x(τ ), x(τ + L), τ , τ + L) − δ

(4.138)

4.9 Proof of Theorem 4.11

137

for all τ = 0, . . . , T − L and T −1 

u(x(t), x(t + 1)) ≥ Lμ − M.

(4.139)

t=T −L

Property (iii), (4.137) and (4.139) imply that there exists

such that

Note that if

t0 ∈ {T − L 1 , . . . , T }

(4.140)

x(t0 ) − x ∗  ≤ δ.

(4.141)

x(T ) − x ∗  ≤ δ,

then we may assume that t0 = T . Assumption (A3), (4.40), (4.137) and (4.138) imply that t0 −1 

u(x(t), x(t + 1))

t=t0 −2L 0 −2L 1

≥ U (x(t0 − 2L 0 − 2L 1 ), x(t0 ), t0 − 2L 0 − 2L 1 , t0 − 1) − δ.

(4.142)

Property (i), (4.135), (4.141) and (4.142) imply that t0 −1 

u(x(t), x(t + 1)) ≥ (2L 0 + 2L 1 )μ − M0 − 1.

(4.143)

t=t0 −2L 0 −2L 1

Property (iii), (4.136) and (4.143) imply that there exists t1 ∈ {t0 − L 0 − L 1 , . . . , t0 − L 0 }

(4.144)

x(t1 ) − x ∗  ≤ δ.

(4.145)

such that

Assume that k ≥ 1 is an integer, we have defined a strictly decreasing sequence of nonnegative integers ti ∈ [0, T ], i = 0, . . . , k such that (4.140) and (4.144) are valid, x(ti ) − x ∗  ≤ δ, i = 0, . . . , k

(4.146)

and for all i = 0, . . . , k − 1, ti+1 ∈ [ti − L 0 − L 1 , ti − L 0 ].

(4.147)

138

4 Turnpike Properties for Autonomous Problems

(Note that by (4.141), (4.142), (4.144) and (4.145), our assumption holds for k = 1.) There are two cases: tk ≥ L 0 + L 1 ;

(4.148)

tk < L 0 + L 1 .

(4.149)

If (4.149) holds and tk = 0, then the construction is completed. If (4.149) is true and tk > 0 and x(0) − x ∗  > δ, then the construction is completed. If tk > 0, x(0) − x ∗  ≤ δ, then we set tk+1 = 0 and the construction is completed. Assume that (4.148) holds. Since T > 2L there exists an integer S ∈ [0, T ] such that (4.150) [tk − L 0 − L 1 , tk ] ⊂ [S, S + L] ⊂ [0, T ]. In view of (4.138), S+L−1 

u(x(t), x(t + 1))

t=S

≥ U (x(S), x(S + L), S, S + L) − δ.

(4.151)

Assumption (A3), (4.150) and (4.151) imply that tk −1 

u(x(t), x(t + 1))

t=tk −L 0 −L 1

≥ U (x(tk − L 0 − L 1 ), x(tk ), tk − L 0 − L 1 , tk ) − δ.

(4.152)

Property (i), (4.135), (4.146) and (4.152) imply that tk −1 

u(x(t), x(t + 1)) ≥ μ(L 0 + L 1 ) − M0 − 1.

t=tk −L 0 −L 1

By property (iii), (4.136) and (4.153), there exists an integer tk+1 ∈ {tk − L 0 − L 1 , . . . , tk − L 0 }

(4.153)

4.9 Proof of Theorem 4.11

139

such that

x(tk+1 ) − x ∗  ≤ δ.

It is not difficult to see that the assumption made for k also holds for k + 1. Thus by induction we constructed a finite strictly decreasing sequence of nonnegative integers ti ∈ [0, T ], i = 0, . . . , q such that t0 ∈ [T − L 1 , T ], tq ≤ L 0 + L 1 , x(ti ) − x ∗  ≤ δ, i = 0, . . . , q, if

(4.154)

x(0) − x ∗  ≤ δ,

then tq = 0, if

x(T ) − x ∗  ≤ δ,

then t0 = T , and for all i = 0, . . . , q − 1, ti − ti+1 ≤ L 0 + L 1 .

(4.155)

Assumption (A3), property (ii), (4.138), (4.154) and (4.155) imply that for each i ∈ {0, . . . , q − 1}, x(t) − x ∗  ≤ , t ∈ {ti+1 , . . . , ti }. Therefore

x(t) − x ∗  ≤ , t ∈ {tq , . . . , t0 }.

The proof of Theorem 4.11 is complete.

4.10 Stability of the Turnpike Phenomenon Let ψ : graph(a) → R 1 . Set ψ = sup{|ψ(z, z )| : (z, z ) ∈ graph(a)}. Let T1 , T2 be integers such that 0 ≤ T1 < T2 and let vt : graph(a) → R 1 , t = T1 , . . . , T2 be bounded on bounded sets functions. For each pair of points z 0 , z 1 ∈ K define

140

4 Turnpike Properties for Autonomous Problems

T2 −1 U ({vt }t=T , z 0 , z 1 ) = sup 1

T 2 −1

T2 vt (x(t), x(t + 1)) : {x(t)}t=T 1

t=T1

 is a program such that x(T1 ) = z 0 , x(T2 ) ≥ z 1 , T2 −1 , z0 ) U ({vt }t=T 1

= sup

T 2 −1

(4.156)

T2 vt (x(t), x(t + 1)) : {x(t)}t=T 1

t=T1

 is a program such that x(T1 ) = z 0 .

(4.157)

Recall that the supremum over an empty set is −∞. In this section we show that the turnpike phenomenon is stable under small perturbations of the objective functions. Theorem 4.14 Let ATP hold, M0 > 0 and  ∈ (0, 1). Then there exist a natural number L and δ0 > 0 such that for each pair of integers T1 ≥ 0, T2 > 2L + T1 , each finite sequence of functions vt : graph(a) → R 1 , t = T1 , . . . , T2 − 1 which satisfy u − vt  ≤ δ0 , t = T1 , . . . , T2 − 1 T2 satisfying and every program {x(t)}t=T 1 τ +L−1 

+L−1 vt (x(t), x(t + 1)) ≥ U ({vt }τt=τ , z 0 , z 1 ) − δ0

t=τ

for all τ ∈ {T1 , . . . , T2 − L}, T −1 

vt (x(t), x(t + 1)) ≥ Lμ − M0

T −L

there exist integers τ1 ∈ [T1 , T1 + L] and τ2 ∈ [T2 − L , T2 ] such that x(t) − x ∗  ≤  holds for all integers t ∈ [τ1 , τ2 ]. Moreover, if then τ1 = T1 and if then τ2 = T2 .

x(T1 ) − x ∗  ≤ δ, x(T2 ) − x ∗  ≤ δ,

4.10 Stability of the Turnpike Phenomenon

141

Proof Theorem 4.14 follows easily from Theorem 4.11. Namely, let M = M0 + 1 and let a natural number L and δ ∈ (0, 1) be as guaranteed by Theorem 4.11. Set δ0 = δ(4L + 4)−1 . Now it is easy to see that the assertion of Theorem 4.14 is true in view of Theorem 4.11. Denote by M the set of all functions v : graph(a) → R 1 such that the following property holds: if (x, y) ∈ graph(a), x˜ ∈ K , x˜ ≥ x, then there exists y˜ ∈ a(x) ˜ such that y˜ ≥ y, u(x, ˜ y˜ ) ≥ u(x, y). By (A3), u ∈ M. Theorem 4.15 Let ATP hold, M > 0 and  > 0. Then there exists a natural number Q and δ > 0 such that for each pair of integers T1 ≥ 0 and T2 ≥ Q + T1 , each finite sequence of functions vt : graph(a) → R 1 , t = T1 , . . . , T2 − 1 which satisfy vt ∈ M, vt − u ≤ δ, t = T1 , . . . , T2 − 1 T2 satisfying and every program {x(t)}t=T 1 T 2 −1

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T , x(T1 )) − M 1

t=T1

the inequality Card({t ∈ {T1 , . . . , T2 } : x(t) − x ∗ (t) > }) ≤ Q holds. Proof By Lemma 4.13, there exist γ0 ∈ (0, 1), M0 > 0 and a natural number L 0 such that the following property holds: (i) For each x ∈ K , each y ∈ K satisfying y − x ∗  ≤ γ0 T and each integer T ≥ L 0 there exists a program {x(t)}t=0 such that

x(0) = x, x(T ) ≥ y

142

4 Turnpike Properties for Autonomous Problems

and

T −1 

u(x(t), x(t + 1)) ≥ T μ − M0 .

t=0

By Lemma 4.8, there exists δ0 ∈ (0, min{, γ0 , 1})

(4.158)

such that the following property holds: S which satisfies (ii) For each integer S ≥ 1 and each program {ξ(t)}t=0 ξ(0) − x ∗  ≤ δ0 , ξ(S) − x ∗  ≤ δ0 and

S−1 

u(ξ(t), ξ(t + 1)) ≥ U (ξ(0), ξ(S), 0, S) − δ0

t=0

the inequality

ξ(t) − x ∗  ≤ 

holds for all integers t = 0, . . . , S. Set M1 > 4 M¯ + M + M0 + 4.

(4.159)

By Theorem 4.10, there exists a natural number L 1 > L 0 such that the following property holds: T satisfying (iii) For each integer T ≥ L 1 and each program {ξ(t)}t=0 T −1 

u(ξ(t), ξ(t + 1)) ≥ T μ − M

t=0

the inequality Card({t ∈ {0, . . . , T } : x(t) − x ∗ (t) > δ0 }) ≤ L 1 . Let

δ ∈ (0, 8−1 δ0 (L 0 + L 1 + 2)−1 ).

(4.160)

Q > (8L 1 + 8L 0 + 8)(L + 2δ0−1 M).

(4.161)

Choose a natural number

4.10 Stability of the Turnpike Phenomenon

143

Assume that integers T1 ≥ 0, T2 ≥ Q + T1 , vt ∈ M, vt − u ≤ δ, t = T1 , . . . , T2 − 1,

(4.162)

T2 is a program satisfying and that {x(t)}t=T 1 T 2 −1

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T , x(T1 )) − M. 1

(4.163)

t=T1

In view of (4.163), T 2 −1

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T , x(T2 − L 1 )) − M. 2 −L 1

(4.164)

t=T2 −L 1

Equations (4.160), (4.162) and (4.164) imply that T 2 −1

u(x(t), x(t + 1)) ≥ U (x(T2 − L 1 ), L 1 ) − M − 1.

(4.165)

t=T2 −L 1

Property (v), (4.159) and (4.165) imply that T 2 −1

u(x(t), x(t + 1)) ≥ μL 1 − μ0 − M − 1 > μL 1 − M1 .

(4.166)

t=T2 −L 1

Properties (iii) and (4.166) imply that there exists

such that

t0 ∈ {T2 − L 1 , . . . , T2 }

(4.167)

x(t0 ) − x ∗  ≤ δ0 .

(4.168)

It follows from (4.161)–(4.163) and (4.167) that t0 −1 

vt (x(t), x(t + 1))

t=t0 −L 0 −L 1 t0 −1 , x(t0 − L 0 − L 1 ), x(t0 ), t0 − L 0 − L 1 , t0 ) − M. ≥ U ({vt }t=t 0 −L 0 −L 1

By (4.160), (4.162), (4.165) and (4.169),

(4.169)

144

4 Turnpike Properties for Autonomous Problems t0 −1 

u(x(t), x(t + 1))

t=t0 −L 0 −L 1

≥ U (x(t0 − L 0 − L 1 ), x(t0 ), t0 − L 0 − L 1 , t0 ) − M − 1. Property (i), (4.158), (4.159), (4.168) and the relation above imply that t0 −1 

u(x(t), x(t + 1))

t=t0 −L 0 −L 1

≥ (L 0 + L 1 )μ − M0 − M − 1 ≥ (L 0 + L 1 )μ − M1 .

(4.170)

Property (iii) and (4.170) imply that there exists an integer

such that

t1 ∈ {t0 − L 0 − L 1 , . . . , t0 − L 0 }

(4.171)

x(t1 ) − x ∗  ≤ δ0 .

(4.172)

Assume that k ≥ 1 is an integer and that we have defined a strictly decsreasing sequence of nonnegative integers ti ∈ [T1 , T2 ], i = 0, . . . , k such that (4.167) and (4.171) hold, (4.173) x(ti ) − x ∗  ≤ δ0 , i = 0, . . . , k and that for all i = 0, . . . , k − 1, ti+1 ∈ [ti − L 0 − L 1 , ti − L 0 ]. (Note that in view of (4.168), (4.171) and (4.172), our assumption holds for k = 1.) If tk < L 0 + L 1 , then the construction of the sequence ti , i = 0, . . . , k is completed. Assume that (4.174) tk ≥ L 0 + L 1 . By (4.162) and (4.163), tk −1 

vt (x(t), x(t + 1))

t=tk −L 0 −L 1 k −1 ≥ U ({vt }tt=t , x(tk − L 0 − L 1 ), x(t0 ), tk − L 0 − L 1 , t0 ) − M. k −t0 −L 1

Together with (4.160) and (4.162) this implies that

4.10 Stability of the Turnpike Phenomenon tk −1 

145

u(x(t), x(t + 1))

t=tk −L 0 −L 1

≥ U (x(tk − L 0 − L 1 ), x(tk ), tk − L 0 − L 1 , tk ) − M − 1.

(4.175)

Property (i), (4.159), (4.173) and (4.175) imply that tk −1 

u(x(t), x(t + 1))

t=tk −L 0 −L 1

≥ (L 0 + L 1 )μ − M0 − M − 1 ≥ (L 0 + L 1 )μ − M1 . By the relation above and property (iii) there exists an integer tk+1 ∈ {tk − L 0 − L 1 , . . . , tk − L 0 } such that

x(tk+1 ) − x ∗  ≤ δ0 .

Clearly, the assumption made for k also holds for k + 1. Therefore, by induction we constructed a finite strictly decreasing sequence of integers q

{ti }i=0 ⊂ {T1 , . . . , T2 } such that (4.167) holds, tq < L 0 + L 1 , x(ti ) − x ∗  ≤ δ0 , i = 0, . . . , q,

(4.176) (4.177)

for all integers i = 0, . . . , q − 1, ti+1 ∈ [ti − L 0 − L 1 , ti − L 0 ].

(4.178)

By induction we define a strictly increasing finite sequence {Si : i = 0, . . . , p} ⊂ {t0 , . . . , tq }. Set S0 = tq . In view of (4.161), (4.167), (4.176), S0 < t0 .

(4.179)

146

4 Turnpike Properties for Autonomous Problems

Assume that p ≥ 0 is an integer and that we have defined a strictly increasing finite sequence p {Si }i=0 ⊂ {t0 , . . . , tq } such that (4.179) holds, if an integer j satisfies 0≤ j < p then S j+1 −1



S

−1

j+1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(S j ), x(S j+1 )) − δ0 /2 j

(4.180)

t=S j

and if an integer S ∈ {t0 , . . . , tq } satisfies S j < S < S j+1 , then

S−1 

S−1 vt (x(t), x(t + 1)) ≥ U ({vt }t=S , x(S j ), x(S)) − δ0 /2. j

t=S j

If S p = t0 , then our construction is completed. If t0 −1  t0 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=S , x(S p ), x(t0 )) − δ/2, p t=S p

then we set S p+1 = t0 and the construction is completed. Assume that t0 −1 

t0 −1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(S p ), x(t0 )) − δ0 /2. p

t=S p

It is not difficult to see that there exists an integer S p+1 ∈ {t0 , . . . , tq }

(4.181)

4.10 Stability of the Turnpike Phenomenon

147

such that S p < S p+1 and

S p+1 −1



S

−1

p+1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(S p ), x(S p+1 )) − δ0 /2 p

t=S p

and if an integer S ∈ {t0 , . . . , tq } satisfies S p < S < S p+1 , then

S−1 

S−1 vt (x(t), x(t + 1)) ≥ U ({vt }t=S , (x(S p ), x(S)) − δ0 /2. p

t=S p

Thus the assumption made for p holds for p + 1. By induction we have constructed a strictly increasing sequence of integers j

{Si }i=0 ⊂ {t0 , . . . , tq } such that S0 = tq , S j = t0 , for each integer p ∈ {0, . . . , j − 1} \ { j − 1} (4.180) holds, and for each p ∈ {0, . . . , j − 1} and each S ∈ {t0 , . . . , tq } satisfying S p < S < S p+1 , (4.181) is true. In view of (4.180), for each integer p satisfying 0 ≤ p < j − 1, S

p+1 such that there exists a program {z p (t)}t=S p

148

4 Turnpike Properties for Autonomous Problems

z p (S p ) = x(S p ), z p (S p+1 ) ≥ x(S p+1 ), S p+1 −1



(4.182)

S p+1 −1

vt (x(t), x(t + 1))
U ({vt }t=S , x(Si ), x(S)) − δ0 /2. i

(4.192)

t=Si

Clearly, there exist k1 , k2 ∈ {0, . . . , q} such that Si = tk1 , S = tk2 , k1 > k2 .

(4.193)

By (4.192) and (4.193), tk2 −1



tk −1

2 vt (x(t), x(t + 1)) ≥ U ({vt }t=t , x(tk1 ), x(tk2 )) − δ0 /2. k1

(4.194)

t=tk1

Let j ∈ {k2 + 1, . . . , k1 }. In view of (4.173), x(t j ) − x ∗  ≤ δ0 , x(t j−1 ) − x ∗  ≤ δ0 .

(4.195)

Equations (4.162) and (4.194) imply that t j−1 −1



t

−1

j−1 vt (x(t), x(t + 1)) ≥ U ({vt }t=t , x(t j ), x(t j−1 )) − δ0 /2. j

t=t j

By the relation above, (4.160), (4.162) and (4.178), t j−1 −1



u(x(t), x(t + 1)) ≥ U (x(t j ), x(t j−1 ), t j , t j−1 ) − δ0 .

(4.196)

t=t j

Property (ii), (4.195) and (4.196) imply that x(t) − x ∗  ≤ , t = t j , . . . , t j−1 . Together with (4.191) and (4.193) this implies that x(t) − x ∗  ≤ , t = Si , . . . , Si+1 − 2L 0 − 2L 1 .

(4.197)

150

4 Turnpike Properties for Autonomous Problems

It follows from (4.189), (4.190) and (4.197) that {t ∈ {T1 , . . . , T2 } : x(t) − x ∗ (t) > } ⊂ {0, . . . , tq } ∪ {t0 , . . . , T } ∪ {{Si , . . . , Si+1 } : i ∈ {0, . . . , j − 1}, Si+1 − Si < 8L 0 + 8L 1 } ∪ {Si+1 − 2L 0 − 2L 1 + 1, . . . , Si+1 } : i ∈ E}. Combined with (4.161), (4.167), (4.176) and (4.188), this implies that Card({t ∈ {T1 , . . . , T2 } : x(t) − x ∗ (t) > }) ≤ 2(L 0 + L 1 + 1) + 2(8L 0 + 8L 1 )(2 + 2δ0−1 M) < Q. Theorem 4.15 is proved.

4.11 A Subclass of Models In this section we consider a subclass of models studied in the previous sections. Note that the RSS model belongs to this subclass. Denote by In the identity self-mapping n , V be a diagonal matrix, of R n . Let D be a nonempty bounded closed subset of R+ n n set V = (vi, j )i, j=1 , vi,i ∈ [0, 1), i = 1, . . . , n. For each x ∈ R+ a(x) = V x + D.

(4.198)

Clearly, (4.1) holds with  ·  =  · 2 and κ = max{vi,i : i = 1, . . . , n}.

(4.199)

There exists c∗ > 0 such that x ≤ c∗ e for all x ∈ D.

Proposition 4.16 Let M > (1 − max{vi,i : i = 1, . . . , n})−1 c∗ , n x ∈ R+ , x ≤ Me, y ∈ a(x). Then y ≤ Me. Proof In view of (4.198), y ∈ V x + D.

(4.200)

4.11 A Subclass of Models

151

This implies that y ≤ max{vi,i : i = 1, . . . , n}x + c∗ e ≤ max{vi,i : i = 1, . . . , n}Me + c∗ e ≤ Me. Proposition 4.16 is proved. n n Let F : R+ → R+ be a continuous function satisfying the following condition: n , x ≤ y, then F(x) ≤ F(y). if x, y ∈ R+

(4.201)

n → [0, ∞) be a continuous function such that Let w : R+ n w(x) ≥ w(0) = 0 for all x ∈ R+

(4.202)

n → [0, ∞) be continuous functions. Assume that Let F1 , F2 : R+ n such that x ≤ y, F1 (x) ≤ F1 (y) for all x, y ∈ R+

(4.203)

F2 (x) = 0 if and only if x = 0

(4.204)

n : F1 (x) ≤ 1}. D = {x ∈ R+

(4.205)

n : F1 (x) + F2 (y) ≤ 1}. (x) = {y ∈ R+

(4.206)

n , for all x ∈ R+

and that n set For every x ∈ R+

n and every For every x ∈ R+ n : F1 (z) ≤ 1} x ∈ a(x) = V x + {z ∈ R+

set u(x, x ) = sup{w(y) : n y ∈ R+ , y ≤ F(x), F2 (y) + F1 (x − V x) ≤ 1}.

(4.207)

It is easy to see that the function u is well-defined and upper semicontinuous on the closed set n , x ∈ a(x)}. graph(a) := {(x, x ) : x ∈ R+

152

4 Turnpike Properties for Autonomous Problems

Assume that the function F1 has the following property: n satisfying (B1) for each r ∈ (0, 1) there exists  > 0 such that for each x ∈ R+ F1 (x) ≤ r , F1 (x + e) ≤ 1. Fix

M∗ > (1 − max{vi,i : i = 1, . . . , n})−1 c∗

(4.208)

n : x ≤ M∗ e}. K = {x ∈ R+

(4.209)

and set

It is clear that we defined the model which belongs to the class of models introduced in Sect. 4.1. Evidently, Eq. (4.2) holds. Proposition 4.17 (A1) holds. Proof Let δ > 0. In view of (4.204), there exists γ ∈ (0, 1) such that the following property holds: n satisfies x ≤ γ, then w(x) ≤ δ/4. (i) If x ∈ R+ By (4.204), there exists γ0 ∈ (0, γ/2) such that the following property holds: n n ) ∩ R+ and x ≥ γ, then F2 (x) ≥ γ0 . (ii) If x ∈ (F(K ) − R+ (B1) implies that there exists λ > 0 such that the following property holds: n satisfying F1 (x) ≤ 1 − γ0 we have (iii) For each x ∈ R+ F1 (x + λe) ≤ 1. Let

x ∈ K , (x, x ) ∈ graph(a)

(4.210)

u(x, x ) ≥ δ.

(4.211)

satisfy

n such that In view of (4.207) and (4.211), there exists y ∈ R+

y ≤ F(x), F2 (y) + F1 (x − V x) ≤ 1

(4.212)

w(y) ≥ δ/2.

(4.213)

y ≥ γ.

(4.214)

and that

Property (i) and (4.213) imply that

4.11 A Subclass of Models

153

Property (ii), (4.212) and (4.214) imply that F2 (y) ≥ γ0 .

(4.215)

Equations (4.212) and (4.215) imply that

By (4.216),

F1 (x − V x) ≤ 1 − γ0 .

(4.216)

F1 (x − V x + λe) ≤ 1.

(4.217)

It follows from (4.205) and (4.217) that x + λe ∈ a(x). Thus (A1) holds. Proposition 4.17 is proved. n Assume that z ∗ ∈ R+ \ {0} satisfies

z ∗ ≤ M∗ e, z ∗ ≤ F(z ∗ ), F2 (z ∗ ) + F1 (z ∗ − V z ∗ ) ≤ 1 and that w(z ∗ ) > 0. It is easy to see that assumptions (A2)–(A4) hold. Therefore all the results obtained in the previous sections of this chapter are true for our model. Note that in this section the functions F, F1 , F2 are fixed but w can vary. n → [0, ∞) which are bounded on Denote by Mb the set of all functions φ : R+ n bounded subsets of R+ . For each φ ∈ Mb and each M > 0 set n , y ≤ M}. φ M = sup{φ(y) : y ∈ R+ n → [0, ∞) such that Denote by M the set of all continuous functions w : R+ n w(x) ≥ w(0) for all x ∈ R+

and set M0 = {w ∈ M : w(0) = 0}. The set M is equipped with the uniformity determined by the base E(M, ) = {(w, v) ∈ M × M : w − v M ≤ }.

154

4 Turnpike Properties for Autonomous Problems

It is not difficult to see that the uniform space M is metrizable (by a metric ρ) and complete and M0 is its closed subset. Set E(z ∗ ) = {w ∈ M : w(z ∗ ) > w(0)},

(4.218)

E0 (z ∗ ) = {w ∈ M0 : w(z ∗ ) > w(0)}.

(4.219)

It is clear that E(z ∗ ) is an open everywhere dense set in the metric space (M, ρ) and E0 (z ∗ ) is an open everywhere dense set in the metric space (M0 , ρ). Let M > (1 − max{vi,i : i = 1, . . . , n})−1 c∗

(4.220)

n : x ≤ Me}. K := K M := {x ∈ R+

(4.221)

w ∈ M.

(4.222)

and

Assume that For every x ∈ K and every x ∈ a(x) define u (w) (x.x ) = sup{w(y) : n y ∈ R+ , y ≤ F(x), F2 (y) + F1 (x − V x) ≤ 1}.

(4.223)

It is easy to see that if the function w ∈ E(z ∗ ), then all the results obtained in the previous sections hold for the model determined by the (F, F1 , F2 , w). Define   T  −1 (w) ∞ μ(w) = sup lim sup T u (x(t), x(t + 1)) : {x(t)}t=0 is a program . T →∞

t=0

(4.224) It is clear that μ(w) does not depend on M if w ∈ E(z ∗ ). Evidently, n , μ(w) ≥ sup{w(x) : x ∈ R+ x ≤ F(x), F2 (x) + F1 (x − V x) ≤ 1}.

(4.225)

n , ν(w) = sup{w(x) : x ∈ R+ x ≤ F(x), F2 (x) + F1 (x − V x) ≤ 1}.

(4.226)

μ(w) ≥ ν(w).

(4.227)

Define

We have

4.11 A Subclass of Models

155

Define Mr = {w ∈ M : μ(w) = ν(w)}

(4.228)

Mr,0 = {w ∈ M0 : μ(w) = ν(w)}.

(4.229)

and Clearly, Mr is a closed subset of the metric space (M, ρ) and Mr,0 is a closed subset of the metric space (M0 , ρ). Define Er = {w ∈ Mr : μ(w) > w(0)}

(4.230)

Er,0 = {w ∈ Mr,0 : μ(w) > 0}.

(4.231)

and Clearly, Er is an open set in the metric space (Mr , ρ) and Er,0 is an open set in the metric space (Mr,0 , ρ). Note that if w ∈ Er , then there exists ξ ∈ R n \ {0} such that ξ ≤ F(ξ), F2 (ξ) + F1 (ξ − V ξ) ≤ 1, w(ξ) > w(0), w ∈ E(ξ) and that all the results obtained in the previous sections hold for the model determined by the (F, F1 , F2 , w). Proposition 4.18 Er is an everywhere dense set in the metric space (Mr , ρ) and Er,0 is an everywhere dense set in the metric space (Mr0 , ρ). Proof Let w ∈ Mr satisfy

μ(w) = w(0).

(4.232)

w(z ∗ ) = w(0).

(4.233)

γ1 ∈ (0, 1), γ0 = γ1 z ∗ .

(4.234)

In view of (4.226) and (4.232),

Let

156

4 Turnpike Properties for Autonomous Problems

Define n . w˜ γ (z) = max{w(z) + γ0 − γ1 z − z ∗ , w(0)}, z ∈ R+

(4.235)

It is clear that w˜ γ ∈ M, if w ∈ M0 , then w˜ γ ∈ M0 , μ(w˜ γ ) ≤ μ(w) + γ0 , ν(w˜ γ ) ≥ w(z ∗ ) + γ0 = μ(w) + γ0 . Therefore μ(w˜ γ ) = ν(w˜ γ ) = w(z ∗ ) + γ0 . On the other hand, w˜ γ (0) = max{w(0) + γ0 − γ1 z − z ∗ , w(0)} = w(0). It is not difficult to see that μ(w˜ γ ) > w˜ γ (0), w˜ γ ∈ Er, and if w ∈ Mr,0 , then w˜ γ ∈ Er,0 . Clearly, w˜ γ → w as γ → 0+ . This completes the proof of Proposition 4.18. Corollary 4.19 Er is an open and everywhere dense set in the metric space (Mr , ρ) and Er,0 is an open an everywhere dense set in the metric space (Mr,0 , ρ). Note that if w ∈ Er then all the results obtained in Chap. 4 are true for our model with u = u (w) . Assume that w ∈ M. n set For each pair of integers T1 , T2 ≥ 0 satisfying T1 , T2 and each z 1 , z 2 ∈ R+

T 2 −1 U (w, z 1 , z 2 , T1 , T2 ) = sup u (w) (x(t), x(t + 1)) : t=T1 T2 {x(t)}t=T is a program and x(T1 ) = z 1 , x(T2 ) ≥ z 2 1



4.11 A Subclass of Models

157

and U (w, z 1 , T1 , T2 ) T 2 −1 T2 u (w) (x(t), x(t + 1)) : {x(t)}t=T is a program such that = sup 1 t=T1

 x(T1 ) = z 1 . Let w ∈ Er . A program {x(t)}∞ t=0 is (w)-good if the sequence T −1 

∞ u (w) t (x(t), x(t

+ 1)) − T μ(w) T =1

t=0

is bounded. We say that the function w possesses the asymptotic turnpike property (ATP) if n such that there exists x ∗ ∈ R+ x ∗ ≤ F(x ∗ ), F1 (x ∗ ) + F2 (x ∗ − V x ∗ ) ≤ 1, x ∗ ∈ a(x ∗ ), μ(w) = u (w) (x ∗ , x ∗ ) and for every w-good program {x(t)}∞ t=0 we have lim x(t) = x ∗ .

t→∞

From now on in this chapter we assume that d = (d1 , . . . , dn ) ∈ R n , d  0, n F2 (z) = dz, z ∈ R+ , vi,i = v ∈ [0, 1), i = 1, . . . , n. We prove the following result which shows that a generic (typical) function w has ATP.

158

4 Turnpike Properties for Autonomous Problems

˜ E) ˜ be either (Mr , Er ) or (Mr,0 , Er,0 ) and the space M ˜ is Theorem 4.20 Let (M, equipped with the metric ρ. Then there exists a set F ⊂ E˜ which is a countable intersection of open and everywhere dense sets in the metric ˜ ρ) such that each w ∈ F has ATP. space (M,

4.12 Auxiliary Results Let w ∈ Er . n such that Then there exist xw ∈ R+

xw ≤ F(xw ),

(4.236)

F1 (xw ) + F2 (xw − V xw ) ≤ 1,

(4.237)

μ(w) = u

(w)

(xw , xw ) = w(xw ).

(4.238)

n Let γ ∈ (0, 1). Define for every z ∈ R+

wγ (z) = max{w(w) − γz − xw  + γxw , w(0)}.

(4.239)

It is not difficult to see that wγ ∈ Er and if w ∈ E0 , then wγ ∈ Er,0 . Clearly, wγ → w as γ → 0+ in (M, ρ).

Lemma 4.21 Let w ∈ Er , γ ∈ (0, 1), {x(t)}∞ t=0 be a (wγ )-good program and let for every integer t ≥ 0, y(t) ≤ F(x(t)), F1 (y(t)) + F2 (x(t + 1) − V x(t)) ≤ 1, wγ (y(t)) = u

(wγ )

(x(t), x(t + 1)).

Then lim y(t) = xw .

t→∞

(4.240) (4.241)

4.12 Auxiliary Results

159

Proof There exists M1 > 0 such that for every integer T ≥ 1, T −1 

w(y(t)) − μ(w)T ≤ M1

(4.242)

t=0

and − M1 ≤

T −1 

wγ (y(t)) − μ(wγ )T.

(4.243)

t=0

By (4.238) and (4.239), μ(wγ ) = μ(w) + γxw  = w(xw ) + γxw  = wγ (xw ).

(4.244)

It follows from (4.242) and (4.243) that for every integer T ≥ 1, −M1 ≤

T −1 

w(y(t)) − T μ(w)

(4.245)

t=0

+

T −1 

(wγ (y(t)) − w(y(t))) − T γxw .

(4.246)

t=0

Equations (4.239), (4.242), (4.244) and (4.246) imply that T −1 

−2M1 ≤ lim

T →∞

= lim

T →∞

T −1 

T →∞

max{−γy(t) − xw , −γxw  + w(0) − w(y(t))}

t=0

T →∞

T −1 

(wγ (y(t)) − w(y(t))) − T γxw 

t=0

= − lim

lim

T −1 

min{γy(t) − xw , w(y(t)) + γxw  − w(0)},

t=0

min{γy(t) − xw , w(y(t)) + γxw  − w(0)} ≤ M1

t=0

and lim

T →∞

T −1  t=0

min{y(t) − xw , γxw } ≤ 2M1 .

160

4 Turnpike Properties for Autonomous Problems

This implies that lim y(t) = xw .

t→∞

Lemma 4.21 is proved. Lemma 4.22 Let w ∈ Er , γ ∈ (0, 1) and let {x(t)}∞ t=0 be a (wγ )-good program Then lim x(t) = xw .

t→∞

n such that Proof For each integer t ≥ 0, there exists y(t) ∈ R+

and

y(t) ≤ F(x(t)), F1 (y(t)) + F2 (x(t + 1) − V x(t)) ≤ 1

(4.247) (4.248)

wγ (y(t)) = u (wγ ) (x(t), x(t + 1)).

(4.249)

Lemma 4.21 implies that lim y(t) = xw .

t→∞

(4.250)

n Let  > 0. There exists 0 ∈ (0, ) such that for each y ∈ R+ satisfying y − xw  ≤ 0 we have (4.251) |F1 (y) − F1 (xw )| ≤ .

Let  > 0. By (4.250), there exists an integer t0 > 0 such that for all integers t ≥ t0 , y(t) − xw  ≤ 0 .

(4.252)

z(t) = x(t + 1) − V x(t).

(4.253)

For every integer t ≥ 0 set

By (4.198), (4.205), (4.247) and (4.253), for all integers t ≥ 0, 0 ≤ y(t) ≤ F(x(t)), z(t) ≥ 0, F(z(t)) ≤ 1, x(t + 1) = V x(t) + z(t).

(4.254) (4.255) (4.256)

Let S ≥ t0 be an integer. In view of (4.253), x(S + 1) = V x(S) + z(S).

(4.257)

4.12 Auxiliary Results

161

By induction we show that for all integers t > S, x(t) = v

t−S

x(S) +

t−1 

v t−i−1 z(i).

(4.258)

i=S

In view of (4.257), (4.258) holds for t = S + 1. Assume that t > S is an integer and (4.258) holds. Equations (4.256) and (4.258) imply that x(t + 1) = vx(t) + z(t) = v t−S+1 x(S) +

t−1 

v t−i z(i) + z(t)

i=S

= v t−S+1 x(S) +

t 

v t−i z(i).

i=S

Hence (4.258) holds for all integers t > S. Proposition 4.16 implies that there exists M1 > 0 such that x(t) ≤ M1 e for all integers t ≥ 0, y(t) ≤ M1 e for all integers t ≥ 0, z(t) ≤ M1 e for all integers t ≥ 0. Recall that n , F2 (z) = dz, z ∈ R+

(4.259)

d = (d1 , . . . , dn ) ∈ R n , d  0, vi,i = v ∈ [0, 1), i = 1, . . . , n.

(4.260)

where

In view of (4.248) and (4.253), F2 (z(t)) ≤ 1 − F1 (y(t))

(4.261)

for all integers t ≥ 0. By (4.258)–(4.260), for all integers t > S, F2 (x(t)) = v t−S F2 (x(S)) +

t−1 

v t−i−1 F2 (z(i)).

(4.262)

i=S

It follows from (4.251) and (4.252) that for all integers t > S, |F1 (y(t)) − F1 (xw )| ≤ .

(4.263)

162

4 Turnpike Properties for Autonomous Problems

By (4.259) and (4.261)–(4.263), d x(t) = F2 (x(t)) ≤v

t−1 

F2 (x(S)) +

t−S

v t−i−1 (1 − F1 (y(t))

i=S

= v t−S

n 

di M1 +

i=1

≤ v t−S M1

t−1 

v t−i−1 (1 − F1 (xw ) + )

i=S

n 

di + (1 − F1 (xw ) + )(1 − v)−1 .

i=1

Since  is any positive number we obtain that lim sup d x(t) ≤ (1 − v)−1 (1 − F1 (xw )). t→∞

Since F1 (xw ) + F2 ((1 − v)xw ) ≤ 1 equations (4.149) and (4.264) imply that lim sup d x(t) ≤ d xw = lim dy(t) ≤ lim inf d x(t). t→∞

t→∞

t→∞

This implies that lim d x(t) = lim dy(t).

t→∞

t→∞

Together with the inequality y(t) ≤ x(t) for all integers t ≥ 0 this implies that lim d(x(t) − y(t)) = 0,

t→∞

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

t→∞

Lemma 4.22 is proved.

t→∞

(4.264)

4.13 Proof of Theorem 4.20

163

4.13 Proof of Theorem 4.20 Let w ∈ E˜ and γ ∈ (0, 1). Consider wγ ∈ E˜ defined by (4.239). By Lemma 4.22, it has ATP. Let M > (1 − v)−1 c∗ be an integer. By Theorem 4.15, there exists a natural number Q(w, γ, M) and an open neigh˜ such that borhood U(w, γ, M) of wγ in M U(w, γ, M) ⊂ E˜ and the following property holds: (v) For each u ∈ U(w, γ, M), each pair of integers T1 ≥ 0, T2 ≥ Q(w, γ, M) + T1 T2 and each program {x(t)}t=T satisfying 1

x(T1 ) ≤ Me, T 2 −1

u(x(t), x(t + 1)) ≥ U (u, x(T1 ), T2 − T1 ) − M

t=T1

the inequality Card({t ∈ {T1 , . . . , T2 } : x(t) − xw  > }) ≤ Q(w, γ, M) holds. Define F=

 {U(w, γ, M) :

 ˜ γ ∈ (0, 1)} : M > (1 − v)−1 c∗ is an integer . w ∈ E,

Clearly,

F ⊂ E˜

˜ ρ). is a countable intersection of open everywhere dense sets in (M,

(4.265)

164

4 Turnpike Properties for Autonomous Problems

Let w ∈ F.

(4.266)

It is sufficient to show that ATP holds for w. Let {x (i) (t)}∞ t=0 , i = 1, 2 be (w)-good programs. There exists M0 > 0 such that for all i = 1, 2 and every integer T ≥ 1, U (w, x (i) (0), T ) − M0 ≤

T −1 

w(x (i) (t), x (i) (t + 1)),

t=0 (i)

x (0) ≤ M0 e, i = 1, 2. Fix an integer

(4.267)

M > M0 + (1 − v)−1 c∗ .

(4.268)

By (4.265) and (4.266), there exist ˜ γ ∈ (0, 1) u ∈ E, such that w ∈ U(u, γ, M).

(4.269)

Let i ∈ {1, 2}. By (4.266) and (4.269), Card({t ∈ {0, 1, . . . , } : x (i) (t) − xu  > M −1 }) ≤ Q(u, γ, M) and there exists an integer Ti > 0 such that x (i) (t) − xu  ≤ M −1 for all integers t ≥ Ti . Since M is any integer satisfying (4.268) we conclude that there exist lim x (i) (t), i = 1, 2, t→∞

and that

lim x (1) (t) = lim x (2) (t).

t→∞

This completes the proof of Theorem 4.20.

t→∞

Chapter 5

The Turnpike Phenomenon for Nonautonomous Problems

Abstract In this chapter we continue to study the class of nonautonomous discrete– time optimal problems analyzed in Chap. 2. We obtain turnpike conditions and establish the stability of the turnpike phenomenon. All the main results of this chapter are new.

5.1 Preliminaries We consider the class of problems studied in Chap. 2. 1 ) be the set of real (nonnegative) numbers and let R n be the nLet R 1 (R+ dimensional Euclidean space with 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 n  xi yi xy = i=1

and let x  y, x > y, x ≥ y have their usual meaning. 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 x2 its Euclidean norm in R n . We assume that  ·  is a norm in R n such that x ≤ y for all x, y ∈ R n satisfying 0 ≤ x ≤ y.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. J. Zaslavski, Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model, Monographs in Mathematical Economics 4, https://doi.org/10.1007/978-981-16-2252-6_5

165

166

5 The Turnpike Phenomenon for Nonautonomous Problems

Recall that for every mapping a : X → 2Y \ {∅}, where X, Y are nonempty sets, graph(a) = {(x, y) ∈ X × Y : y ∈ a(x)}. Let K be a nonempty compact subset of R n . Denote by P(K ) the set of all nonempty closed subsets of K . For every pair of nonempty sets A, B ⊂ R n define H (A, B) = sup{sup inf x − y, sup inf x − y}. x∈A y∈B

y∈B x∈A

For every nonnegative integer t, let at : K → P(K ) be such that graph(at ) is a closed subset of R n × R n . Assume that there exists a number κ ∈ (0, 1) such that for every pair of points x, y ∈ K and every nonnegative integer t, H (at (x), at (y)) ≤ κx − y

(5.1)

and that for every nonnegative integer t the upper semicontinuous function u t : {(x, x ) ∈ K × K , x ∈ at (x)} → [0, ∞) satisfies sup{sup{u t (x, x ) : (x, x ) ∈ graph(at )} : t = 0, 1, . . . } < ∞.

(5.2)

A sequence {x(t)}∞ t=0 ⊂ K is called a program if x(t + 1) ∈ at (x(t)) for every nonnegative integer t. T2 ⊂ K is called a Let T1 , T2 be integers such that T1 < T2 . A sequence {x(t)}t=T 1 program if x(t + 1) ∈ at (x(t)) for every integer t satisfying T1 ≤ t < T2 . We continue to assume that the supremum over an empty set is −∞ and the sum over an empty set is zero. For every point x0 ∈ K and every pair of nonnegative integers T1 < T2 define T 2 −1 u t (x(t), x(t + 1)) : U (x0 , T1 , T2 ) = sup t=T1

 T2 −1 {x(t)}t=T is a program and x(T ) = x 1 0 . 1 Let x0 , x˜0 ∈ K and let T1 < T2 be nonnegative integers. Define

(5.3)

5.1 Preliminaries

167

U (x0 , x˜0 , T1 , T2 ) T 2 −1 T2 u t (x(t), x(t + 1)) : {x(t)}t=T is a program such that = sup 1 t=T1

 x(T1 ) = x0 , x(T2 ) ≥ x˜0 .

(5.4)

Let T1 < T2 be nonnegative integers. Define T  2 −1 T2 −1  u t (x(t), x(t + 1)) : {x(t)}t=T1 is a program . U (T1 , T2 ) = sup

(5.5)

t=T1

We suppose that the following assumptions hold. (A1) For every positive number  there exists a positive number λ such that if an integer t ≥ 0 and if (x, x ) ∈ graph(at ) satisfies u t (x, x ) ≥  and if y, y ∈ K satisfy x − y, x − y  ≤ λ, then there exists z ∈ at (y) satisfying z ≥ y + λe, u t (y, z) ≥ max{/2, u t (x, x ) − }, z − y  ≤ .  (A2) There exist a program { x (t)}∞ t=0 and a positive number Δ ≤ 1 such that  x (t),  x (t + 1)) ≥ Δ u t ( for every nonnegative integer t; (A3) For every nonnegative integer t, every (x, y) ∈ graph(at ) and every x˜ ∈ K ˜ for which which satisfies x˜ ≥ x there exists y˜ ∈ at (x) ˜ y˜ ) ≥ u t (x, y) y˜ ≥ y, u t (x, and y˜ − y ≤ x˜ − x. Note that assumption (A2) was introduced and used in Chap. 2, assumption (A1) is a strong version of assumption (A1) of Chap. 2 and assumption (A3) is a strong version of assumption (A3) of the same chapter. Therefore all the results of Chap. 2 are true for the class of problems considered in this chapter. Fix a number M∗ > 0 such that Theorem 2.4 holds with M = M∗ .

168

5 The Turnpike Phenomenon for Nonautonomous Problems

∞ Theorem 5.1 For every x0 ∈ K there exists a program {x(t)} ¯ t=0 such that

x(0) ¯ = x0 and that for every pair of nonnegative integers T1 , T2 satisfying T1 < T2 , the inequality T −1  2    (T1 , T2 ) ≤ M∗ u t (x(t), ¯ x(t ¯ + 1)) − U    t=T1

holds and for every integer t ≥ 0, ¯ x(t ¯ + 1)) = max{u t (x(t), ¯ z) : z ∈ at (x(t)), ¯ z ≥ x(t ¯ + 1)} u t (x(t), = U x(t), ¯ x(t ¯ + 1)). ∞ ˜ Proof Let x0 ∈ K . By Theorem 2.4, there exists a program {x(t)} t=0 such that

x(0) ˜ = x0

(5.6)

and that for every pair of nonnegative integers T1 , T2 satisfying T1 < T2 , the inequality T −1  2      u t (x(t), ˜ x(t ˜ + 1)) − U (T1 , T2 ) ≤ M∗ .   

(5.7)

t=T1

holds. Set x(0) ¯ = x(0). ˜

(5.8)

There exists ¯ x(1) ¯ ∈ a0 (x(0)) such that x(1) ¯ ≥ x(1) ˜

(5.9)

¯ x(1)) ¯ = max{u 0 (x(0), ¯ z) : z ∈ a0 (x(0)), ¯ z ≥ x(1)}. ˜ u 0 (x(0),

(5.10)

¯ x(1)) ¯ = max{u 0 (x(0), ¯ z) : z ∈ a0 (x(0)), ¯ z ≥ x(1)}. ¯ u 0 (x(0),

(5.11)

and

Clearly,

5.1 Preliminaries

169

Assume that k ≥ 1 is an integer and we defined x(t) ¯ ∈ K , t = 0, . . . , k such that k is a program, (5.8) holds, for all integers t = 1, . . . , k, {x(t)} ¯ t=0 x(t) ¯ ≥ x(t) ˜

(5.12)

¯ x(t ¯ + 1)) ≥ u t (x(t), ˜ x(t ˜ + 1)) u t (x(t),

(5.13)

¯ x(t ¯ + 1)) = max{u t (x(t), ¯ z) : z ∈ at (x(t))}. ¯ u t (x(t),

(5.14)

and for all t = 0, . . . , k − 1,

and

(Note that in view of (5.9)–(5.11), our assumption holds for k = 1.) By assumption (A3) and (5.12), there exists ¯ ξ ∈ ak (x(k)) such that ξ ≥ x(k ˜ + 1) and ¯ ξ) ≥ u k (x(k), ˜ x(k ˜ + 1)). u k (x(k), There exists ¯ x(k ¯ + 1) ∈ ak (x(k)) such that x(k ¯ + 1) ≥ ξ and ¯ x(k ¯ + 1)) = max{u k (x(k), ¯ z) : z ∈ ak (x(k)), ¯ z ≥ ξ}. u k (x(k), Thus the assumption made for k also holds for k + 1. Therefore by induction we ∞ constructed a program {x(t)} ¯ t=0 such that (5.8) holds, for all integers t ≥ 1, (5.12) holds and (5.13), (5.14) are valid for all integers t ≥ 0, This completes the proof of Theorem 5.1.

5.2 A Turnpike Property Recall that a number M∗ > 0 and that Theorem 2.4 holds with M = M∗ . See Theorem ∞ ¯ 5.1. Let a program {x(t)} ¯ t=0 be as guaranteed by Theorem 5.1 with x 0 = x(0). Let T1 ≥ 0 be an integer. A program {x(t)}∞ t=T1 is good if the sequence

170

5 The Turnpike Phenomenon for Nonautonomous Problems

 T −1 

u t (x(t), x(t + 1)) −

t=T1

T −1 

∞ u t (x(t), ¯ x(t ¯ + 1))

t=T1

T =T1

is bounded. If this sequence tends to −∞, then the program is bad. Note that every program which is not good is bad. We say that the sequence {u t }∞ t=0 has the turnpike property (or TP for short) if for each , M > 0 there exist δ > 0 and a natural number L such that for each integer T2 which satisfies T1 ≥ 0, each integer T2 ≥ 2L + T1 and each program {x(t)}t=T 1 T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ

t=T1

and

T 2 −1

(T1 , T2 ) − M u t (x(t), x(t + 1)) ≥ U

t=T1

we have x(t) − x(t) ¯ ≤ , t = T1 + L , . . . , T2 − L . In this chapter we also consider the following properties. (P1) For each integer T1 ≥ 0 and each good program {x(t)}∞ t=T1 we have ¯ = 0. lim (x(t) − x(t))

t→∞

(P2) For each , M > 0 there exists a natural number L ≥ 2 such that for each T +L which satisfies integer T ≥ 0 and each program {x(t)}t=T T +L−1

u t (x(t), x(t + 1)) ≥

t=T

T +L−1

u t (x(t), ¯ x(t ¯ + 1)) − M

t=T

there exists an integer τ ∈ {T, . . . , T + L − 1} such that x(τ ) − x(τ ¯ ), x(τ + 1) − x(τ ¯ + 1) ≤  and

 u τ (x(τ ), x(τ + 1)) ≥ Δ/4.

We will prove the following theorem, which is the main result of this chapter. Theorem 5.2 TP holds if and only if properties (P1) and (P2) hold.

5.3 Examples

171

5.3 Examples We consider the nonstationary RSS model introduced in Sect. 2.3. For every nonnegative integer t, let α(t) = (α1(t) , . . . , αn(t) )  0, b(t) = (b1(t) , . . . , bn(t) )  0, d

(t)

=

(d1(t) , . . . , dn(t) )

∈ ((0, 1])

(5.15)

n

and for every nonnegative integer t, let wt : [0, ∞) → [0, ∞) be a strictly increasing continuous function such that for every integer t ≥ 0, wt (0) = 0, inf{wt (z) : t = 0, 1, . . . } > 0 for all z > 0

(5.16)

and such that the following assumption holds: (A4) for every positive number  there exists a positive number δ such that for every nonnegative integer t and every z ∈ [0, δ] the inequality wt (z) ≤  is valid. n define Let t be a nonnegative integer. For every x ∈ R+ 

n at (x) = y ∈ R+ : yi ≥ (1 − di(t) )xi , i = 1, . . . , n, n 

αi(t) (yi

− (1 −

di(t) )xi )

 ≤1 .

(5.17)

i=1

It is easy to see that for every x ∈ R n , at (x) is a nonempty closed bounded subset n n n and graph(at ) is a closed subset of R+ × R+ . Assume that of R+ inf{di(t) : i = 1, . . . , n, t = 0, 1, . . . } > 0,

(5.18)

inf{eb(t) : t = 0, 1, . . . } > 0,

(5.19)

inf{αi(t) : sup{bi(t) : sup{αi(t) :

i = 1, . . . , n, t = 0, 1, . . . } > 0,

(5.20)

i = 1, . . . , n, t = 0, 1, . . . } < ∞,

(5.21)

i = 1, . . . , n, t = 0, 1, . . . } < ∞

(5.22)

and that for every positive number M, we have sup{wt (M) : t = 0, 1, . . . } < ∞, inf{wt (M) : t = 0, 1, . . . } > 0.

(5.23)

The constraint mappings at , t = 0, 1, . . . have already been defined. Let us now define the cost functions u t , t = 0, 1, . . . .

172

5 The Turnpike Phenomenon for Nonautonomous Problems

For every nonnegative integer t and every (x, x ) ∈ graph(at ) put  u t (x, x ) = sup wt (b(t) y) : 0 ≤ y ≤ x,

ey +

n 

αi(t) (xi

− (1 −

di(t) )xi )

 ≤1 .

(5.24)

i=1

Fix numbers α∗ , α∗ > 0, d∗ > 0 which satisfy α∗ < αi(t) < α∗ , d∗ < di(t) , i = 1, . . . , n, t = 0, 1, . . . .

(5.25)

Therefore all the assumptions made in Sect. 2.3 hold. We suppose that the functions n . Fix wt , t = 0, 1, . . . are uniformly continuous on bounded subsets of R+ M0 > (α∗ d∗ )−1

(5.26)

n : x ≤ M0 e}. K = {x ∈ R+

(5.27)

and set

It was shown in Sect. 2.3 that this model satisfies all the assumptions posed in Sect. 2.1. Now we show that it is also satisfies all the assumptions introduced in Sect. 5.1 with  ·  =  · 2 . Proposition 5.3 Let  > 0. Then there exists a positive number λ such that for each integer t ≥ 0, each (x, y) ∈ graph(at ) which satisfies u t (x, y) ≥  and each x , y ∈ K satisfying x − x 2 , y − y 2 ≤ λ, there exists

z ∈ at (x )

satisfying z ≥ y + λe, u t (x , z) ≥ u t (x, y) − /4 and

z − y2 < , z − y 2 < .

5.3 Examples

173

Proof Assumption (A4) implies that there exists a positive number 0 <  such that 1 such that for every nonnegative integer t and every ξ ∈ R+ wt (ξ) ≥ /4 we have ξ ≥ 0 . Lemma 2.9, (5.26) and (5.27) imply that the following property holds: (i) If x ∈ K , an integer t ≥ 0 and y ∈ at (x), then y ∈ K . Put b∗ = sup{bi(t) : t = 0, 1, . . . , i = 1, . . . , n}.

(5.28)

(5.29)

There exists λ0 > 0 such that λ0 < 0 b∗−1 n −1 , 4λ0 α∗−1 < n −1

(5.30)

and that the following property holds: (ii) For each η1 , η2 ∈ [0, M0 n(b∗ + 1)] satisfying η1 − η2  ≤ b∗ λ0 n we have for all integers t ≥ 0, |wt (η1 ) − wt (η2 )| ≤ /4. Choose a positive number λ < 3−1 λ0 n −1 (α∗ + 1)−1 .

(5.31)

Assume that t ≥ 0 is an integer, x ∈ K , (x, y) ∈ graph(at ),

(5.32)

u t (x, y) ≥ ,

(5.33)

n satisfy and that x , y ∈ R+

x − x 2 ≤ λ, y − y 2 ≤ λ.

(5.34)

n such that By (5.24), (5.32) and (5.33), there exist ξ (1) ∈ R+

ξ (1) ≤ x, eξ (1) +

n  i=1

αi(t) (yi − (1 − di(t) )xi ) ≤ 1

(5.35)

174

5 The Turnpike Phenomenon for Nonautonomous Problems

and that wt (b(t) ξ (1) ) = u t (x, y) ≥ .

(5.36)

It follows from the choice of 0 (see (5.28)), (5.29), (5.35) and (5.36) that b(t) x ≥ b(t) ξ (1) ≥ 0 , b∗

n 

xi ≥ b∗

i=1

n 

ξi(1) ≥ 0 .

(5.37) (5.38)

i=1

In view of (5.35) and (5.38), there exists j0 ∈ {1, . . . , n} such that −1 −1 x j0 ≥ ξ (1) j0 ≥ 0 b∗ n .

(5.39)

Define a vector ξ (2) as follows: for all i = 1, . . . , n, if ξi(1) ≥ λ0 , then ξi(2) = ξi(1) − λ0 otherwise

(5.40)

ξi(2) = 0.

By (5.30), (5.39) and (5.40), n 

ξi(2) =

 (1) {ξi − λ0 : i ∈ {1, . . . , n}, ξi(1) ≥ λ0 }

i=1

=

n 

ξi(1) − λ0 Card{i ∈ {1, . . . , n} : ξi(1) ≥ λ0 }

i=1

≤

n 

ξi(1) − λ0 .

(5.41)

i=1

Equations (5.31), (5.34), (5.35) and (5.40) imply that ξ (2) ≤ x .

(5.42)

z i = yi − (1 − di(t) )xi + (1 − di(t) )xt + λ0 (αi(t) )−1 n −1 .

(5.43)

For i = 1, . . . , n set

It follows from (5.32) and (5.43) that for all i = 1, . . . , n, z i ≥ (1 − di(t) )xi

(5.44)

5.3 Examples

175

and n 

=

αi(t) (z i − (1 − di(t) )xi )

i=1 n 

αi(t) (yi − (1 − di(t) )xi ) + λ0 .

(5.45)

i=1

By (5.35), (5.41) and (5.45), n 

≤

ξi(2) +

i=1 n 

n 

αi(t) (z i − (1 − di(t) )xi )

i=1

αi(t) (yi − (1 − di(t) )xi ) +

i=1

n 

ξi(1) ≤ 1.

(5.46)

i=1

Equations (5.42), (5.44) and (5.46) imply that z ∈ at (x )

(5.47)

u t (x , z) ≥ wt (b(t) ξ (2) ).

(5.48)

and

By (5.25), (5.31), (5.34) and (5.43), z i ≥ yi − λ − |xi − xi | + λ0 n −1 α∗−1

≥ yi − 2λ + λ0 n −1 α∗−1 ≥ yi + λ.

Thus z ≥ y + λe.

(5.49)

|b(t) ξ (2) − b(t) ξ (1) | ≤ b∗ λ0 n.

(5.50)

In view of (5.29) and (5.40),

It follows from property (ii), the choice of λ0 , (5.27) and (5.50) that |w(b(t) ξ (2) ) − w(b(t) ξ (1) )| ≤ /4. By (5.36), (5.48) and (5.51), u t (x , z) ≥ u t (x, y) − /4.

(5.51)

176

5 The Turnpike Phenomenon for Nonautonomous Problems

In view of (5.25), (5.30), (5.31), (5.34) and (5.43), for all i = 1, . . . , n, |z i − yi | ≤ |xi − xi | + λ0 n −1 α∗−1 ≤ λ + λ0 n −1 α∗−1 < 2λ0 n −1 α∗−1 < n −1 , |z i − yi | ≤ |z i − yi | + |yi − yi |

< 2λ0 n −1 α∗−1 + λ < 3λ0 n −1 α∗−1 < n −1 and

z − y2 < , z − y 2 < .

Proposition 5.3 is proved. In view of Proposition 5.3, assumption (A1) holds. Assumption (A2) follows from Proposition 2.12. For the proof of (A3) see Proposition 2.13.

5.4 TP Implies (P1) and (P2) We suppose that all the assumptions introduced in Sects. 5.1 and 5.2 hold. Theorem 5.4 Let TP hold, M > 0 and  > 0. Then there exists a natural number Q such that for each integer T1 ≥ 0, each integer T2 > T1 + Q and each program T2 satisfying {x(t)}t=T 1 T 2 −1

u t (x(t), x(t + 1)) ≥

t=T1

T 2 −1

u t (x(t), ¯ x(t ¯ + 1)) − M

t=T1

the inequality Card({t ∈ {T1 , . . . , T2 } : x(t) − x ∗ (t) > }) ≤ Q holds. Proof Recall that the following property holds: (i) For each pair of nonnegative integers T1 < T2 ,   T 2 −1     u t (x(t), ¯ x(t ¯ + 1)) ≤ M∗ . U (T1 , T2 ) −   t=T1

TP implies that there exist δ ∈ (0, ) and a natural number L such that the following property holds: (ii) For each integer T1 ≥ 0, each integer T2 ≥ 2L + T1 and each program T2 {x(t)}t=T which satisfies 1

5.4 TP Implies (P1) and (P2) T 2 −1

177

u t (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ

t=T1

and T 2 −1

(T1 , T2 ) − M − 3M∗ u t (x(t), x(t + 1)) ≥ U

t=T1

we have x(t) − x(t) ¯ ≤ , t = T1 + L , . . . , T2 − L . Choose a natural number Q > (2L + 4)(4 + δ −1 (M + 3M∗ )).

(5.52)

Assume that an integer T1 ≥ 0, T2 > T1 + Q are integers and that a program T2 satisfies {x(t)}t=T 1 T 2 −1

u t (x(t), x(t + 1)) ≥

t=T1

T 2 −1

u t (x(t), ¯ x(t ¯ + 1)) − M.

t=T1

Let T1 ≤ S1 < S2 ≤ T2 be integers. In view of property (i) and (5.53), S 2 −1

u t (x(t), x(t + 1))

t=S1

=

T 2 −1

u t (x(t), x(t + 1))

t=T1

− − ≥





T 2 −1

{u t (x(t), x(t + 1)) : t ∈ {T1 , . . . , S1 } \ {S1 }} {u t (x(t), x(t + 1)) : t ∈ {S2 , . . . , T2 } \ {S2 }}

u t (x(t), ¯ x(t ¯ + 1)) − M

t=T1

− − =





S 2 −1 t=S1

¯ x(t ¯ + 1)) : t ∈ {T1 , . . . , S1 } \ {S1 }} − M∗ {u t (x(t), {u t (x(t), ¯ x(t ¯ + 1)) : t ∈ {S2 , . . . , T2 } \ {T2 }} − M∗

u t (x(t), ¯ x(t ¯ + 1)) − M − 2M∗ .

(5.53)

178

5 The Turnpike Phenomenon for Nonautonomous Problems

Thus we showed that the following property holds: (iii) For each pair of integers S1 , S2 satisfying T1 ≤ S1 < S2 ≤ T2 , S 2 −1

u t (x(t), x(t + 1)) ≥

t=S1

S 2 −1

u t (x(t), ¯ x(t ¯ + 1)) − M − 2M∗ .

t=S1 q

We construct by induction a finite sequence of nonnegative integers {ti }i=0 . Set t0 = T1 . If

T 2 −1

(5.54)

u t (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ,

t=T1

then we set t1 = T2 and the construction is completed. Assume that T 2 −1

u t (x(t), x(t + 1)) < U (x(T1 ), x(T2 ), T1 , T2 ) − δ.

(5.55)

t=T1

Then there exists an integer t1 > t0 such that t1 ≤ T2 and t1 −1 

u t (x(t), x(t + 1)) < U (x(t0 ), x(t1 ), t0 , t1 ) − δ

t=t0

and if an integer s satisfies t0 < s < t1 , then s−1 

u t (x(t), x(t + 1)) ≥ U (x(t0 ), x(s), t0 , s) − δ.

t=t0

Assume that k ≥ 1 is an integer and that we have defined a strictly increasing sequence of nonnegative integers ti ∈ [T1 , T2 ], i = 0, . . . , k such that (5.54) holds, for every integer p satisfying 0 ≤ p < k, t p+1 −1



u t (x(t), x(t + 1)) < U (x(t p ), x(t p+1 ), t p , t p+1 ) − δ

t=t p

and for each integer s satisfying t p < s < t p+1 , we have

5.4 TP Implies (P1) and (P2) s−1 

179

u t (x(t), x(t + 1)) ≥ U (x(t p ), x(s), t p , s) − δ.

t=t p

(Note that our assumption holds for k = 1.) If tk = T2 , then the construction is completed. Assume that tk < T2 . If T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(tk ), x(T2 ), tk , T2 ) − δ,

t=tk

then we set tk+1 = T2 and the construction is completed. Assume that T 2 −1

u t (x(t), x(t + 1)) < U (x(tk ), x(T2 ), tk , T2 ) − δ.

t=tk

Then there exists an integer tk+1 such that tk < tk+1 ≤ T2 and tk+1 −1



u t (x(t), x(t + 1)) < U (x(tk ), x(tk+1 ), tk , tk+1 ) − δ

t=tk

and for each integer s satisfying tk < s < tk+1 , we have s−1 

u t (x(t), x(t + 1)) ≥ U (x(tk ), x(s), tk , s) − δ.

t=tk

Thus the assumption made for k holds for k + 1. By induction we have constructed q a strictly increasing sequence of integers {ti }i=0 such that t0 = T1 , tq = T2 ,

(5.56)

for each integer p satisfying 0 ≤ p < q − 1, t p+1 −1



u t (x(t), x(t + 1)) < U (x(t p ), x(t p+1 ), t p , t p+1 ) − δ

(5.57)

t=t p

and for each integer p ∈ {0, . . . , q − 1} and each integer s satisfying t p < s < t p+1 , we have s−1  t=t p

u t (x(t), x(t + 1)) ≥ U (x(t p ), x(s), t p , s) − δ.

(5.58)

180

5 The Turnpike Phenomenon for Nonautonomous Problems

We show that q ≤ 4 + δ −1 (M + 3M∗ ).

(5.59)

We may assume without loss of generality that q ≥ 4. 1 such that In view of (5.57), there exists a program {y(t)}tt=t 0

y(t0 ) = x(t0 ), y(t1 ) ≥ x(t1 ), t1 −1 

u t (x(t), x(t + 1))
0 and each  > 0 there exists a natural number Q such that for each integer T1 ≥ 0, each integer T2 > T1 + Q and each program T −2 satisfying {x(t)}t=T 1 T 2 −1 t=T1

u t (x(t), x(t + 1)) ≥

T 2 −1 t=T1

u t (x(t), ¯ x(t ¯ + 1)) − M

5.4 TP Implies (P1) and (P2)

183

the inequality Card({t ∈ {T1 , . . . , T2 } : x(t) − x ∗ (t) > }) ≤ Q holds. In view of Theorem 5.4, TP implies WTP. Proposition 5.5 WTP implies (P1). Proof Assume that T1 ≥ 0 is an integer and {x(t)}∞ t=T1 is a good program. Then there exists M1 > 0 such that for each integer T2 > T1 we have T −1  T 2 2 −1     u t (x(t), x(t + 1)) − u t (x(t), ¯ x(t ¯ + 1)) ≤ M.    t=T1

t=T1

Let  > 0. By WTP there exists an integer Q ≥ 1 such that for each integer T > T1 + Q, Card({t ∈ {T1 , . . . , T } : x(t) − x ∗ (t) > }) ≤ Q. This implies that for all sufficiently large natural numbers t, x(t) − x ∗ (t) ≤ . Since  is an arbitrary positive number, this completes the proof of Proposition 5.5. Proposition 5.6 WTP implies (P2). Proof Assume that WTP holds. Let , M > 0. By WTP, there exists a natural number Q 0 such that the following property holds: T −2 for each integer T1 ≥ 0, each integer T2 > T1 + Q 0 and each program {x(t)}t=T 1 satisfying T T 2 −1 2 −1 u t (x(t), x(t + 1)) ≥ u t (x(t), ¯ x(t ¯ + 1)) − M t=T1

t=T1

the inequality Card({t ∈ {T1 , . . . , T2 } : x(t) − x ∗ (t) > }) ≤ Q 0 holds. Choose a natural number L 0 such that −1 + 4 L 0 > 4(6M∗ + M)Δ (see (A2)). Set L = L 0 Q0.

(5.74)

184

5 The Turnpike Phenomenon for Nonautonomous Problems

T +L Assume that T ≥ 0 is an integer and that a program {x(t)}t=T satisfies T +L−1

u t (x(t), x(t + 1)) ≥

T +L−1

t=T

u t (x(t), ¯ x(t ¯ + 1)) − M.

(5.75)

t=T

Property (i), (5.74) and (5.75) imply that Card({t ∈ {T, . . . , T + L 0 Q 0 } : x(t) − x ∗ (t) > }) ≤ Q 0 .

(5.76)

In view of (5.76), there exists j ∈ {0, . . . , Q 0 − 1} such that x(t) − x ∗ (t) ≤ , t = T + j L 0 + 1, . . . , T + ( j + 1)L 0 .

(5.77)

By Theorem 5.1, the choice of M∗ and (5.75), for each pair of integers S1 , S2 ∈ {T, . . . , T + L} satisfying S1 < S2 , S 2 −1

u t (x(t), x(t + 1))

t=S1

=

T +L−1 

u t (x(t), x(t + 1))

t=T

 {u t (x(t), x(t + 1)) : t ∈ {T, . . . , S1 } \ {S1 }}  − {u t (x(t), x(t + 1)) : t ∈ {S2 , . . . , T + L} \ {T + L}} −

≥

T +L−1 

u t (x(t), ¯ x(t ¯ + 1)) − M

t=T

 {u t (x(t), ¯ x(t ¯ + 1)) : t ∈ {T, . . . , S1 } \ {S1 }} − 2M∗  − {u t (x(t), ¯ x(t ¯ + 1)) : t ∈ {S2 , . . . , T + L} \ {T + L}} − 2M∗ −

=

S 2 −1

u t (x(t), ¯ x(t ¯ + 1)) − M − 4M∗ .

(5.78)

t=S1

It follows from Theorem 5.1, (A2), the choice of M∗ and (5.78) that T +( j+1)L 0 −1



t=T + j L 0 +1

T +( j+1)L 0 −1

u t (x(t), x(t + 1)) ≥



u t (x(t), ¯ x(t ¯ + 1)) − M − 4M∗

t=T + j L 0 +1

(T + j L 0 + 1, T + ( j + 1)L 0 ) − M − 5M∗ ≥U  0 − 1) − M − 5M∗ . ≥ Δ(L

5.4 TP Implies (P1) and (P2)

185

Together with (5.74) this implies that (L 0 − 1) max{u t (x(t), x(t + 1)) : t = T + j L 0 + 1, . . . , T + ( j + 1)L 0 − 1}  0 − 1) − 5M∗ − M ≥ Δ(L and max{u t (x(t), x(t + 1)) : t = T + j L 0 + 1, . . . , T + ( j + 1)L 0 − 1} ¯  − (5M∗ + M)(L 0 − 1)−1 > Δ/2. ≥Δ Thus there exists τ ∈ {T + j L 0 + 1, . . . , T + ( j + 1)L 0 − 1} such that

¯ u τ (x(τ ), x(τ + 1)) ≥ Δ/2.

Therefore property (P2) holds and Proposition 5.6 is proved.

5.5 Auxiliary Results Let c∗ > 1 satisfy c∗−1 z2 ≤ z ≤ c∗ z2 for all z ∈ R n . Lemma 5.7 Assume that properties (P1) and (P2) hold and that 0 ∈ (0, 1). Then there exist δ > 0 and a natural number L such that for each integer T1 ≥ L, each T2 which satisfies integer T2 > T1 and each program {x(t)}t=T 1 ¯ i ) ≤ δ, i = 1, 2, x(Ti ) − x(T x(T1 + 1) − x(T ¯ 1 + 1) ≤ δ, x(T2 − 1) − x(T ¯ 2 − 1) ≤ δ,  u T1 (x(T1 ), x(T1 + 1)) ≥ Δ/4,  u T2 −1 (x(T2 − 1), x(T2 )) ≥ Δ/4 and

T 2 −1 t=T1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ

186

5 The Turnpike Phenomenon for Nonautonomous Problems

the following inequality holds: x(t) − x(t) ¯ ≤ 0 , t = T1 , . . . , T2 . Proof Choose a positive number  < (8n)−1 (l + 1)−1 α∗−1 0 .

(5.79)

Let k ≥ 1 be an integer. Assumption (A1) implies that there exists a positive number  8−k }) δk ∈ (0, min{8−k Δ,

(5.80)

such that the following property holds: (a) For each integer t ≥ 0, each (x, x ) ∈ graph(at ) satisfying  u t (x, x ) ≥ 8−k min{, Δ} and each y, y ∈ K satisfying x − y, x − y  ≤ δk there exists z ∈ at (y) satisfying z ≥ y + δk e,  u t (y, z) ≥ u t (x, x ) − 8−k min{, Δ} and

 z − y  ≤ 8−k min{, Δ}.

We may assume without loss of generality that δk+1 ≤ δk for all integers k ≥ 1.

(5.81)

Assume that the lemma does not hold. Then there exist sequences of natural numbers Tk,2 ∞ (k) {Tk,1 }∞ k=1 , {Tk,2 }k=1 and a sequence of programs {x (t)}t=Tk,1 , k = 1, 2, . . . such that for every integer k ≥ 1, we have Tk,2 > Tk,1 ≥ k, Tk+1,1 > Tk,2 + 4,

(5.82)

x

(k)

(Tk,i ) − x(T ¯ k,i ) ≤ δk , i = 1, 2,

(5.83)

x

(k)

(Tk,1 + 1) − x(T ¯ k,1 + 1) ≤ δk ,

(5.84)

5.5 Auxiliary Results

187

x (k) (Tk,2 − 1) − x(T ¯ k,2 − 1) ≤ δk , (k) (k)  u Tk,1 (x (Tk,1 ), x (Tk,1 + 1)) ≥ Δ/4,

(5.85)

 u Tk,2 −1 (x (k) (Tk,2 − 1), x (k) (Tk,2 )) ≥ Δ/4,

(5.86)

Tk,2 −1



u t (x (k) (t), x (k) (t + 1))

t=Tk,1

≥ U (x (k) (Tk,1 ), x (k) (Tk,2 ), Tk,1 , Tk,2 ) − δk

(5.87)

¯ : t = Tk,1 , . . . , Tk,2 } > 0 . max{x (k) (t) − x(t)

(5.88)

and

In view of (5.82), Tk,2 − Tk,1 ≥ 4, k = 1, 2, . . . .

(5.89)

Let k ≥ 1 be an integer. It follows from property (a), the choice of δk , (5.83), (5.84) and (5.86) that there exists ¯ k,1 )) z k,0 ∈ aTk ,1 (x(T

(5.90)

such that ¯ k,1 + 1) + δk e, z k,0 ≥ x(T ¯ k,1 ), z k,0 ) ≥ u Tk,1 (x u Tk,1 (x(T

(5.91) (k)

(Tk,1 ), x

(k)

−k

(Tk,1 + 1)) − 8

 min{, Δ}.

(5.92)

Theorem 5.1 and (5.90)–(5.92) imply that ¯ k,1 ), x(T ¯ k,1 + 1)) ≥ u Tk,1 (x(T ¯ k,1 ), z k,0 ) u Tk,1 (x(T (k) (k)  ≥ u Tk,1 (x (Tk,1 ), x (Tk,1 + 1)) − 8−k min{, Δ}.

(5.93)

It follows from property (a), (5.83), (5.85), (5.86) and the choice of δk that there exists ¯ k,2 − 1)) ξk ∈ aTk ,2−1 (x(T

(5.94)

such that ¯ k,2 ) + δk e, ξk ≥ x(T u Tk,2 −1 (x(T ¯ k,2 − 1), ξk )  ≥ u Tk,2 −1 (x (k) (Tk,2 − 1), x (k) (Tk,2 )) − 8−k min{, Δ}.

(5.95) (5.96)

188

5 The Turnpike Phenomenon for Nonautonomous Problems

Theorem 5.1, (5.94) and (5.96) imply that ¯ k,2 − 1), x(T ¯ k,2 )) ≥ u Tk,2 −1 (x(T ¯ k,2 − 1), ξk ) u Tk,2 −1 (x(T (k) (k) −k  ≥ u Tk,2 −1 (x (Tk,2 − 1), x (Tk,2 )) − 8 min{, Δ}.

(5.97)

By (5.86), (5.93) and (5.97),  − 8−k min{, Δ}  ≥ Δ/8  ¯ k,1 ), x(T ¯ k,1 + 1)) ≥ Δ/4 u Tk,1 (x(T

(5.98)

 − 8−k min{, Δ}  ≥ Δ/8.  ¯ k,2 − 1), x(T ¯ k,2 )) ≥ Δ/4 u Tk,2 −1 (x(T

(5.99)

and

Property (a), (5.83), (5.84) and (5.86) imply that there exists z k,1 ∈ aTk ,1 (x(Tk,1 ))

(5.100)

such that ¯ k,1 + 1) + δk e, z k,1 ≥ x(T

(5.101)

u Tk,1 (x(Tk,1 ), z k,1 )  ≥ u Tk,1 (x (k) (Tk,1 ), x (k) (Tk,1 + 1)) − 8−k min{, Δ}.

(5.102)

Set z(Tk,1 ) = x(Tk,1 ), z(Tk,1 + 1) = z k,1 .

(5.103)

By (A3), (5.101) and (5.103) there exist z(t) ∈ K , t = Tk,1 + 2, . . . , Tk,2 − 1 T −1

k,2 is a program and that for all integers t = Tk,1 + 1, . . . , Tk,2 − 1, such that {z(t)}t=T k,1

z(t) ≥ x(t), ¯ u t (z(t), z(t + 1)) ≥ u t (x(t), ¯ x(t ¯ + 1)),

(5.104)

t = Tk,1 + 1, . . . , Tk,2 − 2.

(5.105)

Property (a), (5.83), (5.85) and (5.86) imply that there exists ¯ k,2 − 1)) z k,2 ∈ aTk,2 −1 (x(T such that

(5.106)

5.5 Auxiliary Results

189

z k,2 ≥ x (k) (Tk,2 ) + δk e,

(5.107)

u Tk,2 −1 (x(T ¯ k2 ,1 ), z k,2 )  ≥ u Tk,2 −1 (x (k) (Tk,2 − 1), x (k) (Tk,2 )) − 8−k min{, Δ}.

(5.108)

Assumption (A3), (5.104) and (5.106) imply that there exists z(Tk,2 ) ∈ K such that z(Tk,2 ) ∈ aTk,2 −1 (z(Tk,2 − 1)), z(Tk,2 ) ≥ z k,2 , u Tk,2 −1 (z(Tk,2 − 1), z(Tk,2 )) ≥ u Tk,2 −1 (x(T ¯ k,2 − 1), z k,2 )  ≥ u Tk,2 −1 (x (k) (Tk,2 − 1), x (k) (Tk,2 )) − 8−k min{, Δ}.

(5.109) (5.110)

In view of (5.103), (5.107) and (5.109), z(Tk,1 ) = x(Tk,1 ), z(Tk,2 ) ≥ x (k) (Tk,2 ).

(5.111)

T

k,2 is a program. Equations (5.87), (5.102), (5.103), In view of (5.109), {z(t)}t=T k,1 (5.105), (5.110) and (5.111) imply that

Tk,2 −1



u t (x (k) (t), x (k) (t + 1)) + δk

t=Tk,1 Tk,2 −1

≥



u t (z(t), z(t + 1))

t=Tk,1 Tk,2 −2



= u Tk,1 (z(Tk,1 ), z(Tk,1 + 1)) +

u t (z(t), z(t + 1))

t=Tk,1 +1

+ u Tk,2 −1 (z(Tk,2 − 1), z(Tk,2 ))  ≥ u Tk,1 (x (k) (Tk,1 ), x (k) (Tk,1 + 1)) − 8−k min{, Δ} Tk,2 −2

+



u t (x(t), ¯ x(t ¯ + 1))

t=Tk,1 +1

 + u Tk,2 −1 (x (k) (Tk,2 − 1), x (k) (Tk,2 )) − 8−k min{, Δ}. Together with (5.80) this implies that

190

5 The Turnpike Phenomenon for Nonautonomous Problems Tk,2 −2



u t (x (k) (t), x (k) (t + 1))

t=Tk,1 +1 Tk,2 −2

≥



 u t (x(t), ¯ x(t ¯ + 1)) − 3 · 8−k min{, Δ}.

(5.112)

t=Tk,1 +1

Property (a), (5.83), (5.84) and (5.98) imply that there exists ξk,0 ∈ K such that ¯ k,1 )), ξk,0 ∈ aTk,1 (x(T ξk,0 ≥ x

(k)

(5.113)

(Tk,1 + 1) + δk e,

(5.114) −k

u Tk,1 (x(T ¯ k,1 ), ξk,0 ) ≥ u Tk,1 (x(T ¯ k,1 ), x(T ¯ k,1 + 1)) − 8

 min{, Δ}

(5.115)

and  ξk,0 − x (k) (Tk,1 + 1) ≤ 8−k min{, Δ}.

(5.116) T −1

k,2 Assumption (A3), (5.113) and (5.114) imply that there exists a program {y (k) (t)}t=T k,1 such that

¯ k,1 ), y (k) (Tk,1 ) = x(T (k)

(5.117)

y (Tk,1 + 1) = ξk,0 ≥ x

(k)

(Tk,1 + 1)

(5.118)

and that for every t ∈ {Tk,1 + 1, . . . , Tk,2 − 2}, 0 ≤ y (k) (t + 1) − x (k) (t + 1) ≤ y (k) (t) − x (k) (t), (k)

(k)

u t (y (t), y (t + 1)) ≥ u t (x

(k)

(t), x

(k)

(t + 1)).

(5.119) (5.120)

By (5.116), (5.118) and (5.119), for all t = Tk,1 + 1, . . . , Tk,2 − 1, y (k) (t) − x (k) (t) ≤ y (k) (Tk,1 + 1) − x (k) (Tk,1 + 1)  ≤ 8−k min{, Δ}.

(5.121)

Property (a), (5.83), (5.85) and (5.99) imply that there exists ξk,1 ∈ K such that ξk,1 ∈ aTk,2 −1 (x (k) (Tk,2 − 1)),

(5.122)

ξk,1 ≥ x(T ¯ k,2 ) + δk e,

(5.123)

u Tk,2 −1 (x

(k)

(Tk,2 − 1), ξk,1 )

 ≥ u Tk,2 −1 (x(T ¯ k,2 − 1), x(T ¯ k,2 )) − 8−k min{, Δ} and

(5.124)

5.5 Auxiliary Results

191

 ξk,1 − x(T ¯ k,2 ) ≤ 8−k min{, Δ}.

(5.125)

Assumption (A3), (5.119), (5.122) and (5.124) imply that there exists y (k) (Tk,2 ) ∈ K such that y (k) (Tk,2 ) ∈ aTk,2 −1 (y (k) (Tk,2 − 1)),

(5.126)

(k)

¯ k,2 ) + δk e, y (Tk,2 ) ≥ ξk,1 ≥ x(T (k)

(5.127)

(k)

u Tk,2 −1 (y (Tk,2 − 1), y (Tk,2 )) ≥ u Tk,2 −1 (x −k

≥ u Tk,2 −1 (x(T ¯ k,2 − 1), x(T ¯ k,2 )) − δk − 8

(k)

(Tk,2 − 1), ξk,1 )  min{, Δ}, (5.128)

0 ≤ y (k) (Tk,2 ) − ξk,1 ≤ y (k) (Tk,2 − 1) − x (k) (Tk,2 − 1).

(5.129)

T

k,2 is a program, Equations (5.117), (5.126) and (5.127) imply that {y (k) (t)}t=T k,1

y (k) (Tk,1 ) = x(T ¯ k,1 ), ¯ k,2 ). y (k) (Tk,2 ) ≥ x(T

(5.130)

In view of (5.112), (5.115), (5.117), (5.118), (5.120) and (5.128), Tk,2 −1



u t (y (k) (t), y (k) (t + 1))

t=Tk,1

 ≥ u Tk,1 (x(T ¯ k,1 ), x(T ¯ k,1 + 1)) − 8−k min{, Δ}  + {u t (x (k) (t), x (k) (t + 1)) : t ∈ {Tk,1 + 1, . . . , Tk,2 − 2}}  + u Tk,2 −1 (x(T ¯ k,2 − 1), x(T ¯ k,2 )) − 8−k min{, Δ} ≥

Tk,2 

 u t (x(t), ¯ x(t ¯ + 1)) − 5 · 8−k min{, Δ}.

(5.131)

t=Tk,1

By (5.80), (5.85), (5.121), (5.125) and (5.129), y (k) (Tk,2 ) − x (k) (Tk,2 ) ≤ y (k) (Tk,2 ) − ξk,1  + ξk,1 − x (k) (Tk,2 ) ¯ k,2 ) ≤ y (k) (Tk,2 − 1) − x (k) (Tk,2 − 1) + ξk,1 − x(T (k) −k  + x(T ¯ k,2 ) − x (Tk,2 ) ≤ 3 · 8 min{, Δ}.

(5.132)

Equations (5.83), (5.118), (5.121) and (5.132) imply that for all integers t = Tk,1 , . . . , Tk,2 ,

192

5 The Turnpike Phenomenon for Nonautonomous Problems

 y (k) (t) − x (k) (t) ≤ 3 · 8−k min{, Δ}.

(5.133)

Now we construct a program {x(t)}∞ t=0 . For all t = 0, . . . , T1,1 , set x(t) = x(t), ¯

(5.134)

for all integers t = T1,1 , . . . , T1,2 set x(t) = y (1) (t).

(5.135)

¯ 1,2 ). x(T1,2 ) = y (1) (T1,2 ) ≥ x(T

(5.136)

In view of (5.130) and (5.135),

Assumption (A3) and (5.136) imply that there exist x(t) ∈ K , t = T1,2 , . . . , T2,1 such that for all t = T1,2 , . . . , T2,1 − 1 x(t + 1) ∈ at (xt ), x(t + 1) ≥ x(t ¯ + 1), u t (x(t), x(t + 1)) ≥ u t (x(t), ¯ x(t ¯ + 1)),

(5.137) (5.138)

0 ≤ x(t + 1) − x(t ¯ + 1) ≤ x(t) − x(t) ¯ ¯ 1,2 ) = y (1) (T1,2 ) − x(T ¯ 1,2 ). ≤ x(T1,2 ) − x(T

(5.139)

T

2,1 is a program. Assumption (A3), By (5.130), (5.134), (5.135) and (5.137), {x(t)}t=0 (5.130) and (5.139) imply that there exist

x(t), t = T2,1 + 1, . . . , T2,2 T

2,2 such that {x(t)}t=0 is a program,

x(t) ≥ y (2) (t), t = T2,1 , . . . , T2,2 ,

(5.140)

and for all integers t = T2,1 , . . . , T2,2 − 1, u t (x(t), x(t + 1)) ≥ u t (y (2) (t), y (2) (t + 1)), (2)

(5.141)

(2)

0 ≤ x(t + 1) − y (t + 1) ≤ x(t) − y (t) ≤ x(T2,1 ) − y (2) (T2,1 ).

(5.142)

5.5 Auxiliary Results

193 T

k,2 Assume that k ≥ 2 is an integer and we constructed a program {x(t)}t=0 such that for every integer t ∈ {0, . . . , T1,1 } (5.134) holds, for every integer t ∈ {T1,1 , . . . , T1,2 } (5.135) and (5.136) hold, for every integer p ∈ {1, . . . , k − 1} we have

¯ p,2 ), x(T p,2 ) ≥ x(T

(5.143)

for every t ∈ {T p,2 , . . . , T p+1,1 − 1}, 0 ≤ x(t + 1) − x(t ¯ + 1) ¯ p,2 ), ≤ x(t) − x(t) ¯ ≤ x(T p,2 ) − x(T u t (x(t), x(t + 1)) ≥ u t (x(t), ¯ x(t ¯ + 1)),

(5.144) (5.145)

and for every t ∈ {T p+1,1 , . . . , T p+1,2 − 1}, u t (x(t), x(t + 1)) ≥ u t (y ( p+1) (t), y ( p+1) (t + 1)), 0 ≤ x(t + 1) − y ≤ x(t) − y

( p+1)

( p+1)

(5.146)

(t + 1)

(t) ≤ x(T p+1,1 ) − y ( p+1) (T p+1,1 ).

(5.147)

(By (5.134)–(5.136), (5.138), (5.139), (5.141) and (5.142), our assumption holds for k = 2.) In view of (5.130) and (5.147), ¯ k,2 ). x(Tk,2 ) ≥ y (k) Tk,2 ) ≥ x(T

(5.148)

Assumption (A3) and (5.148) imply that there exist x(t) ∈ K , t = Tk,2 + 1, . . . , Tk+1,1 T

k+1,1 such that {x(t)}t=0 is a program, and for all integers t = Tk,2 , . . . , Tk+1,1 − 1,

0 ≤ x(t + 1) − x(t ¯ + 1) ≤ x(t) − x(t) ¯ ¯ k,2 ), ≤ x(Tk,2 ) − x(T

(5.149)

u t (x(t), x(t + 1)) ≥ u t (x(t), ¯ x(t ¯ + 1)).

(5.150)

By (5.117) and (5.149), ¯ k+1,1 ) = y (k) (Tk+1,1 ). x(Tk+1,1 ) ≥ x(T Assumption (A3) and (5.151) imply that there exist x(t) ∈ K , t = Tk+1 + 1, . . . , Tk+1,2

(5.151)

194

5 The Turnpike Phenomenon for Nonautonomous Problems ,2

T

k+1 such that {x(t)}t=0 is a program and for all integers t = Tk+1,1 , . . . , Tk+1,2 − 1,

u t (x(t), x(t + 1)) ≥ u t (y (k+1) (t), y (k+1) (t + 1)), 0 ≤ x(t + 1) − y (k+1) (t + 1) ≤ x(t) − y (k+1) (t) ≤ x(Tk+1,1 ) − y (k+1) (Tk+1,1 ). Therefore the assumption made for k also holds for k + 1. Hence by induction we constructed a program {x(t)}∞ t=0 such that for all integers t = 0, . . . , T1,1 , (5.134) is true, for all t = T1,1 , . . . , T1,2 , Eqs. (5.135) and (5.136) hold, for all integers p ≥ 1, (5.143) holds, for all integers t = T p,2 , . . . , T p+1,1 − 1, (5.144) and (5.145) are true and for all integers t = T p+1,1 , . . . , T p+1,2 − 1, (5.146) and (5.147) are true. Let k ≥ 3 be an integer. By (5.131), (5.134), (5.135), (5.145) and (5.146), Tk,2 −1



Tk,2 −1

u t (x(t), x(t + 1)) −



t=0

≥

u t (x(t), ¯ x(t ¯ + 1))

t=0

p,2 −1 k T 

u t (x(t), x(t + 1)) +

k−1 T p+1,1  −1

p=1 t=T p,1

−

p,2 −1 k T 

u t (x(t), ¯ x(t ¯ + 1)) −

p=1 t=T p,1

≥

k  p=1

≥

k  p=1

≥ −5

⎡ ⎣ ⎡ ⎣

k−1 T p+1  −1

T p,2 −1

u t (x(t), x(t + 1)) −

t=T p,1



p=1

u t (x(t), ¯ x(t ¯ + 1))⎦ T p,2 −1

u t (y

( p)

(t), y

( p)

(t + 1)) −

t=T p,1 k 

⎤

t=T p,1

T p,2 −1



u t (x(t), ¯ x(t ¯ + 1))

p=1 t=T p,2

T p,2 −1



u t (x(t), x(t + 1))

p=1 t=T p,2



⎤ u t (x(t), ¯ x(t ¯ + 1))⎦

t=T p,1

 ≥ −5 8− p Δ

∞ 

 8− p Δ.

p=1

Thus {x(t)}∞ t=0 is a good program. Property (P1) implies that ¯ = 0. lim x(t) − x(t)

t→∞

Then there is an integer τ0 > 0 such that for all integers t ≥ τ0 , x(t) − x(t) ¯ < /4.

(5.152)

Let p ≥ 1 be an integer. By (5.83), (5.130), (5.132), (5.144) and (5.147), ¯ p+1,1 ) ≤ x(T p,2 ) − x(T ¯ p,2 ), 0 ≤ x(T p+1,1 ) − x(T

(5.153)

5.5 Auxiliary Results

195

0 ≤ x(T p+1,2 ) − y ( p+1) (T p+1,2 ) ≤ x(T p+1,1 ) − y ( p+1) (T p+1,1 ) = x(T p+1,1 ) − x(T ¯ p+1,1 ) ≤ x(T p,2 ) − x(T ¯ p,2 ) ¯ p,2 ) = x(T p,2 ) − y ( p) (T p,2 ) + y ( p) (T p,2 ) − x(T = x(T p,2 ) − y ( p) (T p,2 ) + y ( p) (T p,2 ) − x ( p) (T p,2 ) ¯ p,2 ) + x ( p) (T p,2 ) − x(T ≤ x(T p,2 ) − y ( p) (T p,2 ) + 3 · 8− p c∗ e + 8− p c∗ e. Thus for every integer p ≥ 1, 0 ≤ x(T p+1,2 ) − y ( p+1) (T p+1,2 ) ≤ x(T p,2 ) − y ( p) (T p,2 ) + 4 · 8− p c∗ e.

(5.154)

Equations (5.135) and (5.154) imply that for every integer k ≥ 2, 0 ≤ x(Tk,2 ) − y (k) (Tk,2 ) ≤ x(T1,2 ) − y (1) (T1,2) + 4c∗ 

k−1 

8−i e = 4c∗ 

i=1

k−1 

8−i e.

(5.155)

i=1

Let k ≥ 2 be an integer such that Tk,1 > τ0 + 4.

(5.156)

Equations (5.152) and (5.156) imply that x(t) − x(t) ¯ < /4 for all integers t ≥ Tk,1 .

(5.157)

Evidently, (5.155) is true. By (5.83), (5.121), (5.132), (5.153) and (5.155), ¯ k+1,1 ) 0 ≤ x(Tk+1,1 ) − x(T ≤ x(Tk,2 ) − x(T ¯ k,2 ) ¯ k,2 ) ≤ x(Tk,2 ) − y (k) (Tk,2 ) + y (k) (Tk,2 ) − x(T ≤ 4c∗ 

k 

8−i e + y (k) (Tk,2 ) − x (k) (Tk,2 ) + x (k) (Tk,2 ) − x(T ¯ k,2 )

i=1

≤ 4c∗ 

k  i=1

8−i e + 3c∗ 8−k e + 8−k c∗ e.

(5.158)

196

5 The Turnpike Phenomenon for Nonautonomous Problems

By (5.130), (5.147) and (5.158), for all integers t = Tk+1,1 , . . . , Tk+1,2 − 1, 0 ≤ x(t + 1) − y (k+1) (t + 1) ≤ x(Tk+1,1 ) − y (k+1) (Tk+1,1 ) = x(Tk+1,1 ) − x(T ¯ k+1,1 ) ≤ 4c∗ 

k 

8−i e + 4c∗ 8−k e.

i=1

This implies that for all integers t = Tk+1,1 , . . . , Tk+1,2 , 0 ≤ x(t) − y (k+1) (t) ≤ 4c∗ 

k 

8−i e + 4c∗ 8−k e.

(5.159)

i=1

In view of (5.157) and (5.159), for all integers t = Tk+1,1 , . . . , Tk+1,2 , x(t) ¯ − y (k+1) (t) ≤ x(t) ¯ − x(t) + x(t) − y (k+1) (t)  k   −i −k ≤ /4 + 4c∗  8 +8 e.

(5.160)

i=1

It follows from (5.79), (5.121) and (5.160) that for all t = Tk+1,1 + 2, . . . , Tk+1,2 − 1, ¯ x (k+1) (t) − x(t) ¯ ≤ x (k+1) (t) − y (k+1) (t) + y (k+1) (t) − x(t)  k   ≤ 8−k−1  + /4 + 4c∗  8−i + 8−k e i=1

 −k−1

≤ c∗  8

−k

+4·8

+4

−1

+4

k 

 −i

8

e ≤ c∗ 8e < 0 .

i=1

This contradicts (5.88). The contradiction we have reached completes the proof of Lemma 5.7.

5.6 Completion of the Proof of Theorem 5.2 We show that properties (P1) and (P2) imply TP. Assume that (P1) and (P2) hold. Let , M > 0. By Lemma 5.7, there exist δ ∈ (0, ) and a natural number L 0 such that the following property holds: T2 (i) For each integer T1 ≥ L 0 , each integer T2 > T1 and each program {x(t)}t=T 1 which satisfies

5.6 Completion of the Proof of Theorem 5.2

197

x(Ti ) − x(T ¯ i ) ≤ δ, i = 1, 2, x(T1 + 1) − x(T ¯ 1 + 1) ≤ δ, x(T2 − 1) − x(T ¯ 2 − 1) ≤ δ,  u T1 (x(T1 ), x(T1 + 1)) ≥ Δ/4,  u T2 −1 (x(T2 − 1), x(T2 )) ≥ Δ/4 and

T 2 −1

u t (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ

t=T1

we have x(t) − x(t) ¯ ≤ , t = T1 , . . . , T2 . By property (P2), there exists a natural number L 1 ≥ 2 such that the following property holds: T +L 1 which satisfies (ii) For each integer T ≥ 0 and each program {x(t)}t=T T +L 1 −1 

u t (x(t), x(t + 1)) ≥

t=T

T +L 1 −1 

u t (x(t), ¯ x(t ¯ + 1)) − M − 2M∗

t=T

there exists an integer τ ∈ {T, . . . , T + L 1 − 1} such that x(τ ) − x(τ ¯ ), x(τ + 1) − x(τ ¯ + 1) ≤ δ and

 u τ (x(τ ), x(τ + 1)) ≥ Δ/4.

Set L = L 0 + L 1 + 4.

(5.161)

T2 Assume that T1 ≥ 0, T2 ≥ 2L + T1 are integers and that a program {x(t)}t=T 1 satisfies T 2 −1 t=T1

and

u t (x(t), x(t + 1)) ≥ U (x(T1 ), x(T2 ), T1 , T2 ) − δ

(5.162)

198

5 The Turnpike Phenomenon for Nonautonomous Problems T 2 −1

(T1 , T2 ) − M. u t (x(t), x(t + 1)) ≥ U

(5.163)

t=T1

In order to complete the proof of the theorem it is sufficient to show that x(t) − x(t) ¯ ≤ , t = T1 + L , . . . , T2 − L . Let S1 , S2 ∈ {T1 , . . . , T2 } satisfy S1 < S2 . Theorem 5.1 and (5.263) imply that S 2 −1

u t (x(t), x(t + 1))

t=S1

=

T 2 −1

u t (x(t), x(t + 1))

t=T1

− − ≥





T 2 −1

{u t (x(t), x(t + 1)) : t ∈ {T1 , . . . , S1 } \ {S1 }} {u t (x(t), x(t + 1)) : t ∈ {S2 , . . . , T2 } \ {T2 }}

u t (x(t), ¯ x(t ¯ + 1)) − M

t=T1

− − =





S 2 −1

{u t (x(t), ¯ x(t ¯ + 1)) : t ∈ {T1 , . . . , S1 } \ {S1 }} − M∗ {u t (x(t), ¯ x(t ¯ + 1)) : t ∈ {S2 , . . . , T2 } \ {T2 }} − M∗

u t (x(t), ¯ x(t ¯ + 1)) − M − 2M∗ .

t=S1

In view of (5.163), T1 +L 0 +L 1 −1

u t (x(t), x(t + 1)) ≥

t=T1 +L 0

T1 +L 0 +L 1 −1

u t (x(t), ¯ x(t ¯ + 1)) − M − 2M∗

t=T1 +L 0

(5.164) and T 2 −1 t=T2 −L 1

u t (x(t), x(t + 1)) ≥

T 2 −1

u t (x(t), ¯ x(t ¯ + 1)) − M − 2M∗ . (5.165)

t=T2 −L 1

Property (ii), (5.164) and (5.165) imply that there exist integers τ1 ∈ {T1 + L 0 , . . . , T1 + L 0 + L 1 − 1}, τ2 ∈ {T2 − L 1 , . . . , T2 − 1}

5.6 Completion of the Proof of Theorem 5.2

199

such that for all i = 1, 2, ¯ i ), x(τi + 1) − x(τ ¯ i + 1) ≤ δ x(τi ) − x(τ

(5.166)

 u τi (x(τi ), x(τi + 1)) ≥ Δ/4.

(5.167)

and

Property (i), (5.162), (5.166) and (5.167) imply that x(t) − x(t) ¯ ≤ , t = τ1 , . . . , τ2 . Theorem 5.2 is proved.

5.7 A Turnpike Result for Approximate Solutions Assume that the sequence {u t }∞ t=0 has TP. In this chapter we prove the following turnpike result. Theorem 5.8 Let , M > 0. Then there exist δ > 0 and a natural number L such T2 that for each pair of integers T1 ≥ 0, T2 > T1 + 2L and each program {x(t)}t=T 1 satisfying τ +L−1 

u(x(t), x(t + 1)) ≥ U (x(τ ), x(τ + L), τ , τ + L) − δ

t=τ

for all τ = T1 , . . . , T2 − L and T −1 

(T2 − L , T2 ) − M u(x(t), x(t + 1)) ≥ U

t=T −L

there exist integers τ1 ∈ [T, T + L] and τ2 ∈ [T2 − L , T2 ] such that x(t) − x(t) ¯ ≤ , t = τ1 , . . . , τ2 .

200

5 The Turnpike Phenomenon for Nonautonomous Problems

5.8 An Auxiliary Result for Theorem 5.8 Set diam(K ) = sup{x − y : x, y ∈ K }. Lemma 5.9 There exist γ0 ∈ (0, 1), M1 > 0 and a natural number Q 1 ≥ 3 such that for each integer S ≥ 0, each integer T ≥ Q 1 each x ∈ K and each y1 , y2 ∈ K satisfying y2 ∈ a S+T −1 (y1 ),

(5.168)

y1 − x(S ¯ + T − 1) ≤ γ0 , y2 − x(S ¯ + T ) ≤ γ0

(5.169)

and  u S+T −1 (y1 , y2 ) ≥ Δ/4

(5.170)

S+T such that there exists a program { ξ(t)}t=S

 ξ(S) = x,  ξ(T + S) ≥ y2 and

T +S−1

u t ( ξ(t),  ξ(t + 1)) ≥

t=S

T +S−1

¯ + 1)) − M1 . u t (x(t), ¯ ξ(t

t=S

Proof By assumption (A1), there exists a number γ0 ∈ (0, 1) such that the following property holds:  and if (i) If an integer t ≥ 0 and if (x, x ) ∈ graph(at ) satisfies u t (x, x ) ≥ D/8

y, y ∈ K satisfy x − y, x − y  ≤ 2γ0 , then there exists z ∈ at (y) satisfying z ≥ y + γ0 e. Recall that c∗ > 1 satisfies c∗−1 z2 ≤ z ≤ c∗ z2 for all z ∈ R n .

(5.171)

Choose a natural number Q 0 ≥ 3 such that κ Q 0 −1 c∗2 (diam(K ) + 1) < γ0 .

(5.172)

5.8 An Auxiliary Result for Theorem 5.8

201

Property (P2) implies that there exists a natural number L ≥ 2 such that the following property holds: (ii) For each integer S ≥ 0 there exists an integer τ ∈ {S, . . . , S + L − 1} such that

 ¯ ), x(τ ¯ + 1)) ≥ Δ/4. u τ (x(τ

Choose an integer Q1 > Q0 + L + 8

(5.173)

M1 > Q 1 sup{u t (z, z ) : (z, z ) ∈ graph(at ), t = 0, 1, . . . }.

(5.174)

and a number

Let x ∈ K , an integer T ≥ Q 1 , an integer S ≥ 0 and y1 , y2 ∈ K satisfy (5.168)– (5.170). By property (i) applied with t = S + T − 1, x = y1 , x = y2 , y = x(T ¯ + S − 1), y = y2 we obtain that there exists ¯ + S − 1)) x˜ ∈ aT +S−1 (x(T

(5.175)

x˜ ≥ y2 .

(5.176)

satisfying

In view of (5.1), there exists a program {ξ(t)}∞ t=S such that ξ(S) = x

(5.177)

ξ(t + 1) − x(t ¯ + 1) ≤ κξ(t) − x(t) ¯ for all integers t > S.

(5.178)

and that

By (5.171), (5.177) and (5.178), for all integers t > S, ¯ ¯ c∗−1 ξ(t) − x(t) 2 ≤ ξ(t) − x(t) ≤ κt−S ξ(S) − x(S) ¯ ≤ κt−S x(S) ¯ − x ≤ diam(K )κt−S and

202

5 The Turnpike Phenomenon for Nonautonomous Problems

ξ(t) − x(t) ¯ ¯ ≤ c∗ diam(K )κt−S . 2 , ξ(t) − x(t)

(5.179)

Equations (5.172) and (5.179) imply that for all integers t ≥ S + Q 0 − 1, ¯ < γ0 . ξ(t) − x(t) ¯ 2 , ξ(t) − x(t)

(5.180)

Property (ii) implies that there exists an integer S1 ∈ {S + Q 0 , . . . , S + Q 0 + L − 1}

(5.181)

 ¯ 1 ), x(S ¯ 1 + 1)) ≥ Δ/4. u S1 (x(S

(5.182)

such that

Equations (5.180) and (5.181) imply that ¯ 1 )2 , ξ(S + 1) − x(S ¯ + 1)2 , ξ(S1 ) − x(S ξ(S1 ) − x(S ¯ 1 ), ξ(S1 + 1) − x(S ¯ 1 + 1) < γ0 .

(5.183)

By (5.182), (5.183) and property (i) applied with ¯ 1 ), x = x(S ¯ 1 + 1), t = S1 , x = x(S

y = ξ(S1 ), y = x(S ¯ 1 + 1) there exists y˜ ∈ a S1 (ξ(S1 ))

(5.184)

y˜ ≥ x(S ¯ 1 + 1).

(5.185)

 ξ(S1 + 1) = y˜ . ξ(t) = ξ(t), t = S, . . . , S1 , 

(5.186)

such that

Set

S1 +1 In view of (5.184) and (5.186), { ξ(t)}t=S is a program. By (5.175) and (5.181),

S + T − 1 − S1 − 1 ≥ S + T − 1 − (S + Q 0 + L) ≥ Q 1 − Q 0 − L − 1 > 7.

Equations (5.185)–(5.187) imply that there exist  ξ(t) ∈ K , t = S1 + 2, . . . , T + S − 1

(5.187)

5.8 An Auxiliary Result for Theorem 5.8

203

t+S−1 such that { ξ(t)}t=S is a program,

 ξ(t) ≥ x(t), ¯ t = S1 + 2, . . . , T + S − 1, ξ(t),  ξ(t + 1)) ≥ u t (x(t), ¯ x(t ¯ + 1)), t = S1 + 1, . . . , T + S − 2. u t (

(5.188) (5.189)

By (5.175), (5.176) and (5.188), there exists  ξ(T + S − 1)) ξ(T + S) ∈ aT +S−1 (

(5.190)

 ξ(T + S) ≥ x˜ ≥ y2 .

(5.191)

such that

In view of (5.177), (5.186), (5.190) and (5.191), { ξ(t)}t+S t=S is a program and  ξ(S) = x,  ξ(T + S) ≥ y2 . It follows from (5.173), (5.174), (5.181) and (5.189) that T +S−1 

u t ( ξ(t),  ξ(t + 1)) ≥

t=S

T +S−2 

u t (x(t), ¯ x(t ¯ + 1))

t=S+1

≥

T +S−1 

u t (x(t), ¯ x(t ¯ + 1))

t=S

− sup{sup{u t (z, z ) : (z, z ) ∈ graph(at )} : t = 0, 1, . . . }(S1 − S + 3) ≥

T +S−1 

u t (x(t), ¯ x(t ¯ + 1))

t=S

− sup{sup{u t (z, z ) : (z, z ) ∈ graph(at )} : t = 0, 1, . . . }(Q 0 + L + 4) ≥

T +S−1 

u t (x(t), ¯ x(t ¯ + 1)) = M1 .

t=S

Lemma 5.9 is proved.

5.9 Proof of Theorem 5.8 By Lemma 5.9, there exist γ0 ∈ (0, 1), M0 > 0 and a natural number Q 0 ≥ 3 such that the following property holds: (i) For each integer S1 ≥ 0, each integer S2 ≥ S1 + Q 0 , each x ∈ K and each y1 , y2 ∈ K satisfying

204

5 The Turnpike Phenomenon for Nonautonomous Problems

y2 ∈ a S2 −1 (y1 ), y1 − x(S ¯ 2 − 1) ≤ γ0 , y2 − x(S ¯ 2 )) ≤ γ0 and

 u S2 −1 (y1 , y2 ) ≥ Δ/4

S2 such that there exists a program {ξ(t)}t=S 1

ξ(S1 ) = x, ξ(S2 ) ≥ y2 and

S 2 −1

u t (ξ(t), ξ(t + 1)) ≥

t=S1

S 2 −1

u t (x(t), ¯ x(t ¯ + 1)) − M0 .

t=S1

By Lemma 5.7, there exist δ1 ∈ (0, min{γ0 , })

(5.192)

and a natural number L 1 such that the following property holds: S2 which (ii) For each pair of integers S1 ≥ L 1 , S2 > S1 and each program {ξ(t)}t=S 1 satisfies ¯ i ) ≤ δ1 , i = 1, 2, ξ(Si ) − x(S ξ(S1 + 1) − x(S ¯ 1 + 1) ≤ δ1 , ξ(S2 − 1) − x(S ¯ 2 − 1) ≤ δ1 ,  u S1 (ξ(S1 ), ξ(S1 + 1)) ≥ Δ/4,  u S2 −1 (ξ(S2 − 1), ξ(S2 )) ≥ Δ/4 and

S 2 −1

u t (ξ(t), ξ(t + 1)) ≥ U (ξ(S1 ), ξ(S2 ), S1 , S2 ) − δ1

t=S1

we have x(t) − x(t) ¯ ≤ , t = S1 , . . . , S2 . Choose δ ∈ (0, 2−1 δ1 ), M1 > M∗ + M + M0 + 1.

(5.193)

5.9 Proof of Theorem 5.8

205

By property (P2), there exists a natural number L 2 ≥ 2 such that the following property holds: S+L 2 which satisfies (iii) For each integer S ≥ 0 and each program {ξ(t)}t=S S+L 2 −1 

u t (ξ(t), ξ(t + 1)) ≥

t=S

S+L 2 −1 

u t (x(t), ¯ x(t ¯ + 1)) − M1

t=S

there exists an integer τ ∈ {S, . . . , S + L 2 − 1} such that ξ(τ ) − x(τ ¯ ), ξ(τ + 1) − x(τ ¯ + 1) ≤ δ and

 u τ (ξ(τ ), ξ(τ + 1)) ≥ Δ/4.

Choose a natural number L > 8L 1 + 8L 2 + 8Q 0 + 8.

(5.194)

T1 ≥ 0, T2 > T1 + 2L

(5.195)

Assume that

T2 is a program, for all τ = T1 , . . . , T2 − L, are integers, {x(t)}t=T 1 τ +L−1 

u(x(t), x(t + 1)) ≥ U (x(τ ), x(τ + L), τ , τ + L) − δ

(5.196)

t=τ

and T 2 −1

(T2 − L , T2 ) − M. u(x(t), x(t + 1)) ≥ U

t=T2 −L

Theorem 5.1, the choice of M∗ , (5.193), (5.194) and (5.197) imply that

(5.197)

206

5 The Turnpike Phenomenon for Nonautonomous Problems T 2 −1

u(x(t), x(t + 1))

t=T2 −L 2

=

T 2 −1

u(x(t), x(t + 1)) −

t=T2 −L

T2 −L 2 −1

T 2 −1

≥ −M +

u(x(t), x(t + 1)) −

t=T2 −L

=

T 2 −1

u(x(t), x(t + 1))

t=T2 −L T −L 2 −1 

u(x(t), ¯ x(t ¯ + 1)) − M∗

t=T2 −L

u(x(t), ¯ x(t ¯ + 1)) − M1 .

(5.198)

t=T2 −L 2

Property (iii) and (5.198) imply that there exists an integer t0 ∈ {T2 − L 2 , . . . , T2 − 1}

(5.199)

¯ 0 ), x(t0 + 1) − x(t ¯ 0 + 1) ≤ δ x(t0 ) − x(t

(5.200)

 u t0 (x(t0 ), x(t0 + 1)) ≥ Δ/4.

(5.201)

such that

and

Assumption (A3), (5.194)–(5.196) and (5.199) imply that t0 

u(x(t), x(t + 1))

t=t0 −2L 2 −2Q 0

≥ U (x(t0 − 2L 2 − 2Q 0 ), x(t0 + 1), t0 − 2L 2 − 2Q 0 , t0 + 1) − δ.

(5.202)

By property (i), (5.192), (5.200) and (5.201), U (x(t0 − 2L 2 − 2Q 0 ), x(t0 + 1), t0 − 2L 2 − 2Q 0 , t0 + 1) ≥

t0 

u(x(t), ¯ x(t ¯ + 1)) − M0 .

t=t0 −2L 2 −2Q 0

Equations (5.202) and (5.203) imply that

(5.203)

5.9 Proof of Theorem 5.8

207 t0 

u(x(t), x(t + 1))

t=t0 −2L 2 −2Q 0

≥

t0 

u(x(t), ¯ x(t ¯ + 1)) − M0 − 1.

(5.204)

t=t0 −2L 2 −2Q 0

It follows from Theorem 5.1, the choice of M∗ , (5.193) and (5.204) that t0 −L 2 −1−2Q 0

u(x(t), x(t + 1))

t=t0 −2L 2 −2Q 0 t0 

=

u(x(t), x(t + 1)) −

t=t0 −2L 2 −2Q 0 t0 

≥

t0 

u(x(t), ¯ x(t ¯ + 1)) − M0 − 1

t=t0 −2L 2 −2Q 0

≥

t0 −L 2 −2Q 0 −1

u(x(t), x(t + 1))

t=t0 −L 2 −2Q 0 t0 

u(x(t), ¯ x(t ¯ + 1)) − M∗

t=t0 −L 2 −2Q 0

u(x(t), ¯ x(t ¯ + 1)) − M1 .

(5.205)

t=t0 −2L 2 −2Q 0

Property (iii) and (5.205) imply that there exists an integer t1 ∈ {t0 − 2L 2 − 2Q 0 , . . . , t0 − L 2 − 2Q 0 − 1}

(5.206)

¯ 1 ), x(t1 + 1) − x(t ¯ 1 + 1) ≤ δ x(t1 ) − x(t

(5.207)

 u t1 (x(t1 ), x(t1 + 1)) ≥ Δ/4.

(5.208)

such that

and

Assume that k ≥ 1 is an integer and we defined a finite strictly decreasing sequence of nonnegative integers ti ∈ [T1 , T2 ], i = 0, . . . , k such that (5.199) and (5.206) are true, for al i = 0, . . . , k, ¯ i ), x(ti + 1) − x(t ¯ i + 1) ≤ δ x(ti ) − x(t

(5.209)

 u ti (x(ti ), x(ti + 1)) ≥ Δ/4

(5.210)

and

208

5 The Turnpike Phenomenon for Nonautonomous Problems

and for all i = 0, . . . , k − 1, ti+1 ∈ {ti − 2L 2 − 2Q 0 , . . . , ti − L 2 − 2Q 0 − 1}, tk ≥ L 1 + T.

(5.211) (5.212)

(Note that in view of (5.200), (5.201), (5.206)–(5.208), our assumption holds for k = 1.) There are two cases: tk ≥ T1 + 2L 2 + 2Q 0 + L 1 ;

(5.213)

tk < T1 + 2L 2 + 2Q 0 + L 1 .

(5.214)

If (5.214) is true, then the construction of the sequence ti , i = 0, . . . , k is completed. Assume that (5.213) is valid. Assumption (A3), (5.194) and (5.196) imply that tk 

u(x(t), x(t + 1))

t=tk −2L 2 −2Q 0

≥ U (x(tk − 2L 2 − 2Q 0 ), x(tk + 1), tk − 2L 2 − 2Q 0 , tk + 1) − δ.

(5.215)

By property (i), (5.192), (5.193), (5.209) and (5.210), U (x(tk − 2L 2 − 2Q 0 ), x(tk + 1), tk − 2L 2 − 2Q 0 , tk + 1) ≥

tk 

u(x(t), ¯ x(t ¯ + 1)) − M0 .

(5.216)

t=tk −2L 2 −2Q 0

Equations (5.215) and (5.216) imply that tk 

u(x(t), x(t + 1))

t=tk −2L 2 −2Q 0

≥

tk 

u(x(t), ¯ x(t ¯ + 1)) − M0 − 1.

t=tk −2L 2 −2Q 0

It follows from Theorem 5.1, the choice of M∗ , (5.193) and (5.217) that

(5.217)

5.9 Proof of Theorem 5.8 tk −L 2 −1−2Q 0

209

u(x(t), x(t + 1))

t=tk −2L 2 −2Q 0

=

tk  t=tk −2L 2 −2Q 0

≥

tk 

u(x(t), x(t + 1)) −

tk 

tk 

u(x(t), ¯ x(t ¯ + 1)) − M0 − 1 −

t=tk −2L 2 −2Q 0

≥

u(x(t), x(t + 1))

t=tk −L 2 −2Q 0

tk −L 2 −2Q 0 −1

u(x(t), ¯ x(t ¯ + 1)) − M∗

t=tk −L 2 −2Q 0

u(x(t), ¯ x(t ¯ + 1)) − M1 .

(5.218)

t=tk −2L 2 −2Q 0

Property (iii) and (5.218) imply that there exists an integer tk+1 ∈ {tk − 2L 2 − 2Q 0 , . . . , tk − L 2 − 2Q 0 − 1} such that ¯ k+1 ), x(tk+1 + 1) − x(t ¯ k+1 + 1) ≤ δ x(tk+1 ) − x(t and

 u tk+1 (x(tk+1 ), x(tk+1 + 1)) ≥ Δ/4.

Clearly, the assumption made for k also holds for k + 1. Therefore by induction we constructed a finite strictly decreasing sequences of integers q

{ti }i=0 ⊂ [T1 , T2 ] such that (5.199), (5.206) hold, T1 + L 1 ≤ tq < T1 + 2L 2 + 2Q 0 + L 1 , for all i = 0, . . . , q, equations (5.209), (5.210) are true and for all i = 0, . . . , q − 1, (5.211) holds. Property (ii), (5.194), (5.196) and (5.209)–(5.211) imply that for all integers i = 0, . . . , q − 1, x(t) − x(t) ¯ ≤ , t = ti+1 , . . . , ti . This completes the proof of Theorem 5.8.

210

5 The Turnpike Phenomenon for Nonautonomous Problems

5.10 Stability of the Turnpike Phenomenon Let ψ : C → R 1 , where the set C is nonempty. Set ψ = sup{|ψ(z)| : z ∈ C}. Let T1 , T2 be integers such that 0 ≤ T1 < T2 and let vt : graph(at ) → R 1 , t = T1 , . . . , T2 be bounded functions. For each pair of points z 0 , z 1 ∈ K define T 2 −1 T2 −1 T2 z , z ) = sup v(x(t), x(t + 1)) : {x(t)}t=T U ({vt }t=T 1 1 0 1 t=T1



is a program such that x(T1 ) = z 0 , x(T2 ) ≥ z 1 , T 2 −1 T2 −1 T2 U ({vt }t=T z ) = sup vt (x(t), x(t + 1)) : {x(t)}t=T 0 1 1 t=T1

 is a program such that x(T1 ) = z 0 ,

T2 −1 U ({vt }t=T ) 1

= sup

T −1 2 

 vt (x(t), x(t + 1)) :

T2 {x(t)}t=T 1

is a program .

t=T1

Recall that the supremum over an empty set is −∞. We show that the turnpike phenomenon is stable under small perturbations of the objective functions. Theorem 5.10 Let TP hold, M0 > 0 and  > 0. Then there exist a natural number L and δ0 > 0 such that for each pair of integers T1 ≥ 0, T2 ≥ 2L + T1 , each finite sequence of functions vt : graph(at ) → R 1 , t = T1 , . . . , T2 − 1 which satisfy u t − vt  ≤ δ0 , t = T1 , . . . , T2 − 1 T2 satisfying and every program {x(t)}t=T 1 τ +L−1 

+L−1 vt (x(t), x(t + 1)) ≥ U ({vt }τt=τ , x(τ ), x(τ + L)) − δ0

t=τ

for all τ ∈ {T1 , . . . , T2 − L}, T 2 −1 T2 −L

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T ) − M0 2 −L

5.10 Stability of the Turnpike Phenomenon

211

there exist integers τ1 ∈ [T1 , T1 + L] and τ2 ∈ [T2 − L , T2 ] such that x(t) − x ∗ (t) ≤  holds for all integers t ∈ [τ1 , τ2 ]. Proof Theorem 5.10 follows easily from Theorem 5.8. Namely, let M = M0 + 1 and let a natural number L and δ ∈ (0, 1) be as guaranteed by Theorem 5.8. Set δ0 = δ(4L + 4)−1 . Now it is easy to see that the assertion of Theorem 5.8 is true. For every integer t ≥ 0, denote by Mt the set of all functions v : graph(at ) → R 1 such that the following property holds: ˜ such that if (x, y) ∈ graph(at ), x˜ ∈ K , x˜ ≥ x, then there exists y˜ ∈ at (x) y˜ ≥ y, v(x, ˜ y˜ ) ≥ v(x, y).

(5.219)

Theorem 5.11 Let M > 0 and  > 0. Then there exists a natural number Q and δ > 0 such that for each pair of integers T1 ≥ 0 and T2 ≥ Q + T1 , each finite sequence of functions vt ∈ Mt , t = T1 , . . . , T2 − 1

(5.220)

vt − u t  ≤ δ, t = T1 , . . . , T2 − 1

(5.221)

which satisfy

T2 satisfying and every program {x(t)}t=T 1 T 2 −1

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T , x(T1 )) − M 1

t=T1

the inequality Card({t ∈ {T1 , . . . , T2 } : x(t) − x ∗ (t) > }) ≤ Q holds.

212

5 The Turnpike Phenomenon for Nonautonomous Problems

5.11 Proof of Theorem 5.11 By Lemma 5.9, there exist γ0 ∈ (0, 1), M0 > 0 and a natural number L 0 such that the following property holds: (i) For each integer S1 ≥ 0, each integer S2 ≥ S1 + L 0 , each x ∈ K and each y1 , y2 ∈ K satisfying y2 ∈ a S2 −1 (y1 ), y1 − x(S ¯ 2 − 1) ≤ γ0 , y2 − x(S ¯ 2 ) ≤ γ0 and

 u S2 (y1 , y2 ) ≥ Δ/4

S2 such that there exists a program {ξ(t)}t=S 1

ξ(S1 ) = x, ξ(S2 ) ≥ y2 and

S 2 −1

u t (ξ(t), ξ(t + 1)) ≥

t=S1

S 2 −1

u t (x(t), ¯ xi(t ¯ + 1)) − M0 .

t=S1

By Lemma 5.7, there exist δ0 ∈ (0, min{γ0 , })

(5.223)

and a natural number L 1 such that the following property holds: S2 (ii) For each integer S1 ≥ L 1 , each integer S2 > S1 and each program {ξ(t)}t=S 1 which satisfies ξ(τ ) − x(τ ¯ ) ≤ δ1 for every τ ∈ {S1 , S1 + 1, S2 − 1, S2 },  u S1 (ξ(S1 ), ξ(S1 + 1)) ≥ Δ/4,  u S2 −1 (ξ(S2 − 1), ξ(S2 )) ≥ Δ/4 and

S 2 −1

u t (ξ(t), ξ(t + 1)) ≥ U (ξ(S1 ), ξ(S2 ), S1 , S2 ) − δ0

t=S1

the following inequality holds: ξ(t) − x(t) ¯ ≤ , t = S1 , . . . , S2 .

5.11 Proof of Theorem 5.11

213

Set M1 > 4M∗ + M + M0 + 2.

(5.224)

Property (P2) implies that there exists a natural number L 2 ≥ 2 such that the following property holds: S+L 2 which satisfies (iii) For each integer S ≥ 0 and each program {ξ(t)}t=S S+L 2 −1 

u t (ξ(t), ξ(t + 1)) ≥

t=S

S+L−1 

u t (x(t), ¯ x(t ¯ + 1)) − M1

t=S

there exists an integer τ ∈ {S, . . . , S + L 2 − 1} such that ξ(τ ) − x(τ ¯ ), ξ(τ + 1) − x(τ ¯ + 1) ≤ δ0 and

 u τ (ξ(τ ), ξ(τ + 1)) ≥ Δ/4.

Choose δ ∈ (0, 8−1 δ0 (8(L 1 + L 0 + L 2 + 2)−1 )

(5.225)

Q > 4L 0 + 4L 1 + 4L 2 + 80(8 + 4Mδ0−1 )(L 0 + L 2 ).

(5.226)

and a natural number

Assume that T1 ≥ 0, T2 ≥ Q + T1 ,

(5.227)

vt ∈ Mt , t = T1 , . . . , T2 − 1, vt − u t  ≤ δ, t = T1 , . . . , T2 − 1

(5.228) (5.229)

T2 and a program {x(t)}t=T satisfies 1 T 2 −1

T2 −1 vt (x(t), x(t + 1)) ≥ U ({vt }t=T , x(T1 )) − M. 1

(5.230)

t=T1

Property (iii) implies that there exists an integer τ0 ∈ {T2 − L 2 , . . . , T2 − 1}

(5.231)

214

5 The Turnpike Phenomenon for Nonautonomous Problems

such that  ¯ 0 ), x(τ ¯ 0 + 1)) ≥ Δ/4. u τ0 (x(τ

(5.232)

By property (i) applied with S2 = τ0 + 1, S1 = τ0 − L 0 − L 2 , 0 +1 such (5.226), (5.227), (5.231) and (5.232), there exists a program {ξ0 (t)}τt=τ 0 −L 0 −L 2 that

¯ 0 + 1), ξ0 (τ0 + 1) ≥ x(τ ξ0 (τ0 − L 0 − 2) = x(τ0 − L 0 − 2),

(5.233) (5.234)

and τ0 

τ0 

u t (ξ0 (t), ξ0 (t + 1)) ≥

t=τ0 −L 0 −2

u t (x(t), ¯ x(t ¯ + 1)) − M0 . (5.235)

t=τ0 −L 0 −2

Assumption (A3) and (5.233) imply that there exist ξ0 (t) ∈ K , t = τ0 + 2, . . . , T2 + 1 T2 +1 is a program, such that {ξ0 (t)}t=τ 0 −L 0 −L 2

¯ for all integers t satisfying τ0 + 2 ≤ t ≤ T2 + 1, ξ0 (t) ≥ x(t) u t (ξ0 (t), ξ0 (t + 1)) ≥ u t (x(t), ¯ x(t ¯ + 1)), t = τ0 + 1, . . . , T2 . By (5.355) and the relation above, T 2 −1

u t (ξ0 (t), ξ0 (t + 1)) ≥

t=τ0 −L 0 −L 2

T 2 −1

u t (x(t), ¯ x(t ¯ + 1)) − M0 . (5.236)

t=τ0 −L 0 −L 2

It follows from (5.225), (5.229), (5.231) and (5.236) that T 2 −1 t=τ0 −L 0 −L 2

vt (ξ0 (t), ξ0 (t + 1)) ≥

T 2 −1 t=τ0 −L 0 −L 2

Equations (5.230), (5.234) and (5.237) imply that

vt (x(t), ¯ x(t)) ¯ − M0 − 1. (5.237)

5.11 Proof of Theorem 5.11 T 2 −1

215

t=τ0 −L 0 −L 2 T 2 −1

≥

T 2 −1

vt (x(t), x(t + 1)) ≥

vt (ξ0 (t), ξ0 (t)) − M

t=τ0 −L 0 −L 2

vt (x(t), ¯ x(t)) ¯ − M − M0 − 1.

(5.238)

t=τ0 −L 0 −L 2

In view of (5.225), (5.229) and (5.238), T 2 −1

u t (x(t), x(t + 1)) ≥

t=τ0 −L 0 −L 2

T 2 −1

u t (x(t), ¯ x(t)) ¯ − M − M0 − 2.

t=τ0 −L 0 −L 2

(5.239) Theorem 5.1, the choice of M∗ and (5.239) imply that T 2 −1

u t (x(t), x(t + 1))

t=T2 −L 2 T 2 −1

=

u t (x(t), x(t + 1)) −

t=τ0 −L 0 −L 2 T 2 −1

≥

T 2 −1

u t (x(t), x(t + 1))

t=τ0 −L 0 −L 2 T2 −L 2 −1

u t (x(t), ¯ x(t ¯ + 1)) − M − M0 − 2 −

t=τ0 −L 0 −L 2

=

T2 −L 2 −1

u t (x(t), ¯ x(t ¯ + 1)) − M∗

t=τ0 −L 0 −L 2

u t (x(t), x(t + 1)) − M∗ − M − M0 − 2.

(5.240)

t=T2 −L 2

By property (iii), (5.224) and (5.240) there exists an integer t0 ∈ {T2 − L 2 , . . . , T2 − 1}

(5.241)

¯ 0 ), x(t0 + 1) − x(t ¯ 0 + 1) ≤ δ0 x(t0 ) − x(t

(5.242)

 u t0 (x(t0 ), x(t0 + 1)) ≥ Δ/4.

(5.243)

such that

and

Assume that k ≥ 0 is an integer and we defined a strictly decreasing sequence of integers ti ∈ [T1 , T2 ], i = 0, . . . , k such that (5.341) is true, for every integer i ∈ {0, . . . , k}, ¯ i ), x(ti + 1) − x(t ¯ i + 1) ≤ δ0 x(ti ) − x(t

(5.244)

216

5 The Turnpike Phenomenon for Nonautonomous Problems

and  u ti (x(ti ), x(ti + 1)) ≥ Δ/4,

(5.245)

if an integer i satisfies 0 ≤ i < k, then 0 < ti − ti+1 ≤ L 0 + L 2

(5.246)

tk ≥ T1 + L 1 .

(5.247)

and

(Note that in view of (5.242) and (5.243), our assumption holds for k = 0.) If tk < L 0 + L 1 + L 2 , then the construction is completed. Assume that tk ≥ L 0 + L 1 + L 2 .

(5.248)

By property (i) applied with S2 = tk + 1, S1 = tk − L 0 − L 2 , y1 = x(tk ), y2 = x(tk + 1), k +1 such (5.223), (5.244), (5.245) and (5.248), there exists a program {ξk (t)}tt=t k −L 0 −L 2 that

ξk (tk − L 0 − L 2 ) = x(tk − L 0 − L 2 ),

(5.249)

ξk (tk + 1) ≥ x(tk + 1),

(5.250)

and tk 

tk 

u t (ξk (t), ξk (t + 1)) ≥

t=tk −L 0 −L 2

u t (x(t), ¯ xi(t ¯ + 1)) − M0 .

(5.251)

t=tk −L 0 −L 2

In view of (5.225), (5.229) and (5.251), tk  t=tk −L 0 −L 2

vt (ξk (t), ξk (t + 1)) ≥

tk 

vt (x(t), ¯ x(t ¯ + 1)) − M0 − 1.

t=tk −L 0 −L 2

(5.252)

5.11 Proof of Theorem 5.11

217

Equations (5.228) and (5.250) imply that there exist ξk (t) ∈ K

(5.253)

T2 is a program, for all integers t satisfying tk + 2 ≤ t ≤ T2 such that {ξk (t)}t=T 1

ξk (t) ≥ x(t) for all integers t ∈ [tk + 1, T2 ],

(5.254)

vt (ξk (t), ξk (t + 1)) ≥ vt (x(t), x(t + 1))

(5.255)

for all integers t ∈ [tk + 1, T2 − 1]. It follows from (5.230), (5.252), (5.253) and (5.255) that M≥

T 2 −1

vt (ξk (t), ξk (t + 1)) −

t=T1

vt (x(t), x(t + 1))

t=T1

tk 

≥

T 2 −1

vt (ξk (t), ξk (t + 1)) −

t=tk −L 0 −L 2 tk 

≥

tk 

vt (x(t), x(t + 1))

t=tk −L 0 −L 2 tk 

vt (x(t), ¯ x(t ¯ + 1)) −

t=tk −L 0 −L 2

vt (x(t), x(t + 1)) − M0 − 1

t=tk −L 0 −L 2

and in view of (5.225) and (5.229), M + M0 + 2 ≥

tk 

tk 

u t (x(t), ¯ x(t ¯ + 1)) −

t=tk −L 0 −L 2

u t (x(t), x(t + 1)).

t=tk −L 0 −L 2

(5.256) Theorem 5.1, the choice of M∗ and (5.256) imply that tk −L 0 −1 

u t (x(t), x(t + 1))

t=tk −L 0 −L 2

=

tk  t=tk −L 0 −L 2

≥

tk 

u t (x(t), x(t + 1)) −

tk 

u t (x(t), ¯ x(t ¯ + 1)) − M − M0 − 2 −

t=tk −L 0 −L 2

=

tk −L 0 −1 

u t (x(t), x(t + 1))

t=tk −L 0 tk 

u t (x(t), ¯ x(t ¯ + 1)) − M∗

t=tk −L 0

u t (x(t), ¯ x(t ¯ + 1)) − M1 .

t=tk −L 0 −L 2

Property (iii) and (5.257) imply that there exists an integer

(5.257)

218

5 The Turnpike Phenomenon for Nonautonomous Problems

tk+1 ∈ {tk − L 2 − L 0 , . . . , tk − L 0 − 1} such that ¯ k+1 ), x(tk+1 + 1) − x(t ¯ k+1 + 1) ≤ δ0 x(tk+1 ) − x(t and

 u tk+1 (x(tk+1 ), x(tk+1 + 1)) ≥ Δ/4.

It is easy to see that the assumption made for k also holds for k + 1. Therefore by induction we constructed a strictly decreasing sequence of integers ti ∈ [T1 , T2 ], i = 0, . . . , q such that t0 ∈ [T2 − L 2 , T2 − 1],

(5.258)

for every integer i ∈ {0, . . . , q}, ¯ i ), x(ti + 1) − x(t ¯ i + 1) ≤ δ0 x(ti ) − x(t

(5.259)

 u ti (x(ti ), x(ti + 1)) ≥ Δ/4,

(5.260)

and

and for all integers i = 0, . . . , q − 1, 0 < ti − ti+1 ≤ L 0 + L 2

(5.261)

L 1 ≤ tq − T1 ≥ L 0 + L 1 + L 2 .

(5.262)

and

Now we define a strictly increasing finite sequence {Si : i = 0, . . . , p} ⊂ {t0 , . . . , tq }. Set S0 = tq .

(5.263)

Clearly, S0 < t0 . Assume that p ≥ 0 is an integer and that we have defined a strictly increasing finite sequence p {Si }i=0 ⊂ {t0 , . . . , tq }

5.11 Proof of Theorem 5.11

219

such that (5.263) holds; if an integer j satisfies 0≤ j < p then S j+1 −1



S

−1

j+1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(S j ), x(S j+1 )) − δ0 /2 j

(5.264)

t=S j

and if an integer S ∈ {t0 , . . . , tq } satisfies S j < S < S j+1 , then S 

S vt (x(t), x(t + 1)) ≥ U ({vt }t=S , x(S j ), x(S + 1)) − δ/2. j

t=S j

If S p = t0 , then our construction is completed. Assume that S p < t0 . If

t0 

0 vt (x(t), x(t + 1)) ≥ U ({vt }tt=S , x(S p ), x(t0 + 1)) − δ/2, p

t=S p

then we set S p+1 = t0 and the construction is completed. Assume that t0 

0 vt (x(t), x(t + 1)) < U ({vt }tt=S , x(S p ), x(t0 + 1)) − δ0 /2. p

t=S p

It is not difficult to see that there exists an integer S p+1 ∈ {t0 , . . . , tq } such that

(5.265)

220

5 The Turnpike Phenomenon for Nonautonomous Problems

S p < S p+1 and

S p+1 

S

p+1 vt (x(t), x(t + 1)) < U ({vt }t=S , x(S p ), x(S p+1 + 1)) − δ0 /2 p

t=S p

and if an integer S ∈ {t0 , . . . , tq } satisfies S p < S < S p+1 , then

S 

S vt (x(t), x(t + 1)) ≥ U ({vt }t=S , (x(S p ), x(S + 1)) − δ0 /2. p

t=S p

Thus the assumption made for p holds for p + 1. By induction we have constructed a strictly increasing sequence of integers m ⊂ {t0 , . . . , tq } {Si }i=0

such that S0 = tq , Sm = t0 , for each integer j satisfying 0 ≤ j < m − 1 (5.364) holds, and for each j ∈ {0, . . . , m − 1} and each S ∈ {t0 , . . . , tq } satisfying S j < S < S j+1 , (5.365) is true. In view of (5.264), for each integer p satisfying 0 ≤ p < m − 1, S

+1

p+1 there exists a program {z p (t)}t=S p

such that

(5.266)

5.11 Proof of Theorem 5.11

221

z p (S p ) = x(S p ), z p (S p+1 + 1) ≥ x(S p+1 + 1), S p+1 

vt (x(t), x(t + 1))
k2 .

(5.281)

By (5.280) and (5.281), tk2 

tk

2 vt (x(t), x(t + 1)) ≥ U ({vt }t=t , x(tk1 ), x(tk2 + 1)) − δ0 /2. k1

(5.282)

t=tk1

Let j ∈ {k2 + 1, . . . , k1 }. In view of (5.260), (5.279) and (5.283),

(5.283)

5.11 Proof of Theorem 5.11

223

x(t j ) − x(t ¯ j ) ≤ δ0 , x(t j + 1) − x(t ¯ j + 1) ≤ δ0 , x(t j−1 ) − x(t ¯ j−1 ) ≤ δ0 , x(t j−1 + 1) − x(t ¯ j−1 + 1) ≤ δ0 ,  u t j (x(t j ), x(t j + 1)) ≥ Δ/4,

(5.284)

 u t j−1 (x(t j−1 ), x(t j−1 + 1)) ≥ Δ/4.

(5.285)

Equations (5.228), (5.282) and (5.283) imply that t j−1 

t

j−1 vt (x(t), x(t + 1)) ≥ U ({vt }t=t , x(t j ), x(t j−1 + 1)) − δ0 /2. j

(5.286)

t=t j

By (5.225), (5.229), (5.261) and (5.286), t j−1 

t

j−1 u(x(t), x(t + 1)) ≥ U ({u t }t=t , x(t j ), x(t j−1 + 1)) − δ0 /2. j

(5.287)

t=t j

Property (ii), (5.259), (5.260), (5.262) and (5.287) imply that x(t) − x ∗  ≤ , t = t j , . . . , t j−1 . Since the relation above holds for every integer j satisfying (5.283) it follows from (5.279) and (5.281) that x(t) − x(t) ¯ ≤ , t = Si , . . . , Si+1 − L 0 − L 2 . This implies that ¯ > } {t ∈ {T1 , . . . , T2 } : x(t) − x(t) ⊂ {T1 , . . . , tq } ∪ {t0 , . . . , T2 } ∪ {{Si , . . . , Si+1 } : i ∈ {0, . . . , m − 1}, Si+1 − Si < 8L 0 + 8L 2 } ∪ {Si+1 − L 0 − L 2 , . . . , Si+1 } : i ∈ E}. Combined with (5.258), (5.262) and (5.276) this implies that ¯ > }) Card({{t ∈ {T1 , . . . , T2 } : x(t) − x(t) ≤ 4(L 0 + L 1 + L 2 ) + (10L 0 + 10L 2 )m ≤ 4(L 0 + L 1 + L 2 ) + 10(4Mδ0−1 + 8)(L 0 + L 2 ) < Q. Theorem 5.11 is proved.

Chapter 6

Generic Turnpike Results for the One-Dimensional RSS Model

Abstract In this chapter we continue to study the class of one-dimensional nonautonomous discrete–time optimal problems discussed in Sects. 2.3 and 5.3 which determine the one-dimensional Robinson–Solow–Srinivasan model. This class of problems is identified with a complete space of sequences. It is shown that there exists a set in this space which is a countable intersection of open and everywhere dense sets such that for every element the corresponding model has the turnpike property. All the main results of this chapter are new.

6.1 Preliminaries In this chapter we consider the one-dimensional RSS model introduced in studied in Sects. 2.3 and 5.3. For each integer t ≥ 0 let αt > 0, dt ∈ (0, 1].

(6.1)

1 set Let t ≥ 0 be an integer. For each x ∈ R+ 1 : y ≥ (1 − dt )x and αt (y − (1 − dt )x) ≤ 1}. at (x) = {y ∈ R+

(6.2)

1 , It is clear that for each x ∈ R+

at (x) = [(1 − dt )x, αt−1 + (1 − dt )x]

(6.3)

1 1 × R+ . Suppose that and that graph (at ) is a closed subset of R+

d ∗ := sup{dt : t = 0, 1, . . . } < 1, inf{dt : t = 0, 1, . . . } > 0, inf{αt : t = 0, 1, . . . } > 0,

(6.4) (6.5) (6.6)

sup{αt : t = 0, 1, . . . } < ∞.

(6.7)

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. J. Zaslavski, Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model, Monographs in Mathematical Economics 4, https://doi.org/10.1007/978-981-16-2252-6_6

225

226

6 Generic Turnpike Results for the One-Dimensional RSS Model

Choose α∗ , α∗ , d∗ > 0 such that α∗ < αt < α∗ , d∗ < dt for all integers t ≥ 0. Denote by A the set of all sequences of functions {wt }∞ t=0 such that for every integer t ≥ 0, wt : [0, ∞) → [0, ∞) is an increasing and continuous function such that (6.8) wt (0) = 0 and the following assumptions hold. (A1) For each  > 0 there exists δ > 0 such that wt (δ) ≤  for each integer t ≥ 0 and sup{wt (M) : t = 0, 1, . . . } < ∞ for each M > 0. (A2) For each M > 0 and each  > 0 there is δ > 0 such that for each integer t ≥ 0 and each x, y ∈ [0, M] satisfying |x − y| ≤ δ, the inequality |wt (x) − wt (y)| ≤  is true. Denote by A0 the set of all sequences of functions {wt }∞ t=0 ∈ A such that for every integer t ≥ 0, wt is a strictly increasing function, inf{wt (z) : t = 0, 1, . . . } > 0 for each z > 0

(6.9)

and the following property holds. (A3) For each integer k ≥ 1, there is  > 0 such that for each z 1 , z 2 ∈ [0, k] satisfying z 2 ≥ z 1 + 1/k, we have for all integers t ≥ 0, wt (z 2 ) − wt (z 1 ) ≥ . Let {wt }∞ t=0 ∈ A0 . Then all the assumptions posed in Sects. 2.3 and 5.3 hold and all the results of Chap. 5 are true for an arbitrary M0 > (α∗ d∗ )−1 and 1 : x ≤ M0 }. K = {x ∈ R+

The set A is equipped with the uniformity determined by the base ∞ E(N , ) = {({wt }∞ t=0 , {vt }t=0 ) ∈ A × A : |wt (z) − vt (z)| ≤  for all z ∈ [0, N ] and all integers t ≥ 0},

where N ,  > 0.

(6.10)

6.1 Preliminaries

227

It is not difficult to see that the uniform space A is metrizable (by a metric dA ) and complete. We prove the following result. Theorem 6.1 A0 contains a subset of A which is a countable intersection of open everywhere dense sets in (A, dA ). Proof Let {wt }∞ t=0 ∈ A and γ ∈ (0, 1). For every t ∈ {0, 1, . . . , } and every z ≥ 0 define (γ) (6.11) wt (z) = wt (z) + γz. Clearly,

(γ)

{wt }∞ t=0 ∈ A0 and

(γ)

∞ lim {wt }∞ t=0 = {wt }t=0

γ→0+

in (A, dA ). Let {wt }∞ t=0 ∈ A, γ ∈ (0, 1) and k ≥ 1 be an integer. In view of (6.11), for every integer t ≥ 0, (6.12) wγ (k −1 ) ≥ k −1 γ. By (6.11), for each integer t ≥ 0 and each pair of numbers z 1 , z 2 ∈ [0, k] satisfying z 2 − z 1 ≥ k −1 we have

(γ)

(γ)

wt (z 2 ) − wt (z 1 ) = γ(z 2 − z 1 ) ≥ γk −1 .

(6.13)

It follows from (6.10), (6.12) and (6.13) that there exists an open neighborhood U({wt }∞ t=0 , γ, k) (γ)

of {wt }∞ t=0 in A such that the following properties hold: ∞ (i) for each {vt }∞ t=0 ∈ U({wt }t=0 , γ, k) and each integer t ≥ 0, vt (k −1 ) ≥ (2k)−1 γ; (ii) for each integer t ≥ 0 and each pair of numbers z 1 , z 2 ∈ [0, k] satisfying z 2 − z 1 ≥ k −1 , we have vt (z 2 ) − vt (z 1 ) ≥ γ(2k)−1 . Define F=

∞   ∞ {U({wt }∞ t=0 , γ, k) : {wt }t=0 ∈ A, γ ∈ (0, 1)}. k=1

228

6 Generic Turnpike Results for the One-Dimensional RSS Model

Clearly, F is a countable intersection of open everywhere dense sets in (A, dA ). It is not difficult to see that F ⊂ A0 . Theorem 6.1 is proved. Let {wt }∞ t=0 ∈ A. For each integer t ≥ 0 and each (x, x ) ∈ graph(at ) set

u t (x, x ) = sup{wt (y) : 0 ≤ y ≤ x and y + αt (x − (1 − dt )x) ≤ 1}.

(6.14)

Clearly, for each integer t ≥ 0 and each (x, x ) ∈ graph(at ), u t (x, x ) = wt (min{x, 1 − αt (x − (1 − dt )x)}).

(6.15)

For each integer t ≥ 0 and each x, x ≥ 0 set gt (x, x ) = min{x, 1 − αt (x − (1 − dt )x)}.

(6.16)

Then for each integer t ≥ 0 and each (x, x ) ∈ graph(at ), u t (x, x ) = wt (gt (x, x )).

(6.17)

Clearly, for each integer t ≥ 0 the function u t : graph(at ) → [0, ∞) is continuous. T2 1 ⊂ R+ is called a program Let 0 ≤ T1 < T2 be integers. A sequence {x(t)}t=T 1 if x(t + 1) ∈ at (x(t)) for all integers t ∈ [T − 1, T2 − 1]. A sequence {x(t)}∞ t=T1 ⊂ R+ 1 is called a program if x(t + 1) ∈ at (x(t)) for all integers t ≥ T1 . In the sequel we assume that the supremum over an empty set is −∞. 1 and let 0 ≤ T1 < T2 be integers. Set Let x0 , x˜0 ∈ R+ T2 −1 U ({wt }t=T , x0 ) 1

= sup

T 2 −1

T2 wt (gt (x(t), x(t + 1))) : {x(t)}t=T 1

t=T1

 is a program such that x(T1 ) = x0 , T2 −1 U ({wt }t=T , x0 , x˜0 ) 1

= sup

T 2 −1

(6.18)

T2 wt (gt (x(t), x(t + 1))) : {x(t)}t=T 1

t=T1

 is a program such that x(T1 ) = x0 , x(T2 ) ≥ x˜0 .

(6.19)

Let T1 , T2 be nonnegative integers such that T1 < T2 and M > 0. Set T2 −1 M ({wt }t=T ) = sup U 1

T 2 −1

wt (gt (x(t), x(t + 1))) :

t=T1

 T2 {x(t)}t=T is a program and x(T ) ≤ M . 1 1

(6.20)

6.1 Preliminaries

229

Continuity of the function wt ◦ gt , t = 0, 1, . . . , compactness of sets of admissible programs and the optimization theorem of Weierstrass imply the following results. 1 Proposition 6.2 For each x0 ∈ R+ and each pair of integers 0 ≤ T1 < T2 there T2 exists a program {x(t)}t=T1 such that x(T1 ) = x0 and T 2 −1

T2 −1 wt (gt (x(t), x(t + 1))) = U ({wt }t=T , x0 ). 1

t=T1

Proposition 6.3 For each pair of integers 0 ≤ T1 < T2 and each M > 0 there exists T2 such that a program {x(t)}t=T 1 x(0) ≤ M, T 2 −1

T2 −1 M ({wt }t=T wt (gt (x(t), x(t + 1))) = U ). 1

t=T1

Fix

M∗ > (α∗ d∗ )−1 + 1.

It is clear that if

(6.21)

{wt }∞ t=0 ∈ A0 ,

then the model considered here is a particular case of the model discussed in Chap. 5 with n = 1 (see Sects. 5.3 and 2.3). Therefore all the results of Chap. 2 can be applied. Assume that {wt }∞ t=0 ∈ A0 . Theorem 2.4 implies the following result. Theorem 6.4 There exists M¯ > 0 such that for each x0 ∈ [0, M∗ ] there exists a ∞ ¯ = x0 , for each pair of integers T1 , T2 ≥ 0 satisfying program {x(t)} ¯ t=0 such that x(0) T1 < T2 , T −1  2     T −1 2 M∗ ({wt }t=T wt (gt (x(t), ¯ x(t ¯ + 1))) − U )   ≤ M¯ 1   t=T1

and that for each integer T > 0, T −1  t=0

T −1 wt (gt (x(t), ¯ x(t ¯ + 1))) = U ({wt }t=0 , x(0), ¯ x(T ¯ )).

230

6 Generic Turnpike Results for the One-Dimensional RSS Model

∞ Proposition 6.5 Let x0 ∈ [0, M∗ ] and let a program {x(t)} ¯ t=0 be as guaranteed by Theorem 6.4. Assume that S ≥ 0 is an integer and that {x(t)}∞ t=S is a program. Then either the sequence

T −1 

wt (gt (x(t), x(t + 1))) −

t=S

T −1 

∞ wt (gt (x(t), ¯ x(t ¯ + 1))) T =S+1

t=S

is bounded or T −1 

wt (gt (x(t), x(t + 1))) −

t=S

T −1 

wt (gt (x(t), ¯ x(t ¯ + 1))) → −∞ as T → ∞.

t=S

Proof We may assume that x(S) ≤ M∗ .

(6.22)

Assume that the sequence T −1 

wt (gt (x(t), x(t + 1))) −

t=S

T −1 

∞ wt (gt (x(t), ¯ x(t ¯ + 1))) T =S+1

t=S

is not bounded. Let Q > 0. Then there exists a natural number T0 > S which satisfies T 0 −1

wt (gt (x(t), x(t + 1))) −

t=S

T 0 −1

¯ wt (gt (x(t), ¯ x(t ¯ + 1))) < −Q − M.

(6.23)

t=S

In view of (6.23) and Theorem 6.4, for every natural number T > T0 , we have T −1 

wt (gt (x(t), x(t + 1))) −

t=S

=

T 0 −1

T −1 

wt (gt (x(t), ¯ x(t ¯ + 1)))

t=S

wt (gt (x(t), x(t + 1)))

t=S

−

T 0 −1

wt (gt (x(t), ¯ x(t ¯ + 1))) +

t=S

−

T −1 

T −1 

wt (gt (x(t), x(t + 1)))

t=T0

wt (gt (x(t), ¯ x(t ¯ + 1)))

t=T0

< −Q − M¯ + M¯ = Q. Since Q is an arbitrary positive number this completes the proof of Proposition 6.5.

6.2 Main Results

231

6.2 Main Results Let M¯ > 0 be as guaranteed by Theorem 6.4. Theorem 6.4 implies that there exists ∞ a program {x(t)} ¯ t=0 such that (6.24) x(0) ¯ ≤ M∗ , for each integer T > 0, T −1 

T −1 wt (gt (x(t), ¯ x(t ¯ + 1))) = U ({wt }t=0 , x(0), ¯ x(T ¯ ))

(6.25)

t=0

and for each pair of integers T1 , T2 ≥ 0 satisfying T1 < T2 , T −1  2    T −1 2 ¯ M∗ ({wt }t=T ) ≤ M. wt (gt (x(t), ¯ x(t ¯ + 1))) − U  1  

(6.26)

t=T1

For each integer t ≥ 0, set y¯ (t) = min{x(t), ¯ 1 − αt (x(t ¯ + 1) − (1 − dt )x(t))}. ¯

(6.27)

Let γ ∈ (0, 1). For every integer t ≥ 0 set (γ)

wt (y) = wt (y) + γ y, y ∈ [0, y¯ (t)],

(6.28)

(γ) wt (y)

(6.29)

= γwt ( y¯ (t)) + γ y¯ (t) + (1 − γ)wt (y)

for every y > y¯ (t). Clearly,

(γ)

{wt }∞ t=0 ∈ A0 .

(6.30)

In Sect. 6.5 we prove the following turnpike result. Theorem 6.6 Let M,  > 0. Then there exist a natural number L and δ > 0 such 2 that for each pair of integers τ1 ≥ 0, τ2 > 2L + τ1 and each program {x(t)}τt=τ 1 satisfying x(τ1 ) ≤ M∗ , τ 2 −1

(γ)

(γ)

2 −1 wt (gt (x(t), x(t + 1))) ≥ U ({wt }τt=τ , x(τ1 ), x(τ2 )) − δ, 1

t=τ1 τ 2 −1 t=τ1

(γ)

wt (gt (x(t), x(t + 1))) ≥

τ 2 −1 t=τ1

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − M

232

6 Generic Turnpike Results for the One-Dimensional RSS Model

the inequality |x(t) − x(t)| ¯ ≤ holds for all integers t ∈ [τ1 + L , τ2 − L]. It is not difficult to see that Theorem 6.6 is true for every M∗ > (α∗ d∗ )−1 + 1. Therefore Theorem 6.6 implies the following result. Theorem 6.7 Let M,  > 0 and M0 > (α∗ d∗ )−1 + 1. Then there exist a natural number L and δ > 0 such that for each pair of integers 2 satisfying τ1 ≥ 0, τ2 > 2L + τ1 and each program {x(t)}τt=τ 1 x(τ1 ) ≤ M0 , τ 2 −1

(γ)

wt (gt (x(t), x(t + 1)))

t=τ1 (γ)

2 −1 , x(τ1 ), x(τ2 ))) − δ, ≥ max{U ({wt }τt=τ 1

τ 2 −1

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − M}

t=τ1

the inequality |x(t) − x(t)| ¯ ≤ holds for all integers t ∈ [τ1 + L , τ2 − L]. v By Theorem 6.4, for each v = {vt }∞ t=0 ∈ A0 there exist M > 0 and a program ∞ (v) {x (t)}t=0 such that x (v) (0) ≤ 1 + (α∗ d∗ )−1 , (6.31)

for each integer T > 0, T −1 

T −1 vt (gt (x (v) (t), x (v) (t + 1))) = U ({vt }t=0 , x (v) (0), x (v) (T ))

(6.32)

t=0

and for each pair of integers T1 , T2 ≥ 0 satisfying T1 < T2 , T 2 −1 t=T1

T2 −1 vt (gt (x (v) (t), x (v) (t + 1))) ≥ U ({vt }t=T , x (v) (T1 )) − M v . 1

(6.33)

6.2 Main Results

233

Define for all integers t ≥ 0, y (v) (t) = min{x (v) (t), 1 − αt (x (v) (t + 1) − (1 − dt )x (v) (t))}.

(6.34)

For each v = {vt }∞ t=0 ∈ A0 , each γ ∈ (0, 1), each integer t ≥ 0 and each y ≥ 0 set (γ)

and

vt (y) = vt (y) + γ y, y ∈ [0, y (v) (t)]

(6.35)

vt (y) = γvt (y (v) (t)) + γ y (v) (t) + (1 − γ)vt (y).

(6.36)

(γ)

for every y > y (v) (t). Define (γ) ∞ B = {{wt }∞ t=0 : {wt }t=0 ∈ A0 , γ ∈ (0, 1)}. Clearly, B is an everywhere dense subset of (A, dA ). Theorem 6.7, (6.28), (6.29), (6.31) and (6.35) imply the following result. Theorem 6.8 Let v = {vt }∞ t=0 ∈ B. Then there exist Mv > 0 and a program {x (v) (t)}∞ t=0 such that x (v) (0) ≤ 1 + (α∗ d∗ )−1 , for each integer T > 0, T −1 

T −1 vt (gt (x (v) (t), x (v) (t + 1))) = U ({vt }t=0 , x (v) (0), x (v) (T )),

t=0

for each pair of integers T1 , T2 ≥ 0 satisfying T1 < T2 , T 2 −1

T2 −1 vt (gt (x (v) (t), x (v) (t + 1))) ≥ U ({vt }t=T , x (v) (T1 )) − Mv 1

t=T1

and that the following assertion holds. For each M0 ,  > 0 and each M1 > (α∗ d∗ )−1 + 1, there exist a natural number L and δ > 0 such that for each pair of integers τ1 ≥ 0, 2 satisfying τ2 > 2L + τ1 and each program {x(t)}τt=τ 1

234

6 Generic Turnpike Results for the One-Dimensional RSS Model

x(τ1 ) ≤ M1 , τ 2 −1

vt (gt (x(t), x(t + 1)))

t=τ1 2 −1 , x(τ1 ), x(τ2 )) − δ, ≥ max{U ({vt }τt=τ 1

τ 2 −1

vt (gt (x (v) (t), x (v) (t + 1))) − M0 }

t=τ1

the inequality

|x(t) − x (v) (t)| ≤ 

holds for all integers t ∈ [τ1 + L , τ2 − L]. Let

M1 > (α∗ d∗ )−1 + 1, K = [0, M1 ].

Theorem 6.8 means that for every v = {vt }∞ t=0 ∈ B the sequence of functions {vt ◦ has TP, (P1) and (P2). Theorems 5.10 and 6.8 imply the following result. gt }∞ t=0 Proposition 6.9 Let v = {vt }∞ t=0 ∈ B, , M1 > 0, M0 > 1 + (α∗ d∗ )−1 and let a program {x (v) (t)}∞ t=0 satisfy all the conditions of Theorem 6.8. Then there exist δ > 0, a natural number L and an open neighborhood U of v in A such that for each u = {u t }∞ t=0 ∈ U, each pair of integers T1 ≥ 0, T2 > 2L + T1 and each program T2 {x(t)}t=T1 satisfying x(T1 ) ≤ M0 , T 2 −1

u t (gt (x(t), x(t + 1)))

t=T1 T2 −1 ≥ U ({u t }t=T , x(T1 ), x(T2 )) − δ, 1 T 2 −1

T2 −1 u t (gt (x(t), x(t + 1))) ≥ U ({u t }t=T , x(T2 − L)) − M1 , 2 −L

t=T2 −L

the inequality

|x(t) − x (v) (t)| ≤ 

holds for all integers t ∈ [T1 + L , T2 − L].

6.2 Main Results

235

In Sect. 6.6 we prove the following result which shows that for a generic (typical) ∞ sequence of functions v = {vt }∞ t=0 the sequence of functions {vt ◦ gt }t=0 has TP which is stable under small perturbations. Theorem 6.10 There exists a set F ⊂ A0 which is a countable intersection of open everywhere dense sets in (A, dA ) such that for each v = {vt }∞ t=0 ∈ F, each  > 0, each M1 > 0 and each M0 > 1 + (α∗ d∗ )−1 there exist δ > 0, a natural number L and an open neighborhood U of v in A such that the following assertion holds. For each pair of integers T1 ≥ 0, T2 > 2L + T1 , each u = {u t }∞ t=0 ∈ U, and each T2 program {x(t)}t=T1 satisfying x(T1 ) ≤ M0 , T 2 −1

u t (gt (x(t), x(t + 1)))

t=T1 T2 −1 ≥ U ({u t }t=T , x(T1 ), x(T2 )) − δ, 1 T 2 −1

T2 −1 u t (gt (x(t), x(t + 1))) ≥ U ({u t }t=T , x(T2 − L)) − M1 , 2 −L

t=T2 −L

the inequality

|x(t) − x (v) (t)| ≤ 

holds for all integers t ∈ [T1 + L , T2 − L].

6.3 Auxiliary Results Let

−1 v = {vt }∞ t=0 ∈ A0 , M∗ > 1 + (α∗ d∗ ) .

(6.37)

The next result follows from Lemma 2.9 and Corollary 2.20. Proposition 6.11 There exists a positive number M1 such that for every x0 ∈ [0, M∗ ] and every pair of integers T1 ≥ 0, T2 > T1 the inequality T2 −1 T2 −1 , x0 ) − U M∗ ({vt }t=T )| ≤ M1 |U ({vt }t=T 1 1

is valid.

236

6 Generic Turnpike Results for the One-Dimensional RSS Model

Lemma 6.12 Let  > 0 and M ≥ M∗ . Then there exists δ > 0 such that for each integer t ≥ 0 and each (6.38) (x, x ), (y, y ) ∈ graph(at ) satisfying x, y ≤ M, |x − y|, |x − y | ≤ δ the inequality

(6.39)

|vt (gt (x, x )) − vt (gt (y, y ))| ≤ 

holds. Proof By (A2), there is δ > 0 such that for each integer t ≥ 0 and each z, z ∈ [0, M] satisfying |z − z | ≤ 2δ(1 + α∗ ) the following inequality holds: |vt (z) − vt (z )| ≤ /2.

(6.40)

Assume that t ≥ 0 is an integer and (x, x ), (y, y ) ∈ R 2 satisfy (6.38) and (6.39). In order to complete the proof of the lemma it is sufficient to show that vt (gt (y, y )) ≥ vt (gt (x, x )) − . In view of (6.16),

Set

vt (gt (x, x )) = vt (min{x, 1 − αt (x − (1 − dt )x)}).

(6.41)

z = min{x, 1 − αt (x − (1 − dt )x)}.

(6.42)

Equations (6.39) and (6.42) imply that y ≥ x − δ ≥ z − δ.

(6.43)

By (6.4), (6.39) and (6.42), 1 − αt (y − (1 − dt )y) ≥ 1 − αt (x + δ − (1 − dt )(x − δ)) ≥ z − 2α∗ δ.

(6.44)

6.3 Auxiliary Results

237

Set

z = max{0, z − δ(1 + 2α∗ )}.

(6.45)

It follows from (6.38), (6.43), (6.44) and (6.45) that 0 ≤ z ≤ y, 1 − αt (y − (1 − dt )y)) ≥ z . This implies that

vt (gt (y, y )) ≥ vt (z ).

(6.46)

It follows from (6.39), the choice of δ (see (6.40)), (6.42) and (6.45) that |vt (z ) − vt (z)| ≤ /2.

(6.47)

By (6.42), (6.46) and (6.47), vt (gt (y, y )) ≥ vt (z) − /2 ≥ vt (gt (x, x )) − /2. Lemma 6.12 is proved. The next result easily follows from (6.18) and the strict monotonicity of the functions vt , t = 0, 1, . . . . Lemma 6.13 Let t ≥ 0 be an integer, (x, x ) ∈ (graph(at ) and y ∈ [0, x] satisfy y + αt (x − (1 − dt )x) ≤ 1. Then

if and only if

vt (gt (x, x )) = vt (y) y = min{x, 1 − αt (x − (1 − dt )x)}.

 > 0 and a program { Proposition 2.12 implies that there exists  x (t)}∞ t=0 such that

for every integer t ≥ 0. Fix a positive number

 x (0) < 1,

(6.48)

 vt ( x (t),  x (t + 1))) ≥ 

(6.49)

 /4}. γ < min{2−1 , 

(6.50)

Lemmas 2.9 and 2.16 and (6.50) imply the following result.

238

6 Generic Turnpike Results for the One-Dimensional RSS Model

Lemma 6.14 Let M1 be a positive number. Then there exist integers L 1 , L 2 ≥ 4 such T2 that for every pair of integers T1 ≥ 0, T2 ≥ L 1 + L 2 + T1 , every program {x(t)}t=T 1 satisfying x(T1 ) ≤ M1 , T 2 −1

T2 −1 vt (gt (x(t), x(t + 1))) ≥ U ({vt }t=T , x(T1 )) − M1 1

t=T1

and every integer τ ∈ [T1 + L 1 , T2 − L 2 ] the following inequality holds: γ. max{vt (gt (x(t), x(t + 1))) : t = τ , . . . , τ + L 2 − 1} ≥  Lemmas 2.9 and 2.17 imply the following result. Lemma 6.15 Let M1 be a positive number. Then there exist integers L¯ 1 , L¯ 2 ≥ 1 and a positive number M2 such that for every pair of integers τ1 ≥ 0, τ2 ≥ L¯ 1 + L¯ 2 + τ1 2 and every program {x(t)}τt=τ satisfying 1 x(τ1 ) ≤ M∗ , τ 2 −1

2 −1 vt (gt (x(t), x(t + 1))) ≥ U ({vt }τt=τ , x(τ1 )) − M1 1

t=τ1

the following assertion holds. If integers T1 , T2 ∈ [τ1 , τ2 − L¯ 2 ] satisfy L¯ 1 ≤ T2 − T1 , then T 2 −1

T2 −1 vt (gt (x(t), x(t + 1))) ≥ U ({vt }t=T , x(T1 )) − M2 . 1

t=T1

Recall that M¯ is as in Theorem 6.4 with wt = vt , t = 0, 1, . . . . Lemma 6.16 Let , M > 0. Then there exist a natural number L 0 ≥ 4 and δ > 0 such that for each pair of integers τ1 ≥ 0 and τ2 ≥ τ1 + L 0 and each z 1 , z 2 , z 1 , z 2 ∈ [0, M∗ ] satisfying

and

|z i − z i | ≤ δ, i = 1, 2,

(6.51)

2 −1 2 −1 M∗ ({vt }τt=τ , z1, z2 ) ≥ U )−M U ({vt }τt=τ 1 1

(6.52)

6.3 Auxiliary Results

the inequality

239

2 −1 2 −1 ({vt }τt=τ , z 1 , z 2 ) ≥ U , z1, z2 ) −  U ({vt }τt=τ 1 1

holds. Proof By Lemma 6.14, there exist integers L 1 , L 2 ≥ 4 such that the following property holds: T2 (i) For every pair of integers T1 ≥ 0, T2 ≥ L 1 + L 2 + T1 , every program {x(t)}t=T 1 satisfying x(T1 ) ≤ M∗ , T 2 −1

T2 −1 vt (gt (x(t), x(t + 1))) ≥ U ({vt }t=T , x(T1 )) − M − 4 1

t=T1

and every integer τ ∈ [T1 + L 1 , T2 − L 2 ] the following inequality holds: γ. max{vt (gt (x(t), x(t + 1))) : t = τ , . . . , τ + L 2 − 1} ≥  In view of (A1), there exists γ0 > 0 such that γ /4 for every integer t ≥ 0. vt (γ0 ) < 

(6.53)

Assumption (A2) implies that there exists a positive number δ1 < γ0 /4

(6.54)

such that for each integer t ≥ 0 and each y, y ∈ [0, M∗ ] satisfying |y − y | ≤ 2δ1 , the inequality

|vt (y) − vt (y )| ≤ /8

(6.55)

is true. Choose natural numbers L 3 > L 1 + L 2 + 4, L 0 > 8 + 4(L 1 + L 2 + 2L 3 ).

(6.56)

240

6 Generic Turnpike Results for the One-Dimensional RSS Model

By Lemma 6.12, there exists δ2 > 0 such that δ2 α∗ < γ0 /4

(6.57)

and that for each integer t ≥ 0 and each (x, x ), (y, y ) ∈ graph(at ) satisfying x, y ≤ M∗ , |x − x |, |y − y | ≤ 2δ2 the inequality |vt (gt (x, x )) − vt (gt (y, y ))| ≤ 8−1 (L 2 + L 3 + 1)−1

(6.58)

holds. Set δ = min{δ1 , δ2 , 4−1 γ0 (1 + α∗ )−1 , 8−1 (1 − d ∗ ) L 2 +L 3 δ1 (1 + α∗ )−1 }.

(6.59)

Assume that integers τ1 ≥ 0, τ2 ≥ τ1 + L 0 , z 1 , z 2 , z 1 , z 2 ∈ [0, M∗ ]

(6.60) (6.61)

|z i − z i | ≤ δ, i = 1, 2,

(6.62)

2 −1 2 −1 M∗ ({vt }τt=τ , z1, z2 ) ≥ U ) − M. U ({vt }τt=τ 1 1

(6.63)

satisfy

and

By the continuity of the functions vt , t = 0, 1, . . . and (6.63), there exists a program 2 such that {x(t)}τt=τ 1 (6.64) x(τ1 ) = z, x(τ2 ) ≥ z 2 and τ 2 −1

2 −1 vt (gt (x(t), x(t + 1))) = U ({vt }τt=τ , z1, z2 ) 1

t=τ1 2 −1 M∗ ({vt }τt=τ ) − M. ≥U 1

(6.65)

6.3 Auxiliary Results

241

Property (i), (6.54), (6.56), (6.60), (6.61), (6.64) and (6.65) imply that γ max{vt (gt (x(t), x(t + 1))) : t = L 3 + τ1 , . . . , L 3 + τ1 + L 2 − 1} ≥ 

(6.66)

and γ . (6.67) max{vt (gt (x(t), x(t + 1))) : t = τ2 − L 2 − L 3 , . . . , τ2 − L 3 − 1} ≥  Equations (6.66) and (6.67) imply that there exist integers t1 ∈ [L 3 + τ1 , . . . , L 3 + τ1 + L 2 − 1], t2 ∈ [τ2 − L 2 − L 3 , . . . , τ2 − L 3 − 1]

(6.68)

γ , i = 1, 2. vti (gti (x(ti ), x(ti + 1))) ≥ 

(6.69)

such that For every integer t ∈ [τ1 , τ2 − 1] set y(t) = gt (x(t), x(t + 1)).

(6.70)

In view of (6.70), for all integers t = τ1 , . . . , τ2 − 1, vt (gt (x(t), x(t + 1))) = vt (yt ).

(6.71)

It follows from (6.53), (6.69)–(6.71) and the monotonicity of the functions vt , t = 0, 1, . . . that (6.72) y(t1 ), y(t2 ) > γ0 . τ2 Now we construct a program {x(t)} ˜ t=τ1 . Set

x(τ ˜ 1 ) = z 1

(6.73)

and for all integers t = τ1 , . . . , t1 − 1 put ˜ + x(t + 1) − (1 − dt )x(t). x(t ˜ + 1) = (1 − dt )x(t)

(6.74)

t1 Clearly, {x(t)} ˜ t=τ1 is a program. By (6.64), (6.73) and (6.74), for all integers t = τ 1 , . . . , t1 , (6.75) |x(t) ˜ − x(t)| ≤ |x(τ ˜ 1 ) − x(τ1 )| ≤ δ.

Put x(t ˜ 1 + 1) = x(t1 + 1) + δ.

(6.76)

242

6 Generic Turnpike Results for the One-Dimensional RSS Model

In view of (6.70) and (6.72), αt1 (x(t1 + 1) − (1 − dt1 )x(t1 )) ≤ 1 − y(t1 ) ≤ 1 − γ0 .

(6.77)

By (6.4), (6.57) and (6.76)–(6.77), ˜ 1 + 1) − (1 − dt1 )x(t ˜ 1 )) αt1 (x(t ≤ αt1 (x(t1 + 1) − (1 − dt1 )x(t1 ) + 2δ) ≤ 1 − γ0 + 2δα∗ < 1.

(6.78)

Equations (6.75) and (6.76) imply that ˜ 1 ) ≥ x(t1 + 1) − (1 − dt1 )x(t1 ) ≥ 0. x(t ˜ 1 + 1) − (1 − dt1 )x(t

(6.79)

t1 +1 Therefore {x(t)} ˜ t=τ1 is a program. It follows from Lemma 2.9, the choice of δ2 (see (6.67) and (6.68)), (6.61), (6.75) and (6.76) that

˜ x(t ˜ + 1)))| ≤ 8−1 (L 2 + L 3 + 1)−1 (6.80) |vt (gt (x(t), x(t + 1))) − vt (gt (x(t), for all integers t = τ1 , . . . , t1 . Equations (6.68) and (6.80) imply that   t t1 1      vt (gt (x(t), x(t + 1))) − vt (gt (x(t), ˜ x(t ˜ + 1)))    t=τ1

t=τ1

−1

−1

≤ 8 (L 1 + L 2 + L 3 ) (t1 − τ1 + 1) ≤ 8−1 .

(6.81)

For all integers t = t1 + 2, . . . , t2 set ˜ − 1) − (1 − dt )x(t − 1). x(t) ˜ = x(t) + (1 − dt−1 )x(t

(6.82)

Equations (6.76) and (6.82) imply that x(t) ˜ ≥ x(t), t = t1 + 1, . . . , t2 . t2 Clearly, {x(t)} ˜ t=τ1 is a program. Set

δ1 x(t ˜ 2 + 1) = x(t2 + 1) + αt−1 2 + (1 − dt2 )x(t ˜ 2 ) − (1 − dt2 )x(t2 ).

(6.83)

6.3 Auxiliary Results

243

By (6.54), (6.70), (6.72) and (6.83), ˜ 2 + 1) − (1 − dt2 )x(t ˜ 2 )) αt2 (x(t ≤ αt2 (x(t2 + 1) − (1 − dt2 )x(t2 ) + αt−1 δ1 ) 2 ≤ 1 − y(t2 ) + δ1 ≤ 1 − γ0 + δ1 < 1 − γ0 /2.

(6.84)

Equations (6.83) and (6.84) imply that ˜ 2 )). x(t ˜ 2 + 1) ∈ at2 (x(t

(6.85)

It follows from (6.70), (6.72) and (6.83) that x(t ˜ 2 ) ≥ x(t2 ) ≥ y(t2 ) ≥ γ0 .

(6.86)

˜ 2 + 1) − (1 − dt2 )x(t ˜ 2 )). y(t2 ) − δ1 ≤ 1 − αt2 (x(t

(6.87)

In view of (6.84),

It follows from Lemma 2.9, the choice of δ1 (see (6.54) and (6.55)), (6.16), (6.61), (6.86) and (6.87) that ˜ 2 ), x(t ˜ 2 + 1))) ≥ vt2 (y(t2 ) − δ1 ) vt2 (gt2 (x(t ≥ vt2 (y(t2 )) − /8.

(6.88)

For all integers t = t2 + 1, . . . , τ2 − 1 set ˜ − (1 − dt )x(t). x(t ˜ + 1) = x(t + 1) + (1 − dt )x(t)

(6.89)

Equations (6.80) and (6.83) imply that x(t) ˜ ≥ x(t), t = t2 + 1, . . . , τ2 .

(6.90)

τ2 Clearly, {x(t)} ˜ t=τ1 is a program. In view of (6.16), (6.89) and (6.99), for all integers t = t2 + 1, . . . , τ2 − 1,

˜ 2 ), x(t ˜ 2 + 1))) ≥ vt (gt (x(t), x(t + 1))). vt (gt (x(t By (6.4) and (6.89), for all integers t = t2 + 1, . . . , τ2 − 1, ˜ − x(t)) x(t ˜ + 1) − x(t + 1) = (1 − dt )(x(t) ∗ ≥ (1 − d )(x(t) ˜ − x(t)).

(6.91)

244

6 Generic Turnpike Results for the One-Dimensional RSS Model

Combined with (6.68) and (6.83), this implies that x(τ ˜ 2 ) − x(τ2 ) ≥ (1 − d ∗ )τ2 −t2 −1 (x(t ˜ 2 + 1) − x(t2 + 1)) ≥ (1 − d ∗ ) L 2 +L 3 δ1 (α∗ )−1 .

(6.92)

It follows from (6.59), (6.62), (6.64) and (6.92) that x(τ ˜ 2 ) ≥ x(τ2 ) + (1 − d ∗ ) L 2 +L 3 δ1 (α∗ )−1 ≥ z 2 + (1 − d ∗ ) L 2 +L 3 δ1 (α∗ )−1 ≥ z 2 − δ + (1 − d ∗ ) L 2 +L 3 δ1 (α∗ )−1 > z 2 . By the relation above, (6.73), (6.81), (6.82), (6.85), (6.91), and the inequality x(t) ˜ ≥ x(t), t = t1 + 1, . . . , t2 , we have 2 −1 U ({vt }τt=τ , z 1 , z 2 ) ≥ 1

τ 2 −1

vt (gt (x(t), ˜ x(t ˜ + 1)))

t=τ1

≥

τ 2 −1

vt (gt (x(t), x(t + 1)))

t=τ1

+

τ −1 2 

vt (gt (x(t), ˜ x(t ˜ + 1))) −

t=τ1

τ 2 −1

vt (gt (x(t), x(t + 1)))

t=τ1

2 −1 ≥ U ({vt }τt=τ , z1, z2 ) 1

+

t1 

vt (gt (x(t), ˜ x(t ˜ + 1))) −

t=τ1

t1 

vt (gt (x(t), x(t + 1)))

t=τ1

+ vt2 (gt2 (x(t ˜ 2 ), x(t ˜ 2 + 1))) − vt2 (gt2 (x(t2 ), x(t2 + 1))) 2 −1 ≥ U ({vt }τt=τ , z 1 , z 2 ) − 4−1 . 1

Lemma 6.16 is proved. ∞ Recall that by Theorem 6.4, there exist M¯ > 0 and a program {x(t)} ¯ t=0 such that

x(0) ¯ ≤ M∗ , for each pair of integers T1 , T2 ≥ 0 satisfying T1 < T2 ,

(6.93)

6.3 Auxiliary Results

245

T −1  2    T −1 2 M∗ ({vt }t=T ) ≤ M¯ vt (gt (x(t), ¯ x(t ¯ + 1))) − U  1  

(6.94)

t=T1

and that for each integer T > 0, T −1 

T −1 vt (gt (x(t), ¯ x(t ¯ + 1))) = U ({vt }t=0 , x(0), ¯ x(T ¯ )).

(6.95)

t=0

Lemma 6.16 implies the following result. Lemma 6.17 Let , M > 0. Then there exist a natural number L 0 ≥ 4 and δ > 0 such that for each pair of integers τ1 ≥ 0 and τ2 ≥ τ1 + L 0 and each z 1 , z 2 , z 1 , z 2 ∈ [0, M∗ ] satisfying

|z i − z i | ≤ δ, i = 1, 2,

and

2 −1 2 −1 M∗ ({vt }τt=τ , z1, z2 ) ≥ U )−M U ({vt }τt=τ 1 1

the inequality

2 −1 2 −1 ({vt }τt=τ , z 1 , z 2 ) − U , z 1 , z 2 )| ≤  |U ({vt }τt=τ 1 1

holds. Lemma 6.18 Let , M > 0. Then there exist a natural number L 0 ≥ 4 and δ > 0 such that for each pair of integers τ1 ≥ 0 and τ2 ≥ τ1 + L 0 and each program 2 satisfying {x(t)}τt=τ 1 x(τ1 ) ≤ M∗ , τ 2 −1

(6.96)

2 −1 vt (gt (x(t), x(t + 1))) ≥ U ({vt }τt=τ , x(τ1 ), x(τ2 )) − δ 1

(6.97)

t=τ1

and

τ 2 −1 t=τ1

vt (gt (x(t), x(t + 1))) ≥

τ 2 −1

vt (gt (x(t), ¯ x(t ¯ + 1))) − M

(6.98)

t=τ1

for each integer t ∈ [τ1 , τ2 − L 0 ] the following inequality holds: x(t) +  ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)).

(6.99)

246

6 Generic Turnpike Results for the One-Dimensional RSS Model

Proof By Lemma 6.14 and (6.94), there exist integers L 1 , L 2 ≥ 4 such that the following property holds: T2 (i) For every pair of integers T1 ≥ 0, T2 ≥ L 1 + L 2 + T1 , every program {x(t)}t=T 1 satisfying x(T1 ) ≤ M∗ , T 2 −1

vt (gt (x(t), x(t + 1))) ≥

t=T1

T 2 −1

vt (gt (x(t), ¯ x(t ¯ + 1))) − M

t=T1

and every integer τ ∈ [T1 + L 1 , T2 − L 2 ] the following inequality holds: γ. max{vt (gt (x(t), x(t + 1))) : t = τ , . . . , τ + L 2 − 1} ≥  In view of (A1), there exists γ0 > 0 such that γ /4 for every integer t ≥ 0. vt (γ0 ) < 

(6.100)

Choose a positive number γ1 < d∗ γ0 /4

(6.101)

and natural numbers L 0 , L 3 such that L 3 > 4(L 1 + L 2 ), (M∗ + 1)(1 − d∗ ) L 3 < γ0 /4, L 0 ≥ 4(L 1 + L 2 + 2L 3 ).

(6.102)

Choose a positive number γ2 such that γ2 < γ1 /4, γ2 < 8−1 (1 − d ∗ ) L 2 +L 3 (α∗ )−1 .

(6.103)

Assumption (A3) implies that there is δ ∈ (0, 1) such that the following property holds: (ii) For each integer t ≥ 0, each z 1 , z 2 ∈ [0, M∗ ] satisfying z 2 ≥ z 1 + γ2 min{1, α∗ } the inequality vt (z 2 ) − vt (z 1 ) ≥ 4δ holds.

6.3 Auxiliary Results

247

2 Assume that τ1 ≥ 0 and τ2 ≥ τ1 + L 0 are integers, a program {x(t)}τt=τ satisfies 1 (6.96)–(6.98) and that an integer

t0 ∈ [τ1 , τ2 − L 0 ].

(6.104)

We show that (6.99) is true with t = t0 . Assume the contrary. Then x(t0 ) +  < 1 − αt0 (x(t0 + 1) − (1 − dt0 )x(t0 )).

(6.105)

For all integers t = τ1 , . . . , τ2 − 1 set yt = gt (x(t), x(t + 1)).

(6.106)

τ2 Now we define a program {x(t)} ˜ t=τ1 . Set

x(t) ˜ = x(t), t = τ1 , . . . , t0 , x(t ˜ 0 + 1) = x(t0 + 1) + αt−1 . 0

(6.107)

˜ 0 ). x(t ˜ 0 + 1) ≥ (1 − dt0 )x(t

(6.108)

Clearly,

By (6.105)–(6.107), ˜ 0 + 1) − (1 − dt0 )x(t ˜ 0 )) αt0 (x(t = αt0 (x(t0 + 1) − (1 − dt0 )x(t0 )) +  < 1 − x(t0 ) −  +  ≤ 1 − y(t0 ).

(6.109)

t0 +1 Therefore {x(t)} ˜ t=τ1 is a program. Lemma 6.13 and (6.106)–(6.109) imply that

˜ 0 ), x(t ˜ 0 + 1))) ≥ vt0 (y(t0 )) = vt0 (gt0 (x(t0 ), x(t0 + 1))). vt0 (gt0 (x(t

(6.110)

Property (i), (6.96), (6.98) and (6.102)–(6.104) imply that γ . (6.111) max{vt (gt (x(t), x(t + 1))) : t = t0 + 1 + L 3 , . . . , t0 + L 2 + L 3 } ≥  It follows from (6.111) that there exists an integer t1 such that t0 + L 3 + 1 ≤ t1 ≤ t0 + L 2 + L 3

(6.112)

γ. vt1 (gt1 (x(t1 ), x(t1 + 1))) ≥ 

(6.113)

and

248

6 Generic Turnpike Results for the One-Dimensional RSS Model

In view of (6.100), (6.106) and (6.113),  γ ≤ vt1 (gt1 (x(t1 ), x(t1 + 1))) = vt1 (yt1 ), x(t1 ) ≥ y(t1 ) ≥ γ0 .

(6.114)

We show that there exists an integer t2 such that t0 + 1 ≤ t2 ≤ t1 − 1, x(t2 + 1) − (1 − dt2 )x(t2 ) ≥ γ1 .

(6.115) (6.116)

Let us assume the contrary. Then for every integer t ∈ [t0 + 1, t1 − 1], we have x(t + 1) − (1 − dt )x(t) ≤ γ1 .

(6.117)

Lemma 2.9 and (6.96) imply that x(t0 + 1) ≤ M∗

(6.118)

and in view of (6.117), for all integers t ∈ [t0 + 1, t1 − 1], x(t + 1) ≤ (1 − dt )x(t) + γ1 ≤ (1 − d∗ )x(t) + γ1 . Combined with (6.101), (6.102), (6.112) and (6.118) this implies that x(t1 ) ≤ M∗ (1 − d∗ )t1 −t0 −1 + γ1

∞  (1 − d∗ )i i=0

≤ M∗ (1 − d∗ )

L3

+

γ1 d∗−1

< γ0 /2.

This contradicts (6.114). Therefore there exists an integer t2 for which (6.115) and (6.116) hold. For every integer t satisfying t0 + 1 ≤ t < t2 set ˜ + x(t + 1) − (1 − dt )x(t). x(t ˜ + 1) = (1 − dt )x(t)

(6.119)

In view of (6.107) and (6.119), x(t) ˜ ≥ x(t), t = t0 + 1, . . . , t2 ,

(6.120)

t2 {x(t)} ˜ t=τ1 is a program and in view of (6.110) and (6.120), for any integer t satisfying t0 + 1 ≤ t < t2 ,

˜ x(t ˜ + 1)) ≥ vt (gt (x(t), x(t + 1)). vt (gt (x(t),

(6.121)

6.3 Auxiliary Results

249

It follows from (6.119) and (6.120) that for every integer t satisfying t 0 + 1 ≤ t < t2 , we have ˜ − x(t)) x(t ˜ + 1) − x(t + 1) = (1 − dt )(x(t) ≥ (1 − d∗ )(x(t) ˜ − x(t)).

(6.122)

By (6.107), (6.112), (6.115) and (6.122), ˜ 0 + 1) − x(t0 + 1)) x(t ˜ 2 ) − x(t2 ) ≥ (1 − d ∗ )t2 −t0 −1 (x(t ≥ (1 − d∗ ) L 3 +L 2 −1 αt−1  0 ≥ (1 − d∗ ) L 3 +L 2 −1 (α∗ )−1 .

(6.123)

Set ˜ 2 ) + x(t2 + 1) − (1 − dt2 )x(t2 ) − γ2 . x(t ˜ 2 + 1) = (1 − dt2 )x(t

(6.124)

Equations (6.103), (6.116) and (6.124) imply that ˜ 2 ). x(t ˜ 2 + 1) ≥ (1 − dt2 )x(t

(6.125)

It follows from (6.4), (6.103) and (6.123) that ˜ 2 ) − x(t2 )) − γ2 x(t ˜ 2 + 1) = x(t2 + 1) + (1 − dt2 )(x(t ≥ x(t2 + 1) + (1 − d∗ ) L 3 +L 2 (α∗ )−1  − γ2 > x(t2 + 1).

(6.126)

By (6.124), ˜ 2 + 1) − (1 − dt2 )x(t ˜ 2 )) 1 − αt2 (x(t = 1 − αt2 (x(t2 + 1) − (1 − dt2 )x(t2 )) + αt2 γ2 ≥ y(t2 ) + αt2 γ2 ≥ y(t2 ) + α∗ γ2 .

(6.127)

In view of (6.103) and (6.123), x(t ˜ 2 ) ≥ x(t2 ) + γ2 ≥ y(t2 ) + γ2 .

(6.128)

Lemma 2.9, property (ii), (6.96), (6.106), (6.127) and (6.128) imply that ˜ 2 ), x(t ˜ 2 + 1))) ≥ vt2 (y(t2 ) + γ2 min{1, α∗ }) vt2 (gt2 (x(t ≥ vt2 (y(t2 )) + 4δ = vt2 (gt2 (x(t2 ), x(t2 + 1))) + 4δ. (6.129)

250

6 Generic Turnpike Results for the One-Dimensional RSS Model

For every integer t satisfying t 2 + 1 ≤ t ≤ τ2 − 1 set ˜ + x(t + 1) − (1 − dt )x(t). x(t ˜ + 1) = (1 − dt )x(t)

(6.130)

By (6.126) and (6.130), x(t) ˜ ≥ x(t), t = t2 + 1, . . . , τ2 ,

(6.131)

τ2 {x(t)} ˜ t=τ1 is a program and in view of (6.131), for every integer t satisfying

t2 + 1 ≤ t ≤ τ2 − 1, we have ˜ x(t ˜ + 1))) ≥ vt (gt (x(t), x(t + 1))). vt (gt (x(t),

(6.132)

It follows from (6.97), (6.107), (6.110), (6.119), (6.121), (6.129) and (6.132) that 2 −1 U ({vt }τt=τ , x(τ1 ), x(τ2 )) ≥ 1

τ 2 −1

vt (gt (x(t), ˜ x(t ˜ + 1)))

t=τ1

≥

τ 2 −1

vt (gt (x(t), x(t + 1))) + 4δ

t=τ1 2 −1 , x(τ1 ), x(τ2 )) − δ + 4δ. ≥ U ({vt }τt=τ 1

The contradiction we have reached proves (6.99) with t = t0 and Lemma 6.18 itself. Corollary 6.19 For every integer t ≥ 0, x(t) ¯ ≥ 1 − αt (x(t ¯ + 1) − (1 − dt )x(t)). ¯ ¯ let a natural number L 0 ≥ 4 and δ > Lemma 6.20 Let , M0 > 0, M = M0 + 2 M, 0 be as guaranteed by Lemma 6.18 and let L > L 0 be an integer. Then there exists S+k L an integer k ≥ 1 such that for each integer S ≥ 0 and each program {x(t)}t=S satisfying x(S) ≤ M∗ , S+k L−1  t=S

vt (gt (x(t), x(t + 1))) ≥

(6.133) S+k L−1  t=S

vt (gt (x(t), ¯ x(t ¯ + 1))) − M0

(6.134)

6.3 Auxiliary Results

251

there exists an integer j ∈ {0, . . . , k − 1} such that for every integer t ∈ [S + j L , S + ( j + 1)L − L 0 ] the following inequality holds: x(t) +  ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)). Proof Choose a natural number ¯ −1 + 1. k > (M0 + M)δ

(6.135)

S+k L satisfy (6.133) and (6.134). Assume Let S ≥ 0 be an integer and a program {x(t)}t=S that T2 > T1 ≥ 0 are integers and that

T1 T2 ∈ [S, S + k L]. By (6.94), (6.133) and (6.134), T 2 −1

vt (gt (x(t), x(t + 1))) =

t=T1

S+k L−1 

vt (gt (x(t), x(t + 1)))

t=S

 {vt (gt (x(t), x(t + 1))) : t ∈ {S, . . . , T1 } \ {T1 }}  − {vt (gt (x(t), x(t + 1))) : t ∈ {T2 , . . . , S + k L} \ {S + k L}}

−

≥

S+k L−1 

vt (gt (x(t), ¯ x(t ¯ + 1))) − M0

t=S

 {vt (gt (x(t), ¯ x(t ¯ + 1))) : t ∈ {S, . . . , T1 } \ {T1 }} − M¯  − {vt (gt (x(t), ¯ x(t ¯ + 1))) : t ∈ {T2 , . . . , S + k L} \ {S + k L}} − M¯ −

=

T 2 −1

¯ vt (gt (x(t), ¯ x(t ¯ + 1))) − M0 − 2 M.

t=T1

Thus we have shown that for each pair of integers T1 , T2 satisfying S ≤ T1 < T2 ≤ S + k L , we have T 2 −1 t=T1

vt (gt (x(t), x(t + 1))) ≥

T 2 −1

¯ vt (gt (x(t), ¯ x(t ¯ + 1))) − M0 − 2 M.

(6.136)

t=T1

In order to complete the proof it is sufficient to show that there an integer j ∈ {0, . . . , k − 1} such that for every integer t ∈ [S + j L , S + ( j + 1)L − L 0 ], we have x(t) +  ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)).

252

6 Generic Turnpike Results for the One-Dimensional RSS Model

Assume the contrary. Then for each j ∈ {0, . . . , k − 1} we have max{1 − x(t) − αt (x(t + 1) − (1 − dt )x(t)) : t = S + j L , . . . , S + ( j + 1)L − L 0 } > .

(6.137)

Let j ∈ {0, . . . , k − 1}. It follows from Lemmas 2.9 and 6.18, (6.133), (6.134) and the choice of δ that 

S+( j+1)L−1

vt (gt (x(t), x(t + 1)))

t=S+ j L S+( j+1)L−1

< U ({vt }t=S+ j L

, x(S + j L), x(S + ( j + 1)L)) − δ.

By (6.138), there exists a program {x ( j) (t)}t=S+ j L

S+( j+1)L−1

(6.138)

such that

x ( j) (S + j L) = x(S + j L), x ( j) (S + ( j + 1)L) ≥ x(S + ( j + 1)L), 

S+( j+1)L−1

(6.139) 

S+( j+1)L−1

vt (gt (x ( j) (t), x ( j) (t + 1))) ≥

t=S+ j L

vt (gt (x(t), x(t + 1))) + δ.

t=S+ j L

(6.140) S+k L Assumption (A3) and (6.13) imply that there exists a program {x(t)} ˜ t=S such that

x(S) ˜ = x(S), x(S ˜ + j L) ≥ x ( j) (S + j L), j = 0, . . . , k,

(6.141)

for each j ∈ {0, . . . , k − 1}, x(t) ˜ ≥ x ( j) (t), t = S + j L , . . . , S + ( j + 1)L ,

(6.142)

vt (gt (x(t), ˜ x(t ˜ + 1))) ≥ vt (gt (x ( j) (t), x ( j) (t + 1))), t = S + j L , . . . , S + ( j + 1)L − 1. By (6.94), (6.133), (6.134), (6.140), (6.141) and (6.143),

(6.143)

6.3 Auxiliary Results

M¯ +

S+k L−1 

253

vt (gt (x(t), ¯ x(t ¯ + 1))) ≥

S+k L−1 

t=S

vt (gt (x(t), ˜ x(t ˜ + 1)))

t=S

≥

S+k L−1 

vt (gt (x(t), x(t + 1))) + δk

t=S

≥

S+k L−1 

vt (gt (x(t), ¯ x(t ¯ + 1))) − M0 + δk

t=S

and

¯ −1 . k ≤ (M0 + M)δ

This contradicts (6.135). The contradiction we have reached proves Lemma 6.20. ¯ M = M1 + 4M, let a natural numLemma 6.21 Let , M1 > 0, M0 = M1 + 2 M, ber L 0 ≥ 4 and δ > 0 be as guaranteed by Lemma 6.18, let L > L 0 be an integer and let an integer k ≥ 1 be as guaranteed by Lemma 6.18. Assume that τ1 ≥ 0, τ2 ≥ τ1 + k L 2 satisfies are integers, a program {x(t)}τt=τ 1

x(τ1 ) ≤ M∗ , τ 2 −1

vt (gt (x(t), x(t + 1))) ≥

t=τ1

τ 2 −1

vt (gt (x(t), ¯ x(t ¯ + 1))) − M1

(6.144) (6.145)

t=τ1

and that S ∈ {τ1 , . . . , τ2 − k L}.

(6.146)

Then there exists an integer j ∈ {0, . . . , k − 1} such that for every integer t ∈ [S + j L , S + ( j + 1)L − L 0 ] the following inequality holds: x(t) +  ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)). Proof By (6.45) and (6.94),

(6.147)

254

6 Generic Turnpike Results for the One-Dimensional RSS Model

S+k L−1 

vt (gt (x(t), x(t + 1))) =

τ 2 −1

vt (gt (x(t), x(t + 1)))

t=τ1

t=S

 {vt (gt (x(t), x(t + 1))) : t ∈ {τ1 , . . . , S} \ {S}}  − {vt (gt (x(t), x(t + 1))) : t ∈ {S + k L , . . . , τ2 } \ {τ2 }}

−

≥

τ 2 −1

vt (gt (x(t), ¯ x(t ¯ + 1))) − M1

t=τ1

 {vt (gt (x(t), ¯ x(t ¯ + 1))) : t ∈ {τ1 , . . . , S} \ {S}} − M¯  − {vt (gt (x(t), ¯ x(t ¯ + 1))) : t ∈ {S + k L , . . . , τ2 } \ {τ2 }} − M¯ −

=

S+k L−1 

¯ vt (gt (x(t), ¯ x(t ¯ + 1))) − M1 − 2 M.

(6.148)

t=S

Lemma 2.9 and (6.144) imply that x(S) ≤ M∗ . Combined with (6.148) this implies that there exists j ∈ {0, . . . , k − 1} such that for all integers t ∈ [S + j L , S + ( j + 1)L − L 0 ] equation (6.147) is valid. Lemma 6.21 is proved.

6.4 Auxiliary Results for Theorem 6.6 Let {wt }∞ t=0 ∈ A0 , γ ∈ (0, 1) and M∗ > (α∗ d∗ )−1 + 1.

(6.149)

Let M¯ > 0 be as guaranteed by Theorem 6.4. Theorem 6.4 implies that there exists ∞ a program {x(t)} ¯ t=0 such that (6.150) x(0) ¯ ∈ [0, M∗ ], for each pair of integers T1 , T2 ≥ 0 satisfying T1 < T2 , T −1  2    T −1 2 M∗ ({wt }t=T ) ≤ M¯ wt (gt (x(t), ¯ x(t ¯ + 1))) − U  1   t=T1

and for each integer T > 0,

(6.151)

6.4 Auxiliary Results for Theorem 6.6 T −1 

255

T −1 wt (gt (x(t), ¯ x(t ¯ + 1))) = U ({wt }t=0 , x(0), ¯ x(T ¯ )).

(6.152)

t=0

For each integer t ≥ 0, set y¯ (t) = gt (x(t), ¯ x(t ¯ + 1)) ¯ + 1) − (1 − dt )x(t))}. ¯ = min{x(t), ¯ 1 − αt (x(t

(6.153)

For every integer t ≥ 0 and every y ≥ 0 set (see (6.28) and (6.29)) (γ)

wt (y) = wt (y) + γ y, y ∈ [0, y¯ (t)],

(6.154)

(γ) wt (y)

(6.155)

= γwt ( y¯ (t)) + γ y¯ (t) + (1 − γ)wt (y)

for every y > y¯ (t). Clearly,

(γ)

{wt }∞ t=0 ∈ A0

(see (6.30)). Let t ≥ 0 be an integer and (x, x ) ∈ graph(at ). In view of (6.153)–(6.155), (γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) = wt (gt (x(t), ¯ x(t ¯ + 1))) + γgt (x(t), ¯ x(t ¯ + 1)) = wt (gt (x(t), ¯ x(t ¯ + 1))) + γ y¯ (t) and

(γ)

wt (gt (x, x )) ≤ γ y¯ (t) + wt (gt (x, x )).

(6.156)

(6.157)

By (6.151), (6.156) and (6.157), for each pair of integers T1 , T2 ≥ 0 satisfying T1 < T2 , T −1  2    (γ) (γ) T −1 2 M∗ ({wt }t=T ) ≤ M¯ wt (gt (x(t), ¯ x(t ¯ + 1))) − U  1  

(6.158)

t=T1

and for each integer T > 0, T −1 

(γ)

(γ)

T −1 wt (gt (x(t), ¯ x(t ¯ + 1))) = U ({wt }t=0 , x(0), ¯ x(T ¯ )).

(6.159)

t=0

Lemma 6.22 Let  > 0. Then there exist a natural number L 0 ≥ 4 and δ > 0 such that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L 0 , each z 1 , z 2 ∈ [0, M∗ ]

256

6 Generic Turnpike Results for the One-Dimensional RSS Model

2 and each program {x(t)}τt=τ satisfying 1

¯ 1 )|, |z 2 − x(τ ¯ 2 )| ≤ δ, |z 1 − x(τ x(τ1 ) = z 1 , x(τ2 ) ≥ z 2 and

τ 2 −1

γ

(γ)

2 −1 wt (gt (x(t), x(t + 1))) ≥ U ({wt }τt=τ , z1, z2 ) − δ 1

(6.160) (6.161)

(6.162)

t=τ1

the following inequality holds for every integer t ∈ [τ1 , τ2 − 1]: | y¯ (t) − gt (x(t), x(t + 1))| ≤ .

(6.163)

Proof Assumption (A3), (6.154) and (6.155) imply that there exists 1 ∈ (0, /2) such that the following property holds: (i) For each integer t ≥ 0 and each y ∈ [0, M∗ ] satisfying |y − y¯ (t)| ≥ , we have

(γ)

wt (y) − wt (y) − γ y¯ (t) ≤ −1 . By Lemma 6.17, there exist a natural number L 0 ≥ 4 and δ ∈ (0, 1 /4) such that the following property holds: (ii) For each pair of integers τ1 ≥ 0 and τ2 ≥ τ1 + L 0 and each z 1 , z 2 ∈ [0, M∗ ] satisfying ¯ i )| ≤ δ, i = 1, 2, |z i − x(τ we have

and

2 −1 2 −1 , z 1 , z 2 ) − U ({wt }τt=τ , x(τ ¯ 1 ), x(τ ¯ 2 )))| ≤ 1 /4 |U ({wt }τt=τ 1 1

(γ)

(γ)

2 −1 2 −1 , z 1 , z 2 ) − U ({wt }τt=τ , x(τ ¯ 1 ), x(τ ¯ 2 )))| ≤ 1 /4. |U ({wt }τt=τ 1 1

Assume that τ1 ≥ 0, τ2 ≥ τ1 + L 0 are integers, z 1 , z 2 ∈ [0, M∗ ]

(6.164)

6.4 Auxiliary Results for Theorem 6.6

257

2 and a program {x(t)}τt=τ satisfies (6.160)–(6.162). In view of (6.157), for every 1 integer t ∈ {τ1 , . . . , τ2 − 1},

(γ)

wt (gt (x(t), x(t + 1))) ≤ wt (gt (x(t), x(t + 1)) + γ y¯ (t).

(6.165)

Property (i), (6.161) and (6.164) imply that the following property holds: (iii) If an integer t ∈ {τ1 , . . . , τ2 − 1} and | y¯ (t) − gt (x(t), x(t + 1))| ≥ , then

(γ)

wt (gt (x(t), x(t + 1))) ≤ wt (gt (x(t), x(t + 1))) + γ y¯ (t) − 1 . In order to complete the proof it is sufficient to show that (6.163) is true for all integers t = τ1 , . . . , τ2 − 1. Assume the contrary. Then there exists S ∈ {τ1 , . . . , τ2 − 1} such that (6.166) | y¯ (S) − g S (x(S), x(S + 1))| > . In view of (6.166), (γ)

w S (g S (x(S), x(S + 1))) ≤ −1 + w S (g S (x(S), x(S + 1))) + γ y¯ (S).

(6.167)

By (6.152), (6.156) and (6.157), (γ)

2 −1 U ({wt }τt=τ , x(τ ¯ 1 ), x(τ ¯ 2 )) 1 2 −1 , x(τ ¯ 1 ), x(τ ¯ 2 )) + γ = U ({wt }τt=τ 1

τ 2 −1

y¯ (t).

(6.168)

t=τ1

Property (ii), (6.160), (6.164) and (6.168) imply that   τ 2 −1     (γ) τ2 −1 2 −1 , z , z ) − γ y ¯ (t) U ({wt }t=τ1 , z 1 , z 2 ) − U ({wt }τt=τ  1 2 1   t=τ1

≤

(γ) 2 −1 |U ({wt }τt=τ , z1, z2 ) 1

(γ) 2 −1 − U ({wt }τt=τ , x(τ ¯ 1 ), x(τ ¯ 2 ))| 1

 τ 2 −1  2 −1 + U ({wt }τt=τ , x(τ ¯ ), x(τ ¯ )) + γ y¯ (t) 1 2 1 t=τ1

−

2 −1 , z1, z2 ) U ({wt }τt=τ 1

−γ

τ 2 −1 t=τ1

≤ 1 1/4 + 1 /4.

  y¯ (t)

(6.169)

258

6 Generic Turnpike Results for the One-Dimensional RSS Model

By (6.161), (6.162) and (6.169), τ 2 −1

γ

(γ)

2 −1 wt (gt (x(t), x(t + 1))) ≥ U ({wt }τt=τ , z1, z2 ) − δ 1

t=τ1

≥

2 −1 U ({wt }τt=τ , z1, z2 ) 1

+γ

τ 2 −1

y¯ (t) − 1 /2 − δ

t=τ1

≥

τ 2 −1

wt (gt (x(t), x(t + 1))) + γ

t=τ1

τ 2 −1

y¯ (t) − 1 /2 − δ

t=τ1

and τ 2 −1

(γ)

[wt (gt (x(t), x(t + 1))) − wt (gt (x(t), x(t + 1))) − γ y¯ (t)]

t=τ1

≥ −1 /2 − δ.

(6.170)

It follows from (6.165), (6.167) and (6.170) that (γ)

− 1 ≥ w S (g S (x(S), x(S + 1))) − w S (g S (x(S), x(S + 1))) − γ y¯ (S) ≥ −1 /2 − δ ≥ −31 /4. The contradiction we have reached completes the proof of Lemma 6.22. Lemma 6.23 Let , M > 0 and let L be a natural number. Then there exists an S+k L integer k ≥ 1 such that for each integer S ≥ 0 and each program {x(t)}t=S satisfying x(S) ≤ M∗ and S+k L−1 

(γ)

wt (gt (x(t), x(t + 1))) ≥

t=S

S+k L−1 

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − M

t=S

there exists an integer j ∈ {0, . . . , k − 1} such that | y¯ (t) − gt (x(t), x(t + 1))| ≤ , t ∈ [S + j L , S + ( j + 1)L − 1]. Proof Assumption (A3), (6.154) and (6.155) imply that there exists 1 ∈ (0, /2) such that the following property holds:

(6.171)

6.4 Auxiliary Results for Theorem 6.6

259

(i) For each integer t ≥ 0 and each y ∈ [0, M∗ ] satisfying |y − y¯ (t)| ≥ , we have

(γ)

wt (y) − wt (y) − γ y¯ (t)) ≤ −1 . Choose a natural number

¯ −1 k > (M + M) 1 .

(6.172)

S+k L satisfies Assume that S ≥ 0 is an integer and that a program {x(t)}t=S

x(S) ≤ M∗

(6.173)

and S+k L−1 

(γ)

wt (gt (x(t), x(t + 1))) ≥

t=S

S+k L−1 

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − M. (6.174)

t=S

In order to complete the proof it is sufficient to show that there exists an integer j ∈ {0, . . . , k − 1} such that (6.171) holds. Assume the contrary. Then for every integer j ∈ {0, . . . , k − 1}, we have max{| y¯ (t) − gt (x(t), x(t + 1))| : t ∈ {S + j L , . . . , S + ( j + 1)L − 1}} > . (6.175) Let j ∈ {0, . . . , k − 1}. By (6.157), for all integers t ∈ { j L + S, . . . , ( j + 1)L + S − 1}, we have (γ)

wt (gt (x(t), x(t + 1)) ≤ wt (gt (x(t), x(t + 1)) + γ y¯ (t).

(6.176)

Property (i) and (6.175) imply that (γ)

max{|wt (gt (x(t), x(t + 1))) + γ y¯ (t) − wt (gt (x(t), x(t + 1))) : t ∈ {S + j L , . . . , S + ( j + 1)L − 1}} ≥ 1 . (6.177) In view of (6.176) and (6.177), 

S+( j+1)L−1

(γ)

wt (gt (x(t), x(t + 1)))

t= j L+S



S+( j+1)L−1

≤

t= j L+S

(wt (gt (x(t), x(t + 1))) + γ y¯ (t)) − 1 .

(6.178)

260

6 Generic Turnpike Results for the One-Dimensional RSS Model

Equations (6.151), (6.156), (6.173) and (6.178) imply that S+k L−1 

(γ)

wt (gt (x(t), x(t + 1)))

t=S

=

k−1 S+( j+1)L−1   j=0

≤

t= j L+S

k−1 S+( j+1)L−1  

(

j=0

=

(γ)

wt (gt (x(t), x(t + 1))) (wt (gt (x(t), x(t + 1)) + γ y¯ (t)) − 1 )

t= j L+S

S+k L−1 

(wt (gt (x(t), x(t + 1)) + γ y¯ (t)) − 1 k

t=S

≤

S+k L−1 

(wt (gt (x(t), ¯ x(t ¯ + 1))) + γ y¯ (t)) − 1 k + M¯

t=S

=

S+k L−1 

(γ) ¯ wt (gt (x(t), x(t + 1))) − 1 k + M.

(6.179)

t=S

By (6.174) and (6.179) −M+

S+k L−1 

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1)))

t=S

≤

S+k L−1 

(γ)

wt (gt (x(t), x(t + 1)))

t=S

≤

S+k L−1 

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − 1 k + M¯

t=S

and

¯ −1 k ≤ (M + M) 1 .

This contradicts (6.172). The contradiction we have reached completes the proof of Lemma 6.23. Lemma 6.24 Let , M > 0 and let L be a natural number. Then there exists an integer k ≥ 1 such that for each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + k L and each 2 program {x(t)}τt=τ satisfying 1 x(τ1 ) ≤ M∗

6.4 Auxiliary Results for Theorem 6.6

and

τ 2 −1

261

(γ)

wt (gt (x(t), x(t + 1))) ≥

t=τ1

τ 2 −1

(γ)

wt ( y¯ (t)) − M

(6.180)

t=τ1

and each integer τ for which [τ , τ + k L] ⊂ [τ1 , τ2 ] there exists an integer j ∈ {0, . . . , k − 1} such that | y¯ (t) − gt (x(t), x(t + 1))| ≤  for all integers t ∈ {τ + j L , . . . , τ + ( j + 1)L − 1}. Proof By Lemma 6.23, there exists an integer k ≥ 1 such that the following property holds: S+k L (i) For each integer S ≥ 0 and each program {x(t)}t=S satisfying x(S) ≤ M∗ and S+k L−1 

(γ)

wt (gt (x(t), x(t + 1))) ≥

t=S

S+k L−1 

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − M − 2 M¯

t=S

there exists an integer j ∈ {0, . . . , k − 1} such that for every integer t ∈ [S + j L , S + ( j + 1)L − 1], we have | y¯ (t) − gt (x(t), x(t + 1))| ≤ . Assume that τ1 ≥ 0, τ2 ≥ τ1 + k L are integers, an integer τ satisfies [τ , τ + k L] ⊂ [τ1 , τ2 ] 2 satisfies and a program {x(t)}τt=τ 1

x(τ1 ) ≤ M∗ and (6.180). Lemma 2.9, (6.158), (6.180) and (6.181) imply that

(6.181)

262 τ +k L−1 

6 Generic Turnpike Results for the One-Dimensional RSS Model (γ)

wt (gt (x(t), x(t + 1))) =

t=τ

τ 2 −1

(γ)

wt (gt (x(t), x(t + 1)))

t=τ1

 (γ) {wt (gt (x(t), x(t + 1))) : t ∈ {τ1 , . . . , τ } \ {τ }}  (γ) − {wt (gt (x(t), x(t + 1))) : t ∈ {τ + k L , . . . , τ2 } \ {τ2 }} −

≥

τ 2 −1

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − M

t=τ1

 (γ) ¯ x(t ¯ + 1))) : t ∈ {τ1 , . . . , τ } \ {τ }} − M¯ {wt (gt (x(t),  (γ) − {wt (gt (x(t), ¯ x(t ¯ + 1))) : t ∈ {τ + k L , . . . , τ2 } \ {τ2 }} − M¯ −

=

τ +k L−1 

(γ) ¯ wt (gt (x(t), ¯ x(t ¯ + 1))) − M − 2 M.

(6.182)

t=τ

By Lemma 2.9, (6.181) and (6.182), there exists an integer j ∈ {0, . . . , k − 1} such that for every integer t ∈ [τ + j L , τ + ( j + 1)L − 1] we have | y¯ (t) − gt (x(t), x(t + 1))| ≤ . Lemma 6.24 is proved. Lemma 6.25 Let  > 0. Then there exist a natural number L 0 and δ > 0 such that 2 satisfying for each pair of integers τ1 ≥ 0, τ2 ≥ L 0 + τ1 and each program {x(t)}τt=τ 1 x(τ1 ) ≤ M∗ ,

(6.183)

|x(τi ) − x(τ ¯ i )| ≤ δ, i = 1, 2,

(6.184)

and τ 2 −1

(γ)

wt (gt (x(t), x(t + 1)))

t=τ1 (γ)

2 −1 , x(τ1 ), x(τ2 )) − δ ≥ U ({wt }τt=τ 1

(6.185)

the inequality |x(t) − x(t)| ¯ ≤ holds for all integers t ∈ [τ1 , τ2 − L 0 ]. Proof Choose a positive number 1 < 64−1 (α∗−1 + 1)−1 d∗ .

(6.186)

Lemma 6.22 implies that there exist a natural number L 1 ≥ 4 and δ1 > 0 such that the following property holds:

6.4 Auxiliary Results for Theorem 6.6

263

(i) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + L 1 , each z 1 , z 2 ∈ [0, M∗ ] 2 satisfying and each program {x(t)}τt=τ 1

¯ 1 )|, |z 2 − x(τ ¯ 2 )| ≤ δ1 , |z 1 − x(τ x(τ1 ) = z 1 , x(τ2 ) ≥ z 2 and

τ 2 −1

γ

(γ)

2 −1 wt (gt (x(t), x(t + 1))) ≥ U ({wt }τt=τ , z 1 , z 2 ) − δ1 1

t=τ1

the following inequality holds for every integer t ∈ [τ1 , τ2 − 1]: | y¯ (t) − min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}| ≤ 1 . By Lemma 6.18, there exist a natural number L 2 ≥ 4 and δ2 > 0 such that M∗ (1 − d∗ ) L 2 < /8

(6.187)

and the following property holds: 2 (ii) For each pair of integers τ1 ≥ 0 and τ2 ≥ τ1 + L 2 and each program {x(t)}τt=τ 1 satisfying x(τ1 ) ≤ M∗ , τ 2 −1

(γ)

(γ)

2 −1 wt (gt (x(t), x(t + 1))) ≥ U ({wt }τt=τ , x(τ1 ), x(τ2 )) − δ2 1

t=τ1

and τ 2 −1 t=τ1

(γ)

wt (gt (x(t), x(t + 1))) ≥

τ 2 −1

γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − M¯ − 4

t=τ1

for each integer t ∈ [τ1 , τ2 − L 2 ] the following inequality holds: x(t) + 1 ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)). By Lemma 6.17, there exist a natural number L 3 ≥ 4 and δ3 > 0 such that the following property holds: (iii) For each pair of integers τ1 ≥ 0 and τ2 ≥ τ1 + L 3 and each z 1 , z 2 ∈ [0, M∗ ]

264

6 Generic Turnpike Results for the One-Dimensional RSS Model

satisfying ¯ i )| ≤ δ3 , i = 1, 2, |z i − x(τ we have (γ)

(γ)

2 −1 2 −1 , x(τ ¯ 1 ), x(τ ¯ 2 )) − U ({wt }τt=τ , z 1 , z 2 )| ≤ 4−1 . |U ({wt }τt=τ 1 1

Set δ = min{δ1 , δ2 , δ3 , /8}

(6.188)

L 0 > 4(L 1 + L 2 + L 3 ).

(6.189)

and choose a natural number

2 satisfies Assume that τ1 ≥ 0, τ2 ≥ L 0 + τ1 are integers and a program {x(t)}τt=τ 1 (6.183)–(6.185). For all integers t = τ1 , . . . , τ2 − 1 set

y(t) = gt (x(t), x(t + 1)).

(6.190)

Property (i), Lemma 2.9, (6.183)–(6.185) and (6.188)–(6.190) imply that | y¯ (t) − y(t)| ≤ 1 , t = τ1 , . . . , τ2 − 1.

(6.191)

Property (iii), (6.159), (6.183)–(6.185) and (6.189) imply that δ+

τ 2 −1

(γ)

wt (gt (x(t), x(t + 1)))

t=τ1 (γ)

2 −1 , x(τ1 ), x(τ2 )) ≥ U ({wt }τt=τ 1

(γ)

2 −1 , x(τ ¯ 1 ), x(τ ¯ 2 )) − 4−1 ≥ U ({wt }τt=τ 1

≥

τ 2 −1

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − 4−1 .

(6.192)

t=τ1

By property (ii), (6.183)–(6.185), (6.189) and (6.192), for each integer t ∈ {τ1 , . . . , τ2 − L 2 }, x(t) + 1 ≥ 1 − αt (x(t + 1) − (1 − dt )x(t)).

(6.193)

In view of (6.190) and (6.193), for each integer t ∈ [τ1 , τ2 − L 2 ], |y(t) − (1 − αt (x(t + 1) − (1 − dt )x(t)))| ≤ 1 .

(6.194)

6.4 Auxiliary Results for Theorem 6.6

265

Corollary 6.19 and (6.153) imply that for all integers t = τ1 , . . . , τ2 − 1, ¯ + 1) − (1 − dt )x(t)). ¯ y¯ (t) = 1 − αt (x(t

(6.195)

By (6.191), (6.194) and (6.195), for all integers t = τ1 , . . . , τ2 − L 2 , ¯ + 1) − (1 − dt )x(t)) ¯ − αt (x(t + 1) − (1 − dt )x(t))| |αt (x(t ≤ | y¯ (t) − y(t)| + |y(t) − (1 − αt (x(t + 1) − (1 − dt )x(t)))| ≤ 2 + 1 . (6.196) Assume that an integer τ satisfies τ 2 − L 2 ≥ τ > τ1 + L 2 . We have S := τ − L 2 − 1 ≥ τ1 .

(6.197)

By (6.196) and (6.197), for all integers t ∈ [S, S + L 2 ], ¯ − x(t)| |x(t ¯ + 1) − x(t + 1)| ≤ (1 − dt )|x(t) + αt−1 |αt (x(t ¯ + 1) − (1 − dt )x(t)) ¯ − (αt (x(t + 1) − (1 − dt )x(t)))| ≤ (1 − dt )|x(t) ¯ − x(t)| + 2α∗−1 1 .

(6.198)

By Lemma 2.9, the choice of δ, (6.186), (6.187), (6.197) and (6.198), |x(τ ) − x(τ ¯ )| ≤ M∗ (1 − d∗ )

L 2 +1

+

2α∗−1 1

∞  (1 − d∗ )i i=0

= M∗ ((1 − d∗ ) L 2 +1 + 2α∗−1 1 d∗−1 < /4. Thus |x(τ ) − x(τ ¯ )| ≤ /4

(6.199)

for all integers τ satisfying τ1 + L 2 < τ ≤ τ2 − L 2 . Assume that an integer τ satisfies τ1 ≤ τ ≤ τ1 + L 2 . In view of (6.196), for all integers t satisfying τ1 ≤ t < τ ,

(6.200)

266

6 Generic Turnpike Results for the One-Dimensional RSS Model

|x(t ¯ + 1) − x(t + 1)| ≤ (1 − dt )|x(t) ¯ − x(t)| + αt−1 |αt (x(t ¯ + 1) − (1 − dt )x(t)) ¯ − αt (x(t + 1) − (1 − dt )x(t))| ≤ (1 − d∗ )|x(t) ¯ − x(t)| + 2α∗−1 1 . It follows from the relation above, (6.184) and (6.186) that |x(τ ) − x(τ ¯ )| ≤ |x(τ1 ) − x(τ ¯ 1 )| + 2α∗−1 1

 (1 − d∗ )i i=0

≤δ+

2α∗−1 1 d∗−1

< .

Thus |x(τ ) − x(τ ¯ )| <  for all integers τ = τ1 , . . . , τ1 + L 2 . Together with (6.199) this implies that |x(τ ) − x(τ ¯ )| <  for all integers τ = τ1 , . . . , τ2 − L 2 . Lemma 6.25 is proved. Lemma 6.26 Let , M > 0. Then there exists a natural number L such that for each 2 pair of integers τ2 > τ1 ≥ 0 satisfying τ2 ≥ τ1 + L and each program {x(t)}τt=τ 1 satisfying (6.201) x(τ1 ) ≤ M∗ , and τ 2 −1

(γ)

wt (gt (x(t), x(t + 1)))

t=τ1

≥

τ 2 −1

(γ)

wt (gt (x(t), ¯ x(t ¯ + 1))) − M

t=τ1

and each integer S ∈ [τ1 , τ2 − L] the inequality min{|x(t) − x(t)| ¯ : t = S, . . . , S + L} ≤  holds. Proof Choose a positive number 1 < 64−1 (α∗−1 + 1)−1 d∗  and a natural number L¯ ≥ 4 such that

(6.202)

6.4 Auxiliary Results for Theorem 6.6

267 ¯

M∗ (1 − d∗ ) L < /8.

(6.203)

By Lemma 6.21, there exist natural numbers L 0 ≥ 4, L 1 > L¯ + L 0 + 4, k1 such that the following property holds: (i) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + k1 L 1 , 2 satisfies each program {ξ(t)}τt=τ 1

ξ(τ1 ) ≤ M∗ , τ 2 −1

(6.204)

(γ)

wt (gt (ξ(t), ξ(t + 1)))

t=τ1

≥

τ 2 −1

(γ) wt (gt (x(t), ¯ x(t ¯ + 1))) − M − 2 M¯

(6.205)

t=τ1

and each τ ∈ {τ1 , . . . , τ2 − k1 L 1 } there exists j ∈ {0, . . . , k − 1} such that ξ(t) +  ≥ 1 − αt (ξ(t + 1) − (1 − dt )ξ(t)) for all integers t ∈ [τ + j L 1 , τ + ( j + 1)L 1 − L 0 ]. By Lemma 6.24, there exists an integer k2 ≥ 1 such that the following property holds: 2 (ii) For each pair of integers τ1 ≥ 0, τ2 ≥ τ1 + k1 L 1 k2 , each program {x(t)}τt=τ 1 satisfying (6.204) and (6.205) and each integer τ ∈ {τ1 , . . . , τ2 − k2 k1 L 1 } there exists an integer j ∈ {0, . . . , k2 − 1} such that | y¯ (t) − gt (ξ(t), ξ(t + 1))| ≤ 1 for all integers t ∈ {τ + jk1 L 1 , . . . , τ + ( j + 1)k1 L 1 − 1}.

268

6 Generic Turnpike Results for the One-Dimensional RSS Model

Set L = k1 k2 L 1 .

(6.206)

2 satisfy (6.201) and Let τ1 ≥ 0 and τ2 ≥ τ1 + L be integers, a program {x(t)}τt=τ 1 (6.202) and (6.207) S ∈ {τ1 , . . . , τ2 − L}.

Let integers S2 > S1 satisfy S2 , S1 ∈ {τ1 , . . . , τ2 }. By (6.158) and (6.202), S 2 −1

(γ)

wt (gt (x(t), x(t + 1)))

t=S1

=

τ 2 −1

(γ)

wt (gt (x(t), x(t + 1)))

t=τ1

− − ≥





τ 2 −1

(γ)

{wt (gt (x(t), x(t + 1))) : t ∈ {τ1 , . . . , S1 } \ {S1 }} (γ)

{wt (gt (x(t), x(t + 1))) : t ∈ {S2 , . . . , τ2 } \ {τ2 }}

(γ) wt (gt (x(t), ¯ x(t ¯ + 1))) − M − M¯

t=τ1

− − =





S 2 −1

(γ)

{wt (gt (x(t), ¯ x(t ¯ + 1))) : t ∈ {τ1 , . . . , S1 } \ {S1 }} (γ) ¯ x(t ¯ + 1))) : t ∈ {S2 , . . . , τ2 } \ {τ2 }} − M¯ {wt (gt (x(t),

(γ) ¯ wt (gt (x(t), ¯ x(t ¯ + 1))) − M − 2 M.

t=S1

Thus we have shown that for each pair of integers S2 , S1 ∈ {τ1 , . . . , τ2 } satisfying S1 < S2 , S 2 −1 t=S1

(γ)

wt (gt (x(t), x(t + 1))) ≥

S 2 −1

(γ)

¯ (6.208) wt (gt (x(t), ¯ x(t ¯ + 1))) − M − 2 M.

t=S1

Property (ii), Lemma 2.9 (6.201) and (6.206)–(6.208) imply that there exists j2 ∈ {0, . . . , k2 − 1} such that | y¯ (t) − gt (x(t), x(t + 1))| ≤ 1

(6.209)

for all integers t ∈ {S + j2 k1 L 1 , . . . , S + ( j2 + 1)k1 L 1 − 1}. Lemma 2.9, (6.201), ( j2 +1)k1 L 1 (6.208) and property (i) applied to the program {x(t)}t=S+ j2 k1 L 1 and τ = S + j2 k1 L 1 imply that there exists j1 ∈ {0, . . . , k1 − 1} such that

6.4 Auxiliary Results for Theorem 6.6

269

x(t) + 1 ≥ 1 − αt (x(t + 1) − (1 − dt )x(t))

(6.210)

for all integers t ∈ [S + j2 L 1 k1 + j1 L 1 , S + j2 k1 L 1 + ( j1 + 1)L 1 − L 0 ]. Corollary 6.19 implies that for all integers t ≥ 0, ¯ + 1) − (1 − dt )x(t)). ¯ y¯ (t) = 1 − αt (x(t

(6.211)

For every integer t ∈ {τ1 , . . . , τ2 − 1} set y(t) = min{x(t), 1 − αt (x(t + 1) − (1 − dt )x(t))}.

(6.212)

By (6.210) and (6.212), |y(t) − (1 − αt (x(t + 1) − (1 − dt )x(t))| ≤ 1

(6.213)

for all integers t = S + j2 k1 L 1 + j1 L 1 , . . . , S + j2 k1 L 1 + ( j1 + 1)L 1 − L 0 . In view of (6.209) and (6.213), for all integers t = S + j2 k1 L 1 + j1 L 1 , . . . , S + j2 k1 L 1 + ( j1 + 1)L 1 − L 0 , ¯ + 1) − (1 − dt )x(t)) ¯ − (αt (x(t + 1) − (1 − dt )x(t))| |αt (x(t ≤ | y¯ (t) − y(t)| + |y(t) − (1 − αt (x(t + 1) − (1 − dt )x(t)))| ≤ 1 + 1 .

(6.214)

Equation (6.214) implies that for all integers t = S + j2 k1 L 1 + j1 L 1 , . . . , S + j2 k1 L 1 + ( j1 + 1)L 1 − L 0 , ¯ + 2α∗−1 1 . |x(t + 1) − x(t ¯ + 1)| ≤ (1 − dt )|x(t) − x(t)|

(6.215)

It follows from Lemma 2.9, (6.201)–(6.203) and (6.215) that ¯ + j2 k1 L 1 + ( j1 + 1)L 1 − L 0 )| |x(S + j2 k1 L 1 + ( j1 + 1)L 1 − L 0 ) − x(S ≤ M∗ (1 − d∗ ) L 1 −L 0 −2 + 2α∗−1 1

∞  (1 − d∗ )i i=0

L¯

≤ M∗ (1 − d∗ ) + Lemma 6.26 is proved.

2α∗−1 1 d∗−1

< .

270

6 Generic Turnpike Results for the One-Dimensional RSS Model

6.5 Proof of Theorem 6.6 By Lemma 6.25, there exist a natural number L 0 and δ > 0 such that the following property holds: 2 (i) For each pair of integers τ1 ≥ 0, τ2 ≥ L 0 + τ1 and each program {x(t)}τt=τ 1 satisfying x(τ1 ) ≤ M∗ ,

(6.216)

|x(τi ) − x(τ ¯ i )| ≤ δ, i = 1, 2,

(6.217)

and τ 2 −1

(γ)

wt (gt (x(t), x(t + 1)))

t=τ1 (γ)

2 −1 , x(τ1 ), x(τ2 )) − δ ≥ U ({wt }τt=τ 1

(6.218)

the inequality |x(t) − x(t)| ¯ ≤ holds for all integers t ∈ [τ1 , τ2 − L 0 ]. By Lemma 6.26, there exists a natural number L 1 such that the following property holds: 2 (ii) For each pair of integers τ1 ≥ 0, τ2 ≥ τ2 + L 1 and each program {x(t)}τt=τ 1 satisfying (6.202) and (6.216) and each integer S ∈ [τ1 , τ2 − L 1 ] the inequality min{|x(t) − x(t)| ¯ : t = S, . . . , S + L 1 } ≤ δ holds. Set L = 2(L 0 + L 1 ).

(6.219)

2 Assume that τ1 ≥ 0, τ2 ≥ τ1 + 2L are integers and that a program {x(t)}τt=τ 1 satisfies (6.202), (6.216) and (6.218). Property (ii) and (6.219) imply that there exist

S1 ∈ {τ1 , . . . , τ1 + L 1 }, S2 ∈ {τ2 − L 1 , . . . , τ2 }

(6.220)

¯ i )| ≤ δ, i = 1, 2. |x(Si ) − x(S

(6.221)

such that

By Lemma 2.9, (6.216) and (6.218)–(6.221), |x(t) − x(t)| ¯ ≤ , t = τ1 + L 1 , . . . , τ2 − L 0 − L 1 .

6.5 Proof of Theorem 6.6

271

This implies that |x(t) − x(t)| ¯ ≤ , t = τ1 + L , . . . , τ2 − L . Theorem 6.6 is proved.

6.6 Proof of Theorem 6.10 Let v = {vt }∞ t=0 ∈ B and n be a natural number. By Proposition 6.9, there exist an open neighborhood U(v, n) of v in A, a natural number L(v, n) and δ(v, n) > 0 such that the following property holds: (i) For each u = {u t }∞ t=0 ∈ U(v, n), each pair of integers T1 ≥ 0, T2 > 2L(v, n) + T2 satisfying T1 and each program {x(t)}t=T 1 x(T1 ) ≤ n, T 2 −1

u t (gt (x(t), x(t + 1)))

t=T1 T2 −1 ≥ U ({u t }t=T , x(T1 ), x(T2 )) − δ(v, n), 1 T 2 −1

T2 −1 u t (gt (x(t), x(t + 1))) ≥ U ({u t }t=T , x(T2 − L(v, n))) − n, 2 −L(v,n)

t=T2 −L(v,n)

the inequality

|x(t) − x (v) (t)| ≤ n −1

holds for all integers t ∈ [T1 + L(v, n), T2 − L(v, n)]. Define ∞  

 {U(v, n) : v ∈ B} ∩ A0 . F=

(6.222)

n=1

Clearly, F is a countable intersection of open everywhere dense sets in (A, dA ). Let w = {wt }∞ t=0 ∈ F and

 > 0, M1 > 0, M0 > 1 + (α∗ d∗ )−1 .

We may assume without loss of generality that M0 > x (w) (0)

(6.223)

272

6 Generic Turnpike Results for the One-Dimensional RSS Model

and that for each pair of integers T2 > T1 ≥ 0, T 2 −1

T2 −1 M0 ({wt }t=T wt (gt (x (w) (t), x (w) (t + 1))) ≥ U ) − M1 . 1

t=T1

Choose an integer

n > max{M0 + M1 + 1, 4−1 }.

By (6.222) and (6.223), there exists v = {vt }∞ t=0 ∈ B such that

{wt }∞ t=0 ∈ U(v, n).

Property (i) and the inclusion above imply that |x (w) (t) − x (v) (t)| ≤ n −1 for all integers t ≥ L(v, n). Assume that

(6.224)

u = {u t }∞ t=0 ∈ U(v, n),

T2 T1 ≥ 0, T2 ≥ T1 + 2L(v, n) are integers and that a program {x(t)}t=T satisfies 1

x(T1 ) ≤ M0 , T 2 −1

u t (gt (x(t), x(t + 1)))

t=T1 T2 −1 , x(T1 ), x(T2 )) − δ(v, n), ≥ U ({u t }t=T 1 T 2 −1

T2 −1 u t (gt (x(t), x(t + 1))) ≥ U ({u t }t=T , x(T2 − L(v, n))) − M1 . 2 −L(v,n)

t=T2 −L(v,n)

Property (i) and the relation above imply that |x(t) − x (v) (t)| ≤ n −1 for all integers t ∈ [T1 + L(v, n), T2 − L(v, n)]. This implies that |x(t) − x (w) (t)| ≤ 2n −1 <  for all integers t ∈ [T1 + L(v, n), T2 − L(v, n)]. This completes the proof of Theorem 6.10.

Chapter 7

The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

Abstract In this chapter we study the Robinson–Shinkai–Leontief model. We are interested in the turnpike pnenomenon and in the existence of solutions of the corresponding infinite horizon problems.

7.1 The Model Description and Preliminaries 1 Let R 1 (R+ ) be the set of real (non-negative) numbers and let R n be a finitedimensional Euclidean space with non-negative orthant n = {x ∈ R n : xi ≥ 0, i = 1, . . . , n}. R+

For any x, y ∈ R n , let the inner product xy =

n 

xi yi

i=1

and x  y, x > y, x ≥ y have their usual meaning. Let e(i), i = 1, . . . , n, be the n all of whose coordinates are unity. ith unit vector in R n , and e be an element of R+ n For any x ∈ R , let x denote the Euclidean norm of x. Assume that b > 0, a I > 0, aC > 0, d ∈ (0, 1], a I d < b.

(7.1)

We now give a formal description of the technological structure. 2 A sequence {x(t), y(t)}∞ t=0 ⊂ R is called a program if for each integer t ≥ 0

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. J. Zaslavski, Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model, Monographs in Mathematical Economics 4, https://doi.org/10.1007/978-981-16-2252-6_7

273

274

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model 1 1 (x(t), y(t)) ∈ R+ × R+ , x(t + 1) ≥ (1 − d)x(t),

a I b−1 (x(t + 1) − (1 − d)x(t)) + aC y(t) ≤ x(t), b−1 (x(t + 1) − (1 − d)x(t)) + y(t) ≤ 1.

(7.2)

Let T1 , T2 be integers such that 0 ≤ T1 < T2 . A pair of sequences T2 T2 −1 , {y(t)}t=T ) ({x(t)}t=T 1 1 1 is called a program if x(T2 ) ∈ R+ and for each integer t satisfying T1 ≤ t < T2 relations (7.2) hold. Set 1 1 × R+ : x − (1 − d)x ≥ 0 Ω = {(x, x ) ∈ R+

and x ≤ b min{1, a −1 I x} + (1 − d)x}.

(7.3)

1 given by We have a correspondence Λ : Ω → R+ 1 Λ(x, x ) = {y ∈ R+ : y ≤ aC−1 [x − b−1 a −1 I (x − (1 − d)x)]

and y ≤ 1 − b−1 [x − (1 − d)x]}.

(7.4)

The results of this chapter were obtained in [120]. The model was also studied in [20, 82]. Lemma 7.1 Let M0 > bd −1 , (x, x ) ∈ Ω and x ≤ M0 . Then x ≤ M0 . Proof By (7.3), x ≤ (1 − d)x + b min{1, a −1 I x} ≤ (1 − d)M0 + b ≤ (1 − d)M0 + M0 d = M0 . Lemma 7.1 is proved. It is clear that the set Ω is convex and that the graph of the mapping Λ {(x, y, x ) : (x, x ) ∈ Ω, y ∈ Λ(x, x )} is also convex. Let a function w : [0, ∞) → [0, ∞) be a continuous strictly increasing concave and differentiable. For any (x, x ) ∈ Ω define u(x, x ) = max{w(y) : y ∈ Λ(x, x )}. It is clear that the function u is concave.

(7.5)

7.1 The Model Description and Preliminaries

275

1 A golden-rule stock is  x ∈ R+ such that ( x, x ) is a solution to the problem

maximize u(x, x ) subject to (i) x ≥ x; (ii) (x, x ) ∈ Ω. Theorem 7.2 There exists a unique golden-rule stock  x = aC b(b + d(aC − a I ))−1 .

(7.6)

Moreover, the problem w(y) → max, y ∈ Λ( x, x) has a unique solution  y = x (b − a I d)(aC b)−1 .

(7.7)

Proof First we show that ( x, x ) ∈ Ω. Clearly  x > 0. By (7.1),  x − (1 − d) x = d x = aC bd(b + d(aC − a I ))−1 ≤ min{b, ba −1 x }. I  Thus ( x, x ) ∈ Ω. Clearly, the problem w(y) → max, y ∈ Λ( x, x ) has a unique solution  y such that x − b−1 a I ( x − (1 − d) x )], 1 − b−1 [ x − (1 − d) x ]}  y = min{aC−1 [ x b−1 (b − a I d), 1 − b−1 d x }. = min{aC−1 By the relation above and (7.6), x (b − a I d) = 1 − b−1 d x.  y = (aC b)−1

(7.8)

(x, x ) ∈ Ω, x ≥ x,

(7.9)

Assume now that





y ∈ Λ(x, x ), u(x, x ) = w(y) ≥ w( y). Clearly, x = x, y = min{aC−1 [x − b−1 a I (x − (1 − d)x)], 1 − b−1 [x − (1 − d)x]} = min{aC−1 xb−1 (b − a I d), 1 − b−1 d x}.

(7.10)

276

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

If x >  x , then by (7.8) and (7.10), x ) = w( y). w(y) ≤ w(1 − b−1 d x) < w(1 − b−1 d This contradicts (7.9) and therefore x ≤ x.

(7.11)

If x <  x , then by (7.8) and (7.10), x b−1 (b − a I d)) = w( y). w(y) ≤ w(aC−1 xb−1 (b − a I d)) < w(aC−1 This contradicts (7.9) and therefore x ≥  x . Together with (7.11) this implies that  x = x. Together with (7.8) and (7.10) this implies that y =  y. Theorem 7.2 is proved. Put  q = (b + d(aC − a I ))−1 ,  p = w ( y) q.

(7.12)

Lemma 7.3 For any (x, x ) ∈ Ω and any y ∈ Λ(x, x ) px w( y) ≥ w(y) +  px −  and

y ≤ y + q (x − x ).

Proof Let (x, x ) ∈ Ω, y ∈ Λ(x, x ). By the concavity of w, y)(y −  y). w(y) − w( y) ≤ w (

(7.13)

Define a function u 0 (x, x ) = sup{y : y ∈ Λ(x, x )}, (x, x ) ∈ Ω. x, x ) is an interior point of the convex set Ω. Therefore Clearly, u 0 is concave and ( by the definition of  x and the well-known fact of convex analysis [67, 76] there is q ∈ R 1 such that x, x ) + q(x − x ) for all (x, x ) ∈ Ω. u 0 (x, x ) ≤ u( Combined with (7.6) and (7.7) this implies that

7.1 The Model Description and Preliminaries

277

y ≤ y + q(x − x ) = (b − da I )(b + d(aC − a I ))−1 + q(x − x ) for all (x, x ) ∈ Ω and all y ∈ Λ(x, x ).

(7.14)

Applying (7.14) with x = a I , y = 0, x = b + (1 − d)x we obtain that 0 ≤ (b − da I )(b + d(aC − a I ))−1 + q(da I − b), q ≤ (b + d(aC − a I ))−1 .

(7.15)

Applying (7.14) with x = aC , x = (1 − d)x, y = 1 we obtain that 1 ≤ (b − da I )(b + d(aC − a I ))−1 + qdaC , q ≥ (daC )−1 [1 − (b − da I )(b + d(aC − a I ))−1 ] = (b + d(aC − a I ))−1 = q. Together with (7.15) this implies that q. q = (b + d(aC − a I ))−1 =  Combined with (7.14) this implies that for all (x, x ) ∈ Ω and all y ∈ Λ(x, x ) y ≤ y + q (x − x ). Together with (7.13) this implies that for all (x, x ) ∈ Ω and all y ∈ Λ(x, x ) y)(y −  y) ≤ w ( y) q (x − x ) =  p (x − x ). w(y) − w( y) ≤ w ( Lemma 7.3 is proved. For any (x, x ) ∈ Ω and any y ∈ Λ(x, x ) set p (x − x ) − (w(y) − w( y)). δ(x, y, x ) =  By Lemma 7.3,

(7.16)

δ(x, y, x ) ≥ 0

for all (x, x ) ∈ Ω and for all y ∈ Λ(x, x ). Set ξ = 1 − d − b(aC − a I )−1 .

(7.17)

278

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

Remark 7.4 Note that if (x, x ) ∈ Ω and y ∈ Λ(x, x ), then the following equations are equivalent: y = y + q (x − x ); y + d q x. y + q (x − (1 − d)x) =  The following three propositions will be proved in Sect. 7.4. Proposition 7.5 Let aC > a I , (x, x) ¯ ∈ Ω and y¯ ∈ Λ(x, x). ¯ Then y¯ =  y + q (x − x) ¯

(7.18)

if and only if x ∈ [a I , aC ], x¯ = (aC b)(aC − a I )−1 + ξx

(7.19)

and y¯ = min{aC−1 [x − b−1 a I (x¯ − (1 − d)x)], 1 − b−1 [x¯ − (1 − d)x]}. ¯ ∈ Ω and y¯ ∈ Λ(x,  x ). Then if Proposition 7.6 Let aC = a I , (x, x) y¯ =  y + q (x −  x ), then x = a I . Proposition 7.7 Let aC < a I , (x, x) ¯ ∈ Ω and y¯ ∈ Λ(x, x). ¯ Then y¯ =  y + q (x − x) ¯ if and only if x ∈ [aC , a I ], x¯ satisfies (7.19) and y¯ satisfies (7.20). Using Propositions 7.5–7.7 we prove the following result. Theorem 7.8 1. Let aC = a I . Then the von Neumann facet {(x, x ) ∈ Ω : there is y ∈ Λ(x, x ) such that δ(x, y, x ) = 0} is a subset of

and

{(x, x ) ∈ Ω : x ∈ [min{aC , a I }, max{aC , a I }] x = (aC b)(aC − a I )−1 + ξx}.

(7.20)

7.1 The Model Description and Preliminaries

279

2. Let aC = a I . Then the facet is a subset of {(x, x ) ∈ Ω : x = aC }. 3. If the function w is strictly concave, then the facet is the singleton {( x, x )}. Proof By concavity of w, (7.12) and Lemma 7.3, for any (x, x ) ∈ Ω and any y ∈ Λ(x, x ), y) − w(y) +  p (x − x ) δ(x, y, x ) = w( ≥ p (x − x ) − w ( y)(y −  y) = w ( y)[ q (x − x ) − y +  y] ≥ 0 and

q (x − x ) − y +  y = 0. if δ(x, y, x ) = 0, then 

Therefore the von Neumann facet is a subset of y + q (x − x )}, {(x, x ) ∈ Ω : there is y ∈ Λ(x, x ) such that y =  which in its turn by Lemma 7.1 and Proposition 7.5 is a subset of {(x, x ) ∈ Ω : x ∈ [min{a I , aC }, max{a I , aC }] and

x = (aC b)(aC − a I )−1 + ξx}.

If aC = a I , then assertion 2 follows from Proposition 7.6. Clearly, if w is strictly concave, then the facet is { x, x )}. Theorem 7.8 is proved. 1 A program {x(t), y(t)}∞ t=0 is called good if there is M ∈ R such that T −1 

w(y(t)) − T w( y) ≥ M for all integers T ≥ 1.

t=0

A program {x(t), y(t)} is called bad if T −1  lim [w(y(t)) − T w( y)] = −∞.

T →∞

t=0

Proposition 7.9 (i) Any program that is not good is bad. 1 (ii) For every initial state x0 ∈ R+ \ {0} there exists a good program {x(t), y(t)}∞ t=0 such that x(0) = x0 . Moreover, for any Δ > 0 there is M > 0 such that for each x ≥ Δ there is a program {x(t), y(t)}∞ t=0 such that x(0) = x and that for all integers T ≥ 1,

280

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model T −1 

w(y(t)) − T w( y) ≥ −M.

(7.21)

t=0

(iii) Any program {x(t), y(t)}∞ t=0 is good if and only if ∞ 

δ(x(t), y(t), x(t + 1)) < ∞.

t=0

Proof Assertions (i) and (iii) follow from the fact that δ(·, ·, ·) is nonnegative and Lemma 7.1. Namely, assume that {x(t), y(t)}∞ t=0 is a program. Then for each natural number T it follows from (7.16) that T −1 

w(y(t)) − T w( y) = −

t=0

T −1 

δ(x(t), y(t), x(t + 1)) +  p (x(0) − x(T )).

t=0

Since by Lemma 7.1 sup{x(t) : t = 0, 1, . . . } < ∞ we conclude that assertions (i) and (iii) hold. Let us prove assertion (ii). Let Δ > 0. We may assume that Δ < a I . For each z > 0 set x(z, ˜ 0) = z and ˜ t)} x(z, ˜ t + 1) = (1 − d)x(z, ˜ t) + b min{1, a −1 I x(z,

(7.22)

for all integers t ≥ 0. Clearly, for each z > 0 and each integer t ≥ 0, (x(z, ˜ t), x(z, ˜ t+ 1)) ∈ Ω and the following property holds: (P1) If 0 < z 1 < z 2 , then for all integers t ≥ 0 ˜ 2 , t). x(z ˜ 1 , t) ≤ x(z ˜ 0) < x(Δ, ˜ 1). Since Δ < a I it follows from (7.1) that x(Δ, is increasing. This implies that the sequence {x(Δ, ˜ t)}∞ t=0 By Lemma 7.1 this sequence is bounded and therefore it converges to a number Δ˜ > Δ which satisfies ˜ Δ˜ = (1 − d)Δ˜ + b min{1, a −1 I Δ}.

(7.23)

Δ˜ ≥ a I and Δ˜ = bd −1 .

(7.24)

By (7.21) and (7.23),

By (7.1) and (7.6),

7.1 The Model Description and Preliminaries

281

bd −1 −  x = bd −1 − aC b(b + d(aC − a I ))−1 = d −1 (b + d(aC − a I ))−1 [b(b + d(aC − a I )) − aC bd] = d −1 (b + d(aC − a I ))−1 [b(b − da I )] > 0. By (7.24) and the relation above there is a natural number τ such that x(Δ, ˜ τ) >  x.

(7.25)

Let x0 ≥ Δ. We construct a trajectory {x(t), y(t)}∞ t=0 such that x(0) = x 0 and (7.21) holds for all integers T ≥ 1 with M = 1 + τ w( y).

(7.26)

x. x(x ˜ 0, τ ) > 

(7.27)

By (7.25) and (P1),

Define x(t) = x(x ˜ 0 , t), t = 0, . . . , τ , y(t) = 0, t = 0, . . . , τ − 1.

(7.28)

−1 Clearly, ({x(t)}τt=0 , {y(t)}τt=0 ) is a program and

x(τ ) >  x. For all integers t ≥ τ set y(t) =  y, x(t + 1) = (1 − d)x(t) + d x. It is not difficult to verify by induction that {x(t), y(t)}∞ t=0 is a program. It is easy to see that for any natural number T T −1 

w(y(t)) − T w( y) ≥ −τ w( y) > −M.

t=0

Thus assertion (ii) holds. Proposition 7.9 is proved. Let x0 > 0. Set Δ(x0 ) = inf

 ∞

δ(x(t), y(t), x(t + 1)) : {x(t), y(t)}∞ t=0 is a

t=0

 program starting from x0 .

(7.29)

282

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

Since there exists a good program from x0 , it follows from Proposition 7.9 that Δ(x0 ) < ∞.

(7.30)

∞ Proposition ∞7.10 Let x0 > 0. Then there exists a program {x(t), y(t)}t=0 from x0 such that t=0 δ(x(t), y(t), x(t + 1)) = Δ(x0 ).

Proof The proposition is proved as Proposition 8 of [36]. (See also Proposition 2.10 of [126].) For any natural number i there is a program {x (i) (t), y (i) (t)}∞ t=0 from x 0 such that ∞  δ(x (i) (t), y (i) (t), x (i) (t + 1)) ≤ Δ(x0 ) + i −1 . t=0

Extracting a subsequence, re-indexing and using the diagonalization process we may assume without loss of generality that there is a program {x(t), y(t)}∞ t=0 such that for all integers t ≥ 0, x(t) = lim x (i) (t), y(t) = lim y (i) (t). i→∞

i→∞

It is easy to see that ∞ 

δ(x(t), y(t), x(t + 1)) = Δ(x0 ).

t=0

Proposition 7.10 is proved. Theorem 7.11 For each good program {x(t), y(t)}∞ x, y) t=0 , lim t→∞ (x(t), y(t)) = ( if at least one of the following assumptions holds: (i) the function w is strictly concave; (ii) ξ = −1. Theorem 7.11 is proved exactly as Theorems 2 and 3 of [92]. (See also Theorems 3.1 and 3.2 of [126].) A program {x ∗ (t), y ∗ (t)}∞ t=0 is called overtaking optimal if lim sup T →∞

T  [w(y(t)) − w(y ∗ (t))] ≤ 0 t=0

∗ for every program {x(t), y(t)}∞ t=0 satisfying x(0) = x (0). ∞ ∗ ∗ A program {x (t), y (t)}t=0 is called weakly optimal if for any program

{x(t), y(t)}∞ t=0 satisfying x(0) = x ∗ (0) the following inequality holds:

7.1 The Model Description and Preliminaries

lim inf T →∞

T 

283

[w(y(t)) − w(y ∗ (t))] ≤ 0.

t=0

Analogously to Theorem 4 of [92] (see also Theorem 3.3 of [126]) we can prove the following result. Theorem 7.12 Assume that for each good program {x(t), y(t)}∞ t=0 lim (x(t), y(t)) = ( x, y).

t→∞

1 Then for each initial stock x0 ∈ R+ there exists an overtaking optimal program ∞ {x(t), y(t)}t=0 which satisfies x(0) = x0 .

Analogously to Proposition 9 of [36] (see also Theorem 2.12 of [126]) we can prove the following result. Theorem 7.13 For each initial stock x0 ≥ 0 there exists a weakly optimal program {x(t), y(t)}∞ t=0 which satisfies x(0) = x 0 .

7.2 Turnpike Results for a General Model 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 T Ω˜ is understood) if (xt , xt+1 ) ∈ Ω˜ for all integers t ≥ 0. A sequence {xt }t=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]. Set v = sup{|v(x, y)| : x, y ∈ X }. For each x ∈ X and each integer T ≥ 1 set σ(v, T, x) = sup

T −1 

 v(xi , xi+1 ) :

T {xi }i=0

i=0

σ(v, T ) = sup

T −1 

v(xi , xi+1 ) :

is a program and x0 = x , 

T {xi }i=0

is a program .

i=0

(Here we use the convention that the supremum of an empty set is −∞). In this section we suppose that there exist x¯ ∈ X and a constant c¯ > 0 such that the following assumptions hold. (A1) (x, ¯ x) ¯ is an interior point of Ω˜ (there is  > 0 such that {(x, y) ∈ X × X : ˜ and v is continuous at (x, ρ(x, x), ¯ ρ(y, x) ¯ ≤ } ⊂ Ω) ¯ x). ¯ (A2) σ(v, T ) ≤ T v(x, ¯ x) ¯ + c¯ for all integers T ≥ 1. In this section we present the results of [96]. They are also discussed in [121].

284

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

Proposition 7.14 For each program {xt }∞ t=0 either the sequence T −1 

∞ v(xt , xt+1 ) − T v(x, ¯ x) ¯ T =1

t=0

is bounded or lim T →∞ [

T −1 t=0

v(xt , xt+1 ) − T v(x, ¯ x)] ¯ = −∞.

A program {xt }∞ t=0 is called (v)-good if the sequence T −1 

∞ v(xt , xt+1 ) − T v(x, ¯ x) ¯ T =1

t=0

is bounded. In this section we suppose that the following assumption holds. (A3) (the asymptotic turnpike property) For every (v)-good program {xt }∞ t=0 , ¯ = 0. limt→∞ ρ(xt , x) In view of (A3), v > 0. For each M > 0 denote by X M the set of all x ∈ X for which there exists a program {xt }∞ t=0 such that x 0 = x and that for all integers T ≥ 1 T −1 

v(xt , xt+1 ) − T v(x, ¯ x) ¯ ≥ −M.

t=0

We state the following turnpike result. Theorem 7.15 Let , M be positive numbers. Then there exist a natural number L T and a positive number δ such that for each integer T > 2L and each program {xt }t=0 which satisfies T −1  x0 ∈ X M , v(xt , xt+1 ) ≥ σ(v, T, x0 ) − δ t=0

there exist nonnegative integers τ1 , τ2 ≤ L such that ρ(xt , x) ¯ ≤  for all t = τ1 , . . . , ¯ ≤ δ, then τ1 = 0. T − τ2 and if ρ(x0 , x) In the sequel we use a notion of an overtaking optimal program. A program {xt }∞ optimal if for each program {yt }∞ t=0 is called (v)-overtaking t=0 T −1 satisfying y0 = x0 the inequality lim supT →∞ t=0 [v(yt , yt+1 ) − v(xt , xt+1 )] ≤ 0 holds. The following result establishes the existence of an overtaking optimal program. Theorem 7.16 Assume that x ∈ X and that there exists a (v)-good program {xt }∞ t=0 such that x0 = x. Then there exists an (v)-overtaking optimal program {xt∗ }∞ t=0 such that x0∗ = x.

7.2 Turnpike Results for a General Model

285

The next theorem is a refinement of Theorem 7.15. According to Theorem 7.15 we have τ2 ≤ L, where the constant L depends on M and . The next theorem shows that τ2 ≤ L 0 , where the constant L 0 depends only on . Theorem 7.17 Let  > 0. Then there exists a natural number L 0 such that for each M > 0 there exist a natural number L > L 0 and δ > 0 such that the following assertion holds: T which satisfies x0 ∈ X M and For each integer T > 2L and each program {xt }t=0 T −1 

v(xt , xt+1 ) ≥ σ(v, T, x0 ) − δ

t=0

there exist integers τ1 ∈ [0, L], τ2 ∈ [0, L 0 ] such that ρ(xt , x) ¯ ≤  for all t = ¯ ≤ δ, then τ1 = 0. τ1 , . . . , T − τ2 and if ρ(x0 , x) The following result provides necessary and sufficient conditions for overtaking optimality. Theorem 7.18 Let {xt }∞ t=0 be a program such that x0 ∈ ∪{X M : M ∈ (0, ∞)}. Then the program {xt }∞ t=0 is (v)-overtaking optimal if and only if the following condi¯ = 0; (ii) for each natural number T and each program tions hold: (i) limt→∞ ρ(xt , x) T satisfying y0 = x0 , yT = x T the inequality {yt }t=0 T −1  t=0

v(yt , yt+1 ) ≤

T −1 

v(xt , xt+1 )

t=0

holds. The next two theorems establish uniform convergence of overtaking optimal programs to x. ¯ Theorem 7.19 Assume that the function v is continuous and let  > 0. Then there exists δ > 0 such that for each (v)-overtaking optimal program {xt }∞ t=0 satisfying ¯ ≤ δ the inequality ρ(xt , x) ¯ ≤  holds for all integers t ≥ 0. ρ(x0 , x) Theorem 7.20 Assume that the function v is continuous and let M,  > 0. Then there exists a natural number L such that for each (v)-overtaking optimal program {xt }∞ ¯ ≤  holds for all integers t ≥ L. t=0 satisfying x 0 ∈ X M the inequality ρ(x t , x) Example 7.21 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 , x¯ ∈ X , (x, ¯ x) ¯ be an interior point of Ω˜ 1 and let v : X × X → R be a strictly concave continuous function such that

286

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

˜ v(x, ¯ x) ¯ = sup{v(z, z) : z ∈ X and (z, z) ∈ Ω}. We assume that there is r¯ > 0 such that ˜ ¯ y − x ¯ ≤ r¯ } ⊂ Ω. {(x, y) ∈ R n × R n : x − x, Then (A1), (A2) and (A3) hold.

7.3 Main Results In this section we consider the Robinson–Shinkai–Leontief model considered in Sect. 7.1. We use the notation from the previous sections. Fix D > bd −1 and set X = [0, D]. Put Ω˜ = Ω ∩ (X × X ) and consider the function u defined by (7.5). Since Ω is convex the set Ω˜ is also convex. Clearly, Ω˜ is closed. It was mentioned in Sect. 7.1 that u is concave. Clearly, u is bounded. It is not difficult to see that u is upper semicontinuous. First we show that the results of Sect. 7.2 hold for the model considered in this section with v = u and x¯ =  x. Proposition 7.22 (A1) holds with x¯ =  x , v = u. Namely, ( x, x ) is an interior point of Ω˜ and u is continuous at ( x, x ). ˜ Let us show that u is continuous at Proof Clearly, ( x, x ) is an interior point of Ω. ( x, x ). We already know that u is upper semicontinuous. Let  > 0. Therefore it is x |, |x − sufficient to show that there is δ > 0 such that if (x, x ) ∈ Ω˜ satisfies |x −   x | ≤ δ, then x, x ) − . u(x, x ) ≥ u( y) such that There is 0 ∈ (0,  y) − /4. w( y − 0 ) > w( It is not difficult to see that there is δ > 0 such that for each (x, x ) ∈ R 2 satisfying x | ≤ δ we have |x −  x | ≤ δ and |x −  ˜  y − 0 ∈ Λ(x, x ). (x, x ) ∈ Ω, x | ≤ δ, |x −  x | ≤ δ, then This implies that if (x, x ) ∈ R 2 satisfies |x −  y − 0 ) > w( y) − /4. u(x, x ) ≥ w( This completes the proof of Proposition 7.22.

7.3 Main Results

287

Proposition 7.23 (A2) holds. Proof For each natural number T ≥ 1 and each program T −1 T ({x(t)}t=0 , {y(t)}t=0 )

by (7.16) and Lemma 7.3, T −1

w(y(t)) − T u( x, x) =

t=0

T −1

w(y(t)) − T w( y)

t=0

= p (x(0) − x(T )) −

T −1

δ(x(t), y(t), x(t + 1))

t=0

≤ p (x(0) − x(T )) ≤  p (x(0) + x(T )) ≤ 2n D( p ).

This implies the validity of Proposition 7.23. In the sequel in this section we assume that for each good program {x(t), y(t)}∞ t=0 lim (x(t), y(t)) = ( x, y).

t→∞

Then we can see that the assumptions (A1)–(A3) hold and Theorems 7.15–7.18 also hold for the model. In this section we will establish a refinement of Theorem 7.17. In order to meet this goal we need the following lemma. Lemma 7.24 Let  > 0. Then there exists δ > 0 such that for each (x, x ) ∈ Ω and each y ∈ Λ(x, x ) satisfying x| ≤ δ |x −  x |, |x −  and

w(y) ≥ u(x, x ) − δ

the inequality |y −  y| ≤  holds. Proof Assume the contrary. Then for each integer k ≥ 1 there are (xk , xk ) ∈ Ω, yk ∈ Λ(xk , xk ) such that x | ≤ k −1 , |xk −  x | ≤ k −1 , |xk −  w(yk ) ≥ u(xk , xk ) − 1/k, |yk −  x | > .

(7.31)

288

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

Extracting a subsequence and re-indexing we may assume without loss of generality that there exist y˜ = lim yk . k→∞

Clearly, | y˜ −  y| ≥ .

(7.32)

In view of (7.31), (7.32) and (A1), y˜ ∈ Λ( x, x ), x, x ) = w( y). w( y˜ ) = lim w(yk ) ≥ lim sup u(xk , xk ) = u( k→∞

k→∞

Together with Theorem 7.2 this implies that y˜ =  y. This contradicts (7.32). The contradiction we have reached proves Lemma 7.24. Now we prove the following turnpike result. Theorem 7.25 Let 0 > 0. Then there exists a natural number L 0 such that for each γ ∈ (0,  x /4) there exist a natural number L > L 0 and δ > 0 such that for each T −1 T , {y(t)}t=0 ) which satisfies integer T > 2L and each program ({x(t)}t=0 x(0) ∈ [γ, D], T −1 

w(y(t)) ≥ σ(u, T, x(0)) − δ

(7.33)

t=0

there are integers τ2 ∈ [0, L 0 ], τ1 ∈ [0, L 1 ] such that |x(t) −  x | ≤ 0 for all t = τ1 , . . . , T − τ2 , |y(t) −  y| ≤ 0 for all t = [τ1 , T − τ2 − 1] and if |x(0) −  x | ≤ δ, then τ1 = 0. Proof By Lemma 7.24, there is  ∈ (0, 0 /4) such that the following property holds: (P2) If (x, x ) ∈ Ω and y ∈ Λ(x, x ) satisfy x| ≤  |x −  x |, |x −  and

w(y) ≥ u(x, x ) − ,

(7.34)

7.3 Main Results

289

then |y −  y| ≤ 0 /2. Let a natural number L 0 be as guaranteed by Theorem 7.17. Assume that γ ∈ (0,  x /4).

(7.35)

By Proposition 7.6 (ii) there is a number M > 0 such that for each x0 ∈ [γ, D] there is a good program {x(t), y(t)}∞ t=0 satisfying x(0) = x 0 we have T −1 

w(y(t)) − T w( y) ≥ −M for all integers T ≥ 1.

t=0

This implies that [γ, D] ⊂ X M .

(7.36)

By the choice of L 0 and Theorem 7.17, there exist a natural number L > L 0 and δ ∈ (0, /4) such that the following assertion holds: T which satisfies (C) For each integer T > 2L and each program {x(t), y(t)}t=0 x(0) ∈ X M and

T −1 

u(x(t), x(t + 1)) ≥ σ(u, T, x(0)) − δ

t=0

there exist integers τ1 ∈ [0, L], τ2 ∈ [0, L 0 ] such that |x(t) −  x | ≤  for all t = τ1 , . . . , T − τ2

(7.37)

if |x(0) −  x | ≤ δ then τ1 = 0.

(7.38)

and

T −1 T Assume that an integer T > 2L and a program ({x(t)}t=0 , {y(t)}t=0 ) satisfies (7.33). By (7.33), (7.36) and (C) there are integers τ1 ∈ [0, L], τ2 ∈ [0, L 0 ] such that (7.37) and (7.38) hold. Then for any integer t ∈ [τ1 , T − τ2 − 1]

|x(t) −  x | ≤ , |x(t + 1) −  x| ≤ 

(7.39)

290

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

and in view of (7.33) and (P2) w(y(t)) ≥ u(x(t), x(t + 1)) − δ ≥ u(x(t), x(t + 1)) − , |y(t) −  y| < 0 .

(7.40)

In view of (7.39) and (7.40), Theorem 7.25 is proved.

7.4 Proofs of Propositions 7.5–7.7 Proof of Proposition 7.5 Let x > 0. For any z ∈ [0, b min{1, a −1 I x}]

(7.41)

x(z) = z + (1 − d)x.

(7.42)

{(x, x(z)) : z ∈ [0, b min{1, a −1 I x}]} = {(x, x ) : (x, x ) ∈ Ω}.

(7.43)

set

Clearly,

By (7.4) and (7.42), for any z satisfying (7.41) sup{y : y ∈ Λ(x, x(z))} = min{aC−1 [x − b−1 a I z], 1 − b−1 z} and sup{y : y ∈ Λ(x, x(z))} +  qz qz = min{aC−1 [x − b−1 a I z], 1 − b−1 z} +  = min{φ1 (z), φ2 (z)},

(7.44)

where for any z satisfying (7.41) q )z, φ1 (z) = aC−1 x − (aC−1 b−1 a I −  φ2 (z) = 1 − (b−1 −  q )z. By Lemma 7.3, for any x ∈ R 1 satisfying (x, x ) ∈ Ω and any y ∈ Λ(x, x )

(7.45)

7.4 Proofs of Propositions 7.5–7.7

291

y + q (x − (1 − d)x) ≤  y + q d x.

(7.46)

By (7.42)–(7.44) and (7.46), y + qd x min{φ1 (z), φ2 (z)} ≤ 

(7.47)

for all z satisfying (7.41). We are interested in the point of maximum of the function min{φ1 (·), φ2 (·)} on the interval [0, b min{1, a −1 I x}]. By (7.1), (7.12) and (7.45), q − (aC b)−1 a I φ 1 (·) =  = (b + d(aC − a I ))−1 − (aC b)−1 a I = (aC b)−1 (b + d(aC − a I ))−1 [aC b − a I (b + d(aC − a I ))] = (aC b)−1 (b + d(aC − a I ))−1 (aC − a I )(b − da I ) > 0.

(7.48)

By (7.1), (7.12) and (7.45), q = −b−1 + (b + d(aC − a I ))−1 < 0. φ 2 (·) = −b−1 + 

(7.49)

We consider the following cases separately: x = aC ; x > aC , x = a I , x < a I , x ∈ (a I , aC ). Assume that x = aC .

(7.50)

Then by (7.45) and (7.50), φ1 (0) = φ2 (0) = 1, z¯ = 0 is a unique point of the maximum of min{φ1 (·), φ2 (·)} on [0, b min{1, a −1 I x}] and combined with (7.6), (7.7) and (7.17) this implies that y + d q aC . φ1 (0) = φ2 (0) = 1 = 

(7.51)

By Lemma 7.3, Remark 7.4, (7.42), (7.44), (7.46) and (7.47) the relation (7.18) holds if and only if x¯ = (1 − d)x = (1 − d)aC

292

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

and y¯ satisfies (7.20). Note that x¯ = (1 − d)aC is equivalent to (7.19) when x = aC . Thus the assertion of the proposition holds for x = aC . Assume that x > aC .

(7.52)

φ2 (0) < φ1 (0).

(7.53)

By (7.42) and (7.45),

By (7.48), (7.49) and (7.53), for all z ∈ [0, b min{1, a −1 I x}] we have φ1 (z) > φ2 (z) and zero is a unique point of maximum of the function min{φ1 , φ2 } on [0, b min{1, a −1 I x}]. By (7.6), (7.7), (7.12), (7.45), (7.52), y + d q aC <  y + d q x. φ2 (0) = 1 =  This implies that for all z ∈ [0, b min{1, a −1 I x}], y + d q x. min{φ2 (z), φ2 (z)} <  Together with (7.42)–(7.44) this implies that there are no x, ¯ y¯ satisfying (x, x) ¯ ∈ Ω, y ∈ Λ(x, x) ¯ and (7.18). Assume that x = aI . Then

(7.54)

b min{1, a −1 I x} = b.

By (7.12), (7.45) and (7.54), q, φ1 (b) = aC−1 a I − aC−1 a I + b(b + d(aC − a I ))−1 = b

(7.55)

φ2 (b) = b q , φ1 (b) = φ2 (b). By (7.48), (7.49) and (7.55), b is a unique point of maximum of the function min{φ1 , φ2 } on [0, b] and q = y + d q aI . min{φ1 (b), φ2 (b)] = b

7.4 Proofs of Propositions 7.5–7.7

293

By Lemma 7.3, Remark 7.4, (7.42)–(7.44), (7.46), (7.47) and (7.55) the relation (7.18) holds if and only if x¯ = a I (1 − d) + b and y¯ satisfies (7.20). Clearly, x¯ = a I (1 − d) + b if and only if (7.19) holds. Thus the assertion of the proposition holds if x = a I . Assume that x < aI . Then

(7.56)

−1 b min{1, a −1 I x} = ba I x

and by (7.12) and (7.45) −1 −1 −1 q )ba −1 φ1 (ba −1 I x) = aC x − (aC b a I −  I x −1 −1 − aC−1 b−1 a I + (b + d(aC − a I ))−1 ) = ba −1 I x(a I aC b −1 xb(aC−1 b−1 − aC−1 b−1 + a −1 I (b + d(aC − a I )) ).

By the relation above, (7.45) and (7.56), −1 −1 − q )ba −1 φ2 (ba −1 I x) − φ1 (ba I x) = 1 − (b I x − xb −1 = 1 − a −1 + a −1 I (b + d(aC − a I )) I x > 0.

(7.57)

By (7.48), (7.49) and (7.57) the function min{φ1 , φ2 } on [0, ba −1 I x] has a unique point of maximum ba −1 I x and −1 −1 min{φ1 (ba −1 I x), φ2 (ba I x)} = φ1 (ba I x) −1 = a −1 I xb(b + d(aC − a I )) .

By (7.1), (7.6), (7.7), (7.12) and (7.56),  y + q d x − φ1 (ba −1 I x) q + d q x − a −1 q = (b − da I ) I xb = q [b − da I + d x − a −1 I xb] = q (a I − x)(−d + (ba I )−1 ) > 0. Therefore

 y + q d x > min{φ1 (z), φ2 (z)} for all z ∈ [0, ba −1 I x]

(7.58)

294

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

and there are no x, ¯ y¯ satisfying (x, x) ¯ ∈ Ω, y¯ ∈ Λ(x, x) ¯ and (7.18). Assume that x ∈ (a I , aC ).

(7.59)

z¯ = (aC − x)(aC − a I )−1 b.

(7.60)

z¯ > 0, z¯ < b min{1, a −1 I x} = b.

(7.61)

Define

By (7.1), (7.59) and (7.60),

By (7.45) and (7.60), q )(aC − x)(aC − a I )−1 b, φ1 (¯z ) = aC−1 x − (aC−1 b−1 a I −  φ2 (¯z ) = 1 − (b−1 −  q )(aC − x)(aC − a I )−1 b.

(7.62)

In view of (7.48), (7.49) and (7.62), φ1 (¯z ) − φ2 (¯z ) = aC−1 x − aC−1 b−1 a I (aC − x)(aC − a I )−1 b − 1 + (aC − x)(aC − a I )−1 = aC−1 x − 1 + (aC − x)(aC − a I )−1 (1 − aC−1 a I ) = aC−1 x − 1 + aC−1 (aC − x) = 0.

(7.63)

By (7.63), z¯ is a unique point of maximum of the function min{φ1 (·), φ2 (·)} on [0, b min{1, a −1 I x}]. By (7.6), (7.7), (7.12) and (7.62), y + q d x) φ1 (¯z ) − ( = aC−1 x − aC−1 a I (aC − x)(aC − a I )−1 +  q b(aC − x)(aC − a I )−1 −  q (b − da I + d x) = aC−1 (aC − a I )−1 [(aC − a I )x − a I (aC − x)] +  q (aC − a I )−1 [b(aC − x) − (aC − a I )(b − da I + d x)] = aC−1 (aC − a I )−1 aC (x − a I ) +  q (aC − a I )−1 (baC − bx − baC + ba I + (aC − a I )(a I − x)d) = (x − a I )(aC − a I )−1 +  q (aC − a I )−1 (a I − x)[b + d(aC − a I )] = 0 and

7.4 Proofs of Propositions 7.5–7.7

295

min{φ1 (¯z ), φ2 (¯z )} =  y + d x q. By the relation above, Lemma 7.3, Remark 7.4, (7.42), (7.43), (7.46) and (7.47), the relation (7.18) holds if and only if x¯ = (1 − d)x + z¯ and (7.20) holds. It is clear that x¯ = (1 − d)x + z¯ is equivalent to the relation x¯ = (1 − d)x + (aC − x)(aC − a I )−1 b = (aC b)(aC − a I )−1 + (1 − d − b(aC − a I ))−1 = (ac b)(aC − a I )−1 + ξx and the assertion of the proposition holds. Summarizing all the cases we conclude that Proposition 7.5 holds. Proof of Proposition 7.6 By (7.6) and (7.7), y = (b − a I d)b−1 .  x = aC = a I , 

(7.64)

For all z satisfying (7.41) define x(z) by (7.42). Clearly, (7.43) holds and for any z satisfying (7.41) the relation (7.44) holds with φ1 , φ2 defined by (7.45). Clearly, (7.46) holds and (7.47) holds for all z satisfying (7.41). By (7.12), (7.64) and (7.45) q − b−1 = 0, φ2 (·) = −b−1 +  q = 0. φ 1 (·) = 

(7.65)

By (7.45) and (7.65), sup{min{φ1 (z), φ2 (z)} : z satisfies (A1)} = min{φ1 (0), φ2 (0)} = min{a −1 I x, 1}. By (7.12) and (7.64)–(7.66), for any z satisfying (7.41), y + qd x min{φ1 (z), φ2 (z)} =  if and only if −1 + b−1 d x = 1 − b−1 d(a I − x), min{a −1 I x, 1} = (b − a I d)b

(7.66)

296

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

which holds only if and only if a I = x. Proposition 7.6 is proved. Proof of Proposition 7.7 For any z satisfying (7.41), define x(z) by (7.42). Clearly, (7.43) holds for any z satisfying (7.41) and (7.44) holds with φ1 , φ2 defined by (7.45) for any z satisfying (7.41). As in the proof of Proposition 7.5, we can show that (7.47) holds for all z satisfying (7.41) and that φ1 (·) < 0, φ2 (·) > 0.

(7.67)

z¯ ∈ [0, b min{1, a −1 I x}], ¯ φ1 (¯z ) = φ2 (¯z ),

(7.68)

By (7.65) if

(7.69)

then z¯ is a unique point of maximum of the function min{φ1 (·), φ2 (·)} on the interval [0, b min{1, a −1 I x}]. We verify when there is z¯ satisfying (7.68) and (7.69). Let z¯ satisfy (7.68). Then in view of (7.45) the equality (7.69) holds if and only if q )¯z − 1 + (b−1 −  q )¯z ) 0 = aC−1 x − (aC−1 b−1 a I −  = aC−1 x − 1 − z¯ (aC−1 b−1 a I − b−1 ) = aC−1 b−1 [b(x − aC ) − z¯ (a I − aC )], which holds if and only if b(x − aC ) = z¯ (a I − aC ), which in its turn is equivalent to z¯ = b(x − aC )(a I − aC )−1 .

(7.70)

We conclude that there exists z¯ satisfying (7.68) and (7.69) if and only if b(x − aC )(a I − aC )−1 ∈ [0, b min{1, a −1 I x}], which is equivalent to the following relations: x ≥ aC , (x − aC )(a I − aC )−1 ≤ min{1, a −1 I x}.

(7.71)

If x ≥ a I , then (7.71) holds if and only if x ≤ a I . Thus if x ≥ a I and (7.71) holds, then x = a I . If x ≤ a I , then (7.71) holds if and only if

7.4 Proofs of Propositions 7.5–7.7

297

x ≥ aC , 0 ≤ −a I (x − aC ) + (a I − aC )x = aC (a I − x) which is equivalent to the inclusion x ∈ [aC , a I ]. Therefore there is z¯ satisfying (7.68) and (7.69) if and only if x ∈ [aC , a I ]. In this case (7.70) holds, z¯ = (baC )(aC − a I )−1 + xb(a I − aC )−1 , (1 − d)x + z¯ = (baC )(aC − a I )−1 + xξ and q )¯z = 1 − d(aC − a I )b−1 q z¯ φ1 (¯z ) = φ2 (¯z ) = 1 − (b−1 −  = 1 − aC d q + xd q = y + xd q. In this case the assertion of Proposition 7.7 holds. Assume that for all z satisfying (7.41) φ1 (z) < φ2 (z).

(7.72)

By (7.45), this holds if and only if for all z satisfying (7.41), q )z − 1 + (b−1 −  q )z 0 > aC−1 x − (aC−1 b−1 a I −  = aC−1 x − aC−1 b−1 a I z − 1 + b−1 z = aC−1 b−1 [b(x − aC ) − z(a I − aC )], which is equivalent to the inequality z(a I − aC ) > b(x − aC ) for all z satisfying (7.41), which is true if and only if x < aC . Thus (7.72) holds for all z satisfying (7.41) if and only if x < aC . In this case the unique point maximum of the function min{φ1 , φ2 } on [0, b min{1, a −1 I x}] is zero, φ1 (0) = aC−1 x and  y + qd x = (b − da I )(b + d(aC − a I ))−1 + (b + d(aC − a I ))−1 d x = (b − da I + d x)(b + daC − da I )−1 .

298

7 The Turnpike Phenomenon for the Robinson–Shinkai–Leontief Model

Clearly, φ1 (0) = aC−1 x < (b − da I + d x)(b + daC − da I )−1 = y + q d x. Thus y + d qx sup{min{φ1 (z), φ2 (z)} : z satisfies (7.41)} <  and in this case there are no x, ¯ y¯ such that (x, x) ¯ ∈ Ω, y¯ ∈ Λ(x, x) ¯ and (7.18) holds. Assume that for all z satisfying (7.41) φ1 (z) > φ2 (z). This is equivalent to the fact that for all z satisfying (7.41), z(a I − aC ) < b(x − aC ), which is true if and only if −1 b min{1, a −1 I x} < b(x − aC }(ai − aC ) ,

which is equivalent to the following inequality, −1 min{1, a −1 I x} < (x − aC )(a I − aC ) ,

which holds if and only if x > a I . Thus φ1 (z) > φ2 (z) for all z satisfying (7.41) if and only if x > a I . In this case b = b min{1, a −1 I x} is a unique point of maximum q. of min{φ1 , φ2 } on [0, b] and φ2 (b) = b Clearly,  y + d x q = (b − da I )(b + d(aC − a I ))−1 + (b + d(aC − a I ))−1 d x = q (b − da I + d x) > b q = φ2 (b). This implies that in this case there is no z satisfying (7.41) such that q + d x q min{φ1 (z), φ2 (z)} =  and there are no x, ¯ y¯ such that (x, x) ¯ ∈ Ω, y¯ ∈ Λ(x, x) ¯ and (7.18) holds. Summarizing all the cases, we conclude that Proposition 7.7 is proved.

Chapter 8

Discrete Dispersive Dynamical Systems

Abstract In this chapter we study turnpike properties for a discrete dispersive dynamical system generated by set-valued mappings which was introduced by A. M. Rubinov in 1980. This dispersive dynamical system has a prototype in mathematical economics. In particular, it is an abstract extension of the classical von Neumann– Gale model. Our dynamical system is described by a compact metric space of states and a transition operator which is set-valued. Our goal is to study the asymptotic behavior of the trajectories of this dynamical system.

8.1 Uniform Convergence to Global Attractors In [77, 78] A. M. Rubinov introduced a discrete dispersive dynamical system generated by a set-valued mapping acting on a compact metric space, which was studied in [22, 77, 78, 90, 95, 97, 99, 110]. This dispersive dynamical system has a prototype in mathematical economics [58, 77, 93]. In particular, it is an abstract extension of the classical von Neumann–Gale model [58, 77, 93]. Our dynamical system is described by a compact metric space of states and a transition operator which is set-valued. Dynamical systems theory has been a rapidly growing area of research which has various applications to physics, engineering, biology and economics. In this theory, one of the goals is to study the asymptotic behavior of the trajectories of a dynamical system. Usually in the dynamical systems theory a transition operator is single-valued. In [22, 77, 78, 90, 95, 97, 99, 110] and in this chapter we study dynamical systems with a set-valued transition operator. Such dynamical systems correspond to certain models of economic dynamics [58, 77, 93]. Let (X, ρ) be a compact metric space and let a : X → 2 X \ {∅} be a set-valued mapping whose graph graph(a) = {(x, y) ∈ X × X : y ∈ a(x)}

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. J. Zaslavski, Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model, Monographs in Mathematical Economics 4, https://doi.org/10.1007/978-981-16-2252-6_8

299

300

8 Discrete Dispersive Dynamical Systems

is a closed subset of X × X . For each nonempty subset E ⊂ X set a(E) =

 {a(x) : x ∈ E} and a 0 (E) = E.

By induction we define a n (E) for any natural number n and any nonempty subset E ⊂ X as follows: a n (E) = a(a n−1 (E)). In this chapter we study convergence of trajectories of the dynamical system generated by the set-valued mapping a. Following [77, 78] this system is called a discrete disperse dynamical system. First we define a trajectory of this system. A sequence {xt }∞ t=0 ⊂ X is called a trajectory of a (or just a trajectory if the mapping a is understood) if xt+1 ∈ a(xt ) for all integers t ≥ 0. Put Ω(a) = {z ∈ X : for each  > 0 there is a trajectory {xt }∞ t=0 such that lim inf ρ(z, xt ) ≤ }. t→∞

(8.1)

Clearly, Ω(a) is closed subset of (X, ρ). In the dynamical system theory the set Ω(a) is called a global attractor of a. Note that in [77, 78] Ω(a) is called a turnpike set of a. This terminology was motivated by mathematical economics [58, 77, 93]. For each x ∈ X and each nonempty closed subset E ⊂ X put ρ(x, E) = inf{ρ(x, y) : y ∈ E}. It is clear that for each trajectory {xt }∞ t=0 we have lim ρ(xt , Ω(a)) = 0.

t→∞

It is not difficult to see that if for a nonempty closed set B ⊂ X lim ρ(xt , B) = 0

t→∞

for each trajectory {xt }∞ t=0 , then Ω(a) ⊂ B. We study uniform convergence of trajectories to the global attractor Ω(a). The following useful result will be proved in Sect. 8.2. Proposition 8.1 Let  > 0. Then there exists a natural number T () such that for each trajectory {xt }∞ t=0 min{ρ(xt , Ω(a)) : t = 0, . . . , T ()} ≤ .

8.1 Uniform Convergence to Global Attractors

301

The following theorem provides necessary and sufficient conditions for uniform convergence of trajectories to the global attractor. Theorem 8.2 The following properties are equivalent: (1) For each  > 0 there exists a natural number T () such that for each trajectory {xt }∞ t=0 and each integer t ≥ T () we have ρ(x t , Ω(a)) ≤ . (2) If a sequence {xt }∞ t=−∞ ⊂ X satisfies x t+1 ∈ a(x t ) for all integers t, then ⊂ Ω(a). {xt }∞ t=−∞ (3) For each  > 0 there exists δ > 0 such that for each trajectory {xt }∞ t=0 satisfying ρ(x0 , Ω(a)) ≤ δ the inequality ρ(xt , Ω(a)) ≤  holds for all integers t ≥ 0. Theorem 8.2 will be proved in Sect. 8.3. The following two theorems show that convergence of trajectories to the global attractor holds even in the presence of computational errors. These theorems will be proved in Sect. 8.5. Theorem 8.3 Let  > 0. Then there exist δ > 0 and a natural number T () such that for each sequence {xt }∞ t=0 ⊂ X satisfying ρ(x t+1 , a(x t )) ≤ δ for each integer t ≥ 0 the following inequality holds: min{ρ(xt , Ω(a)) : t = 0, . . . , T ()} ≤ . Theorem 8.4 Assume that property (2) of Theorem 8.2 holds. Then for each  > 0 there exist δ > 0 and a natural number T () such that for each sequence {xt }∞ t=0 ⊂ X satisfying ρ(xt+1 , a(xt )) ≤ δ for all integers t ≥ 0 the inequality ρ(xt , Ω(a)) ≤  holds for each integer t ≥ T (). Some examples of set-valued mappings are considered in Sect. 8.6. In Sect. 8.7 we prove generic convergence results for certain classes of set-valued mappings. The results of this section were obtained in [95].

8.2 Proof of Proposition 8.1 Let us assume the contrary. Then for each natural number n there exists a trajectory {xt(n) }∞ t=0 such that min{ρ(xt(n) , Ω(a)) : t = 0, . . . , n} ≥ .

(8.2)

It is easy to see that there exists a strictly increasing sequence of natural numbers {n k }∞ k=1 such that for each integer t ≥ 0 there exists

302

8 Discrete Dispersive Dynamical Systems

xt := lim xt(n k ) . k→∞

(8.3)

Since graph(a) is a closed subset of X × X , equality (8.3) implies that {xt }∞ t=0 is a trajectory. It follows from (8.2) and (8.3) that for each integer t ≥ 0 the inequality ρ(xt , Ω(a)) ≥  holds. This contradicts the definition of Ω(a). The contradiction we have reached proves Proposition 8.1.

8.3 Proof of Theorem 8.2 We will show that property (1) implies property (2). Assume that property (1) holds. Let a sequence {xt }∞ t=−∞ ⊂ X satisfy xt+1 ∈ a(xt ) for all integers t. Let τ be an integer,  be a positive number and let a natural number T () be as guaranteed by property (1). Define yt = xt+τ −T () for each integer t ≥ 0.

(8.4)

It is clear that {yt }∞ t=0 is a trajectory. By property (1), the choice of T () and (8.4), ρ(xτ , Ω(a)) = ρ(yT () , Ω(a)) ≤ . Since  is an arbitrary positive number, we conclude that xτ ∈ Ω(a) for each integer τ . Thus property (1) implies property (2). Let us show that property (2) implies property (3). Assume that property (2) holds. Let  ∈ (0, 1). We show that there exists δ > 0 such that for each trajectory {xt }∞ t=0 satisfying ρ(x0 , Ω(a)) ≤ δ the inequality ρ(xt , Ω(a)) ≤  holds for all integers t ≥ 0. Let us assume the contrary. Then for each integer n ≥ 1 there exists a trajectory {xt(n) }∞ t=0 such that ρ(x0(n) , Ω(a)) ≤ (2n)−1  and sup{ρ(xt(n) , Ω(a)) : t ≥ 0 is an integer} > . (8.5) In view of (8.5) for each natural number n there exists a natural number Tn such that , Ω(a)) > . ρ(x T(n) n

(8.6)

Assume that the sequence {Tn }∞ n=1 is not bounded. Extracting a subsequence and re-indexing if necessary we may assume without loss of generality that Tn → ∞ as n → ∞. For each integer n ≥ 1 set

8.3 Proof of Theorem 8.2

303 (n) yt(n) = xt+T for all integers t ≥ −Tn . n

(8.7)

Evidently, there exists a strictly increasing sequence of natural numbers {n k }∞ k=1 such that for each integer t there exists yt = lim yt(n k ) .

(8.8)

k→∞

Since the graph of a is closed, it follows from (8.7) and (8.8) that yt+1 ∈ a(yt ) for each integer t. By property (2), {yt }∞ t=−∞ ⊂ Ω(a). On the other hand, by (8.6)–(8.8), ρ(y0 , Ω(a)) = lim ρ(y0(n k ) , Ω(a)) = lim ρ(x T(nn k ) , Ω(a)) ≥ . k→∞

k→∞

k

The contradiction we have reached proves that our assumption is wrong and the sequence {Tn }∞ n=1 is bounded. Extracting a subsequence and re-indexing we may assume without loss of generality that Tn = T1 for all integers n ≥ 1.

(8.9)

Let n be a natural number. It follows from (8.5) that there is z n ∈ Ω(a) such that ρ(x0(n) , z n ) ≤ (2n)−1 .

(8.10)

By the definition of Ω(a) there exists a trajectory {yt(n) }∞ t=0 such that lim inf ρ(yt(n) , z n ) ≤ (8n)−1 . t→∞

(8.11)

In view of (8.11), there exists a natural number Sn > n such that , z n ) < (4n)−1 . ρ(yS(n) n

(8.12)

Relations (8.10) and (8.12) imply that , x0(n) ) ≤ ρ(yS(n) , z n ) + ρ(z n , x0(n) ) < /n. ρ(yS(n) n n Set

(8.13)

304

8 Discrete Dispersive Dynamical Systems (n) ξt(n) = yt+S , t = −Sn , . . . , −1, 0, ξt(n) = xt(n) , t = 1, 2, . . . n

(8.14)

Clearly, there exists a strictly increasing sequence of natural numbers {n k }∞ k=1 such that for each integer t there exists ξt := lim ξt(n k )

(8.15)

x0 := lim x0(n k ) .

(8.16)

k→∞

and also there exists k→∞

Since the graph of a is closed it follows from (8.14) and (8.15) that ξt+1 ∈ a(ξt ) for each integer t ≥ 1 and for each integer t ≤ −1. We will show that ξ1 ∈ a(ξ0 ). Since the graph of a is closed it follows from (8.14)–(8.16) that ξ1 ∈ a(x0 ). By (8.13)–(8.16) and the inclusion above, ρ(x0 , ξ0 ) = lim ρ(x0(n k ) , ξ0(n k ) ) = lim ρ(x0(n k ) , yS(nn k ) ) = 0, k→∞

k→∞

k

x0 = ξ0 and ξ1 ∈ a(ξ0 ). Thus we have shown that ξt+1 ∈ a(ξt ) for all integers t.

(8.17)

In view of property (2), ξt ∈ Ω(a) for all integers t. On the other hand, it follows from (8.4), (8.5), (8.6) and (8.9) that ρ(ξT1 , Ω(a)) = lim ρ(ξT(n1 k ) , Ω(a)) = lim ρ(x T(n1 k ) , Ω(a)) ≥ . k→∞

k→∞

The contradiction we have reached proves that there exists δ > 0 such that for each trajectory {xt }∞ t=0 satisfying ρ(x 0 , Ω(a)) ≤ δ the inequality ρ(x t , Ω(a)) ≤  holds for all integers t ≥ 0. Thus property (2) implies property (3). Let us show that property (3) implies property (1). Assume that property (3) holds. Let  > 0 and let δ > 0 be as guaranteed by property (3). By Proposition 8.1 there exists a natural number T0 such that for each trajectory {xt }∞ t=0 min{ρ(xt , Ω(a)) : t = 0, . . . , T0 } ≤ δ.

(8.18)

8.3 Proof of Theorem 8.2

305

Let {xt }∞ t=0 be a trajectory. By the choice of T0 there is an integer j ∈ [0, T0 ] such that ρ(x j , Ω(a)) ≤ δ. In view of this inequality and the choice of δ, ρ(xt , Ω(a)) ≤  for all integers t ≥ j and property (1) holds. Thus property (3) implies property (1). Theorem 8.2 is proved.

8.4 An Auxiliary Result Lemma 8.5 Let T be a natural number and let  > 0. Then there exists a number T ⊂ X satisfying δ > 0 such that for each sequence {xt }t=0 ρ(xt+1 , a(xt )) ≤ δ, t = 0, . . . , T − 1 T ⊂ X such that there is a sequence {yt }t=0

yt+1 ∈ a(yt ), t = 0, . . . , T − 1, ρ(yt , xt ) ≤ , t = 0, . . . , T.

(8.19) (8.20)

Proof Let us assume the contrary. Then for each natural number n there exists a T ⊂ X such that sequence {xt(n) }t=0 (n) , a(xt(n) )) ≤ 1/n, t = 0, . . . , T − 1 ρ(xt+1

(8.21)

T ⊂ X satisfying (8.19) and that for each sequence {yt }t=0

sup{ρ(yt , xt(n) ) : t = 0, . . . , T } > .

(8.22)

Extracting a subsequence and re-indexing if necessary we may assume without loss of generality that for t = 0, . . . , T there exists xt := lim xt(n) . n→∞

(8.23)

By (8.21), for t = 0, . . . , T − 1 and each integer n ≥ 1 there is (n) ∈ a(xt(n) ) z t+1

such that

(8.24)

306

8 Discrete Dispersive Dynamical Systems (n) (n) ρ(xt+1 , z t+1 ) ≤ 1/n.

(8.25)

Extracting a subsequence and re-indexing if necessary we may assume without loss of generality that for t = 0 . . . , T − 1 there is (n) . z t+1 := lim z t+1 n→∞

(8.26)

Since the graph of a is closed it follows from (8.23), (8.24) and (8.26) that for each t = 0, . . . , T − 1 z t+1 ∈ a(xt ).

(8.27)

By (8.23), (8.25) and (8.26), for each t = 0, . . . , T − 1 we have xt+1 = z t+1 . Together with (8.27) this equality implies that xt+1 ∈ a(xt ) for t = 0, . . . , T − 1. In view of (8.23) there is a natural number n 0 such that ρ(xt , xt(n 0 ) ) ≤ /4, t = 0, . . . , T. T (see (8.22)). The contradiction we have This contradicts the choice of {xt(n 0 ) }t=0 reached proves Lemma 8.5.

8.5 Proofs of Theorems 8.3 and 8.4 Proof of Theorem 8.3 By Proposition 8.1 there exists a natural number T () such that for each trajectory {xt }∞ t=0 of a, min{ρ(xt , Ω(a)) : t = 0, . . . , T ()} ≤ /4.

(8.28)

T () By Lemma 8.5, there exists a number δ > 0 such that for each sequence {xt }t=0 ⊂X satisfying

ρ(xt+1 , a(xt )) ≤ δ, t = 0, . . . , T () − 1

(8.29)

T () there exists a sequence {yt }t=0 ⊂ X such that

yt+1 ∈ a(yt ), t = 0, . . . , T () − 1, ρ(yt , xt ) ≤ /4, t = 0, . . . , T (). Assume that a sequence {xt }∞ t=0 ⊂ X satisfies

(8.30) (8.31)

8.5 Proofs of Theorems 8.3 and 8.4

307

ρ(xt+1 , a(xt )) ≤ δ for all integers t ≥ 0.

(8.32)

T () It follows from (8.32) and the choice of δ that there exists a sequence {yt }t=0 ⊂X such that (8.30) and (8.31) hold. By (8.30) and the choice of T () (see (8.28)) there is j ∈ {0, . . . , T ()} such that

ρ(y j , Ω(a)) ≤ /4. Combined with (8.37) this inequality implies that ρ(x j , Ω(a)) ≤ ρ(x j , y j ) + ρ(y j , Ω(a)) ≤ /2. Theorem 8.3 is proved. Proof of Theorem 8.4 Let  > 0. By Theorem 8.2, property (1) holds and there exists a natural number T () ≥ 4 such that for each trajectory {xt }∞ t=0 of a and each integer t ≥ T (), ρ(xt , Ω(a)) ≤ /8.

(8.33)

() By Lemma 8.5 there exists a number δ > 0 such that for each sequence {yt }4T t=0 ⊂ X satisfying

ρ(yt+1 , a(yt )) ≤ δ, t = 0, . . . , 4T () − 1

(8.34)

() there is a sequence {z t }4T t=0 ⊂ X such that

z t+1 ∈ a(z t ), t = 0, . . . , 4T () − 1, ρ(yt , z t ) ≤ /8, t = 0, . . . , 4T ().

(8.35) (8.36)

Assume that a sequence {xt }∞ t=0 ⊂ X satisfies ρ(xt+1 , a(xt )) ≤ δ for each integer t ≥ 0.

(8.37)

() In view of (8.37) and the choice of δ (see (8.34)–(8.36)) there is a sequence {z t }4T t=0 ⊂ X such that (8.35) is true and

ρ(xt , z t ) ≤ /8, t = 0, . . . , 4T ().

(8.38)

By (8.35) and the choice of T () (see (8.33)), ρ(z t , Ω(a)) ≤ /8, t = T (), . . . , 4T (). Relations (8.38) and (8.39) imply that for t = T (), . . . , 4T (),

(8.39)

308

8 Discrete Dispersive Dynamical Systems

ρ(xt , Ω(a)) ≤ ρ(xt , z t ) + ρ(z t , Ω(a)) ≤ /4.

(8.40)

We show that ρ(xt , Ω(a)) ≤  for all integers t ≥ T (). Let us assume the contrary. Then there is an integer j≥T () such that ρ(x j , Ω(a)) > ,

and if an integer t satisfies T () ≤ t < j, then ρ(xt , Ω(a)) ≤ .

(8.41)

In view of (8.40), j > 4T ().

(8.42)

yt = xt+ j−2T () .

(8.43)

For t = 0, . . . , 4T () set

By (8.37) and (8.43), for t = 0, . . . , 4T () − 1 ρ(yt+1 , a(yt )) = ρ(xt+ j−2T ()+1 , a(xt+ j−2T () )) ≤ δ. In view of this relation and the choice of δ (see (8.34)–(8.36)) there is a sequence () {ξt }4T t=0 ⊂ X such that ξt+1 ∈ a(ξt ), t = 0, . . . , 4T () − 1,

(8.44)

ρ(ξt , yt ) ≤ /8, t = 0, . . . , 4T ().

(8.45)

It follows from (8.44) and the choice of T () (see (8.33)) that ρ(ξt , Ω(a)) ≤ /8, t = T (), . . . , 4T (). Together with (8.45) this inequality implies that for t = T (), . . . , 4T () ρ(yt , Ω(a)) ≤ ρ(yt , ξt ) + ρ(ξt , Ω(a)) ≤ /4. Together with (8.43) this inequality implies that ρ(x j , Ω(a)) = ρ(y2T () , Ω(a)) ≤ /4. This relation contradicts (8.41). The contradiction we have reached proves that ρ(xt , Ω(a)) ≤  for all integers t ≥ T (). Theorem 8.4 is proved.

8.6 Examples

309

8.6 Examples Denote by Π (X ) the set of all nonempty closed subsets of (X, ρ). For each A, B ∈ Π (X ) set H (A, B) = max{sup ρ(x, B), sup ρ(y, A)}. x∈A

y∈B

Clearly, the space (Π (X ), H ) is a complete metric space. Example 8.6 Let a : X → X satisfy ρ(a(x), a(y)) ≤ ρ(x, y) for each x, y ∈ X . Since the mapping a is single-valued it is not difficult to see that a(Ω(a)) ⊂ Ω(a) and property (3) of Theorem 8.2 holds. Example 8.7 Let a : X → X satisfy the following condition: (C1) for each  > 0 there exists δ ∈ (0, ) such that for each x, y ∈ X satisfying ρ(x, y) ≤ δ we have ρ(a n x, a n y) ≤  for all natural numbers n. Define ρ1 (x, y) = sup{ρ(a n x, a n y) : n = 0, 1, . . . }, x, y ∈ X. Clearly, (X, ρ1 ) is a complete metric space and for each x, y ∈ X we have ρ(x, y) ≤ ρ1 (x, y). Let  > 0 and let δ ∈ (0, ) be as guaranteed by (C1). It is clear that ρ1 (x, y) ≤  for each x, y ∈ X satisfying ρ(x, y) ≤ δ. Thus the metrics ρ and ρ1 induce in X the same topology. It is clear that ρ1 (a(x), a(y)) ≤ ρ1 (x, y) for each x, y ∈ X . Thus in view of Example 8.6, property (3) of Theorem 8.2 holds. Example 8.8 Let a : X → 2 X \ ∅ have a closed graph. Assume that H (a(x), a(y)) ≤ cρ(x, y) for all x, y ∈ X with a constant c ∈ (0, 1). We will show that property (3) of Theorem 8.2 holds. Clearly, it is sufficient to show that a(Ω(a)) =

 {a(z) : z ∈ Ω(a)} ⊂ Ω(a).

Assume that E 1 , E 2 are nonempty closed sets in (X, ρ). Let z ∈ a(E 1 ) and let  be a positive number. There exist x ∈ E 1 such that z ∈ a(x) and y ∈ E 2 such that

310

8 Discrete Dispersive Dynamical Systems

ρ(x, y) ≤ ρ(x, E 2 ) + . It is not difficult to see that ρ(z, a(E 2 )) ≤ ρ(z, a(y)) ≤ H (a(x), a(y)) ≤ cρ(x, y) ≤ cρ(x, E 2 ) + c ≤ cH (E 1 , E 2 ) + c. Since  is an arbitrary positive number we conclude that ρ(z, a(E 2 )) ≤ cH (E 1 , E 2 ) for all z ∈ a(E 1 ). Analogously we can show that ρ(y, a(E 1 )) ≤ cH (E 1 , E 2 ) for all y ∈ a(E 2 ). Hence H (a(E 1 ), a(E 2 )) ≤ cH (E 1 , E 2 ) for all E 1 , E 2 ∈ Π (X ). By this inequality and the Banach fixed point theorem there is a unique Ω∗ ∈ Π (X ) such that a(Ω∗ ) = Ω∗ and that for each E ∈ Π (X ) a n (E) → Ω∗ as n → ∞ in (Π (X ), H ).

(8.46)

Clearly, Ω(a) ⊂ Ω∗ . It is sufficient to show that Ω∗ ⊂ Ω(a). Let z ∈ Ω∗ and  > 0. In view of (8.46), for each x ∈ X there exists an integer n 0 ≥ 1 such that ρ(z, a n 0 (x)) ≤ . This implies that z ∈ Ω(a). Therefore Ω∗ ⊂ Ω(a). Example 8.9 Let X = [0, 1], a(x) = x 2 , x ∈ [0, 1]. It is clear that Ω(a) = {0, 1} and that a(Ω(a)) = Ω(a). It is not difficult to see that for any z ∈ (0, 1) there exists ∞ ⊂ (0, 1) such that x0 = z and xi+1 = a(xi ) for all integers i. a sequence {xi }i=−∞ Therefore property (2) of Theorem 8.2 does not hold.

8.7 Spaces of Set-Valued Mappings In this section we consider classes of discrete disperse dynamical systems whose global attractors are a singleton.

8.7 Spaces of Set-Valued Mappings

311

Denote by A the set of all mappings a : X → Π (X ) with closed graphs. For each a1 , a2 ∈ A set dA (a1 , a2 ) = sup{H (a1 (x), a2 (x)) : x ∈ X }.

(8.47)

It is clear that the metric space (A, dA ) is complete. Denote by Ac the set of all continuous mappings a : X → Π (X ) which belong to A, by A f the set of all a ∈ A such that a(x) is a singleton for each x ∈ X and set A f c = A f ∩ Ac . Clearly, A f , Ac and A f c are closed subsets of (A, dA ). Let M be one of the following spaces: A; Ac ; A f ; A f c . The space M is equipped with the metric dA . Denote by Mreg the set of all a ∈ M such that Ω(a) is a singleton and that properties (1)–(3) of Theorem 8.2 hold. ¯ reg the closure of Mreg in (M, dA ). In this section we prove the Denote by M ¯ reg (in the following result obtained in [95] which shows that most elements of M sense of Baire category) belong to Mreg . Theorem 8.10 The set Mreg contains a countable intersection of open and every¯ reg , dA ). where dense subsets of (M Proof For each a ∈ Mreg there is xa ∈ X such that Ω(a) = {xa }.

(8.48)

Let a ∈ Mreg and let n be a natural number. Since the mapping a has property (2) of Theorem 8.2 it follows from Theorem 8.4 that there exist a natural number T (a, n) and δ(a, n) > 0 such that the following property holds: (P1) For each sequence {xt }∞ t=0 ⊂ X satisfying ρ(xt+1 , a(xt )) ≤ δ(a, n), t = 0, 1, . . . and each integer t ≥ T (a, n) we have ρ(xt , xa ) ≤ 1/n. ¯ reg , dA ) such that Let U(a, n) be an open neighborhood of a in (M H (a(x), b(x)) ≤ δ(a, n)/2 for each x ∈ X an each b ∈ U(a, n).

(8.49)

It follows from property (P1) and (8.49) that the following property holds: (P2) For each b ∈ U(a, n) and each sequence {xt }∞ t=0 ⊂ X satisfying x t+1 ∈ b(x t ), t = 0, 1, . . . ρ(xt , xa ) ≤ 1/n for all integers t ≥ T (a, n). Define

312

8 Discrete Dispersive Dynamical Systems

F=

∞   {U(a, n) : a ∈ Mreg }. n=1

Clearly, F is a countable intersection of open everywhere dense subsets of ¯ reg , dA ). In order to complete the proof it is sufficient to show that F ⊂ Mreg . (M Let b ∈ F and  > 0. Choose a natural number n such that n > 8(min{1, })−1 .

(8.50)

By the definition of F there exists a ∈ Mreg such that b ∈ U(a, n).

(8.51)

Let {xt }∞ t=0 be a trajectory of b. By (8.51) and property (P2), ρ(xt , xa ) ≤ 1/n < 8−1  for all integers t ≥ T (a, n).

(8.52)

Since  is an arbitrary positive number we conclude that {xt }∞ t=0 is a Cauchy sequence. Therefore there exists limt→∞ xt ∈ X . By (8.52), ρ( lim xt , xa ) ≤ 8−1 . t→∞

(8.53)

Since  is an arbitrary positive number and {xt }∞ t=1 is an arbitrary trajectory of b we conclude that there exists xb ∈ X such that lim xt = xb

t→∞

for each trajectory {xt }∞ t=0 of b. By (8.53), ρ(xa , xb ) ≤ 8−1 .

(8.54)

By (8.52) and (8.54), for each trajectory {xt }∞ t=0 of b and all integers t ≥ T (a, n), ρ(xt , xb ) ≤ ρ(xt , xa ) + ρ(xa , xb ) ≤ /4. Theorem 8.10 is proved.

8.8 Attracting Sets Let (X, ρ) be a compact metric space and let a : X → 2 X \ {∅} be a set-valued mapping whose graph

8.8 Attracting Sets

313

graph(a) = {(x, y) ∈ X × X : y ∈ a(x)} is a closed subset of X × X . We continue to use the notation and definitions introduced in Sect. 8.1. Recall that a sequence {xt }∞ t=0 ⊂ X is called a trajectory of a (or just a trajectory if the mapping a is understood) if xt+1 ∈ a(xt ) for all integers t ≥ 0. T2 ⊂ X is called Let T1 , T2 be integers such that 0 ≤ T1 ≤ T2 . A sequence {xt }t=T 1 a trajectory of a (or just a trajectory if the mapping a is understood) if xt+1 ∈ a(xt ) for all integers t = T1 , . . . , T2 − 1. Consider the turnpike Ω(a) defined by (8.1). A point x ∈ X is called stable (with respect to a) if there exists a trajectory {xt }∞ t=0 such that x0 = x and lim inf ρ(xt , x) = 0. t→∞

Denote by Π (a) the set of all stable points. Clearly, Π (a) ⊂ Ω(a). Denote by S(X ) the set of all nonempty closed subsets of (X, ρ) equipped with the Hausdorff metric H (A, B) = max{sup ρ(x, B), sup ρ(y, A)} x∈A

y∈B

which is defined for each pair of nonempty sets A, B ⊂ X . It is well-known that (S(X ), H ) is a complete metric space. We assume that the mapping a : X → S(X ) is continuous and study the convergence of trajectories to the set Π (a). We prove the following results obtained in [110]. Theorem 8.11 For each x ∈ X there exists a trajectory {xt }∞ t=0 such that x 0 = x and that lim inf ρ(xt , Π (a)) = 0. t→∞

Theorem 8.11 is proved in Sect. 8.9. Theorem 8.12 Assume that F is a nonempty subset of X and that for each x ∈ X there exists a trajectory {xt }∞ t=0 such that x0 = x and lim inf ρ(xt , F) = 0. t→∞

Then for each  > 0 there exists a natural number q such that for each x ∈ X there q exists a trajectory {xt }t=0 such that x0 = x and min{ρ(xt , F) : t = 1, . . . , q} ≤ . Theorem 8.12 is proved in Sect. 8.10. Theorems 8.11 and 8.12 imply the following result.

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8 Discrete Dispersive Dynamical Systems

Theorem 8.13 For each  > 0 there exists a natural number q such that for each q x ∈ X there exists a trajectory {xt }t=0 such that x0 = x and min{ρ(xt , Π (a)) : t = 1, . . . , q} ≤ . Theorem 8.14 Assume that F is a nonempty subset of X and that for each x ∈ X there exists a trajectory {xt }∞ t=0 such that x0 = x and lim inf ρ(xt , F) = 0. t→∞

Let  > 0. Then there exist a natural number q and a positive number δ such that for each sequence of mappings at : X → 2 X \ {∅}, t = 0, . . . , q − 1 which satisfy H (at (x), a(x)) ≤ δ for all x ∈ X and all t = 0, . . . , q − 1 q

and each x ∈ X there exists a sequence {xt }t=0 ⊂ X such that x0 = x, xt+1 ∈ at (xt ), t = 0, . . . , q − 1 and min{ρ(xt , F) : t = 1, . . . , q} ≤ . Theorem 8.14 is proved in Sect. 8.11. Theorems 8.11 and 8.14 imply the following result. Theorem 8.15 Let  > 0. Then there exist a natural number q and a positive number δ such that for each sequence of mappings at : X → 2 X \ {∅}, t = 0, . . . , q − 1 which satisfy H (at (x), a(x)) ≤ δ for all x ∈ X and all t = 0, . . . , q − 1 q

and each x ∈ X there exists a sequence {xt }t=0 ⊂ X such that x0 = x, xt+1 ∈ at (xt ), t = 0, . . . , q − 1 and min{ρ(xt , Π (a)) : t = 1, . . . , q} ≤ . Theorem 8.14 implies the following result. Theorem 8.16 Assume that F is a nonempty subset of X and that for each x ∈ X there exists a trajectory {xt }∞ t=0 such that x0 = x and lim inf ρ(xt , F) = 0. t→∞

8.8 Attracting Sets

315

Let {δt }∞ t=0 be a sequence of positive numbers such that lim δt = 0

t→∞

and let at : X → 2 X \ {∅}, t = 0, 1, . . . be a sequence of mappings which satisfy H (at (x), a(x)) ≤ δt for all x ∈ X and all integers t ≥ 0. Then for each x ∈ X there exists a sequence {xt }∞ t=0 ⊂ X such that x0 = x, xt+1 ∈ at (xt ), t = 0, 1, . . . and lim inf ρ(xt , F) = 0. t→∞

Theorems 8.11 and 8.16 imply the following result. Theorem 8.17 Let {δt }∞ t=0 be a sequence of positive numbers such that lim δt = 0

t→∞

and let at : X → 2 X \ {∅}, t = 0, 1, . . . be a sequence of mappings which satisfy H (at (x), a(x)) ≤ δt for all x ∈ X and all integers t ≥ 0. Then for each x ∈ X there exists a sequence {xt }∞ t=0 ⊂ X such that x0 = x, xt+1 ∈ at (xt ), t = 0, 1, . . . and lim inf ρ(xt , Π (a)) = 0. t→∞

8.9 Proof of Theorem 8.11 Denote by M the collection of all nonempty closed sets D ⊂ X such that a(D) ⊂ D. Clearly, M = ∅ because X ∈ M. Let D1 , D2 ∈ M. We say that D1 ≤ D2 if D1 ⊂ D2 . Zorn’s lemma implies the following result. Lemma 8.18 For each D ∈ M there is a minimal element D0 of M such that D0 ⊂ D. ∞ i For each x ∈ X denote by E(x) the closure of ∪i=0 a (x).

316

8 Discrete Dispersive Dynamical Systems

The next lemma follows from the continuity of a. Lemma 8.19 Let x ∈ X . Then E(x) ∈ M. Lemmas 8.18 and 8.19 imply the following result. Lemma 8.20 Let D be a minimal element of M. Then for each x ∈ D, D = E(x). Corollary 8.21 Let D be a minimal element of M, x, y ∈ D and  > 0. Then there q exist an integer q ≥ 1 and a trajectory {xt }t=0 such that x0 = x and ρ(xq , y) ≤ . Corollary 8.22 Let D be a minimal element of M. Then D ⊂ Π (a). Lemma 8.23 Let x ∈ X and  > 0. Then there exist an integer q ≥ 1 and a trajectory q {xt }t=0 ⊂ X such that x0 = x and ρ(xq , Π (a)) ≤ . Proof Fix x1 ∈ a(x).

(8.55)

By Lemmas 8.18 and 8.19, there is a minimal element D∗ ∈ M such that D∗ ⊂ E(x1 ).

(8.56)

z ∈ D∗ .

(8.57)

Fix

q

By (8.55)–(8.57) there exist a natural number q and a trajectory {xt }t=0 such that x0 = x and ρ(xq , z) ≤ .

(8.58)

In view of (8.57), (8.58) and Corollary 8.22, ρ(xq , Π (a)) ≤ . Lemma 8.23 is proved. Theorem 8.11 follows from Lemma 8.23.

8.10 Proof of Theorem 8.12 q

Let  > 0 and x ∈ X . There exist an integer q ≥ 1 and a trajectory {xt }t=0 such that x0 = x and ρ(xq , F) < /4.

(8.59)

8.10 Proof of Theorem 8.12

317

Put δq = /8.

(8.60)

By induction it is not difficult to show the existence of a sequence of positive numbers q {δi }i=0 such that for each integer i ∈ [0, q − 1] δi < 2−1 δi+1

(8.61)

and H (a(xi ), a(z)) < δi+1 for each z ∈ X satisfying ρ(xi , z) ≤ δi .

(8.62)

Let y ∈ X and ρ(x, y) < δ0 .

(8.63)

y0 = y.

(8.61)

Set

By (8.59), (8.61)–(8.63) and the inclusion x1 ∈ a(x0 ) there is y1 ∈ a(y0 )

(8.62)

ρ(y1 , x1 ) < δ1 .

(8.63)

such that

Assume that an integer k satisfies 1 ≤ k ≤ q and we defined y0 , . . . , yk ∈ X such that yi+1 ∈ a(yi ) for all integers i satisfying 0 ≤ i < k

(8.64)

ρ(xi , yi ) ≤ δi , i = 0, . . . , k.

(8.65)

and

(In view of (8.61)–(8.63) our assumption holds for k = 1.) If k = q, then our construction is completed. Assume that k < q. By (8.62) and (8.65), H (a(xk ), a(yk )) < δk+1 .

318

8 Discrete Dispersive Dynamical Systems

Combined with the inclusion xk+1 ∈ a(xk ) this implies that there is yk+1 ∈ a(yk ) such that ρ(xk+1 , yk+1 ) < δk+1 and the assumption made for k holds also for k + 1. q Therefore by induction we constructed a sequence {yi }i=0 such that y0 = y, yi+1 ∈ a(yi ), i = 0, . . . , q − 1, ρ(xq , yq ) < δq .

(8.66)

By (8.59), (8.60) and (8.66), ρ(yq , F) ≤ ρ(xq , yq ) + ρ(xq , F) < . Thus we have shown that for each x ∈ X there exist an open neighborhood Vx of x in X and a natural number q(x) such that for each y ∈ Vx there is a trajectory q(x) {yi }i=0 such that y0 = y and ρ(yq(x) , F) < .

(8.67)

Clearly, X ⊂ ∪x∈X Vx . Since X is compact there exists a finite sequence x1 , . . . , x p ∈ X such that  (8.68) {Vxi : i = 1, . . . , p} = X. Put q = max{q(xi ) : i = 1, . . . , p}.

(8.69)

Let x ∈ X . By (8.68) there is a natural number i ≤ p such that x ∈ Vxi .

(8.70) q(x )

By (8.70) and the choice of Vxi and q(xi ) there exists a trajectory {x j } j=0i such that x0 = x, ρ(xq(xi ) , F) < . This completes the proof of Theorem 8.12.

8.11 Proof of Theorem 8.14 Let  > 0. By Theorem 8.12 there exists a natural number q such that the following property holds:

8.11 Proof of Theorem 8.14

319 q

(P1) For each x ∈ X there exists a trajectory {xi }i=0 such that x0 = x and min{ρ(xt , F) : t = 1, . . . , q} ≤ /8. Put δq = /8.

(8.71)

Since X is compact and the mapping a is continuous there exists a sequence of q positive numbers {δi }i=0 such that for each integer i ∈ [1, q] δi−1 < 2−1 δi

(8.72)

and H (a(y), a(z)) ≤ 2−1 δi for each y, z ∈ X satisfying ρ(y, z) < δi−1 .

(8.73)

Put δ = δ0 /4.

(8.74)

ai : X → 2 X \ {∅}, i = 0, . . . , q − 1,

(8.75)

H (ai (x), a(x)) ≤ δ for all x ∈ X and all t = 0, . . . , q − 1

(8.76)

Let

and x ∈ X . By (P1) there exists a sequence q

{yi }i=0 ⊂ X

(8.77)

such that y0 = x,

(8.78)

yi+1 ∈ a(yi ), i = 0, . . . , q − 1

(8.79)

min{ρ(yi , F) : i = 1, . . . , q} ≤ /8.

(8.80)

and

Assume that an integer j ∈ [0, q − 1] and we defined x0 , . . . , x j ∈ X such that

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8 Discrete Dispersive Dynamical Systems

x0 = x,

(8.81)

xi+1 ∈ ai (xi ) for all integers i satisfying 0 ≤ i ≤ j − 1,

(8.82)

ρ(xi , yi ) ≤ δi , i = 0, . . . , j.

(8.83)

and

(Clearly, for j = 0 the assumption holds.) By (8.83), ρ(x j , y j ) ≤ δ j . Combined with (8.73) this implies H (a(x j ), a(y j )) < 2−1 δ j+1 . Together with (8.79) this implies that ρ(y j+1 , a(x j )) < 2−1 δ j+1 .

(8.84)

By (8.72), (8.74) and (8.76) H (a j (x j ), a(x j )) < 4−1 δ j+1 . Combined with (8.84) this implies that ρ(y j+1 , a j (x j )) < (3/4)δ j+1 and there is x j+1 ∈ a j (x j ) such that ρ(x j+1 , y j+1 ) < δ j+1 . Therefore the assumption made for j holds also for j + 1. q Thus by induction we constructed a sequence {xi }i=0 ⊂ X such that x0 = x, xi+1 ∈ ai (xi ), i = 0, . . . , q − 1, ρ(xi , yi ) ≤ /8, i = 0, . . . , q.

(8.85) (8.86)

By (8.80) there is j ∈ {1, . . . , q} such that ρ(y j , F) ≤ /8. Combined with (8.86) this implies that ρ(x j , F) ≤ /2. This completes the proof of Theorem 8.14.

8.12 Extensions Let (X, ρ) be a compact metric space, K be a nonempty closed subset of (X, ρ) and let a : K → 2 X \ {∅} be a set-valued mapping whose graph

8.12 Extensions

321

graph(a) = {(x, y) ∈ K × X : y ∈ a(x)} is a closed subset of X × X . Recall that for each x ∈ X and each nonempty closed set E ⊂ X we put ρ(x, E) = inf{ρ(x, y) : y ∈ E}. A sequence {xt }∞ t=0 ⊂ X is called a trajectory of a (or just a trajectory if the mapping a is understood) if {xi }∞ t=0 ⊂ K and if x t+1 ∈ a(x t ) for all integers t ≥ 0. T ⊂ X , where T is a natural number, is called a trajectory of a A sequence {xt }t=0 T −1 ⊂ K and if xt+1 ∈ a(xt ) (or just a trajectory if the mapping a is understood) if {xt }t=0 for all integers t ∈ [0, T − 1]. In this section we use the following assumption: (A) For each integer n ≥ 1 there exists a sequence {xt }nt=0 ⊂ X such that for each integer t satisfying 0 ≤ t ≤ n − 1 there is (yt , z t ) ∈ graph(a) such that ρ(xt , yt ) ≤ 1/n and ρ(xt+1 , z t ) ≤ 1/n. In Sect. 8.13 we prove the following result. Proposition 8.24 Assume that (A) holds. Then there exists a trajectory {xt }∞ t=0 of a. In this section we assume that (A) holds. Put Ω(a) = {z ∈ X : for each  > 0 there is a trajectory {xt }∞ t=0 such that lim inf ρ(z, xt ) ≤ }. t→∞

(8.87)

Clearly, Ω(a) is a nonempty closed subset of K . It is clear that for each trajectory {xt }∞ t=0 we have lim ρ(xt , Ω(a)) = 0. t→∞

It is not difficult to see that if for a nonempty closed set B ⊂ X lim ρ(xt , B) = 0

t→∞

for each trajectory {xt }∞ t=0 , then Ω(a) = B. In this section we study uniform convergence of trajectories to the global attractor Ω(a). The following useful result will be proved in Sect. 8.13. Theorem 8.25 Let  > 0. Then there exist δ > 0 and a natural number T such that T −1 T if {xt }t=0 ⊂ X and if {(yt , z t )}t=0 ⊂ graph(a) satisfies ρ(xt , yt ) ≤ δ, ρ(xt+1 , z t ) ≤ δ for all t = 0, . . . , T − 1, then

322

8 Discrete Dispersive Dynamical Systems

min{ρ(xt , Ω(a)) : t = 0, . . . , T } ≤ . The following theorem, which will be proved in Sect. 8.14, provides necessary and sufficient conditions for uniform convergence of trajectories to the global attractor. It also shows that convergence of trajectories to the global attractor holds even in the presence of computational errors. Theorem 8.26 The following properties are equivalent: (1) If a sequence {xt }∞ t=−∞ ⊂ K satisfies x t+1 ∈ a(x t ) for all integers t, then ⊂ Ω(a). {xt }∞ t=−∞ (2) For each  > 0 there is a natural number L such that for each trajectory {xt }∞ t=0 , ρ(xt , Ω(a)) ≤  for all integers t ≥ L . (3) Let  > 0. Then there exist δ > 0 and a natural number L such that for T ⊂ X which satisfies for all each integer T > 2L and each sequence {xt }t=0 t = 0, . . . , T − 1 inf{ρ(xt , y) + ρ(xt+1 , z) : (y, z) ∈ graph(a)} ≤ δ, the inequality ρ(xt , Ω(a)) ≤  holds for all integers t = L , . . . , T − L . The following result will be proved in Sect. 8.15. Theorem 8.27 Assume that the property (1) of Theorem 8.26 holds and let  > 0. Then there exist δ > 0 and a natural number L such that for each integer T > L and T each sequence {xt }t=0 ⊂ X which satisfies ρ(x0 , Ω(a)) < δ

(8.88)

inf{ρ(xt , y) + ρ(xt+1 , z) : (y, z) ∈ graph(a)} ≤ δ

(8.89)

and

for t = 0, . . . , T − 1, the following inequality holds: ρ(xt , Ω(a)) ≤ , t = 0, . . . , T − L . In Sect. 8.16 we obtain generic convergence results for certain classes of set-valued mappings. The results of this section we obtained in [97].

8.13 Proof of Proposition 8.24 and Theorem 8.25

323

8.13 Proof of Proposition 8.24 and Theorem 8.25 Proof of Proposition 8.24 Let n ≥ 1 be an integer. By (A) there exist a sequence {xt(n) }nt=0 ⊂ X and a sequence {(yt(n) , z t(n) )}n−1 t=0 ⊂ graph(a)

(8.90)

(n) , z t(n) ) ≤ 1/n, t = 0, . . . , n − 1. ρ(xt(n) , yt(n) ), ρ(xt+1

(8.91)

such that

By extracting subsequences and using the diagonalization process we obtain that there exists a strictly increasing sequence of natural numbers {n j }∞ j=1 such that for any integer t ≥ 0 there exist (n j )

xt = lim xt j→∞

(n j )

, yt = lim yt j→∞

(n j )

, z t = lim z t j→∞

.

(8.92)

In view of (8.91) and (8.92), xt = yt , xt+1 = z t for all integers t ≥ 0.

(8.93)

Since graph(a) is a closed set, relations (8.90) and (8.92) imply that (yt , z t ) ∈ graph(a) for all integers t ≥ 0. Combined with (8.93) this implies that {xt }∞ t=0 is a trajectory of a. Proposition 8.24 is proved. Proof of Theorem 8.25 Let us assume that the theorem does not hold. Then for each natural number n there exist {xt(n) }nt=0 ⊂ X and (yt(n) , z t(n) ) ∈ graph(a), t = 0, . . . , n − 1

(8.94)

such that (n) , z t(n) ) ≤ 1/n for t = 0, . . . , n − 1 ρ(xt(n) , yt(n) ) ≤ 1/n and ρ(xt+1

(8.95)

and min{ρ(xt(n) , Ω(a)) : t = 0, . . . , n} > .

(8.96)

By extracting subsequences and using the diagonalization process we obtain that there exists a strictly increasing sequence of natural numbers {n j }∞ j=1 such that for each integer t ≥ 0 there exist

324

8 Discrete Dispersive Dynamical Systems (n j )

xt = lim xt j→∞

(n j )

, yt = lim yt j→∞

(n j )

, z t = lim z t j→∞

.

(8.97)

Since graph(a) is a closed set it follows from (8.94) and 8.97 that (yt , z t ) ∈ graph(a) for any integer t ≥ 0.

(8.98)

By (8.95) and (8.97), xt = yt , xt+1 = z t+1 for all integers t ≥ 0. Together with (8.99) this implies that {xt }∞ t=0 is a trajectory of a. By (8.96) and (8.97), ρ(xt , Ω(a)) ≥  for all integers t ≥ 0. This contradicts the definition of Ω(a). The contradiction we have reached proves Theorem 8.25.

8.14 Proof of Theorem 8.26 It is clear that the property (1) follows from the property (2) and that the property (2) follows from the property (3). Assume that the property (1) holds. We show that the property (3) holds. Let  > 0. Assume that the property (3) does not hold with . Then for each natural number n Tn ⊂ X such that for all there exist a natural number Tn ≥ 4 and a sequence {xt(n) }t=0 t = 0, . . . , Tn − 1 (n) , z) : (y, z) ∈ graph(a)} ≤ 1/n inf{ρ(xt(n) , y) + ρ(xt+1

(8.99)

and that sup{ρ(xt(n) , Ω(a)) : t = n, . . . , Tn − n} > .

(8.100)

Let n ≥ 1 be an integer. By (8.99), for any integer t = 0, . . . , Tn − 1 there are (n) , z t(n) ) ≤ 1/n. (yt(n) , z t(n) ) ∈ graph(a) such that ρ(xt(n) , yt(n) ) + ρ(xt+1

(8.101)

In view of (8.100), there is an integer τn ∈ [n, Tn − n] such that

(8.102)

8.14 Proof of Theorem 8.26

325

ρ(xτ(n) , Ω(a)) > . n

(8.103)

For each integer t ∈ [−τn , Tn − τn ] put (n) x˜t(n) = xt+τ n

(8.104)

and for any t ∈ [−τn , Tn − τn − 1] set (n) (n) , z˜ t(n) = z t+τ . y˜t(n) = yt+τ n n

(8.105)

and ρ(x˜0(n) , Ω(a)) > . x˜0(n) = xτ(n) n

(8.106)

By (8.103) and (8.104),

By (8.101) and (8.105), for all t ∈ [−τn , Tn − τn − 1] we have ( y˜t(n) , z˜ t(n) ) ∈ graph(a)

(8.107)

and in view of (8.101), (8.104) and (8.105), for all t ∈ [−τn , Tn − τn − 1] we have (n) (n) (n) (n) (n) , z˜ t(n) ) = ρ(xt+τ , yt+τ ) + ρ(xt+1+τ , z t+τ ) ≤ 1/n. (8.108) ρ(x˜t(n) , y˜t(n) ) + ρ(x˜t+1 n n n n

By extracting subsequences and using the diagonalization process we obtain that there exists a strictly increasing sequence of natural numbers {n j }∞ j=1 such that for any integer t there exist (n j )

xt = lim x˜t j→∞

(n j )

, yt = lim y˜t j→∞

(n j )

, z t = lim z˜ t j→∞

.

(8.109)

Since graph(a) is a closed set, relations (8.107) and (8.109) imply that (yt , z t ) ∈ graph(a) for all integers t.

(8.110)

By (8.108) and (8.109), xt = yt , xt+1 = z t for all integers t.

(8.111)

It follows from (8.110), (8.109) and the property (1) that {xt }∞ t=−∞ ⊂ Ω(a). On the other hand, by (8.106) and (8.109), ρ(x0 , Ω)) ≥ . The contradiction we have reached proves that the property (3) holds. This completes the proof of Theorem 8.26.

326

8 Discrete Dispersive Dynamical Systems

8.15 Proof of Theorem 8.27 Let δ > 0 and a natural number L be as guaranteed by the property (3) of Theorem 8.26. T ⊂ X satisfies (8.88) Assume that an integer T > L and that a sequence {xt }t=0 and (8.89) for all t = 0, . . . , T − 1. By (8.88) there is z 0 ∈ Ω(a) such that ρ(x0 , z 0 ) < δ.

(8.112)

In view of (8.112) and the definition of Ω(a)) (see (8.87)) there exists a trajectory {yt }∞ t=0 such that lim inf ρ(yt , z 0 ) < δ − ρ(x0 , z 0 ). t→∞

(8.113)

By (8.113) there is an integer s > L + 2 such that ρ(ys , z 0 ) < δ − ρ(x0 , z 0 ).

(8.114)

T +s Define a sequence {x˜t }t=0 ⊂ X by

x˜t = yt , t = 0, . . . , s − 1, x˜t = xt−s , t = s, . . . , T + s.

(8.115)

Since {yt }∞ t=0 is a trajectory it follows from (8.89) and (8.115) that inf{ρ(x˜t , y) + ρ(x˜t+1 , z) : (y, z) ∈ graph(a)} ≤ δ for all t = 0, . . . , s − 2 and all t = s, . . . , T + s − 1.

(8.116)

Since {yt }∞ t=0 is a trajectory, relations (8.114) and (8.115) imply that inf{ρ(x˜s−1 , y) + ρ(x˜s , z) : (y, z) ∈ graph(a)} ≤ ρ(x˜s−1 , ys−1 ) + ρ(x˜s , ys ) = ρ(x˜s , ys ) = ρ(x0 , ys ) ≤ ρ(x0 , z 0 ) + ρ(z 0 , ys ) < ρ(x0 , z 0 ) + δ − ρ(x0 , z 0 ) = δ. Thus (8.116) holds for all t = 0, . . . , T + s − 1. Combined with the choice of L , δ this implies that ρ(x˜t , Ω(a)) ≤ , t = L , . . . , T + s − L . Together with (8.115) this implies that ρ(xt , Ω(a)) ≤ , t = 0, . . . , T − L . Theorem 8.27 is proved.

8.16 Generic Results

327

8.16 Generic Results In this section we consider classes of discrete disperse dynamical systems whose global attractors are a singleton. Let (Y, ρ) be a compact space. Denote by Π (Y ) the set of all nonempty closed subsets of Y . For each x ∈ Y and each A ∈ Π (Y ) put ρ(x, A) = inf{ρ(x, y) : y ∈ A}. For each A, B ∈ Π (Y ) set HY (A, B) = max{sup ρ(x, B), sup ρ(y, A)}. x∈A

y∈B

It is known that the space (Π (Y ), HY ) is a complete metric space. Let (X, ρ) be a compact metric space. For each (x1 , x2 ), (y1 , y2 ) ∈ X × X set ρ1 ((x1 , x2 ), (y1 , y2 )) = ρ(x1 , y1 ) + ρ(x2 , y2 ). Then (X × X, ρ1 ) is a complete metric space. We consider the complete metric space (Π (X × X ), H X ×X ). It is clear that any set-valued mapping defined on a subset of X with values in Π (X ) is identified with its graph. For each S ∈ Π (X × X ) set K S = {x ∈ X : there is y ∈ X such that (x, y) ∈ S},

(8.117)

a S (x) = {y ∈ X : (x, y) ∈ S}, x ∈ K S .

(8.118)

Denote by M the set of all S ∈ Π (X × X ) such that a S possesses a trajectory {xt }∞ t=0 . ¯ the closure of M in the space (Π (X × X ), H X ×X ). Denote by M Let K be a nonempty closed subset of X . Denote by Π K (X × X ) the set of all S ∈ Π (X × X ) such that K S = K . Clearly, Π K (X × X ) is a closed subset of (Π (X × X ), H X ×X ). Set M K = M ∩ Π K (X × X ). ¯ K the closure of M K in (Π (X × X ), H X ×X ). Denote by M ¯ and M ¯ K with the metric H X ×X . We equip the spaces M In this section we prove the following result obtained in [97] which shows that ¯ (respectively M ¯ K ) (in the sense of Baire category) belong to most elements of M M (respectively, M K ). Theorem 8.28 The set M (respectively, M K ) contains a countable intersection of ¯ (respectively, M ¯ K ). open everywhere dense subsets of M

328

8 Discrete Dispersive Dynamical Systems

Proof Let L be either M or M K . Denote by L¯ the closure of L in (Π (X × X ), H X ×X ). Let S ∈ L and n be a natural number. There is an open neighborhood U(S, n) of S in L¯ such that for each D ∈ U(S, n) we have H X ×X (D, S) ≤ (4n)−1 .

(8.119)

Set F=

∞  

{U(s, n) : S ∈ L, n = 1, 2, . . . }.

(8.120)

n=1

¯ Clearly, F is a countable intersection of open everywhere dense subsets of L. Let D ∈ F. We show D ∈ L. By Proposition 8.24 it is sufficient to show that the assumption (A) holds for a D . Let n ≥ 1 be an integer. By (8.120) there is S ∈ L such that D ∈ U(S, n).

(8.121)

There is a trajectory {xt }∞ t=0 of a S . Let t ≥ 0 be an integer. Then (x t , x t+1 ) ∈ graph(a S ) = S. Together with (8.119) and (8.121) this implies that there is (yt , z t ) ∈ D such that ρ(xt , yt ), ρ(xt+1 , z t ) ≤ (4n)−1 . Thus the assumption (A) holds for a D . Theorem 8.28 is proved. Denote by Mr the set of all S ∈ Π (X × X ) such that a S possesses a trajectory {xt }∞ t=0 , Ω(a S ) is a singleton and the property (1) of Theorem 8.26 holds with a = a S . Set Mr,K = Mr ∩ Π K (X × X ). ¯ r,K ) the closure of Mr (respectively, Mr,K ) in ¯ r (respectively, M Denote by M the metric space (Π (X × X ), H X ×X ). ¯ r,K with the metric H X ×X . ¯ r and M We equip the spaces M In this section we prove the following result obtained in [97] which shows that ¯ r,K ) (in the sense of Baire category) belong to ¯ r (respectively M most elements of M Mr (respectively, Mr,K ). Theorem 8.29 The set Mr (respectively, Mr,K ) contains a countable intersection ¯ r,K ). ¯ r (respectively, M of open everywhere dense subsets of M Proof Let L be either Mr or Mr,K . Denote by L¯ the closure of L in Π (X × X ). Let S ∈ L and n ≥ 1 be an integer. By Theorem 8.26 there exist a positive number δ(S, n) < (8n)−1 and a natural number L(S, n) such that the following property holds:

8.16 Generic Results

329

(P) If a sequence {xt }∞ t=0 ⊂ X satisfies inf{ρ(xt , y) + ρ(xt+1 , z) : (y, z) ∈ S} ≤ 4δ(S, n) for all integers t ≥ 0, then ρ(xt , Ω(a S )) ≤ (8n)−1 for all integers t ≥ L(s, n). There is an open neighborhood U(S, n) of S in L¯ such that for each D ∈ U(S, n) we have H X ×X (D, S) ≤ 4−1 δ(S, n).

(8.122)

∞   {U(S, n) : S ∈ L, n = 1, 2, . . . }. F=

(8.123)

Set

n=1

¯ Clearly, F is a countable intersection of open everywhere dense subsets of L. Let D ∈ F. In order to prove the theorem it is sufficient to show that D ∈ L. Let n ≥ 1 be an integer. By (8.123) there is Sn ∈ L such that D ∈ U(Sn , n).

(8.124)

There is a trajectory {xt }∞ t=0 of a Sn . Let t ≥ 0 be an integer. Then (xt , xt+1 ) ∈ Sn and by (8.122) and (8.124) there is (yt , z t ) ∈ D such that ρ(xt , yt ), ρ(xt+1 , z t ) ≤ 4/n. Since n is an arbitrary natural number, Proposition 8.24 implies that D ∈ M. Assume that {xt }∞ t=0 is a trajectory of a D . Then for each integer t ≥ 0 we have (xt , xt+1 ) ∈ D. Together with (8.122) and (8.124) this implies that for each integer t ≥ 0 there is (yt , z t ) ∈ Sn such that ρ(xt , yt ), ρ(xt+1 , z t ) ≤ 4−1 δ(Sn , n). Together with the property (P) this implies that

330

8 Discrete Dispersive Dynamical Systems

ρ(xt , Ω(a Sn )) ≤ (8n)−1 for all integers t ≥ L(Sn , n).

(8.125)

This implies that H X ×X (Ω(a Sn ), Ω(a D )) ≤ 1/n.

(8.126)

For each integer n ≥ 0 there is ξn ∈ X such that Ω(a Sn ) = {ξn }.

(8.127)

By (8.126), {ξn }∞ n=1 is a Cauchy sequence. There is ξ = lim ξn

(8.128)

Ω(a D )) = ξ,

(8.129)

n→∞

in (X ρ). By (8.126)–(8.128),

ρ(ξn , ξ) ≤ 1/n for any integer n ≥ 1. Assume that {xt }∞ t=−∞ is a trajectory of a D and let n ≥ 1 be an integer. By (8.125) (which holds for any trajectory of a D ), ρ(xt , ξn ) ≤ (8n)−1 for all integers t. Together with (8.129) this implies that ρ(xt , ξ) ≤ 2/n for all integers t. Since n is an arbitrary natural number, we conclude that xt = ξ for all integers t. Theorem 8.29 is proved.

8.17 Dynamical Systems with a Lyapunov Function The results of this section were obtained in [99]. Let (X, ρ) be a compact metric space and let a : X → 2 X \ {∅} be a set-valued mapping whose graph graph(a) = {(x, y) ∈ X × X : y ∈ a(x)} is a closed subset of X × X . We continue to use the notation and definitions introduced in Sect. 8.1. Recall that

8.17 Dynamical Systems with a Lyapunov Function

331

Ω(a) = {z ∈ X : for each  > 0 there is a trajectory {xt }∞ t=0 such that lim inf ρ(z, xt ) ≤ }. t→∞

(8.130)

Let φ : X → R 1 be a continuous function such that φ(z) ≥ 0 for all z ∈ X, φ(y) ≤ φ(x) for all x ∈ X and all y ∈ a(x).

(8.131) (8.132)

It is clear that the function φ is a Lyapunov function for the dynamical system generated by the mapping a. It should be mentioned that in mathematical economics usually X is a subset of the finite-dimensional Euclidean space and φ is a linear functional on this space [58, 77, 93]. The following theorem will be proved in Sect. 8.18. Theorem 8.30 The following properties are equivalent: (1) If a sequence {xt }∞ t=−∞ ⊂ X satisfies x t+1 ∈ a(x t ) and φ(x t+1 ) = φ(x t ) for all integers t, then {xt }∞ t=−∞ ⊂ Ω(a). (2) For each  > 0 there exists a natural number T () such that for each trajectory {xt }∞ t=0 ⊂ X satisfying φ(x t ) = φ(x t+1 ) for all integers t ≥ 0 the inequality ρ(xt , Ω(a)) ≤  holds for all integers t ≥ T (). For each x ∈ X set π(x) = sup{ lim φ(xt ) : {xt }∞ t=0 is a trajctory and x 0 = x}. t→∞

(8.133)

The following two results will be proved in Sect. 8.19. Proposition 8.31 Let x ∈ X . Then there is a trajectory {xt }∞ t=0 such that x 0 = x and π(x) = limt→∞ φ(xt ). Proposition 8.32 The function π : X → R 1 is upper semicontinuous. It is clear that for each x ∈ X and each y ∈ a(x), π(y) ≤ π(x),

(8.134)

π(x) ≤ φ(x)

(8.135)

for each x ∈ X ,

and that for each x ∈ X and each natural number n, π(x) ≤ sup{φ(y) : y ∈ a n (x)}.

(8.136)

332

8 Discrete Dispersive Dynamical Systems

It is easy to see that the following proposition holds. Proposition 8.33 Let x ∈ X and {xt }∞ t=0 ⊂ X be a trajectory such that x 0 = x. Then lim φ(xt ) = π(x)

t→∞

if and only if for each integer t ≥ 0, π(xt+1 ) = max{π(z) : z ∈ a(xt )}. The following result will also be proved in Sect. 8.19. Proposition 8.34 Let x ∈ X . Then π(x) = lim sup{φ(y) : ∈ a n (x)}. n→∞

The following theorem will be proved in Sect. 8.20. Theorem 8.35 Assume that the property (1) of Theorem 8.30 holds. Let  > 0 and x ∈ X . Then there exist δ > 0 and a natural number L such that T for each integer T > 2L and each trajectory {xt }t=0 satisfying x0 = x and φ(x T ) ≥ π(x0 ) − δ the following inequality holds: ρ(xt , Ω(a)) ≤ , t = L , . . . , T − L . In this section we use the following property. (P) If x1 , x2 ∈ Ω(a) and φ(x1 ) = φ(x2 ), then x1 = x2 . Note that the property (P) holds for many models of economic dynamics for which Ω(a) is a subinterval of a line [58, 77, 93]. The following theorem will be proved in Sect. 8.21. Theorem 8.36 Assume that the property (P) holds. Then each trajectory of a converges to an element of Ω(a). It is not difficult to see that the following result holds. Proposition 8.37 Assume that the property (P) holds and that {xt }∞ t=0 is a trajectory of a such that limt→∞ φ(xt ) = π(x). Then by Theorem 8.36 there exists F(x) = lim xt , t→∞

8.17 Dynamical Systems with a Lyapunov Function

333

the equality φ(F(x)) = lim φ(xt ) = π(x) t→∞

holds and moreover, F(x) is a unique element of Ω(a) belonging to φ−1 (π(x)). In the sequel if the property (P) holds, then for each x ∈ X we denote by F(x) the unique element of Ω(a) ∩ φ−1 (π(x)). The following turnpike result describes the structure of optimal (with respect to the functional φ) trajectories of a. It will be proved in Sect. 8.22. Theorem 8.38 Assume that the property (P) and the property (1) of Theorem 8.30 hold. Let  > 0 and x ∈ X . Then there exist δ > 0 and a natural number L such that T for each integer T > 2L and each trajectory {xt }t=0 satisfying x0 = x and φ(x T ) ≥ π(x) − δ the following inequality holds: ρ(xt , F(x)) ≤ , t = L , . . . , T − L .

8.18 Proof of Theorem 8.30 It is clear that the property (2) implies the property (1). Let us show that the property (1) implies the property (2). Assume that the property (1) holds and assume that the property (2) does not hold. Then there is  > 0 such that for each natural number n there exist a trajectory {xt(n) }∞ t=0 and an integer τn ≥ n such that , Ω(a)) > , ρ(xτ(n) n φ(xt(n) ) = φ(x0(n) ), t = 0, 1, . . . .

(8.137)

Let n ≥ 1 be an integer. Define (n) for all integers t ≥ −τn . yt(n) = xt+τ n

(8.138)

, n = 1, 2, . . . . y0(n) = xτ(n) n

(8.139)

Clearly,

Extracting subsequences and using the diagonalization process we obtain that there exists a strictly increasing sequence of natural numbers {n j }∞ j=1 such that for each integer t there is

334

8 Discrete Dispersive Dynamical Systems (n j )

yt = lim yt j→∞

.

(8.140)

It is not difficult to see that yt+1 ∈ a(yt ) and φ(yt ) = φ(yt+1 ) for all integers t.

(8.141)

In view of (8.137), (8.139) and (8.140), ρ(y0 , Ω(a)) ≥ .

(8.142)

By (8.141) and property (1), {yt }∞ t=−∞ ⊂ Ω(a). This contradicts (8.142). The contradiction we have reached proves that (1) implies (2). Theorem 8.30 is proved.

8.19 Proofs of Propositions 8.31, 8.32 and 8.34 Proof of Proposition 8.31. It is clear that for each integer n ≥ 1 there is a trajectory {xt(n) }∞ t=0 such that x0(n) = x, lim φ(xt(n) ) ≥ π(x) − 1/n.

(8.143)

t→∞

Extracting subsequences and using the diagonalization process we obtain that there exists a strictly increasing sequence of natural numbers {n j }∞ j=1 such that for each integer t ≥ 0 there is (n j )

yt = lim xt j→∞

.

(8.144)

Clearly, {yt }∞ t=0 is a trajectory and y0 = x. By (8.143) and (8.144) for each integer s≥0 (n j )

φ(ys ) = lim φ(xs j→∞

(n j )

) ≥ lim sup lim φ(xt j→∞ t→∞

)

≥ lim π(x) − n −1 j = π(x). j→∞

Proposition 8.31 is proved. Proof of Proposition 8.32. Let (n) x ∈ X , {x (n) }∞ = x. n=1 ⊂ X, lim x n→∞

We show that

π(x) ≥ lim sup π(x (n) ). n→∞

(8.145)

8.19 Proofs of Propositions 8.31, 8.32 and 8.34

335

We may assume without loss of generality that there is limn→∞ π(x (n) ). By Proposition 8.31, for each integer n ≥ 1 there is a trajectory {xt(n) }∞ t=0 such that for each integer n ≥ 1 x0(n) = x (n) , lim φ(xt(n) ) = π(x0(n) ) = π(x (n) ). t→∞

(8.146)

Extracting subsequences and using the diagonalization process we obtain that there is a strictly increasing sequence of natural numbers {n j }∞ j=1 such that for each integer t ≥ 0 there is (n j )

xt = lim xt

.

(8.147)

= lim x (n j ) = x.

(8.148)

j→∞

Clearly, {xt )∞ t=0 is a trajectory and (n j )

x0 = lim x0 j→∞

j→∞

By (8.146) and (8.147), for each integer t ≥ 0 (n j )

φ(xt ) = lim φ(xt j→∞

(n j )

) ≥ lim sup lim φ(xs j→∞ s→∞

) = lim sup π(x (n) ). j→∞

Together with (8.133) and (8.148) this implies π(x) ≥ lim φ(xt ) ≥ lim π(x (n) ). t→∞

n→∞

Proposition 8.32 is proved. Proof of Proposition 8.34. Clearly, the sequence {sup{φ(z) : z ∈ a n (x)}}∞ n=1 is monotone decreasing and its limit is larger or equal than π(x). Assume that the proposition does not hold. Then l := lim sup{φ(z) : z ∈ a n (x)} > π(x). n→∞

(8.149)

For each natural number n there is a trajectory {xt(n) : t = 0, . . . , n} such that x0(n) = x, φ(xn(n) ) ≥ l.

(8.150)

Extracting a subsequence and using the diagonalization process we obtain that there exists a strictly increasing sequence of natural numbers {n j }∞ j=1 such that for each integer t ≥ 0 there is

336

8 Discrete Dispersive Dynamical Systems (n j )

xt = lim xt j→∞

.

(8.151)

Clearly, {xt }∞ t=0 is a trajectory and x0 = x.

(8.152)

By (8.149)–(8.151) for each integer t ≥ 0, (n j )

φ(xt ) = lim φ(xt j→∞

)≥l

and lim φ(xt ) ≥ l > π(x).

t→∞

This contradicts the definition of π(x) (see (8.133)). The contradiction we have reached proves Proposition 8.34.

8.20 Proof of Theorem 8.35 Assume that the theorem does not hold. Then for each natural number n there exist Tn satisfying an integer Tn > 4n, a trajectory {xt(n) }t=0 ) ≥ π(x) − 1/n, x0(n) = x φ(x T(n) n

(8.153)

τn ∈ [n, Tn − n]

(8.154)

, Ω(a)) > . ρ(xτ(n) n

(8.155)

and an integer

such that

By (8.132) and (8.153), for each integer n ≥ 1 and each each integer t ∈ [0, Tn ], φ(x) ≥ φ(xt(n) ) ≥ π(x) − 1/n.

(8.156)

Let n ≥ 1 be an integer. Set (n) for all integers t = −τn , . . . , Tn − τn . yt(n) = xt+τ n

In view of (8.153), (8.155) and (8.157),

(8.157)

8.20 Proof of Theorem 8.35

337

ρ(y0(n) , Ω(a)) > ,

(8.158)

) φ(yT(n) n −τn

(8.159)

≥ π(x) − 1/n.

By (8.156) and (8.157), for each integer n ≥ 1 and each integer t ∈ [−τn , Tn − τn ], φ(x) ≥ φ(yt(n) ) ≥ π(x) − 1/n.

(8.160)

Extracting subsequences and using the diagonalization process we obtain that there exists a strictly increasing sequence of natural numbers {n j }∞ j=1 such that for each integer t there exists (n j )

yt = lim yt

.

(8.161)

ρ(y0 , Ω(a)) ≥ .

(8.162)

j→∞

By (8.158) and (8.161),

In view of (8.160) and (8.161), for each integer t ≥ 0, (n j )

φ(yt ) = lim φ(yt j→∞

) ≥ π(x).

(8.163)

Let t be an integer. By (8.153), (8.157), (8.161), (8.163) and Proposition 8.34, (n j )

φ(yt ) = lim φ(yt j→∞

(n )

) = lim φ(xt+τj n j ) j→∞

≤ lim sup sup{φ(z) : z ∈ a t+τn j (x)} = π(x). j→∞

Together with (8.163) this implies that φ(yt ) = π(x) for all integers t.

(8.164)

By (8.157) and (8.161), yt+1 ∈ a(yt ) for all integers t. Combined with (8.164) and the property (1) this implies that {yt }∞ t=−∞ ⊂ Ω(a). This contradicts (8.162). The contradiction we have reached proves Theorem 8.35.

338

8 Discrete Dispersive Dynamical Systems

8.21 Proof of Theorem 8.36 Let {xt }∞ t=0 be a trajectory of a and y be its limit point. Then y ∈ Ω(a) and φ(y) = lim φ(xt ). t→∞

If z is also a limit point of {xt }∞ t=0 , then z ∈ Ω(a) and φ(z) = lim t→∞ φ(x t ). By the property (P) y = z. This implies that y = lim xt . t→∞

Theorem 8.36 is proved.

8.22 Proof of Theorem 8.38 Recall that F(x) is as guaranteed by Proposition 8.37. Namely, {F(x)} = Ω(a) ∩ φ−1 (π(x)). By the property (P) there is δ1 > 0 such that the following property holds: (P1) If z 1 , z 2 ∈ Ω(a) satisfy |φ(z 1 ) − φ(z 2 )| ≤ 2δ1 , then ρ(z 1 , z 2 ) ≤ /4. Since φ is uniformly continuous on X there is 1 ∈ (0, /4) such that the following property holds: (P2) For each z 1 , z 2 ∈ X satisfying ρ(z 1 , z 2 ) ≤ 41 , |φ(z 1 ) − φ(z 2 )| ≤ δ1 /4. By Proposition 8.34, there is a natural number L 0 such that | sup{φ(z) : z ∈ a L 0 (x)} − π(x)| ≤ δ1 /2.

(8.165)

By Theorem 8.35, there exist δ ∈ (0, δ1 ) and a natural number L > 2L 0 such that the following property holds: T satisfying (P3) For each integer T > 2L and each trajectory {xt }t=0 φ(x T ) ≥ π(x0 ) − δ, x0 = x the inequality ρ(xt , Ω(a)) ≤ 1 , t = L , . . . , T − L holds. T satisfies Assume that an integer T > 2L and that a trajectory {xt }t=0

8.22 Proof of Theorem 8.38

339

x0 = x, φ(x T ) ≥ π(x) − δ.

(8.166)

ρ(xt , Ω(a)) ≤ 1 , t = L , . . . , T − L .

(8.167)

Then by (8.166) and (P3),

Assume that an integer t ∈ [L , T − L]. By (8.166) and (8.167), there is z such that z ∈ Ω(a), ρ(xt , z) ≤ 1 .

(8.168)

In view of (8.165), (8.166) and the relation L 0 < L ≤ t ≤ T , φ(xt ) ≥ φ(x T ) ≥ π(x) − δ ≥ π(x) − δ1 , φ(xt ) ≤ φ(x L 0 ) ≤ π(x) + δ1 /2 and |φ(xt ) − π(x)| ≤ δ1 .

(8.169)

|φ(z) − φ(xt )| ≤ δ1 /4.

(8.170)

By (8.168) and (P2),

It follows from Proposition 8.37, (8.169) and (8.170) that |φ(z) − φ(F(x))| = |φ(z) − π(x)| ≤ |φ(z) − φ(xt )| + |φ(xt ) − π(x)| ≤ (3/2)δ1 . Together with the definition of F(x), (P1) and the inclusions z, F(x) ∈ Ω(a) this implies that ρ(F(x), z) ≤ /4. Together with (8.168) this implies that ρ(xt , F(x)) ≤ ρ(xt , z) + ρ(z, F(x)) ≤ 1 + /4 < /2. Theorem 8.38 is proved.

References

1. Artstein Z, Leizarowitz A (1985) Tracking periodic signals with the overtaking criterion. IEEE Trans Autom Control 30:1123–1126 2. Aseev SM, Krastanov MI, Veliov VM (2017) Optimality conditions for discrete-time optimal control on infinite horizon. Pure Appl Funct Anal 2:395–409 3. Aseev SM, Veliov VM (2012) Maximum principle for infinite-horizon optimal control problems with dominating discount. Dyn Contin Discrete Impuls Syst Ser B 19:43–63 4. Aseev SM, Veliov VM (2012) Necessary optimality conditions for improper infinite-horizon control problems. Oper Res Proc 21–26 5. Atsumi H (1965) Neoclassical growth and the efficient program of capital accumulation. Rev Econ Stud 3:127–136 6. Aubry S, Le Daeron PY (1983) The discrete Frenkel-Kontorova model and its extensions I. Physica D 8:381–422 7. Bachir M, Blot J (2015) Infinite dimensional infinite-horizon Pontryagin principles for discrete-time problems. Set-Valued Var Anal 23:43–54 8. Bachir M, Blot J (2017) Infinite dimensional multipliers and Pontryagin principles for discretetime problems. Pure Appl Funct Anal 2:411–426 9. Baumeister J, Leitao A, Silva GN (2007) On the value function for nonautonomous optimal control problem with infinite horizon. Syst Control Lett 56:188–196 10. Blot J (2009) Infinite-horizon Pontryagin principles without invertibility. J Nonlinear Convex Anal 10:177–189 11. Blot J, Cartigny P (2000) Optimality in infinite-horizon variational problems under sign conditions. J Optim Theory Appl 106:411–419 12. Blot J, Hayek N (2000) Sufficient conditions for infinite-horizon calculus of variations problems. ESAIM Control Optim Calc Var 5:279–292 13. Blot J, Hayek N (2014) Infinite-horizon optimal control in the discrete-time framework. Springer Briefs in Optimization, New York 14. Brock WA (1970) On existence of weakly maximal programmes in a multi-sector economy. Rev Econ Stud 37:275–280 15. Carlson DA (1990) The existence of catching-up optimal solutions for a class of infinite horizon optimal control problems with time delay. SIAM J Control Optim 28:402–422 16. Carlson DA, Haurie A, Leizarowitz A (1991) Infinite horizon optimal control. Springer, Berlin 17. Cartigny P, Michel P (2003) On a sufficient transversality condition for infinite horizon optimal control problems. Automatica J IFAC 39:1007–1010

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. J. Zaslavski, Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model, Monographs in Mathematical Economics 4, https://doi.org/10.1007/978-981-16-2252-6

341

342

References

18. Coleman BD, Marcus M, Mizel VJ (1992) On the thermodynamics of periodic phases. Arch Rational Mech Anal 117:321–347 19. Damm T, Grune L, Stieler M, Worthmann K (2014) An exponential turnpike theorem for dissipative discrete time optimal control problems. SIAM J Control Optim 52:1935–1957 20. Deng L, Fujio M, Khan MA (2019) Optimal growth in the Robinson–Shinkai–Leontief model: the case of capital-intensive consumption goods. Stud Nonlinear Dyn Econom 23:18 21. De Oliveira VA, Silva GN (2009) Optimality conditions for infinite horizon control problems with state constraints. Nonlinear Anal 71:1788–1795 22. Dzalilov Z, Zaslavski AJ (2009) Global attractors for discrete disperse dynamical systems. J Nonlinear Convex Anal 10:191–198 23. Gaitsgory V, Grune L, Thatcher N (2015) Stabilization with discounted optimal control. Syst Control Lett 82:91–98 24. Gaitsgory V, Mammadov M, Manic L (2017) On stability under perturbations of long-run average optimal control problems. Pure Appl Funct Anal 2:461–476 25. Gaitsgory V, Parkinson A, Shvartsman I (2017) Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time. Discrete Contin Dyn Syst Ser B 22:3821–3838 26. Gaitsgory V, Rossomakhine S, Thatcher N (2012) Approximate solution of the HJB inequality related to the infinite horizon optimal control problem with discounting. Dyn Contin Discret Impuls Syst Ser B 19:65–92 27. Gale D (1967) On optimal development in a multi-sector economy. Rev Econ Stud 34:1–18 28. Glizer VY, Kelis O (2017) Upper value of a singular infinite horizon zero-sum linear-quadratic differential game. Pure Appl Funct Anal 2:511–534 29. Gugat M, Trelat E, Zuazua E (2016) Optimal Neumann control for the 1D wave equation: finite horizon, infinite horizon, boundary tracking terms and the turnpike property. Systems Control Lett 90:61–70 30. Guo X, Hernandez-Lerma O (2005) Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates. Bernoulli 11:1009–1029 31. Hammond PJ (1974) Consistent planning and intertemporal welfare economics. University of Cambridge, Cambridge 32. Hammond PJ (1975) Agreeable plans with many capital goods. Rev Econ Stud 42:1–14 33. Hammond PJ, Mirrlees JA (1973) Agreeable plans. In: Mirrlees JA, Stern NH (eds) Models of economic growth. Wiley, New York, pp 283–299 34. Hayek (2011) Infinite horizon multiobjective optimal control problems in the discrete time case. Optimization 60:509–529 35. Jasso-Fuentes H, Hernandez-Lerma O (2008) Characterizations of overtaking optimality for controlled diffusion processes. Appl Math Optim 57:349–369 36. Khan MA, Mitra T (2005) On choice of technique in the Robinson–Solow–Srinivasan model. Int J Econ Theory 1:83–110 37. Khan MA, Mitra T (2006). Optimal growth in the two-sector RSS model: a continuous time analysis. Electronic Publication 38. Khan MA, Mitra T (2006) Undiscounted optimal growth in the two-sector Robinson–Solow– Srinivasan model: a synthesis of the value-loss approach and dynamic programming. Econom Theory 29:341–362 39. Khan MA, Mitra T (2006) Discounted optimal growth in the two-sector RSS model: a geometric investigation. Adv Math Econ 8:349–381 40. Khan MA, Mitra T (2007) Optimal growth in a two-sector RSS model without discounting: a geometric investigation. Jpn Econ Rev 58:191–225 41. Khan MA, Mitra T (2007) Optimal growth under discounting in the two-sector Robinson– Solow–Srinivasan model: a dynamic programming approach. J Differ Equ Appl 13:151–168 42. Khan MA, Mitra T (2008) Growth in the Robinson–Solow–Srinivasan model: undiscounted optimal policy with a strictly concave welfare function. J Math Econom 44:707–732

References

343

43. Khan MA, Mitra T (2012) Impatience and dynamic optimal behavior: a bifurcation analysis of the Robinson–Solow–Srinivasan model. Nonlinear Anal 75:1400–1418 44. Khan MA, Mitra T (2013) Discounted optimal growth in a two-sector RSS model: a further geometric investigation. Adv Math Econ 17:39–70 45. Khan MA, Piazza A (2010) On the non-existence of optimal programs in the Robinson– Solow–Srinivasan (RSS) model. Econom Lett 109:94–98 46. Khan MA, Piazza A (2011) The concavity assumption on felicities and asymptotic dynamics in the RSS model. Set-Valued Var Anal 19:135–156 47. Khan MA, Piazza A (2011) An overview of turnpike theory: towards the discounted deterministic case. Adv Math Econ 14:39–67 48. Khan MA, Zaslavski AJ (2007) On a uniform turnpike of the third kind in the Robinson– Solow–Srinivasan model. J Econ 92:137–166 49. Khan MA, Zaslavski AJ (2009) On existence of optimal programs: the RSS model without concavity assumptions on felicities. J Math Econ 45:624–633 50. Khan MA, Zaslavski AJ (2010) On two classical turnpike results for the Robinson–Solow– Srinivisan (RSS) model. Adv Math Econom 13:47–97 51. Khan MA, Zaslavski AJ (2010) On locally optimal programs in the RSS (Robinson–Solow– Srinivasan) model. J Econ 99:65–92 52. Khlopin DV (2017) On Lipschitz continuity of value functions for infinite horizon problem. Pure Appl Funct Anal 2:535–552 53. Kolokoltsov V, Yang W (2012) The turnpike theorems for Markov games. Dyn Games Appl 2:294–312 54. Leizarowitz A (1985) Infinite horizon autonomous systems with unbounded cost. Appl Math Opt 13:19–43 55. Leizarowitz A (1986) Tracking nonperiodic trajectories with the overtaking criterion. Appl Math Opt 14:155–171 56. Leizarowitz A, Mizel VJ (1989) One dimensional infinite horizon variational problems arising in continuum mechanics. Arch Rat Mech Anal 106:161–194 57. Lykina V, Pickenhain S, Wagner M (2008) Different interpretations of the improper integral objective in an infinite horizon control problem. J Math Anal Appl 340:498–510 58. Makarov VL, Rubinov AM (1977) Mathematical theory of economic dynamics and equilibria. Springer, New York 59. Malinowska AB, Martins N, Torres DFM (2011) Transversality conditions for infinite horizon variational problems on time scales. Optimization Lett 5:41–53 60. Mammadov M (2014) Turnpike theorem for an infinite horizon optimal control problem with time delay. SIAM J Control Optim 52:420–438 61. Marcus M, Zaslavski AJ (1999) On a class of second order variational problems with constraints. Isr J Math 111:1–28 62. Marcus M, Zaslavski AJ (1999) The structure of extremals of a class of second order variational problems. In: Annales de l’Institut Henri Poincare (C) non linear analysis, vol 16, pp 593–629 63. Marcus M, Zaslavski AJ (2002) The structure and limiting behavior of locally optimal minimizers. In: Annales de l’Institut Henri Poincare (C) non linear analysis, vol 19, pp 343–370 64. McKenzie LW (1976) Turnpike theory. Econometrica 44:841–866 65. Mordukhovich BS (1990) Minimax design for a class of distributed parameter systems. Autom Remote Control 50:1333–1340 66. Mordukhovich BS (2011) Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions. Appl Anal 90:1075–1109 67. Mordukhovich BS, Nam NM (2014) An easy path to convex analysis and applications. Morgan Claypool Publishers, San Rafael, CA 68. Mordukhovich BS, Shvartsman I (2004) Optimization and feedback control of constrained parabolic systems under uncertain perturbations. Optimal control, stabilization and nonsmooth analysis. Springer, Lecture Notes Control Information Science, pp 121–132 69. Ocana AE, Cartigny P, Loisel P (2009) Singular infinite horizon calculus of variations. Applications to fisheries management. J Nonlinear Convex Anal 10:157–176

344

References

70. Okishio N (1966) Technical choice under full employment in a socialist economy. Econ J 76:585–592 71. Pickenhain S, Lykina V, Wagner M (2008) On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems. Control Cybernet 37:451–468 72. Porretta A, Zuazua E (2013) Long time versus steady state optimal control. SIAM J Control Optim 51:4242–4273 73. Reich S, Zaslavski AJ (2014) Genericity in nonlinear analysis. Springer, New York 74. Robinson J (1960) Exercises in economic analysis. MacMillan, London 75. Robinson J (1969) A model for accumulation proposed by J.E. Stiglitz. Econ J 79:412–413 76. Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton, NJ 77. Rubinov AM (1980) Magistral sets in discrete, disperse dynamical systems. Sib Math J 21:583–590 78. Rubinov AM (1980) Multivalued mappings and their applications in economic mathematical problems. Nauka, Leningrad 79. Rubinov AM (1984) Economic dynamics. J Soviet Math 26:1975–2012 80. Sagara N (2018) Recursive variational problems in nonreflexive Banach spaces with an infinite horizon: an existence result. Discrete Contin Dyn Syst Ser S 11:1219–1232 81. Samuelson PA (1965) A catenary turnpike theorem involving consumption and the golden rule. Amer Econom Rev 55:486–496 82. Shinkai Y (1960) On equilibrium growth of capital and labor. Int Econ Rev 1:107–111 83. Solow RM (1962) Substitution and fixed proportions in the theory of capital. Rev Econ Stud 29:207–218 84. Srinivasan TN (1962) Investment criteria and choice of techniques of production. Yale Econ Essays 1:58–115 85. Stiglitz JE (1968) A note on technical choice under full employment in a socialist economy. Econ J 78:603–609 86. Stiglitz JE (1970) Reply to Mrs. Robinson on the choice of technique. Econ J 80:420–422 87. Stiglitz JE (1973) Recurrence of techniques in a dynamic economy. In: Mirrlees JA, Stern NH (eds) Models of economic growth. Wiley, New York, pp 283–299 88. Trelat E, Zhang C, Zuazua E (2018) Optimal shape design for 2D heat equations in large time. Pure Appl Funct Anal 3:255–269 89. von Weizsacker CC (1965) Existence of optimal programs of accumulation for an infinite horizon. Rev Econ Stud 32:85–104 90. Zaslavski AJ (1982) Turnpike sets of continuous transformations in compact metric spaces. Sib Math J 23:136–146 91. Zaslavski AJ (1987) Ground states in Frenkel-Kontorova model. Math USSR Izv 29:323–354 92. Zaslavski AJ (2005) Optimal programs in the RSS model. Int J Econ Theory 1:151–165 93. Zaslavski AJ (2006) Turnpike properties in the calculus of variations and optimal control. Springer, New York 94. Zaslavski AJ (2006) Good programs in the RSS model with a nonconcave utility function. J Ind Manag Optim 2:399–423 95. Zaslavski AJ (2007) Uniform convergence to global attractors for discrete disperse dynamical systems. Nonlinear Dyn Syst Theory 7:315–325 96. Zaslavski AJ (2007) Turnpike results for a discrete-time optimal control systems arising in economic dynamics. Nonlinear Anal 67:2024–2049 97. Zaslavski AJ (2008) Convergence of trajectories of discrete dispersive dynamical systems. Commun Math Anal 4:10–19 98. Zaslavski AJ (2008) A turnpike result for a class of problems of the calculus of variations with extended-valued integrands. J Convex Anal 15:869–890 99. Zaslavski AJ (2009) Structure of trajectories of discrete dispersive dynamical systems. Commun Math Anal 6:1–9 100. Zaslavski AJ (2009) Structure of approximate solutions of variational problems with extendedvalued convex integrands. ESAIM Control Optim Calc Var 15:872–894

References

345

101. Zaslavski AJ (2009) The structure of good programs in the RSS Model. In: Proceedings of the annual meeting at the Kyoto University, vol 1654. Departmental Bulletin Paper, pp 166–178. repository.kulib.kyoto-u.ac.jp 102. Zaslavski AJ (2009) Good solutions for a class of infinite horizon discrete-time optimal control problems. Taiwan J Math 13:1637–1669 103. Zaslavski AJ (2009) Turnpike results for a class of infinite horizon discrete-time optimal control problems arising in economic dynamics. Set-Valued Var Anal 17:285–318 104. Zaslavski AJ (2009) Existence and structure of solutions for an infinite horizon optimal control problem arising in economic dynamics. Adv Differ Equ 14:477–496 105. Zaslavski AJ (2010) Optimal solutions for a class of infinite horizon variational problems with extended-valued integrands. Optimization 59:181–197 106. Zaslavski AJ (2010) Good locally maximal programs for the Robinson–Solow–Srinivasan (RSS) model. Adv Math Econ 13:161–176 107. Zaslavski AJ (2010) Overtaking optimal solutions for a class of infinite horizon discrete-time optimal control problems. Dyn Contin Discret Impuls Syst Ser B 17:607–620 108. Zaslavski AJ (2010) Locally maximal solutions of control systems arising in economic dynamics. Comm Appl Nonlinear Anal 17:61–68 109. Zaslavski AJ (2010) Structure of approximate solutions for discrete-time control systems arising in economic dynamics. Nonlinear Anal 73:952–970 110. Zaslavski AJ (2011) Convergence to attracting sets for discrete dispersive dynamical systems. Nonlinear Funct Anal Appl 16:313–321 111. Zaslavski AJ (2011) Two turnpike results for a continuous-time optimal control systems. In: Proceedings of an international conference, complex analysis and dynamical systems IV: function theory and optimization 553:305–317 112. Zaslavski AJ (2011) Stability of a turnpike phenomenon for the Robinson–Solow–Srinivasan model. Dyn Syst Appl 20:25–44 113. Zaslavski AJ (2011) A turnpike property of approximate solutions of an optimal control problem arising in economic dynamics. Dynam Syst Appl 20(2011):395–422 114. Zaslavski AJ (2012) Weakly agreeable programs for the Robinson–Solow–Srinivasan (RSS) model. Optim Theory Relat Top Contemp Math 568:259–271 115. Zaslavski AJ (2013) Structure of solutions of variational problems. SpringerBriefs in Optimization, New York 116. Zaslavski AJ (2013) Structure of approximate solutions of optimal control problems. SpringerBriefs in Optimization, New York 117. Zaslavski AJ (2013) On turnpike properties of approximate optimal programs of the discretetime Robinson–Solow–Srinivasan model. Commun Appl Anal 17:129–145 118. Zaslavski AJ (2013) On a class of discrete-time optimal control problems arising in economic dynamics. Pan-Am Math J 23:1–12 119. Zaslavski AJ (2013) Stability of the turnpike phenomenon for a convex optimal control problem arising in economic dynamics. Commun Appl Anal 17:271–288 120. Zaslavski AJ (2013) Existence and structure of solutions for a class of optimal control problems arising in economic dynamics. Pan-Am Math J 23(4):1–22 121. Zaslavski AJ (2014) Turnpike phenomenon and infinite horizon optimal control. Springer Optimization and Its Applications, Springer, Cham 122. Zaslavski AJ (2016) Structure of solutions of optimal control problems on large intervals: a survey of recent results. Pure Appl Funct Anal 1:123–158 123. Zaslavski AJ (2017) Discrete-time optimal control and games on large intervals. Springer Optimization and Its Applications, Springer, Cham 124. Zaslavski AJ (2018) Equivalence of optimality criterions for discrete time optimal control problems. Pure Appl Funct Anal 3:505–517 125. Zaslavski AJ (2019) Turnpike conditions in infinite dimensional optimal control. Springer Optimization and Its Applications, Springer, Cham 126. Zaslavski AJ (2021) Turnpike theory for the Robinson–Solow–Srinivasan model. Springer Optimization and Its Applications, Springer, Cham

Index

A Admissible program, 1 Admissible trajectory, 1 Approximate solution, 1, 2, 5 Asymptotic turnpike property, 2, 4, 108 Attractor, 300 Autonomous discrete-time control system, 1 Average turnpike property, 12

B Bad program, 12 Baire category, 311 Banach fixed point theorem, 310

C Cardinality of a set, 71 Cauchy sequence, 58 Compact metric space, 1 Compact set, 2, 3, 28 Complete metric space, 309 Concave function, 10, 274 Constrained problem, 3 Convex set, 6, 274

D Differentiable function, 10 Discrete disperse dynamical system, 300 Discrete-time problem, 1 Dynamic optimization problem, 1

E Euclidean space, 2, 6, 27

G Golden-rule stock, 11 Good program, 5, 11 Gross investment sequence, 10

I Increasing function, 171, 226 Infinite horizon, 1 Infinite horizon optimal control problem, 27, 33 Inner product, 8 Interior point, 6

L Lyapunov function, 331

N Norm, 6

O Objective function, 2, 7, 71 Overtaking optimal program, 5 Overtaking optimal solution, 59

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. J. Zaslavski, Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model, Monographs in Mathematical Economics 4, https://doi.org/10.1007/978-981-16-2252-6

347

348 P Program, 3 S Set-valued mapping, 299 Strictly concave function, 2, 13, 74 Strictly increasing function, 10, 171 T Trajectory, 300 Turnpike, 2, 74 Turnpike phenomenon, 2 Turnpike property, 1, 2, 108 Turnpike result, 125, 132

Index U Unconstrained problem, 3 Uniform concavity, 75 Uniform equicontinuity, 75 Uniformity, 226 Uniform space, 227 Upper semicontinuous function, 1, 3, 28 Utility function, 27

V von Neumann facet, 13 von Neumann–Gale model, 299