Turnpike Phenomenon in Metric Spaces 3031272072, 9783031272073

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Springer Optimization and Its Applications 201

Alexander J. Zaslavski

Turnpike Phenomenon in Metric Spaces

Springer Optimization and Its Applications Volume 201

Series Editors Panos M. Pardalos , University of Florida My T. Thai , University of Florida Honorary Editor Ding-Zhu Du, University of Texas at Dallas Advisory Editors Roman V. Belavkin, Middlesex University John R. Birge, University of Chicago Sergiy Butenko, Texas A&M University Vipin Kumar, University of Minnesota Anna Nagurney, University of Massachusetts Amherst Jun Pei, Hefei University of Technology Oleg Prokopyev, University of Pittsburgh Steffen Rebennack, Karlsruhe Institute of Technology Mauricio Resende, Amazon (United States) Tamás Terlaky, Lehigh University Van Vu, Yale University Michael N. Vrahatis, University of Patras Guoliang Xue, Arizona State University Yinyu Ye, Stanford University

Aims and Scope Optimization has continued to expand in all directions at an astonishing rate. New algorithmic and theoretical techniques are continually developing and the diffusion into other disciplines is proceeding at a rapid pace, with a spot light on machine learning, artificial intelligence, and quantum computing. Our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in areas not limited to applied mathematics, engineering, medicine, economics, computer science, operations research, and other sciences. The series Springer Optimization and Its Applications (SOIA) aims to publish state-of-the-art expository works (monographs, contributed volumes, textbooks, handbooks) that focus on theory, methods, and applications of optimization. Topics covered include, but are not limited to, nonlinear optimization, combinatorial optimization, continuous optimization, stochastic optimization, Bayesian optimization, optimal control, discrete optimization, multi-objective optimization, and more. New to the series portfolio include Works at the intersection of optimization and machine learning, artificial intelligence, and quantum computing. Volumes from this series are indexed by Web of Science, zbMATH, Mathematical Reviews, and SCOPUS.

Alexander J. Zaslavski

Turnpike Phenomenon in Metric Spaces

Alexander J. Zaslavski Haifa, Israel

ISSN 1931-6828 ISSN 1931-6836 (electronic) Springer Optimization and Its Applications ISBN 978-3-031-27210-3 ISBN 978-3-031-27208-0 (eBook) https://doi.org/10.1007/978-3-031-27208-0 Mathematics Subject Classification: 49J20, 49J99, 49K20, 49K27, 49K40 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In this book we study the turnpike phenomenon arising in the optimal control theory. The term was first coined by P. Samuelson in 1948 when he showed that an efficient expanding economy would spend most of the time in the vicinity of a balanced equilibrium path (also called a von Neumann path). To have the turnpike property means, roughly speaking, that the approximate solutions of the problems are determined mainly by the objective function and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. The turnpike property discovered by P. Samuelson is well known in the economic literature, where it was studied for various models of economic growth. Usually for these models, a turnpike is a singleton. In our previous research, it was shown that the turnpike property is a general phenomenon which holds for large classes of finite-dimensional variational and optimal control problems. In that research, using the Baire category (generic) approach, we showed that the turnpike property holds for a generic (typical) variational problem [117] and for a generic optimal control problem [130]. According to the generic approach, we say that a property holds for a generic (typical) element of a complete metric space (or the property holds generically) if the set of all elements of the metric space possessing this property contains a Gδ everywhere dense subset of the metric space which is a countable intersection of open everywhere dense sets. This means that the property holds for most elements of the metric space. In this book, we are interested in turnpike results, in sufficient and necessary conditions for the turnpike phenomenon and in its stability under small perturbations of objective functions. The most important feature of this book is that we study a large, general class of optimal control problems in metric space. Probably, at the first time in the literature to our knowledge. All the main results obtained in the book are new. The monograph contains nine chapters. Chapter 1 is an introduction. In Chap. 2 we discuss Banach space valued functions, set-valued mappings in infinite dimensional spaces, and related continuous-time dynamical systems. Some convergence results are obtained. In Chap. 3, we study a discrete-time dynamical system with a Lyapunov function in a metric space induced by a set-valued mapping. For v

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this discrete-time dynamical system, we establish a turnpike result. Chapter 4 is devoted to the study of a class of continuous-time dynamical systems which is an analog of the class of discrete-time dynamical systems considered in Chap. 3. This class also contains the class of dynamical systems studied in Chap. 2. In Chap. 5, we develop a turnpike theory for a class of general dynamical systems in a metric space with a Lyapunov function. This class of dynamical systems contains as particular cases the classes of dynamical systems studied in Chaps. 3 and 4. We obtain several turnpike results and show that the turnpike phenomenon is stable under small perturbations of objective functions. Chapter 6 contains our study of the turnpike phenomenon for discrete-time nonautonomous problems on subintervals of half-axis in metric spaces, which are not necessarily compact. For these optimal control problems, the turnpike is not a singleton. We show that the turnpike phenomenon is stable under small perturbations of an objective function. Using the Baire category approach, we show that for some classes of problems a typical (generic) problem has a turnpike property. Chapter 7 contains preliminaries which we need in order to study turnpike properties of infinite dimensional optimal control problems. We discuss unbounded operators, semigroups of linear operators, evolution equations, and admissible control operators. In Chap. 8, we establish sufficient and necessary conditions for the turnpike phenomenon for continuoustime optimal control problems on subintervals of half-axis in metric spaces. For these optimal control problems, the turnpike is not a singleton. The results of this chapter will be obtained for three large classes of problems which will be treated simultaneously. In Chap. 9, we continue to study the turnpike phenomenon for the continuous-time optimal control problems on subintervals of half-axis in metric spaces which are discussed in Chap. 8. We show that the turnpike phenomenon is stable under small perturbations of an objective function. Using the Baire category approach, we show that for some classes of problems a typical (generic) problem has a turnpike property. Rishon LeZion, Israel October 19, 2022

Alexander J. Zaslavski

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Turnpike Property for Variational Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 General Dynamical Systems with a Lyapunov Function . . . . . . . . . . . . 1.4 Continuous-Time Nonautonomous Problems on Half-Axis . . . . . . . . .

1 1 6 7 11

2

Differential Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Banach Space-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Set-Valued Mappings and a Convergence Result . . . . . . . . . . . . . . . . . . . . 2.3 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Proof of Proposition 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Proof of Proposition 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Proof of Theorem 2.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 24 27 30 33 35 36 38

3

Discrete-Time Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proofs of Propositions 3.2 and 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Proof of Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 42 43 44 45 47

4

Continuous-Time Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Proof of Theorem 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Proof of Theorem 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 54 55 56 57 59

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5

General Dynamical Systems with a Lyapunov Function . . . . . . . . . . . . . . . . 5.1 Preliminaries and Two Turnpike Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Two Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Proof of Proposition 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Proof of Proposition 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Proof of Theorem 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 An Auxiliary Result for Theorem 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Proof of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Proofs of Propositions 5.6 and 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 A Weak Turnpike Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Turnpike Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Proof of Theorem 5.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Auxiliary Results for Theorem 5.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Proofs of Theorems 5.16 and 5.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Extensions of Theorems 5.16 and 5.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16 An Auxiliary Result for Theorem 5.24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Proof of Theorem 5.24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.18 Generalizations of Theorems 5.16 and 5.17 . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Proof of Theorem 5.28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20 Continuity of the Function π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.21 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 69 72 73 74 77 78 81 84 91 93 94 97 100 105 106 108 112 114 123 126

6

Discrete-Time Nonautonomous Problems on Half-Axis . . . . . . . . . . . . . . . . . 6.1 Preliminaries and Boundedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Turnpike Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Perturbed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Space of Objective Functions and the Stability Result. . . . . . . . . . 6.6 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Proof of Theorem 6.22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Problems with Discount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Auxiliary Results for Theorem 6.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Proof of Theorem 6.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Genericity Results for Discrete-Time Problems. . . . . . . . . . . . . . . . . . . . . . 6.12 Proof of Theorem 6.29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Examples of the Space M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Extensions of the Generic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Smooth Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 134 138 139 141 145 151 158 159 171 180 182 190 191 194

7

Infinite Dimensional Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Unbounded Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 C0 Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 C0 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Admissible Control Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197 198 200 200 201 206

Contents

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8

Continuous-Time Nonautonomous Problems on Half-Axis . . . . . . . . . . . . . 8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The First Class of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Second Class of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 A0 and the First and Second Classes of Problems . . . . . . . . . . . . . . . . . . . 8.5 The Third Class of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Boundedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Turnpike Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Lower Semicontinuity Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Perturbed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Auxiliary Results for Theorems 8.5 and Its Proof . . . . . . . . . . . . . . . . . . . 8.11 Proof of Theorem 8.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.12 Proof of Theorem 8.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.13 Proof of Theorem 8.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.14 Proof of Theorem 8.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.15 An Auxiliary Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.16 Proof of Proposition 8.28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.17 Auxiliary Results for Theorem 8.29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.18 Proof of Theorem 8.29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.19 Proof of Theorem 8.33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.20 The Strong Turnpike Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.21 Auxiliary Results for Theorem 8.39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.22 Proof of Theorem 8.39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.23 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

213 213 216 218 220 224 228 231 236 238 238 241 242 243 249 254 255 265 269 272 274 275 280 293

9

Stability and Genericity Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Stability of TP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Auxiliary Results for Theorem 9.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Proof of Theorem 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Problems with Discount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Auxiliary Results for Theorem 9.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Proof of Theorem 9.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Genericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Proof of Theorem 9.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 A Turnpike Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 Spaces od Smooths Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.12 Assumption (A5) for the First Class of Problems . . . . . . . . . . . . . . . . . . . . 9.13 Assumption (A5) for the Second Class of Problems . . . . . . . . . . . . . . . . . 9.14 Assumption (A5) for the Third Class of Problems . . . . . . . . . . . . . . . . . . .

297 297 300 301 308 314 315 325 333 335 342 346 348 350 353

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Chapter 1

Introduction

In this chapter, we discuss turnpike properties in the calculus of variations. To have the turnpike property means, roughly speaking, that the approximate solutions of the problems are determined mainly by the integrand and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. We also consider some classes of optimal control problems studied in the book.

1.1 Turnpike Property for Variational Problems The study of the existence and structure of solutions of optimal control problems and dynamic games defined on infinite intervals and sufficiently large intervals has been a rapidly growing area of research [9, 10, 24, 31, 39, 43, 44, 46, 60, 63, 64, 76, 88, 117, 119, 121, 124, 129, 133, 138, 139, 142, 144, 147] that has various applications in engineering [6, 31, 70, 117], models of economic growth [7, 11, 29–32, 42, 47, 58, 59, 62, 69, 75, 81, 90, 97, 99, 105, 117, 127, 134, 143, 144], infinite discrete models of solid-state physics related to dislocations in one dimensional crystals [14, 94, 107], model predictive control [38, 50], and the theory of thermodynamical equilibrium for materials [36, 71, 78–80]. Discrete-time optimal control problems were considered in [8, 15, 16, 27, 45, 55, 108, 109, 114, 116, 120, 123, 125, 127, 131, 132, 135, 140, 141, 143], finite dimensional continuous-time problems were analyzed in [23, 25, 26, 28, 33, 69, 72, 74, 77, 91, 112, 115, 118, 130, 145, 146], infinite dimensional optimal control was studied in [31, 32, 51– 53, 84, 86, 87, 92, 101, 102, 111, 113, 122], while solutions of dynamic games were discussed in [22, 48, 49, 54, 56, 66, 93, 126, 128, 136, 137]. Sufficient and necessary conditions for the turnpike phenomenon were obtained in our previous research [115, 116, 118, 131].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0_1

1

2

1 Introduction

In this section, which is based on [115], we discuss the structure of approximate solutions of variational problems with continuous integrands .f : [0, ∞) × R n × R n → R 1 that belong to a complete metric space of functions. We do not impose any convexity assumption. The main result of this section, obtained in [115], deals with the turnpike property of variational problems. We consider the variational problems 

T2

.

f (t, z(t), z (t))dt → min, z(T1 ) = x, z(T2 ) = y,

(P )

T1

z : [T1 , T2 ] → R n is an absolutely continuous function,

.

where .T1 ≥ 0, .T2 > T1 , .x, y ∈ R n , and .f : [0, ∞) × R n × R n → R 1 belongs to a space of integrands described below. It is well known that the solutions of the problems (P) exist for integrands f that satisfy two fundamental hypotheses concerning the behavior of the integrand as a function of the last argument (derivative): one that the integrand should grow superlinearly at infinity and the other that it should be convex [100]. Moreover, certain convexity assumptions are also necessary for properties of lower semicontinuity of integral functionals that are crucial in most of the existence proofs, although there are some interesting theorems without convexity [34, 35, 83, 85]. For integrands f that do not satisfy the convexity assumption, the existence of solutions of the problems (P) is not guaranteed, and in this situation, we consider .δ-approximate solutions. Let .T1 ≥ 0, .T2 > T1 , .x, y ∈ R n , .f : [0, ∞) × R n × R n → R 1 be an integrand, and let .δ be a positive number. We say that an absolutely continuous (a.c.) function n .u : [T1 , T2 ] → R satisfying .u(T1 ) = x, .u(T2 ) = y is a .δ-approximate solution of the problem (P) if 

T2

.

T1



f (t, u(t), u (t))dt ≤



T2

f (t, z(t), z (t))dt + δ

T1

for each a.c. function .z : [T1 , T2 ] → R n satisfying .z(T1 ) = x, z(T2 ) = y. The main result of [115] deals with the turnpike property of the variational problems (P). As usual, to have this property means, roughly speaking, that the approximate solutions of the problems (P) are determined mainly by the integrand and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. In the classical turnpike theory, it was assumed that a cost function (integrand) is convex. The convexity of the cost function played a crucial role there. In [115], we get rid of convexity of integrands and establish necessary and sufficient conditions for the turnpike property for a space of nonconvex integrands .M described below. Let us now define the space of integrands. Denote by .| · | the Euclidean norm in n .R . Let a be a positive constant, and let .ψ : [0, ∞) → [0, ∞) be an increasing

1.1 Turnpike Property for Variational Problems

3

function such that .ψ(t) → +∞ as .t → ∞. Denote by .M the set of all continuous functions .f : [0, ∞) × R n × R n → R 1 that satisfy the following assumptions: A(i) The function f is bounded on .[0, ∞)×E for any bounded set .E ⊂ R n ×R n . A(ii) .f (t, x, u) ≥ max{ψ(|x|), ψ(|u|)|u|} − a for each .(t, x, u) ∈ [0, ∞) × R n × Rn. A(iii) For each .M,  > 0, there exist .Γ, δ > 0 such that |f (t, x1 , u) − f (t, x2 , u)| ≤  max{f (t, x1 , u), f (t, x2 , u)}

.

for each .t ∈ [0, ∞) and each .u, x1 , x2 ∈ R n that satisfy |xi | ≤ M, i = 1, 2, |u| ≥ Γ,

.

|x1 − x2 | ≤ δ.

A (iv) For each .M,  > 0, there exists .δ > 0 such that .|f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤  for each .t ∈ [0, ∞) and each .u1 , u2 , x1 , x2 ∈ R n that satisfy |xi |, |ui | ≤ M, i = 1, 2,

.

max{|x1 − x2 |, |u1 − u2 |} ≤ δ.

It is easy to show that an integrand .f = f (t, x, u) ∈ C 1 ([0, ∞) × R n × R n ) belongs to .M if f satisfies assumption A(ii), and if .sup{|f (t, 0, 0)| : t ∈ [0, ∞)} < ∞ and also there exists an increasing function .ψ0 : [0, ∞) → [0, ∞) such that .

sup{|∂f/∂x(t, x, u)|, |∂f/∂u(t, x, u)|} ≤ ψ0 (|x|)(1 + ψ(|u|)|u|)

for each .t ∈ [0, ∞) and each .x, u ∈ R n . For the set .M, we consider the uniformity that is determined by the following base: E(N, , λ) = {(f, g) ∈ M × M : |f (t, x, u) − g(t, x, u)| ≤ 

.

.

for each t ∈ [0, ∞) and each x, u ∈ R n satisfying |x|, |u| ≤ N and (|f (t, x, u)| + 1)(|g(t, x, u)| + 1)−1 ∈ [λ−1 , λ]

.

.

for each t ∈ [0, ∞) and each x, u ∈ R n satisfying |x| ≤ N },

where .N > 0, . > 0, .λ > 1. It is not difficult to show that the space .M with this uniformity is metrizable (by a metric .ρw ). It is known (see [115]) that the metric space .(M, ρw ) is complete. The metric .ρw induces in .M a topology.

4

1 Introduction

We consider functionals of the form  I f (T1 , T2 , x) =

T2

.

f (t, x(t), x  (t))dt,

T1

where .f ∈ M, .0 ≤ T1 < T2 < ∞, and .x : [T1 , T2 ] → R n is an a.c. function. For .f ∈ M, .y, z ∈ R n , and numbers .T1 , T2 satisfying .0 ≤ T1 < T2 , we set U f (T1 , T2 , y, z) = inf{I f (T1 , T2 , x) : x : [T1 , T2 ] → R n

.

.

is an a.c. function satisfying x(T1 ) = y, x(T2 ) = z}.

It is easy to see that .−∞ < U f (T1 , T2 , y, z) < ∞ for each .f ∈ M, each .y, z ∈ R n , and all numbers .T1 , T2 satisfying .0 ≤ T1 < T2 . Let .f ∈ M. A locally absolutely continuous (a.c.) function .x : [0, ∞) → R n is called an .(f )-good function [117] if for any a.c. function .y : [0, ∞) → R n there is a number .My such that I f (0, T , y) ≥ My + I f (0, T , x) for each T ∈ (0, ∞).

.

The following result was proved in [115]. Proposition 1.1 Let .f ∈ M, and let .x : [0, ∞) → R n be a bounded a.c. function. Then the function x is .(f )-good if and only if there is .M > 0 such that I f (0, T , x) ≤ U f (0, T , x(0), x(T )) + M for any T > 0.

.

Let us now give the precise definition of the turnpike property. Assume that .f ∈ M. We say that f has the turnpike property, or briefly TP, if there exists a bounded continuous function .Xf : [0, ∞) → R n that satisfies the following condition: For each .K,  > 0, there exist constants .δ, L > 0 such that for each .x, y ∈ R n satisfying .|x|, |y| ≤ K, each .T1 ≥ 0, .T2 ≥ T1 + 2L, and each a.c. function .v : [T1 , T2 ] → R n that satisfy v(T1 ) = x, v(T2 ) = y, I f (T1 , T2 , v) ≤ U f (T1 , T2 , x, y) + δ,

.

the inequality .|v(t) − Xf (t)| ≤  holds for all .t ∈ [T1 + L, T2 − L]. The function .Xf is called the turnpike of f . Assume that .f ∈ M and .X : [0, ∞) → R n is a bounded continuous function. How to verify if the integrand f has TP and X is its turnpike? In [115], we introduced two properties (P1) and (P2) and show that f has TP if and only if f possesses the properties (P1) and (P2). The property (P2) means that all .(f )-good functions have the same asymptotic behavior, while the property (P1) means that if

1.1 Turnpike Property for Variational Problems

5

an a.c. function .v : [0, T ] → R n is an approximate solution and T is large enough, then there is .τ ∈ [0, T ] such that .v(τ ) is close to .X(τ ). The next theorem is the main result [115]. Theorem 1.2 Let .f ∈ M and .Xf : [0, ∞) → R n be a bounded absolutely continuous function. Then f has the turnpike property with .Xf being the turnpike if and only if the following two properties hold: (P1) For each .K,  > 0, there exist .γ , l > 0 such that for each .T ≥ 0 and each a.c. function .w : [T , T + l] → R n that satisfy |w(T )|, |w(T +l)| ≤ K, I f (T , T +l, w) ≤ U f (T , T +l, w(T ), w(T +l))+γ ,

.

there is .τ ∈ [T , T + l] for which .|Xf (τ ) − v(τ )| ≤ . (P2) For each .(f )-good function .v : [0, ∞) → R n , |v(t) − Xf (t)| → 0 as t → ∞.

.

In [115], we proved the following theorem that is an extension of Theorem 1.2. Theorem 1.3 Let .f ∈ M, and .Xf : [0, ∞) → R n be an .(f )-good function. Assume that the properties (P1), (P2) hold. Then for each .K,  > 0, there exist .δ, L > 0, and a neighborhood .U of f in .M such that for each .g ∈ U , each .T1 ≥ 0, n .T2 ≥ T1 + 2L, and each a.c. function .v : [T1 , T2 ] → R that satisfy |v(T1 )|, |v(T2 )| ≤ K, I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + δ,

.

the inequality .|v(t) − Xf (t)| ≤  holds for all .t ∈ [T1 + L, T2 − L]. In [148], we studied sufficient and necessary conditions for the turnpike phenomenon, using the approach developed in [115, 116, 118, 131], for discrete-time optimal control problems in metric spaces, which are not necessarily compact, and for continuous-time infinite dimensional optimal control problems. The main results of [148] have Theorem 1.2 as their prototype. Since the discovery of the turnpike phenomenon by Paul Samuelson in 1948, different versions of the turnpike property were considered in the literature. In this book as well as in [115, 116, 118, 131, 148], we study the turnpike property introduced and used in our previous research [112, 117, 130, 131, 142]. This turnpike property differs from other versions and has important features. Our turnpike property is a property of approximate solutions, and the turnpike is a nonstationary trajectory. As it was shown in [112, 130], our turnpike property holds for most problems belonging to large classes of variational and optimal control problems. The main features of this book are that we study optimal control problems in metric spaces, obtain sufficient and necessary conditions for the turnpike phenomenon, and show that the turnpike phenomenon is stable under small perturbations of objective function.

6

1 Introduction

1.2 Notation In this section, we collect the notation that will be used in the book. Let .(X, ρX ) be a metric space equipped with the metric .ρX that induces the topology in X. For each .x ∈ X and each .r > 0, set BX (x, r) = {y ∈ X : ρ(x, y) ≤ r}.

.

Let .(Xi , ρXi ), .i = 1, 2, be metric spaces. The set .X1 × X2 is equipped with the metric ρX1 ×X2 ((x1 , x2 ), (y1 , y2 )) = ρX1 (x1 , y1 ) + ρX2 (x2 , y2 ),

.

(x1 , x2 ), (y1 , y2 ) ∈ X1 × X2 .

.

We denote by mes.(Ω) the Lebesgue measure of a Lebesgue measurable set .Ω ⊂ R 1 and define χΩ (x) = 1 for all x ∈ Ω, χΩ (x) = 0 for all x ∈ R 1 \ Ω.

.

For each function .f : X → [−∞, ∞], where X is nonempty, we set .

inf(f ) = inf{f (x) : x ∈ X}.

For each .s ∈ R 1 , set .s+ = max{s, 0}, .s− = min{s, 0}, . s = max{z : z is an integer and z ≤ s} and assume that .s ≥ −∞, .s > −∞. If .(X,  · ) is a normed space, then X is equipped with the metric .ρX (x, y) = x − y, .x, y ∈ X. Let .(X, ρX ) be a metric space and T be a Lebesgue measurable subset of .R 1 . A function .f : T → X is called Lebesgue measurable if for any Borel set .D ⊂ X the set .f −1 (D) is a Lebesgue measurable set. In the sequel, we denote by Card.(D) the cardinality of a set D and suppose that the sum over an empty set is zero, the infimum over an empty set is .∞, and the supremum over an empty set is .−∞. Let .(X, ·, · X ) be a Hilbert space equipped with an inner product . ·, · X that induces the norm . · X . If the space X is understood, we use the notation . ·, · = ·, · X and . ·  =  · X . Let X be a Banach space equipped with the norm . · X and .X∗ be its dual space with the norm . · X∗ . If the space X is understood, we use the notation . ·  =  · X . For .x ∈ X and .l ∈ X∗ , we set .l(x) = l, x X∗ ,X . The symbol . ·, · X∗ ,X is referred to as the duality pairing between .X∗ and X. When the pair .X, X∗ is understood, we use the notation . ·, · = ·, · X∗ ,X .

1.3 General Dynamical Systems with a Lyapunov Function

7

1.3 General Dynamical Systems with a Lyapunov Function In Chap. 5, we develop a turnpike theory for a class of general dynamical systems in a metric space with a Lyapunov function. This class of dynamical systems contains as particular cases the classes of dynamical systems studied in Chaps. 3 and 4. Our goal is to obtain several turnpike results and show that the turnpike phenomenon is stable under small perturbations of objective functions. Now we describe this class of problems. Assume that .(X, ρ) is a metric space. For each .x ∈ X and each .r > 0, set B(x, r) = {y ∈ X : ρ(x, y) ≤ r} and B 0 (x, r) = {y ∈ X : ρ(x, y) < r}.

.

For each .x ∈ X and each pair of nonempty sets .A, B ⊂ X, define ρ(x, B) = inf{ρ(x, y) : y ∈ B}

.

and dist(A, B) = sup{ρ(x, B) : x ∈ A}.

.

Let .Γ be either .R 1 or the set of all integers .Z = {0, ±1, ±2, . . . }. Assume that for each pair .T1 , T2 ∈ Γ satisfying .T1 < T2 , .Y (T1 , T2 ) is a nonempty set of functions .x : [T1 , T2 ] ∩ Γ → X such that the following properties hold: (A1) For each pair .T1 , T2 ∈ Γ , each pair .S1 , S2 ∈ Γ satisfying .T1 ≤ S1 < S2 ≤ T2 , and each .x ∈ Y (T1 , T2 ), the restriction of x to .[S1 , S2 ] belongs to .Y (S1 , S2 ). (A2) For each pair .T1 , T2 ∈ Γ satisfying .T1 < T2 , each .S ∈ Γ , and each .x ∈ Y (T1 , T2 ), the function .x(t +S), .t ∈ [T1 , T2 ]∩Γ , belongs to .Y (T1 +S, T2 +S). (A3) For each triplet .T1 , T2 , T3 ∈ Γ satisfying .T1 < T2 < T3 , each .x1 ∈ Y (T1 , T2 ), and each .x2 ∈ Y (T2 , T3 ) satisfying .x1 (T2 ) = x2 (T2 ), the function .x3 : [T1 , T3 ] ∩ Γ → X defined by x3 (t) = x1 (t), t ∈ [T1 , T2 ] ∩ Γ, x3 (t) = x2 (t), t ∈ (T2 , T3 ] ∩ Γ

.

belongs to .Y (T1 , T3 ). If .Γ = Z, then we set A = {(y, z) ∈ X × X : there is x ∈ Y (0, 1) such that x(0) = y, x(1) = z}},

.

define a(x) = {y ∈ X : (x, y) ∈ A}, x ∈ X,

.

8

1 Introduction

and assume that .A is closed in the metric space .X × X. For each pair of numbers .T1 , T2 ∈ Γ satisfying .T1 < T2 , elements of the set .Y (T1 , T2 ) are called trajectories. Let .T1 ∈ Γ . A function .x : [T1 , ∞) ∩ Γ → X is a trajectory if for every .T2 ∈ (T1 , ∞) ∩ Γ , the restriction of x to .[T1 , T2 ] ∩ Γ belongs to .Y (T1 , T2 ). The set of all trajectories .x : [T1 , ∞) ∩ Γ → X is denoted by .Y (T1 , ∞). Denote by .Y (Γ ) the set of all functions .x : Γ → X such that for each pair of numbers .T1 , T2 ∈ Γ satisfying .T1 < T2 the restriction of x to .[T1 , T2 ] ∩ Γ belongs to .Y (T1 , T2 ). The elements of .Y (Γ ) are called trajectories too. θ ∈ X. Fix . Assume that .φ : X → R 1 is a continuous function such that for each pair of numbers .T1 ∈ Γ , .T2 ∈ (T1 , ∞) ∩ Γ , each .x ∈ Y (T1 , T2 ), and each .S1 , S2 ∈ [T1 , T2 ] ∩ Γ satisfying .S1 < S2 , φ(x(S2 )) ≤ φ(x(S1 )).

.

(1.1)

Assume that .X0 is nonempty, closed subset of X such that for each .r > 0 the set X0 ∩ B( θ , r) is compact

.

(1.2)

and that the following assumptions hold: (A4) For each pair of numbers .T1 , T2 ∈ Γ satisfying .T2 > T1 , each .M > 0, and each sequence .{xn }∞ n=1 ⊂ Y (T1 , T2 ) that satisfy .

lim sup{ρ(xn (t), X0 ∩ B( θ , M)) : t ∈ [T1 , T2 ] ∩ Γ } = 0,

n→∞

there exists a subsequence .{xnk }∞ k=1 that converges uniformly on .[T1 , T2 ] ∩ Γ to a trajectory .x ∈ Y (T1 , T2 ) as .k → ∞. (A5) For each bounded, nonempty set B, there exists .M > 0, and for each . > 0, there exists a positive number .T (B, ) ∈ Γ such that for each .T ∈ Γ ∩ [T (B, ), ∞) and each .x ∈ Y (0, T ) satisfying .x(0) ∈ B, ρ(x(t), X0 ∩ B( θ , M)) ≤ , t ∈ [T (B, ), T ] ∩ Γ.

.

(Note that in the case .Γ = Z, (A4) follows from the compactness of the set .X0 ∩ B( θ , M) and the closedness of .A.) We suppose that .Y (0, ∞) = ∅. Here we consider a general and abstract dynamical system and treat discrete-time and continuous-time case simultaneously. For examples of such systems, see Sects. 2.8 and 3.1. There are also infinite dimensional examples considered in [18, 82]. Define Ω = {z ∈ X : for very  > 0 there exists x ∈ Y (0, ∞)

.

1.3 General Dynamical Systems with a Lyapunov Function

9

for which lim inf ρ(z, x(t)) ≤ }.

.

t→∞

Clearly, .Ω is a closed subset of the metric space .(X, ρ). We will show that Ω = ∅, Ω ⊂ X0 ,

.

and for each .x ∈ Y (0, ∞), .

lim ρ(x(t), Ω) = 0.

t→∞

We also assume that the following assumption holds: (A6) If .x ∈ Y (Γ ) is bounded and for all .t ∈ Γ , x(t) ∈ X0 and φ(x(t)) = φ(x(0)),

.

then .x(t) ∈ Ω for every .t ∈ Γ . If .Γ = Z and .E ⊂ Γ , then we set mes(E) = Card(E).

.

For each .x ∈ X, set π(x) = lim sup{φ(x(T )) : x ∈ Y (0, T ) and x(0) = x}.

.

T →∞

(We assume that the supremum over an empty set is .−∞.) It is clear that π(x) ≤ φ(x), x ∈ X.

.

The function .π plays an important role in our study. Evidently, the following result is true. Proposition 1.4 Let .T > 0 and .x ∈ Y (0, T ). Then .π(x(s)) ≤ π(x(0)) for all .s ∈ [0, T ] ∩ Γ . The following result is proved in Sect. 5.3. Proposition 1.5 Let .x ∈ X. The value .π(x) is finite if and only if for every T ∈ Γ ∩ (0, ∞), there exists .y ∈ Y (0, T ) satisfying .y(0) = x.

.

In Sect. 5.4, we prove the following result. Proposition 1.6 Let .M0 > 0. Then there exists .M > 0 such that for each x ∈ B( θ , M0 ) satisfying .π(x) > −∞,

.

B( θ , M) ∩ Ω ∩ φ −1 (π(x)) = ∅.

.

10

1 Introduction

The following turnpike result is proved in Sect. 5.5. Theorem 1.7 Let . > 0 and . x ∈ X satisfy .π( x ) > −∞. Then there exist δ > 0 and .L > 0 such that for every .T ∈ Γ ∩ (2L, ∞) and every .x ∈ Y (0, T ) that satisfy

.

x(0) =  x and φ(x(T )) ≥ π( x ) − δ,

.

the inequality ρ(x(t), Ω) ≤ 

.

holds for all .t ∈ [L, T − L] ∩ Γ. Clearly, the theorem above establishes the turnpike property for approximate optimal trajectories that have a fixed starting point with respect to the objective function .φ. The turnpike is the set .Ω, and .δ, L depend on . and the initial point . x. For each .x ∈ X satisfying .π(x) > −∞, set F (x) = Ω ∩ φ −1 (π(x)),

.

which is nonempty. The following result, which is a generalization of the previous theorem, is proved in Sect. 5.7. According to the previous theorem, for all .t ∈ [L, T − L] ∩ Γ , the distance between .x(t) and the turnpike .Ω does not exceed .. According to our next result, for all .t ∈ [L, T − L] ∩ Γ , the distance between .x(t) and the set .F ( x ) ∩ B( θ , M) does not exceed ., where a positive constant M depends on . and the initial point . x and .F ( x ) is the set of all points of the turnpike .Ω where the value of the Lyapunov function .φ is .π( x ). Theorem 1.8 Let . > 0 and . x ∈ X satisfy .π( x ) > −∞. Then there exist M, L, δ > 0 such that for every .T ∈ Γ ∩ (2L, ∞) and every .x ∈ Y (0, T ) that satisfy

.

x(0) =  x and φ(x(T )) ≥ π( x ) − δ,

.

the inequality ρ(x(t), F ( x ) ∩ B( θ , M)) ≤ 

.

holds for all .t ∈ [L, T − L] ∩ Γ. In Chap. 5, we obtain several extensions and generalizations of these two turnpike results. In particular, we show that if the function .π is continuous on the turnpike set .Ω, then the constants .L, δ do not depend on the starting point . x . We also show that the turnpike phenomenon is stable under small perturbations of objective functions.

1.4 Continuous-Time Nonautonomous Problems on Half-Axis

11

1.4 Continuous-Time Nonautonomous Problems on Half-Axis In Chapter 8, we establish sufficient and necessary conditions for the turnpike phenomenon for continuous-time optimal control problems on subintervals of halfaxis in metric spaces. For these optimal control problems, the turnpike is not a singleton. The results of this chapter are obtained for three large classes of problems that will be treated simultaneously. In Chap. 9, for the same classes of problems, we show that the turnpike phenomenon is stable under small perturbations of objective functions. Let us describe these classes of problems. Let .(E, ρE ) be a complete metric space and .(F, ρF ) be a metric space. We suppose that .A is a nonempty subset of .[0, ∞) × E, and .U : A → 2F is a point to set mapping with a graph M = {(t, x, u) : (t, x) ∈ A, u ∈ U (t, x)}.

.

We suppose that .M is a Borel measurable subset of .[0, ∞) × E × F . Assume that for each pair of numbers .T1 , T2 satisfying .0 ≤ T1 < T2 < ∞, we are given a set .X(T1 , T2 ) of pairs .(x, u) (called as trajectory–control pairs) such that .x : [T1 , T2 ] → E is a continuous function, and .u : [T1 , T2 ] → F is a Lebesgue function satisfying (t, x(t)) ∈ A, t ∈ [T1 , T2 ],

.

u(t) ∈ U (t, x(t)), t ∈ [T1 , T2 ] almost everywhere (a.e.).

.

We suppose that the following property holds: if .0 ≤ T1 ≤ S1 < S2 ≤ T2 < ∞, .(x, u) ∈ X(T1 , T2 ), .x˜ is the restriction of x to .[S1 , S2 ], and .u ˜ is the restriction of u to .[S1 , S2 ], then .(x, ˜ u) ˜ ∈ X(S1 , S2 ); for each triplet of nonnegative numbers .T1 < T2 < T3 , each .(x1 , u1 ) ∈ X(T1 , T2 ), and each .(x2 , u2 ) ∈ X(T2 , T3 ) satisfying x1 (T2 ) = x2 (T2 )

.

a pair .(x3 , u3 ) ∈ X(T1 , T3 ), where x3 (t) = x1 (t), t ∈ [T1 , T2 ], x3 (t) = x2 (t), t ∈ (T2 , T3 ],

.

u3 (t) = x1 (t), t ∈ [T1 , T2 ], u3 (t) = u2 (t), t ∈ (T2 , T3 ].

.

Let .a0 > 0, .θ0 ∈ E, .θ1 ∈ F , and let .ψ : [0, ∞) → [0, ∞) be an increasing function such that ψ(t) → ∞ as t → ∞,

.

12

1 Introduction

and let .ψ1 : M → [0, ∞) satisfy ψ1 (t, x, u) ≥ ψ(ρE (x, θ0 )), (t, x, u) ∈ M.

.

Denote by .Mψ the set of all Borelian functions .g : M → R 1 such that for each .(t, x, u) ∈ M, g(t, x, u) ≥ ψ1 (t, x, u) − a0 , (t, x, u) ∈ M.

.

For each pair of numbers .T2 > T1 ≥ 0, each .(x, u) ∈ X(T1 , T2 ), and each .g ∈ Mψ , define  I g (T1 , T2 , x, u) =

T2

.

g(t, x(t), u(t))dt ∈ (−∞, ∞].

T1

We consider functionals of the form .I g (T1 , T2 , x, u), where .0 ≤ T1 < T2 , .(x, u) ∈ X(T1 , T2 ), and .g ∈ Mψ . Let .g ∈ Mψ . For each pair of numbers .T2 > T1 ≥ 0 and each pair of points .(T1 , y), (T2 , z) ∈ A, we define U g (T1 , T2 , y, z) = inf{I g (T1 , T2 , x, u) :

.

(x, u) ∈ X(T1 , T2 ), x(T1 ) = y, x(T2 ) = z},

.

U g (T1 , T2 , y) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 ), x(T1 ) = y},

.

g (T1 , T2 , z) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 ), x(T2 ) = z}, U

.

U g (T1 , T2 ) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 )}.

.

We equip the space .Mψ with the uniformity determined by the base E(N, , λ) = {(f, g) ∈ Mψ × Mψ : |f (t, x, u) − g(t, x.u)| ≤ 

.

.

.

for each (t, x, u) ∈ M satisfying ρE (x, θ0 ), ρ(u, θ1 ) ≤ N }

∩ {(f, g) ∈ Mψ × Mψ : (|f (t, x, u)| + 1)(|g(t, x, u)| + 1)−1 ∈ [λ−1 , λ] .

for each (t, x, u) ∈ M satisfying ρE (x, θ0 ) ≤ N },

where .N,  > 0, λ > 1. Clearly, the uniform space .Mψ is Hausdorff and has a countable base. Therefore, .Mψ is metrizable (by a metric .dψ ). It is not difficult to see that the uniform space .Mψ is complete. It is equipped with a topology induced by the uniformity. In this chapter, we assume that the following assumption holds:

1.4 Continuous-Time Nonautonomous Problems on Half-Axis

13

(A0) For each .L0 > 0, there exists .L1 > 0 such that for each .T1 ≥ 0, each T2 ∈ (T1 , T1 + L0 ], each .(x, u) ∈ X(T1 , T2 ), and each .g ∈ Mψ that satisfy

.

ρE (x(T1 ), θ0 ) ≤ L0 , I g (T1 , T2 , x, u) ≤ L0 ,

.

the inequality ρE (θ0 , x(t)) ≤ L1

.

is true for all .t ∈ [T1 , T2 ]. Let .f ∈ Mψ . In some cases, we will use the following assumption: (A0)’ For each .L0 > 0, there exists .L1 > 0 such that for each .T1 ≥ 0, each .T2 ∈ (T1 , T1 + L0 ], and each .(x, u) ∈ X(T1 , T2 ) that satisfy I f (T1 , T2 , x, u) ≤ L0

.

and .

min{ρE (θ0 , x(t)) : t ∈ [T1 , T2 ]} ≤ L0 ,

the inequality ρE (θ0 , x(t)) ≤ L1

.

is true for all .t ∈ [T1 , T2 ]. Assumption (A0) easily implies the following result. Proposition 1.9 Assume that .M1 > 0, .0 < τ < τ0 < τ1 . Then there exists .M2 > 0 such that for each .T1 ≥ 0, each .T2 ∈ [T1 + τ0 , T1 + τ1 ], each .g ∈ Mψ , and each .(x, u) ∈ X(T1 , T2 ) that satisfy I g (T1 , T2 , x, u) ≤ M1 ,

.

the inequality ρE (θ0 , x(t)) ≤ M2

.

is true for all .t ∈ [T1 + τ, T2 ]. Usually, a control system is governed by a differential equation, and its trajectories are solutions of this equation. Therefore, properties of trajectories depend on the equation. In this book, we use a different framework. We assume that sets of trajectory–control pairs are given and that they satisfy some natural assumptions that hold in the control theory. No equation is involved in the definition of our control system. Therefore, our results can be implied to different classes of control systems.

14

1 Introduction

Now we describe three classes of optimal control problems that are particular cases of the class of problems introduced above. The First Class of Problems We begin with the description of the first class of problems. Let .(E,  · ) be a Banach space and .E ∗ be its dual. Let .{A(t) : t ∈ [0, ∞)} be the family of closed densely defined linear operators with the domain and range in the Banach space E. Let .(F, ρF ) be a metric space, θ0 = 0, θ1 ∈ F,

.

ρE (x, y) = x − y, x, y ∈ E. We suppose that .A is a nonempty subset of [0, ∞) × E, and .U : A → 2F is a point to set mapping with a graph

. .

M = {(t, x, u) : (t, x) ∈ A, u ∈ U (t, x)}.

.

We suppose that .M is a Borel measurable subset of .[0, ∞)×E×F and .G : M → E is a Borelian function. We consider the homogeneous Cauchy problem x  (t) = A(t)x(t), t ∈ [0, ∞).

.

We assume that there exists a function .U : {(t, s) ∈ R 2 : 0 ≤ s ≤ t < ∞} → L(E) that has the following properties [21]: (i) For each .x0 ∈ E, the function .(t, s) → U (t, s)x0 is continuous on the set 2 .{(t, s) ∈ R : 0 ≤ s ≤ t < ∞}. (ii) .U (s, s) = I d for all .s ∈ [0, ∞), where I d is the identity operator. (iii) .U (t, s)U (s, τ ) = U (t, τ ) for all numbers .t ≥ s ≥ τ ≥ 0. (iv) For each .s ≥ 0, there exists a densely linear subspace .Es of E such that for each .x0 ∈ Es the function .t → U (t, s)x0 is continuously differentiable on .[s, ∞) and (∂/∂t)U (t, s)x0 = A(t)U (t, s)x0 , t ∈ [s, ∞).

.

(v) There exists an increasing function .τ → Δτ > 0, .τ > 0 such that for each .τ > 0, each .s ≥ 0, and each .t ∈ [s, s + τ ], U (t, s) ≤ Δτ .

.

In this case problem, homogeneous Cauchy problem is called well-posed [21]. Let .0 ≤ T1 < T2 , and consider the following equation: x  (t) = A(t)x(t) + f (t), t ∈ [T1 , T2 ], x(0) = xT1 ,

.

where .xT1 ∈ E and .f ∈ L1 (T1 , T2 ; E).

1.4 Continuous-Time Nonautonomous Problems on Half-Axis

15

A continuous function .x : [T1 , T2 ] → E is a solution of the equation above if 

t

x(t) = U (t, T1 )x(T1 ) +

.

U (t, s)f (s)ds, t ∈ [T1 , T2 ].

T1

Assume that the equation above holds and .τ ∈ [T1 , T2 ]. It is not difficult to see (p. 386, [148]) that .x : [τ, T2 ] → E is a solution of the equation y  (t) = A(t)y(t) + f (t), t ∈ [τ, T2 ], y(τ ) = x(τ ).

.

Let .0 ≤ T1 < T2 < T3 , .z0 ∈ E, .f ∈ L1 (T1 , T3 ; E), a continuous function .x1 : [T1 , T2 ] → E is a solution of the equation x  (t) = A(t)x(t) + f (t), t ∈ [T1 , T2 ], x(T1 ) = z0 ,

.

and a continuous function .x2 : [T2 , T3 ] → E is a solution of the equation x  (t) = A(t)x(t) + f (t), t ∈ [T2 , T3 ], x(T2 ) = x1 (T2 ).

.

Set x(t) = x1 (t), t ∈ [T1 , T2 ], x(t) = x2 (t), t ∈ [T2 , T3 ].

.

Clearly, the function .x : [T1 , T3 ] → E is continuous. It is not difficult to see (p. 387, [148]) that .x(·) is a solution of the equation x  (t) = A(t)x(t) + f (t), t ∈ [T1 , T3 ], x(T1 ) = z0 .

.

Let .0 ≤ T1 < T2 . We consider the following equation: x  (t) = A(t)x(t) + G(t, x(t), u(t)), t ∈ [T1 , T2 ].

.

A pair of functions .x : [T1 , T2 ] → E, .u : [T1 , T2 ] → F is called a (mild) solution of the equation above if .x : [T1 , T2 ] → E is a continuous function, .u : [T1 , T2 ] → F is a Lebesgue measurable function, (t, x(t)) ∈ A, t ∈ [T1 , T2 ],

.

u(t) ∈ U (t, x(t)), t ∈ [T1 , T2 ] almost everywhere (a.e.),

.

G(s, x(s), u(s)), s ∈ [T1 , T2 ] is Bochner integrable, and for every .t ∈ [T1 , T2 ],

.

 x(t) = U (t, T1 )x(T1 ) +

t

U (t, s)G(s, x(s), u(s))ds.

.

T1

16

1 Introduction

The set of all solutions .(x, u) is denoted by .X(T1 , T2 ). Let .T1 ≥ 0. A pair of functions .x : [T1 , ∞) → E, .u : [T1 , ∞) → F is called a (mild) solution of the system x  (t) = A(t)x(t) + G(t, x(t), u(t)), t ∈ [T1 , ∞)

.

if for every .T2 > T1 , the pair .x : [T1 , T2 ] → E, .u : [T1 , T2 ] → F belongs to .X(T1 , T2 ). The set of all such pairs .(x, u), which are solutions of the equation above, is denoted by .X(T1 , ∞). A function .x : I → E, where I is either .[T1 , T2 ] or .[T1 , ∞) (.0 ≤ T1 < T2 ), is called a trajectory if there exists a Lebesgue measurable function .u : I → F (referred to as a control) such that .(x, u) ∈ X(T1 , T2 ) or .(x, u) ∈ X(T1 , ∞), respectively. For this class of problem, for each .(t, x, u) ∈ M, set ψ1 (t, x, u) = max{ψ(x), ψ(u),

.

ψ((G(t, x, u) − a0 x)+ )(G(t, x, u) − a0 x)+ }.

.

In Sect. 8.4, we show that assumption (A0) holds for this class of problems. The Second Class of Problems Let .(E, ·, · )E be a Hilbert space equipped with an inner product . ·, · E that induces the norm .·E , and let .(F, ·, · F ) be a Hilbert space equipped with an inner product . ·, · F that induces the norm .·F . For simplicity, we set . ·, · E = ·, · , .·E = ·, . ·, · F = ·, · , . · F =  · , if .E, F are understood. Set θ0 = 0, θ1 = 0,

.

ρE (x, y) = x − y, x, y ∈ E, .ρF (x, y) = x − y, x, y ∈ F . We suppose that .A0 is a nonempty subset of E, and .U0 : A0 → 2F is a point to set mapping with a graph

.

M0 = {(x, u) : x ∈ A0 , u ∈ U0 (x)}.

.

We suppose that .M0 is a Borel measurable subset of .E × F . Define A = [0, ∞) × A0 ,

.

U : A → 2F by

.

U (t, x) = U0 (x), (t, x) ∈ A,

.

M = [0, ∞) × M0 .

.

1.4 Continuous-Time Nonautonomous Problems on Half-Axis

17

Let a linear operator .A : D(A) → E generate a .C0 semigroup .S(t) = eAt , ∗ ∗ .t ∈ [0, ∞) on E. As usual, we denote by .S(t) the adjoint of .S(t). Then .S(t) , t ∈ ∗ [0, ∞), is .C0 semigroup, and its generator is the adjoint .A of A. The domain .D(A∗ ) is a Hilbert space equipped with the graph norm . · D(A∗ ) : z2D(A∗ ) = z2E + A∗ z2E , z ∈ D(A∗ ).

.

Let .D(A∗ ) be the dual of .D(A∗ ) with the pivot space E. In particular, E1d := D(A∗ ) ⊂ E ⊂ D(A∗ ) = E−1 .

.

(Here we use the notation of Chap. 7.) Let .G : M → D(A∗ ) = E−1 , .B ∈ L(F, E−1 ) is an admissible control operator for .eAt , .t ≥ 0, and for all .(t, x, u) ∈ M, G(t, x, u) = Bu.

.

Let .0 ≤ T1 < T2 . We consider the following equation: x  (t) = Ax(t) + Bu(t), t ∈ [T1 , T2 ] a.e.

.

A pair of functions .x : [T1 , T2 ] → E, .u : [T1 , T2 ] → F is called a (mild) solution of the equation above if .x : [T1 , T2 ] → E is a continuous function, .u : [T1 , T2 ] → F is a Lebesgue measurable function, .u ∈ L2 (T1 , T2 ; F ), (t, x(t)) ∈ A, t ∈ [T1 , T2 ],

.

u(t) ∈ U (t, x(t)), t ∈ [T1 , T2 ] a.e.,

.

and for each .t ∈ [T1 , T2 ],  x(t) = eA(t−T1 ) x(T1 ) +

t

.

eA(t−s) Bu(s)ds

T1

in .E−1 . The set of all solutions .(x, u) is denoted by .X(T1 , T2 , A, G). In the sequel for simplicity, we use the notation .X(T1 , T2 ) = X(T1 , T2 , A, G) if the pair .(A, G) is understood. Let .T1 ≥ 0. A pair of functions .x : [T1 , ∞) → E, .u : [T1 , ∞) → F is called a (mild) solution of the system x  (t) = Ax(t) + Bu(t), t ∈ [T1 , ∞))

.

if for every .T2 > T1 , .x : [T1 , T2 ] → E, .u : [T1 , T2 ] → F belongs to .X(T1 , T2 ). The set of all such pairs .(x, u), which are solutions of the equation above, is denoted by .X(T1 , ∞).

18

1 Introduction

A function .x : I → E, where I is either .[T1 , T2 ] or .[T1 , ∞) (.0 ≤ T1 < T2 ), is called a trajectory if there exists a Lebesgue measurable function .u : I → F (referred to as a control) such that .(x, u) ∈ X(T1 , T2 ) or .(x, u) ∈ X(T1 , ∞), respectively. For this class of problem for each .(t, x, u) ∈ M, set ψ1 (t, x, u) = max{ψ(x), K0 u2 } − a0 ,

.

where .K, a0 are positive constants. In Sect. 8.4, we show that assumption (A0) holds for this class of problems. The Third Class of Problems Let .(E, ρE ) be a complete metric space, .(F, ρF ) be a metric space, θ0 ∈ E, θ1 ∈ F.

.

We suppose that .A is a nonempty subset of .[0, ∞) × E, and .U : A → 2F is a point to set mapping with a graph M = {(t, x, u) : (t, x) ∈ A, u ∈ U (t, x)}.

.

We suppose that .M is a Borel measurable subset of .[0, ∞) × E × F and .G : M → [0, ∞) is a Borelian function. Let .0 ≤ T1 < T2 < ∞. Denote by .X(T1 , T2 ) the set of all pairs of functions .x : [T1 , T2 ] → E and .u : [T1 , T2 ] → F such that x is continuous on .[T1 , T2 ], u is Lebesgue measurable on function .[T1 , T2 ], (t, x(t)) ∈ A, t ∈ [T1 , T2 ],

.

u(t) ∈ U (t, x(t)), t ∈ [T1 , T2 ] almost everywhere (a.e.),

.

G(s, x(s), u(s)), s ∈ [T1 , T2 ] is Bochner integrable, and for every pair .S1 , S2 ∈ [T1 , T2 ] satisfying .S1 < S2 ,

.

 ρE (x(S1 ), x(S2 )) ≤

S2

G(t, x(t), u(t))dt.

.

S1

(See Section 1.1 of [5].) Pairs .(x, u) ∈ X(T1 , T2 ) are called trajectory–control pairs. Let .T1 ≥ 0. Denoted by .X(T1 , ∞) the set of all pairs of functions .x : [T1 , ∞) → E, .u : [T1 , ∞) → F such that for every .T2 > T1 , the pair .x : [T1 , T2 ] → E, .u : [T1 , T2 ] → F belongs to .X(T1 , T2 ). Pairs .(x, u) ∈ X(T1 , ∞) are called trajectory– control pairs. A function .x : I → E, where I is either .[T1 , T2 ] or .[T1 , ∞) (.0 ≤ T1 < T2 ), is called a trajectory if there exists a Lebesgue measurable function .u : I → F (referred to as a control) such that .(x, u) ∈ X(T1 , T2 ) or .(x, u) ∈ X(T1 , ∞), respectively).

1.4 Continuous-Time Nonautonomous Problems on Half-Axis

19

For this class of problem for each .(t, x, u) ∈ M, set ψ1 (t, x, u) = max{ψ(ρE (x, θ0 )), ψ(ρF (u, θ1 )),

.

ψ((G(t, x, u) − a0 ρE (x, θ0 ))+ )(G(t, x, u) − a0 ρE (x, θ0 ))+ }.

.

For each pair of numbers .T2 > T1 ≥ 0, each .(x, u) ∈ X(T1 , T2 ), and each g ∈ Mψ , define

.

 I (T1 , T2 , x, u) =

.

g

T2

g(t, x(t), u(t))dt ∈ (−∞, ∞].

T1

In Chap. 8, we show now that assumption (A0) holds for this class of problems. Let us consider our general class of problems. We say that the integrand f possesses the turnpike property (or TP for short) if for each . > 0 and each .M > 0 there exist .δ > 0 and .L > 0 such that for each .T1 ≥ 0, each .T2 ≥ T1 + 2L, and each .(x, u) ∈ X(T1 , T2 ) that satisfy I f (T1 , T2 , x, u) ≤ min{U f (T1 , T2 ) + M, U f (T1 , T2 , x(T1 ), x(T2 )) + δ}

.

there exist .τ1 , τ2 ∈ [0, L] such that ρE (x(t), xf (t)) ≤  for all t ∈ [T1 + τ1 , T2 − τ2 ].

.

Moreover, if .ρE (x(T2 ), xf (T2 )) ≤ δ, then .τ2 = 0, and if .T1 ≥ L and ρE (x(T1 ), xf (T1 )) ≤ δ, then .τ1 = 0. The next theorem is the main result of Chap. 8. It is proved in Sect. 8.13.

.

Theorem 1.10 f has TP if and only if the following properties hold: (P1) For each .(f )-good pair .(x, u) ∈ X(0, ∞), .

lim ρE (x(t), xf (t)) = 0.

t→∞

(P2) For each . > 0 and each .M > 0, there exist .δ > 0 and .L > 0 such that for each .T ≥ 0 and each .(x, u) ∈ X(T , T + L) that satisfy I f (T , T + L, x, u) ≤ min{U f (T , T + L, x(T ), x(T + L)) + δ,

.

.

I f (T , T + L, xf , uf ) + M}

there exists .s ∈ [T , T + L] such that .ρE (x(s), xf (s)) ≤ . In Chap. 9, we study the stability of the turnpike phenomenon. In particular, it is shown that if TP holds, then it is stable under small perturbations of the objective function.

Chapter 2

Differential Inclusions

In this chapter, we discuss Banach space-valued functions, set-valued mappings in infinite dimensional spaces, and related continuous-time dynamical systems. Some convergence results are obtained.

2.1 Banach Space-Valued Functions Let (X,  · ) be a Banach space and a < b be real numbers. For any set E ⊂ R 1 , define χE (t) = 1 for all t ∈ E and χE (t) = 0 for all t ∈ R 1 \ E. If a set E ⊂ R 1 is Lebesgue measurable, then its Lebesgue measure is denoted by |E| or by mes(E). A function f : [a, b] → X is called a simple function if there exists a finite collection of Lebesgue measurable sets Ei ⊂ [a, b], i ∈ I , mutually disjoint, and xi ∈ X, i ∈ I such that f (t) =



χEi (t)xi , t ∈ [a, b].

i∈I

A function f : [a, b] → X is strongly measurable if there exists a sequence of simple functions φk : [a, b] → X, k = 1, 2, . . . , such that lim φk (t) − f (t) = 0, t ∈ [a, b] almost everywhere (a.e.).

k→∞

For every simple function f (·) = Bochner integral by

 i∈I

(2.1)

χEi (·)xi , where the set I is finite, define its

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0_2

21

22

2 Differential Inclusions



b

f (t)dt =



a

|Ei |xi .

i∈I

Let f : [a, b] → X be a strongly measurable function. We say that f is Bochner integrable if there exists a sequence of simple functions φk : [a, b] → X, k = b 1, 2, . . . , such that (2.1) holds and the sequence { a φk (t)dt}∞ k=1 strongly converges in X. In this case, we define the Bochner integral of the function f by  b  b f (t)dt = lim φk (t)dt. k→∞ a

a

It is known that the integral defined above is independent of the choice of the sequence {φk }∞ k=1 [73]. Similar to the Lebesgue integral, for any measurable set E ⊂ [a, b], the Bochner integral of f over E is defined by  b  f (t)dt = χE (t)f (t)dt. E

a

The following result is true (see Proposition 3.4, Chapter 2 of [73]). Proposition 2.1 Let f : [a, b] → X be a strongly measurable function. Then f is Bochner integrable if and only if the function f (·) is Lebesgue integrable. Moreover, in this case,  b  b  f (t)dt ≤ f (t)dt. a

a

The Bochner integral possesses almost the same properties as the Lebesgue integral. If f : [a, b] → X is strongly measurable and f (·) ∈ Lp (a, b), for some p ∈ [1, ∞), then we say that f (·) is Lp Bochner integrable. For every p ≥ 1, the set of all Lp Bochner integrable functions is denoted by Lp (a, b; X), and for every f ∈ Lp (a, b; X),  b f Lp (a,b;X) = ( f (t)p dt)1/p . a

Clearly, the set of all Bochner integrable functions on [a, b] is L1 (a, b; X). Let a < b be real numbers. A function x : [a, b] → X is absolutely continuous (a.c.) on [a, b] if for each  > 0 there exists δ > 0 such that for each pair of q q sequences {tn }n=1 , {sn }n=1 ⊂ [a, b] satisfying tn < sn , n = 1, . . . , q,

q 

(sn − tn ) ≤ δ,

n=1

(tn , sn ) ∩ (tm , sm ) = ∅ for all m, n ∈ {1, . . . , q} such that m = n,

2.1 Banach Space-Valued Functions

23

we have q 

x(tn ) − x(sm ) ≤ .

n=1

The following result is true (see Theorem 1.124 of [21]). Proposition 2.2 Let X be a reflexive Banach space. Then every a.c. function x : [a, b] → X is a.e. differentiable on [a, b] and  x(t) = x(a) +

t

(dx/dt)(s)ds, t ∈ [a, b],

a

where dx/dt ∈ L1 (a, b; X) is the strong derivative of x. Let −∞ < τ1 < τ2 < ∞. Denote by W 1,1 (τ1 , τ2 ; X) (or W 1,1 (τ1 , τ2 ) if the space X is understood) the set of all functions x : [τ1 , τ2 ] → X for which there exists a Bochner integrable function u : [τ1 , τ2 ] → X such that for all t ∈ (τ1 , τ2 ],  x(t) = x(τ1 ) +

t

u(s)ds. τ1

For each x ∈ X and each r > 0, set B(x, r) = {y ∈ X : x − y ≤ r} and B 0 (x, r) = {y ∈ X : x − y < r}. For each nonempty set Y and each function h : Y → R 1 , set inf(h; Y ) = inf{h(y) : y ∈ Y }. Let Y be a topological space and Z ⊂ Y . Then Z is equipped with the relative topology. The set Z is relatively compact if its closure is compact. In the sequel, we use the following result (see Theorem 4 on page 13 of [12]). Theorem 2.3 Let −∞ < a < b < ∞ (X,  · ) be a reflexive Banach space, and let for every integer k ≥ 1, xk ∈ W 1,1 (a, b; X). Assume that for each real number t ∈ [a, b], the sequence {xk (t)}∞ k=1 is a relatively compact subset of X and that there exists a positive function c ∈ L1 (a, b; X) such that xk (t) ≤ c(t) for almost every (a.e.) t ∈ [a, b] and every integer k ≥ 1. Then there are a 1,1 (a, b; X) such that x converges to the function subsequence {xki }∞ ki i=1 and x ∈ W x uniformly on [a, b] and that the function xk i converges weakly to the function x in L1 (a, b; X) as i → ∞.

24

2 Differential Inclusions

2.2 Set-Valued Mappings and a Convergence Result Let X be a Hausdorff topological space. Suppose that a+∞ = ∞ and a−∞ = −∞ for every a ∈ R 1 . Let f : X → [−∞, ∞] and x0 ∈ X. We say that f is upper semicontinuous at x0 if for every  > 0, there exists a neighborhood V of x0 in X such that for every x ∈ V, f (x) ≤ f (x0 ) + . Let X, Y be nonempty sets and F : X → 2Y . Define the domain of F Dom(F ) = {x ∈ X : F (x) = ∅} and the graph graph(F ) = {(x, y) ∈ X × Y : y ∈ F (x)}. The mapping F is proper if Dom(F ) = ∅. Assume that X, Y are Hausdorff topological spaces. We say that F is upper semicontinuous (u.s.c.) at x0 ∈ X if for any open set N ⊂ Y containing F (x0 ) there exists a neighborhood M of x0 in X such that F (M) ⊂ N. We say that F is upper semicontinuous if it is upper semicontinuous at every x0 ∈ X. For the proof of the following result, see Proposition 2 on page 41 of [12]). Proposition 2.4 If F is upper semicontinuous and F (z) is closed for every z ∈ X, then graph(F ) is closed. Assume that X is a Hausdorff topological space and Y is a locally convex Hausdorff space. Denote by Y ∗ its dual. The bilinear form p, y = p(y) is a duality pairing. We supply the space Y with the weak topology σ (Y, Y ∗ ). For every K ⊂ Y , define σK (p) = σ (K, p) = sup{p(y) : y ∈ K} ∈ [−∞, ∞], p ∈ Y ∗ . (Here we assume that the supremum over empty set is −∞.) We say that F is upper hemicontinuous (u.h.c.) at x0 ∈ X if for every p ∈ Y ∗ the function x → σ (F (x), p), x ∈ X is upper semicontinuous at x0 . We say that F is upper hemicontinuous if it is upper hemicontinuous at every x0 ∈ X. Now we present the proof of the following result (see Theorem 1 on page 60 of [12]). Theorem 2.5 Let (X,  · ) be a normed space, (Y,  · ) be a Banach space, F : X → 2Y be a proper hemicontinuous mapping such that F (x) is closed and convex for every x ∈ X, −∞ < a < b < ∞, and let for every integer k ≥ 1, xk : [a, b] →

2.2 Set-Valued Mappings and a Convergence Result

25

X and yk : [a, b] → Y be strongly measurable functions such that for almost every t ∈ [a, b] and for every neighborhood N of zero in X × Y there exists an integer k0 = k0 (t, N ) such that for each integer k ≥ k0 , (xk (t), yk (t)) ∈ graph(F ) + N . Assume that xk converges almost everywhere to a function x : [a, b] → X as k → ∞ and that yk ∈ L1 (a, b; Y ) converges weakly to y in L1 (a, b; Y ) as k → ∞. Then for almost every t ∈ [a, b], y(t) ∈ F (x(t)). Proof For a convex subset of a Banach space (here L1 (a, b; Y )), the strong closure coincides with the weak closure (here in the weak topology σ (L1 , L∞ )). This fact implies that for every integer h ≥ 1, y belongs to the strong closure of the convex hull co({yk : k ≥ h is an integer }). Hence, for any integer h ≥ 1, there exist an integer k(h) ≥ h, numbers αhk ≥ 0, k = h, . . . , k(h) satisfying k(h) 

αhk = 1

k=h

such that vh =

k(h) 

αhk yk

k=h

converges to y as h → ∞ in L1 (a, b; Y ). This implies that there exists a subsequence (again denoted by vh , h = 1, 2, . . . ) that converges to y for almost every t ∈ [a, b]. Let t ∈ [a, b], xh (t) converge to x(t) in X, and vh (t) converge to y(t) in Y as h → ∞. In order to complete the proof of the theorem, it is sufficient to show that y(t) ∈ F (x(t)). Assume the contrary. Then y(t) ∈ F (x(t)), and there exists p ∈ Y ∗ such that p(y(t)) > sup{p(z) : z ∈ F (x(t))} = σ (F (x(t), p).

(2.2)

(If F (x(t)) = ∅, then the right-hand side of (2.2) is −∞.) Fix λ > σ (F (x(t)), p) and η > 0. Since the mapping F is upper hemicontinuous, there exists a neighborhood M0 of zero in X such that

26

2 Differential Inclusions

σ (F (u), p) ≤ λ for all u ∈ x(t) + M0 . There exists an integer k1 ≥ 1 such that xk (t) ∈ x(t) + 2−1 M0 for all integers k ≥ k1 . In view of our assumptions, there exists an integer k0 ≥ k1 such that for each integer k ≥ k0 there exists (uk , wk ) ∈ graph(F ) such that uk ∈ xk (t) + 2−1 M0 , yk (t) − wk  ≤ η. Therefore, for every integer k ≥ k0 , uk ∈ x(t) + M0 , p(yk (t)) ≤ p(wk ) + ηp ≤ σ (F (uk ), p) + ηp ≤ λ + ηp, and p(yk (t)) ≤ λ + ηp for all integers k ≥ k0 . Let h ≥ k0 be a fixed integer. By multiplying the above inequality by αhk and summing them, we deduce that p(vh (t)) ≤ λ + ηp. Letting h → ∞, we obtain that p(y(t)) ≤ λ + pη. Since η is any positive number and λ is an arbitrary number satisfying λ > σ (F (x), p), we conclude that p(y(t)) ≤ σ (F (x), p). This contradicts (2.2). The contradiction we have reached completes the proof of Theorem 2.5.

2.3 Dynamical Systems

27

2.3 Dynamical Systems Let (X, ·, ·) be a Hilbert space equipped with an inner product ·, · that induces the norm x = x, x1/2 , x ∈ X. For each x ∈ X and each pair of nonempty sets A, B ⊂ X, define ρ(x, B) = inf{x − y : y ∈ B} and dist(A, B) = sup{ρ(x, B) : x ∈ A}. Assume that a mapping F : X → 2X is proper (Dom(F ) = ∅) and satisfies the following assumptions: (A1) The mapping F is upper hemicontinuous, and F (x) is a closed, convex set in X for every point x ∈ X. (A2) For each x ∈ X, there exist r, M > 0 such that F (y) ⊂ B(0, M) for each y ∈ B(x, r). Let −∞ < T1 < T2 < ∞. A function x ∈ W 1,1 (T1 , T2 ; X) is called a trajectory if x (t) ∈ F (x(t)), t ∈ [T1 , T2 ] almost everywhere (a.e.).

(2.3)

Denote by Y (T1 , T2 ) the collection of all trajectories x : [T1 , T2 ] → X. Let T1 ∈ R 1 . A function x : [T1 , ∞) → X is a trajectory if for every T2 > T1 its restriction to the interval [T1 , T2 ] belongs to Y (T1 , T2 ). Denote by Y (T1 , ∞) the set of all trajectories x : [T1 , ∞) → X. A function x : R 1 → X is a trajectory if for each pair of numbers T2 > T1 its restriction to the interval [T1 , T2 ] belongs to Y (T1 , T2 ). Denote by Y (−∞, ∞) the collection of all trajectories x : R 1 → X. Suppose that the function φ : X → [0, ∞) is locally Lipschitz and satisfies lim φ(z) = ∞

z→∞

and that the following assumption holds: (A3) For each x ∈ X and each h ∈ F (x), if there exists lim t −1 (φ(x + th) − φ(x)),

t→0

then this limit is non-positive.

(2.4)

28

2 Differential Inclusions

Proposition 2.6 Assume that −∞ < T1 < T2 < ∞ and that x ∈ Y (T1 , T2 ). Then the functions x and φ ◦ x are Lipschitz, the function φ ◦ x is decreasing on [T1 , T2 ], and for a.e. τ ∈ [T1 , T2 ], there exist x (τ ) and (φ ◦ x) (τ ) and (φ ◦ x) (τ ) = lim h−1 [φ(x(τ ) + hx (τ )) − φ(x(τ ))] ≤ 0. h→0

Proof In view of (A2), for every τ ∈ [T1 , T2 ], there exist Mτ , rτ > 0 such that F (x(t)) ⊂ B(0, Mτ ) for each y ∈ B(x(τ ), rτ ).

(2.5)

Clearly, the set x([T1 , T2 ]) is compact. In view of (2.5), x([T1 , T2 ]) ⊂ ∪{B 0 (x(τ ), rτ ) : τ ∈ [T1 , T2 ]}. Therefore, there exist τi ∈ [T1 , T2 ], i = 1, . . . , q such that q

x([T1 , T2 ]) ⊂ ∪i=1 B 0 (x(τi ), rτi ).

(2.6)

Set M = max{Mτi : i = 1, . . . , q}. By (2.3), (2.5) and (2.6), for a.e. τ ∈ [T1 , T2 ], there exists j ∈ {1, . . . , q} such that x(τ ) ∈ B 0 (x(τj ), rτj ), x (τ ) ∈ F (x(τ )) ⊂ B(0, Mτj ) ⊂ B(0, M). Thus the function x : [T1 , T2 ] → X is Lipschitz. Clearly, φ ◦ x is locally Lipschitz on the compact set [T1 , T2 ]. Therefore, the function φ ◦ x is Lipschitz on [T1 , T2 ] too, and for a.e. t ∈ [T1 , T2 ], there exist x (t) and (φ ◦ x) (t). Let τ ∈ (0, T ), and there exist x (τ ) and (φ ◦ x) (τ ). The function φ is Lipschitz in a neighborhood of x(τ ) with a Lipschitz constant L > 0. Therefore, x (τ ) = lim h−1 [x(τ + h) − x(τ )] h→0

and h−1 |[(φ ◦ x)(τ + h) − (φ ◦ x)(τ )] − [φ(x(τ ) + hx (τ )) − φ(x(τ ))]| = h−1 |(φ ◦ x)(τ + h) − φ(x(τ ) + hx (τ ))| ≤ Lh−1 x(τ + h) − x(τ ) − hx (τ ) → 0 as h → 0.

2.3 Dynamical Systems

29

Together with (A3), this completes the proof of Proposition 2.6. Assume that X0 is nonempty, closed subset of X such that for each r > 0 the set X0 ∩ B(x, r) is compact

(2.7)

and that the following assumption holds: (A4) For each bounded nonempty set B and each  > 0, there exists a natural number T (B, ) such that for each T ≥ T (B, ), dist({x(T ) : x ∈ Y (0, T ) and x(0) ∈ B}, X0 ) ≤ . In Sect. 2.5, we prove the following result. Proposition 2.7 The following properties are equivalent: (1) There exists x ∈ Y (0, ∞) such that x(t) ∈ X0 for every t ≥ 0. (2) There exists x ∈ Y (0, ∞). (3) There exist M > 0 and xk ∈ Y (0, Tk ), k = 1, 2, . . . , such that limk→∞ Tk → ∞ and φ(xk (0)) ≤ M, k = 1, 2, . . . . In this chapter, in the sequel, we suppose that there exists x ∈ Y (0, ∞). Define Ω(F ) = {z ∈ X : for every positive number  there is a function x ∈ Y (0, ∞) for which lim inf z − x(t) ≤ }. t→∞

(2.8)

Evidently, Ω(F ) is a closed subset of X. In Sect. 2.6, we prove the following result. Proposition 2.8 Ω(F ) = ∅ and Ω(F ) ⊂ X0 . It is not difficult to see that the following two results hold. Proposition 2.9 For every function x ∈ Y (0, ∞), lim ρ(x(t), Ω(F )) = 0.

t→∞

Proposition 2.10 Assume that B ⊂ X be a nonempty, closed set such that for every x ∈ Y (0, ∞), lim ρ(x(t), B) = 0.

t→∞

Then Ω(F ) ⊂ B. The following theorem will be proved in Sect. 2.7.

30

2 Differential Inclusions

Theorem 2.11 Let , M be positive real numbers. Then there is a positive number T such that for every function x ∈ Y (0, T ) that satisfies φ(x(0)) ≤ M the inequality inf{ρ(x(t), Ω(F )) : t ∈ [0, T ]} ≤  holds. Note that the dynamical system considered here was studied in [150, 151] when the space X was finite dimensional.

2.4 Auxiliary Results Lemma 2.12 Let M > 0. Then there exist r, L > 0 such that for each z ∈ ∪{B(x, r) : x ∈ X0 ∩ B(0, M)} the inclusion F (z) ⊂ B(0, L) holds. Proof In view of (A2), for each x ∈ X0 ∩ B(0, M), there exist Lx , rx > 0 such that F (y) ⊂ B(0, Lx ) for each y ∈ B(x, 2rx ).

(2.9)

Clearly, X0 ∩ B(0, M) ⊂ ∪{B 0 (x, rx ) : x ∈ X0 ∩ B(0, M)}.

(2.10)

Since the set X0 ∩ B(0, M) is compact, it follows from (2.10) that there exists a finite set xi ∈ X0 ∩ B(0, M), i = 1, . . . , q, such that q

X0 ∩ B(0, M) ⊂ ∪i=1 B 0 (xi , rxi ).

(2.11)

L = max{Lxi : i = 1, . . . , q},

(2.12)

r = min{rxi : i = 1, . . . , q}.

(2.13)

Set

2.4 Auxiliary Results

31

Assume that z ∈ ∪{B(x, r) : x ∈ X0 ∩ B(0, M)}. By the inclusion above, there exists x ∈ X0 ∩ B(0, M) ∩ B(z, r).

(2.14)

Equations (2.11) and (2.14) imply that there exists i ∈ {1, . . . , q}

(2.15)

x ∈ B 0 (xi , rxi ).

(2.16)

such that

It follows from (2.13), (2.14), and (2.16) that z ∈ B(xi , 2rxi ).

(2.17)

In view of (2.9), (2.13), and (2.17), F (z) ⊂ B(0, Lxi ) ⊂ B(0, L). Lemma 2.12 is proved. Lemma 2.13 Let M > 0. Then there exist r, L > 0 such that for each z1 , z2 ∈ ∪{B(x, r) : x ∈ X0 ∩ B(0, M)}

(2.18)

z1 − z2  ≤ r

(2.19)

satisfying

the inequality |φ(z1 ) − φ(z2 )| ≤ Lz1 − z2  holds. Proof Since the function φ is locally Lipschitz for each x ∈ X0 ∩ B(0, M), there exist Lx , rx > 0 such that |φ(z1 ) − φ(z2 )| ≤ Lx z1 − z2  for all z1 , z2 ∈ B(x, 4rx ). Since the set X0 ∩ B(0, M) is compact, there exists a finite set

(2.20)

32

2 Differential Inclusions

xi ∈ X0 ∩ B(0, M), i = 1, . . . , q such that q

X0 ∩ B(0, M) ⊂ ∪i=1 B 0 (xi , rxi ).

(2.21)

L = max{Lxi : i = 1, . . . , q},

(2.22)

r = min{rxi : i = 1, . . . , q}.

(2.23)

Set

Assume that z1 , z2 ∈ X satisfy (2.18) and that z1 − z2  ≤ r.

(2.24)

x ∈ X0 ∩ B(0, M) ∩ B(z1 , r).

(2.25)

By (2.18), there exists

It follows from (2.24) and (2.25) that x − z1  ≤ r, z2 − x ≤ z2 − z1  + z1 − x ≤ 2r.

(2.26)

In view of (2.25), there exists p ∈ {1, . . . , q} such that x ∈ B 0 (xp , rxp ).

(2.27)

Equations (2.23), (2.26), and (2.27) imply that zi − xp  ≤ zi − x + x − xp  ≤ 2r + rxp ≤ 3rxp .

(2.28)

By (2.20), (2.22), and (2.28), |φ(z1 ) − φ(z2 )| ≤ Lxp z1 − z2  ≤ Lz1 − z2 . Lemma 2.13 is proved. Theorem 2.14 Let −∞ < a < b < ∞, M, L, r > 0, and F (z) ⊂ (0, L) for each

(2.29)

2.5 Proof of Proposition 2.7

33

z ∈ E := ∪{B(x, z) : x ∈ X0 ∩ B(0, M)}. Assume that for each integer k ≥ 1, xk ∈ Y (a, b) satisfies xk (t) ∈ E, t ∈ [a, b]

(2.30)

lim ρ(xk (t), X0 ) = 0.

(2.31)

and that for each t ∈ [a, b], k→∞

Then there exist a subsequence {xki }∞ i=1 and x ∈ Y (a, b) such that xki converges to x uniformly on [a, b]

(2.32)

xk i converges to x weakly in L1 (a, b; X)

(2.33)

and

as i → ∞. Proof By (2.3), (2.29), and (2.30), for each integer k ≥ 1 and almost every t ∈ [a, b], xk (t) ≤ L.

(2.34)

It follows from (2.30), (2.31), and compactness of bounded, closed subsets of X0 that for every t ∈ [a, b], the sequence {xk (t)}∞ k=1 is relatively compact in X. Together with (2.34) and Theorem 2.3, this implies that there exist a subsequence 1,1 (a, b; X) such that (2.32) and (2.33) hold. Theorem 2.5 {xki }∞ i=1 and x ∈ W implies that x (t) ∈ F (x(t)) for almost every t ∈ [a, b]. Thus x ∈ Y (a, b). Theorem 2.14 is proved.

2.5 Proof of Proposition 2.7 Clearly, (1) implies (2) and (2) implies (3). In order to complete the proof of the proposition, it is sufficient to show that (3) implies (1). Assume that xk ∈ Y (0, Tk ), k = 1, 2, . . . , lim Tk = ∞, k→∞

φ(xk (0)) ≤ M, k = 1, 2, . . . .

(2.35)

34

2 Differential Inclusions

We may assume that Tk > k, k = 1, 2, . . . .

(2.36)

Proposition 2.6 and (2.35) imply that φ(xk (t)) ≤ φ(xk (0)) ≤ M for all integers k ≥ 1 and all t ∈ [0, Tk ].

(2.37)

Set C = {xk (t) : k = 1, 2, . . . , t ∈ [0, Tk ]}.

(2.38)

In view of (2.4) and (2.37), the set C is bounded, and there exists M0 > 1 such that C ⊂ B(0, M0 ). Lemmas 2.12 and 2.13 imply that there exist L > 1, r ∈ (0, 1) such that the following properties hold: (i) For each z ∈ ∪{B(x, r) : x ∈ X0 ∩ B(0, M0 + 1)}, we have F (z) ⊂ B(0, L). (ii) For each z1 , z2 ∈ ∪{B(x, r) : x ∈ X0 ∩ B(0, M0 + 1)} satisfying z1 − z2  ≤ r, we have |φ(z1 ) − φ(z2 )| ≤ Lz1 − z2 . Assumption (A4) and Eqs. (2.36) and (2.38) imply that for every natural number k there exists an integer n(k) > k such that for each integer n ≥ n(k) and each t ∈ [n(k), Tn ], ρ(xn (t), X0 ) ≤ (2k)−1 r.

(2.39)

2.6 Proof of Proposition 2.8

35

For each integer k ≥ 1, define yk ∈ Y (0, n(k)) by yk (t) = x2n(k) (t + nk ), t ∈ [0, n(k)].

(2.40)

It follows from (2.38)–(2.40) that for each integer k ≥ 1 and each t ∈ [0, nk ], ρ(yk (t), X0 ) = ρ(x2n(k) (n(k) + t), X0 ) ≤ (2k)−1 r, yk (t) ≤ M0 , and ρ(yk (t), X0 ∩ B(0, M0 + 2−1 )) ≤ (2k)−1 r.

(2.41)

Property (i) and (2.41) imply that for each t ≥ 0, lim ρ(yk (t), X0 ∩ B(0, M0 + 1)) = 0.

k→∞

(2.42)

By Theorem 2.14, property (i), (2.41) and (2.42), extracting subsequences and using the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers {kj }∞ j =1 and x ∈ Y (0, ∞) such that ykj converges to x uniformly on [0, p] as j → ∞ for any integer p ≥ 1. In view of (2.42), x(t) ∈ X0 for all t ≥ 0. Thus (1) holds and Proposition 2.7 is proved.

2.6 Proof of Proposition 2.8 Let x ∈ Y (−∞, ∞). In view of (A4), lim ρ(x(t), X0 ) = 0.

(2.43)

φ(x(t)) ≤ φ(x(0)), t ∈ [0, ∞).

(2.44)

t→∞

Proposition 2.6 implies that

From Equations (2.4) and (2.44), there exists M0 > 0 such that x(t) ≤ M0 , t ∈ [0, ∞).

(2.45)

lim ρ(x(t), X0 ∩ B(0, M0 + 1)) = 0.

(2.46)

By (2.43) and (2.45), t→∞

36

2 Differential Inclusions

In view of (2.46), for each integer k ≥ 1, there exists zk ∈ X0 ∩ B(0, M0 + 1)

(2.47)

and a number tk ≥ k such that x(tk ) − zk  ≤ k −1 .

(2.48)

Since the set X0 ∩ B(0M0 + 1) is compact, there exists a convergent subsequence {zki }∞ i=1 . By (2.48), lim x(tki ) = lim zki ∈ Ω(F )

i→∞

i→∞

and Ω(F ) = ∅. Let z ∈ Ω(F ) and  > 0. There exists x ∈ Y (0, ∞) such that lim inf z − x(t) ≤ . t→∞

(2.49)

Assumption (A4) implies that lim ρ(x(t), X0 ) = 0.

t→∞

Together with (2.49), this implies that ρ(z, X0 ) ≤ 2. Since the set X0 is closed and  is an arbitrary positive number, we conclude that z ∈ X0 and Ω(F ) ⊂ X0 . Proposition 2.8 is proved.

2.7 Proof of Theorem 2.11 Assume that the theorem does not hold. Then for each integer k ≥ 1, there exists xk ∈ Y (0, k) such that φ(xk (0)) ≤ M,

(2.50)

inf{ρ(x(t), Ω(F )) : t ∈ [0, k]} > .

(2.51)

2.7 Proof of Theorem 2.11

37

Proposition 2.6 and (2.49) imply that φ(xk (t)) ≤ M, t ∈ [0, k], k = 1, 2, . . . .

(2.52)

Equations (2.4) and (2.52) imply that there exists M1 > 0 such that xk (t) ≤ M1 , t ∈ [0, k], k = 1, 2, . . . .

(2.53)

Lemma 2.12 implies that there exist r ∈ (0, 1) and L > 1 such that the following property holds: (i) For each z ∈ ∪{B(x, r) : x ∈ X0 ∩ B(0, M1 + 1)}, we have F (z) ⊂ B(0, L). Assumption (A4) and (2.53) imply that for each integer k ≥ 1 there exists an integer n(k) > k such that for each integer n ≥ n(k) and each t ∈ [n(k), n] we have ρ(xn (t), X0 ) ≤ (2k)−1 r.

(2.54)

For each integer k ≥ 1, define yk ∈ Y (0, n(k)) by yk (t) = x2n(k) (n(k) + t), t ∈ [0, n(k)].

(2.55)

By (2.53)–(2.55), for each integer k ≥ 1 and each t ∈ [0, nk ], ρ(yk (t), X0 ) = ρ(x2n(k) (n(k) + t), X0 ) ≤ (2k)−1 r,

(2.56)

yk (t) ≤ M1 , ρ(yk (t), X0 ∩ B(0, M1 + 2−1 )) ≤ (2k)−1 r.

(2.57)

In view of (2.57), for each integer t ≥ 0, lim ρ(yk (t), X0 ∩ B(0, M1 + 1)) = 0.

k→∞

(2.58)

By Theorem 2.14, property (i), (2.57) and (2.58), extracting subsequences and using the diagonalization process, we obtain that there exist a subsequence s {yki }∞ i=1 and y ∈ Y (0, ∞) such that yki converges to y uniformly on [0, p]

(2.59)

as i → ∞ for any integer p ≥ 1. It follows from (2.51), (2.55), and (2.59) that for every integer t ≥ 0,

38

2 Differential Inclusions

ρ(y(t), Ω(F )) = lim ρ(ykj (t), Ω(F )) ≥ . j →∞

(2.60)

Equations (2.58) and (2.59) imply that for each t ≥ 0, y(t) ∈ X0 ∩ B(0, M1 + 1), which is a compact set. Therefore, there exists a strictly increasing sequence of natural numbers {pj }∞ j =1 such that there exists lim y(pj ) ∈ Ω(F ).

j →∞

This contradicts (2.60). The contradiction we have reached proves Theorem 2.11.

2.8 An Example Assume that n X = R+ := {x = (x1 , . . . , xn ) ∈ R n : xi ≥ 0, i = 1, . . . , n},

a : X → 2X \ {∅}, the graph of a is convex, a(λx) = λa(x) for each x ∈ X and each λ ≥ 0, n n a(x) = (a(x) − R+ ) ∩ R+ , x ∈ X.

This set-valued map describes the von Neumann–Gale model of economic dynamics [75, 95, 97, 117] and is a convex process. It is known that there exists a triplet n satisfy ( α,  x, p ) such that  α ≥ 0,  x, p  ∈ R+  x  = 1,  p  = 1,  α x ∈ a( x ),  αp , x ≥ p , y n and each y ∈ a(x). The study of this mapping and the for each x ∈ R+ corresponding model can be reduced to the case when  α = 1. In this case, n n F (x) = a(x) − x, x ∈ R+ , F (x) = ∅, x ∈ R n \ R+ ,

and this is a particular case of set-valued mappings and the corresponding differential inclusions considered in this chapter with Ω(F ) = {λ x : λ ∈ [0∞)}, φ(x) =

n  i=1

|xi | pi , x ∈ X.

Chapter 3

Discrete-Time Dynamical Systems

In this chapter, we study a discrete-time dynamical system with a Lyapunov function in a metric space induced by a set-valued mapping. For this discrete-time dynamical system, we establish a turnpike result.

3.1 Preliminaries and Main Results Assume that .(X, ρ) is a metric space and .A is a nonempty, closed subset of the metric space .X × X equipped with the metric .ρ1 : X × X → [0, ∞) defined by ρ1 ((x1 , x2 ), (y1 , y2 )) = ρ(x1 , y1 ) + ρ(x2 , y2 ), x1 , x2 , y1 , y2 ∈ X.

.

For each .x ∈ X and each .r > 0, set B(x, r) = {y ∈ X : ρ(x, y) ≤ r} and B 0 (x, r) = {y ∈ X : ρ(x, y) < r}.

.

Set XA = {x ∈ X : {x} × X ∩ A = ∅}.

(3.1)

a(x) = {y ∈ X : (x, y) ∈ A}.

(3.2)

.

For each .x ∈ XA , set .

For each .x ∈ X and each pair of nonempty sets .A, B ⊂ X, define ρ(x, B) = inf{ρ(x, y) : y ∈ B}

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0_3

39

40

3 Discrete-Time Dynamical Systems

and dist(A, B) = sup{ρ(x, B) : x ∈ A}.

.

(3.3)

For each nonempty subset .D ⊂ X, set a(D) = ∪{a(x) : x ∈ D}.

.

θ ∈ X. Assume that .φ : X → R 1 is a continuous function such that Fix . φ(y) ≤ φ(x) for each (x, y) ∈ A.

.

(3.4)

Assume that .X0 is nonempty, closed subset of X such that for each .r > 0 the set X0 ∩ B( θ , r) is compact .

.

(3.5)

2 ⊂ X is called a trajectory of a (or Let .T2 > T1 be integers. A sequence .{xt }Tt=T 1 just a trajectory if the mapping a is understood) if .(xt , xt+1 ) ∈ A for all integers .t ∈ {T1 , . . . , T2 − 1}. A sequence .{xt }∞ t=T1 ⊂ X is called a trajectory of a (or just a trajectory if the mapping a is understood) if .(xt , xt+1 ) ∈ A for all nonnegative integers .t ≥ T1 . 2 and by .Y (T1 , ∞) Denote by .Y (T1 , T2 ) the collection of all trajectories .{yt }Tt=T 1 ∞ the set of all trajectories .{yt }t=T1 . We assume that the following assumption holds: (A) For each bounded, nonempty set C, there exists .MC > 0, and for each . > 0, there exists a natural number .n(C, ) such that for each integer .n ≥ n(C, ),

dist({x ∈ X : there is a trajectory {xt }nt=0 such that x0 and xn = x},

.

X0 ∩ B( θ , MC )}) ≤ .

.

In Sect. 3.2, we prove the following result. Proposition 3.1 The following properties are equivalent: (1) There exists a trajectory .{xt }∞ t=0 ⊂ X0 . (2) There exists a trajectory .{xt }∞ t=0 . (3) There exists .M > 0, and for each integer .n ≥ 1, there exists a trajectory .{xt }nt=0 satisfying .ρ(x0 ,  θ ) ≤ M. In the sequel, we suppose that there exists a trajectory .{xt }∞ t=0 . Define Ω(a) = {z ∈ X : for every  > 0 there exists a trajectory {xt }∞ t=0

.

for which lim inf ρ(z, xt ) ≤ }.

.

t→∞

(3.6)

3.1 Preliminaries and Main Results

41

Clearly, .Ω(a) is a closed subset of the metric space .(X, ρ). In Sect. 3.3, we prove the following two results. Proposition 3.2 .Ω(a) = ∅ and .Ω(a) ⊂ X0 . Proposition 3.3 For every trajectory .{xt }∞ t=0 , .

lim ρ(x(t), Ω(a)) = 0.

t→∞

It is not difficult to see that the following result holds. Proposition 3.4 Assume that .B ⊂ X be a nonempty, closed set such that for every trajectory .{xt }∞ t=0 , .

lim ρ(x(t), B) = 0.

t→∞

Then .Ω(a) ⊂ B. The following theorem will be proved in Sect. 3.4. Theorem 3.5 Let ., M be positive real numbers. Then there is a positive integer T such that for every trajectory .{xt }Tt=0 that satisfies ρ(x0 ,  θ) ≤ M

.

the inequality .

min{ρ(xt , Ω(a)) : t = 0, . . . , T } ≤ 

holds. The next theorem is proved in Sect. 3.5. Theorem 3.6 The following properties are equivalent: (1) If a bounded sequence .{xt }∞ t=−∞ ⊂ X satisfies xt+1 ∈ a(xt ) and φ(xt+1 ) = φ(xt )

.

for all integers t, then the inclusion .xt ∈ Ω(a) holds for all integers t. (2) If a bounded sequence .{xt }∞ t=−∞ ⊂ X0 satisfies xt+1 ∈ a(xt ) and φ(xt+1 ) = φ(xt )

.

for all integers t, then the inclusion .xt ∈ Ω(a) holds for all integers t. (3) For each .M,  > 0, there exist .δ > 0 and a natural number L such that for each integer .T > 2L and each trajectory .{xt }Tt=0 that satisfy

42

3 Discrete-Time Dynamical Systems

ρ(x0 ,  θ ) ≤ M and φ(x0 ) − φ(xT ) ≤ δ

.

the inequality ρ(xt , Ω(a)) ≤ 

.

holds for all integers .t = L, . . . , T − L. Assume that properties of Theorem 3.6 hold. The next theorem is a weak turnpike result. It is proved in Sect. 3.6. Theorem 3.7 Let ., M > 0. Then there exist natural numbers .L, Q such that for every natural number .T > L and every trajectory .{xt }Tt=0 that satisfy ρ(x0 ,  θ ) ≤ M,

.

q

q

there exist an integer .q ∈ [1, Q] and finite sequences of integers .{ai }i=1 , .{bi }i=1 ⊂ [0, T ] such that ai ≤ bi , i = 1, . . . , q, ai+1 > bi , i ∈ {1, . . . , q} \ {q},

.

ρ(xt , Ω(a)) ≤ , t ∈ {ai , . . . , bi }, i = 1, . . . , q,

.

and q

Card({0, . . . , T } \ ∪i=1 {ai , . . . , bi }) ≤ L.

.

It is not difficult to see that the set-valued mapping a from Sect. 2.8 and the discrete-time dynamical system induced by a satisfy all the assumptions posed in this section. It should be mentioned that a particular case of our system, when a is a selfmapping of a compact metric space, was introduced in [96]. Several results related to this particular case are presented in [149].

3.2 Proof of Proposition 3.1 In the proof of Proposition 3.1, we use the following simple auxiliary result that is proved easily. Lemma 3.8 Assume that C is a nonempty, compact set in X. Then every sequence {xi }∞ i=0 ⊂ X, which satisfies .limi→∞ ρ(xi , C) = 0, has a subsequence that converges to a point of C.

.

3.3 Proofs of Propositions 3.2 and 3.3

43

Proof of Proposition 3.1 Evidently, (1) implies (2) and (2) implies (3). Assume that (3) holds. There exist .M > 0 and trajectories .{xt(n) }nt=0 ⊂ X, .n = 1, 2, . . . , such that ρ( θ , x0(n) ) ≤ M, n = 1, 2, . . . .

.

(3.7)

By (3.7) and assumption (A), there exists .M0 > 0 such that for every natural number k there exists an integer .n(k) > k such that for each integer .n ≥ n(k) and each integer .t ∈ [n(k), n] we have ρ(xt , X0 ∩ B( θ , M0 )) ≤ k −1 . (n)

.

(3.8)

For each integer .k ≥ 1, define a trajectory 2n(k) yt(k) = xt+n(k) , t = 0, . . . , n(k).

.

(3.9)

Equations (3.8) and (3.9) imply that for each integer .k ≥ 1 and each .t ∈ {0, . . . , n(k)}, 2n(k) ρ(yt(k) , X0 ∩ B( θ , M0 )) = ρ(xt+n(k) , X0 ∩ B( θ , M0 )) ≤ k −1 .

.

(3.10)

In view of (3.10), for each integer .t ≥ 0, lim ρ(yt , X0 ∩ B( θ , M0 )) = 0. (k)

.

t→∞

(3.11)

By Lemma 3.8 and (3.11), extracting subsequences and using the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers ∞ such that for each integer .t ≥ 0 there exists .{kj } j =1 (kj )

xt = lim yt

.

j →∞

∈ X0 ∩ B( θ , M0 ).

Since the set .A is closed, Eq. (3.9) implies that .{xt }∞ t=0 is a trajectory. Thus property (1) holds and (3) implies (1). This completes the proof of Proposition 3.1.

3.3 Proofs of Propositions 3.2 and 3.3 Let .{xt }∞ t=0 be a trajectory. Assumption (A) implies that there exists .M0 > 0 such that .

lim ρ(xt , X0 ∩ B( θ , M0 )) = 0.

t→∞

(3.12)

44

3 Discrete-Time Dynamical Systems

Lemma 3.8 and (3.5) and (3.12) imply that the sequence .{xt }∞ t=0 has a subsequence that converges to a point of .Ω(a). Hence, Ω(a) = ∅.

.

Let .z ∈ Ω(a) and . > 0. There exists a trajectory .{xt }∞ t=0 such that lim inf ρ(z, xt ) ≤ ..

.

t→∞

(3.13)

In view of (A), .

lim ρ(xt , X0 ) = 0.

t→∞

By the equation above and (3.13), ρ(z.X0 ) ≤ .

.

Since the set .X0 is closed and . is an arbitrary positive number, we conclude that z ∈ X0 and .Ω(a) ⊂ X0 . Proposition 3.2 is proved. Proposition 3.3 follows from Lemma 3.8 and assumption (A).

.

3.4 Proof of Theorem 3.5 Assume that Theorem 3.5 does not hold. Then for every integer n, there exists a (n) trajectory .{xt }nt=0 such that (n) ρ( θ , x0 ) ≤ M,

(3.14)

ρ(xt , Ω(a)) > , t = 0, , . . . , n.

(3.15)

.

(n)

.

Assumption (A) and (3.14) imply that there exists .M1 > 0 such that for every integer k ≥ 1 there exists an integer .n(k) ≥ k such that for each integer .n ≥ n(k),

.

ρ(xt , X0 ∩ B(θ, M1 )) ≤ k −1 , t = n(k), . . . , n.

.

(n)

(3.16)

For each integer .k ≥ 1, define a trajectory (k)

yt

.

(2n(k))

= xt+n(k) , t = 0, . . . , n(k).

(3.17)

It follows from (3.15)–(3.17) that for each integer .k ≥ 1 and .t = 0, . . . , n(k),

3.5 Proof of Theorem 3.6

45 (k)

ρ(yt , Ω(a)) > ,

(3.18)

ρ((yt , X0 ∩ B( θ , M1 )) ≤ k −1 .

(3.19)

.

(k)

.

By Lemma 3.8 and (3.19), extracting subsequences and using the diagonalization process we obtain that there exists a strictly increasing sequence of natural numbers ∞ such that for each integer .t ≥ 0 there exists .{kj } j =1 (kj )

yt = lim yt

.

j →∞

.

(3.20)

Since the set .A is closed, it follows from (3.20) that .{yt }∞ t=0 is a trajectory. Equations (3.18)–(3.20) imply that for every integer .t ≥ 0, ρ(yt ,  θ ) ≤ M1 , ρ(yt , Ω(a)) ≥ ,

.

(3.21)

yt ∈ X0 ∩ B( θ , M1 ).

.

θ , M1 ) is compact, there exists a subsequence Since in view of (3.5), the set .X0 ∩ B( ∞ that converges to a point of .Ω(a). This contradicts (3.21). The contradiction .{yti } i=1 we have reached proves Theorem 3.5.

3.5 Proof of Theorem 3.6 Clearly, (1) implies (2) and (3) implies (1). Therefore, in order to prove the theorem, it is sufficient to show that (2) implies (3). Assume that (2) holds and (3) does not hold. Then there exist ., M > 0 such that (k) k for each integer .k ≥ 1 there exist an integer .Tk > 2k and a trajectory .{xt }Tt=0 such that ρ(x0 ,  θ ) ≤ M,

(3.22)

|φ(x0 ) − φ(xTk )| ≤ k −1

(3.23)

max{ρ(xt(k) , Ω(a)) : t = k, . . . , Tk − k} > .

(3.24)

.

.

(k)

(k)

(k)

and .

By (A) and (3.22), there exists .M1 > 0 such that for each integer .p ≥ 1 there exists an integer .k(p) > p such that for each integer .k ≥ k(p) and all .t = k(p), . . . , Tk we have

46

3 Discrete-Time Dynamical Systems

ρ(xt , X0 ∩ B(θ, M1 )) ≤ p−1 . (k)

.

(3.25)

Let .p ≥ 1 be an integer. In view of (3.24), there exists an integer τp ∈ {2k(p), . . . , T2k(p) − 2k(p)}

.

such that ρ(xτ(2k(p)) , Ω(a)) > . p

.

(3.26)

Define a trajectory (p)

(2k(p))

= xt+τp , t = −τp , . . . , T2k(p) − τp .

yt

.

(3.27)

Equations (3.23) and (3.25)–(3.27) imply that (p)

ρ(y0 , Ω(a)) = ρ(xτ(2k(p)) , Ω(a)) >  p

.

(3.28)

and that for all .t = −τp , . . . , T2k(p) − τp , (p) (2k(p)) ρ(yt , X0 ∩ B( θ , M1 )) = ρ(xt+τp , X0 ∩ B( θ , M1 )) ≤ p−1 ,

(3.29)

φ(y−τp ) − φ(yT2k(p) −τp ) ≤ (2k(p))−1 ≤ (2p)−1 .

(3.30)

.

.

(p)

(p)

In view of (3.29), for each integer t, .

(p) lim ρ(yt , X0 ∩ B( θ , M1 )) = 0.

p→∞

(3.31)

By Lemma 3.8, (3.5) and (3.31), extracting subsequences and using the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers .{pi }∞ i=1 such that for each integer t there exists (pi )

yt = lim yt

.

i→∞

.

(3.32)

Since the set .A is closed, it follows from (3.30), (3.31), and (3.32) that for each integer t we have φ(yt ) = φ(yt+1 ), yt+1 ∈ a(yt ), yt ∈ X0 .

.

Property (2) implies that yt ∈ Ω(a) for each integer t.

.

3.6 Proof of Theorem 3.7

47

On the other hand, it follows from (3.28) and (3.32) that ρ(y0 , Ω(a)) ≥ .

.

The contradiction we have reached completes the proof of Theorem 3.6.

3.6 Proof of Theorem 3.7 We may assume without loss of generality that . < 1 and .M > 0. Assumption (A) implies that there exist .M1 > M and an integer .L1 ≥ 1 such that the following property holds: (a) For each integer .n ≥ L1 and each trajectory .{xt }nt=0 satisfying .ρ(x0 ,  θ ) ≤ M, we have ρ(xt , X0 ∩ B( θ , M1 )) ≤ 1, |φ(xt )| ≤ M1

.

for all .t = L1 , . . . , n. Property (3) of Theorem 3.6 implies there exist .δ ∈ (0, 1) and a natural number .L0 such that the following property holds: (b) For each integer .T > 2L0 and each trajectory .{xt }Tt=0 that satisfy ρ(x0 ,  θ ) ≤ M1 + 2 and φ(x0 ) − φ(xT ) ≤ δ

.

the inequality ρ(xt , Ω(a)) ≤ 

.

holds for all integers .t = L0 , . . . , T − L0 . Choose integers Q > 2M1 δ −1 + 2

(3.33)

L > (4L0 + 8)(2δ −1 M1 + 2) + L1 + 4.

(3.34)

.

and .

Suppose that .T > L is a natural number and that a trajectory .{xt }Tt=0 satisfies ρ(x0 ,  θ ) ≤ M, φ(x0 ) ≤ M.

.

Property (a) and (3.35) imply that

(3.35)

48

3 Discrete-Time Dynamical Systems

ρ(xt ,  θ ) ≤ M1 + 2, |φ(xt )| ≤ M1

.

(3.36)

for all .t = L1 , . . . , T . By induction, we define a strictly increasing finite sequence .ti ∈ {0, . . . , T }, .i = 0, . . . , q. Set t 0 = L1 .

.

If φ(xT ) ≥ φ(xt0 ) − δ,

.

then set .t1 = T and complete to construct the sequence. Assume that φ(xT ) < φ(xt0 ) − δ.

.

Evidently, there is an integer .t1 ∈ (t0 , T ] satisfying φ(xt1 ) < φ(x0 ) − δ

.

and that if an integer S satisfies .t0 < S < t1 , then φ(xS ) ≥ φ(x0 ) − δ.

.

If .t1 = T , then we complete to construct the sequence. Assume that .k ≥ 1 is an integer and that we defined a strictly increasing sequence .t0 , . . . , tk ∈ {0, 1, . . . , T } such that t 0 = L1 , t k ≤ T

.

and that for each .i ∈ {0, . . . , k − 1}, φ(xti+1 ) < φ(xti ) − δ,

.

and if an integer S satisfies .ti < S < ti+1 , then φ(xS ) ≥ φ(xti ) − δ.

.

(It is not difficult to see that the assumption is true with .k = 1). If .tk = T , then we complete to construct the sequence. Assume that .tk < T . If φ(xT ) ≥ φ(xtk ) − δ,

.

then we set .tk+1 = T and complete to construct the sequence.

3.6 Proof of Theorem 3.7

49

Assume that φ(xT ) < φ(xtk ) − δ.

.

Evidently, there is a natural number .tk+1 ∈ (tk , T ] for which φ(xtk+1 ) < φ(xtk ) − δ

.

and that if an integer S satisfies .tk < S < tk+1 , then φ(xS ) ≥ φ(xtk ) − δ.

.

Evidently, the assumption made for k is true for .k + 1 too. Therefore by induction, we constructed the strictly increasing finite sequence of integers .ti ∈ [0, T ], .i = 0, . . . , q, such that t 0 = L1 , t q = T

.

(3.37)

and that for every i satisfying .0 ≤ i < q − 1, φ(xti+1 ) < φ(xti ) − δ

.

(3.38)

and for each .i ∈ {0, . . . , q − 1} and each integer S satisfying .ti < S < ti+1 , we have φ(xS ) ≥ φ(xti ) − δ.

.

(3.39)

By (3.4), (3.33),(3.35), (3.36), and (3.38), 2M1 ≥ φ(xt0 ) − φ(xtq−1 )

.

.

≥



{φ(xti ) − φ(xti+1 ) : i is an integer, 0 ≤ i ≤ q − 2} ≥ δ(q − 1)

and q ≤ 1 + 2δ −1 M1 < Q.

.

(3.40)

Define E = {i ∈ {0, . . . , q − 1} : ti+1 − ti ≥ 2L0 + 4}.

.

(3.41)

Let i ∈ E.

.

(3.42)

50

3 Discrete-Time Dynamical Systems

By (3.41) and (3.42), ti+1 − 1 − ti ≥ 2L0 + 3.

.

(3.43)

Equations (4.43) and (4.47) imply that φ(xti+1 −1 ) ≥ φ(xti ) − δ.

.

(3.44)

Equations (3.36), (3.37), (3.43), (3.44) and property (b) applied to the trajectory ti+1 −1 {xt }t=t imply that i

.

ρ(xt , Ω(a)) ≤ , t = ti + L0 , . . . , ti+1 − 1 − L0 .

.

(3.45)

Set ai = ti + L0 , bi = ti+1 − L0 − 1.

.

By (3.37) and (3.45), {0, . . . , T } \ ∪i∈E {ai , . . . , bi }

.

.

.

⊂ ∪{{ti , . . . , ti+1 } : i ∈ {0, . . . , q − 1} \ E} ∪ {0, . . . , L1 − 1}

∪ {{ti , . . . , ti + L0 − 1} ∪ {ti+1 − L0 , . . . , ti+1 } : i ∈ E}.

It follows from (3.40), (3.41), (3.44), and (3.46) that Card({0, . . . , T } \ ∪i∈E {ai , . . . , bi })

.

.

≤ q(2L0 + 5) + (2L0 + 2)q + L1 = q(4L0 + 7) + L1 .

≤ (4L0 + 7)(1 + 2M1 δ −1 ) + L1 < L.

Theorem 3.7 is proved.

(3.46)

Chapter 4

Continuous-Time Dynamical Systems

In this chapter, we study a class of continuous-time dynamical systems that is an analog of the class of discrete-time dynamical systems considered in Chap. 3. This class also contains the class of dynamical systems studied in Chap. 2.

4.1 Preliminaries and Main Results Assume that .(X, ρ) is a metric space. For each .x ∈ X and each .r > 0, set B(x, r) = {y ∈ X : ρ(x, y) ≤ r} and B 0 (x, r) = {y ∈ X : ρ(x, y) < r}.

.

For each .x ∈ X and each pair of nonempty sets .A, B ⊂ X, define ρ(x, B) = inf{ρ(x, y) : y ∈ B}

.

and dist(A, B) = sup{ρ(x, B) : x ∈ A}.

.

Assume that for each pair of real numbers .T1 < T2 , .Y (T1 , T2 ) is a nonempty set of functions .x : [T1 , T2 ] → X such that the following properties hold: (A1) For each pair of numbers .T1 < T2 , each .S1 , S2 ∈ [T1 , T2 ] satisfying .S1 < S2 , and each .x ∈ Y (T1 , T2 ), the restriction of x to .[S1 , S2 ] belongs to .Y (S1 , S2 ). (A2) For each pair of real numbers .T1 < T2 , each .S ∈ R 1 , and each .x ∈ Y (T1 , T2 ), the function .x(t + S), .t ∈ [T1 , T2 ], belongs to .Y (T1 + S, T2 + S). (A3) For each triplet of numbers .T1 < T2 < T3 , each .x1 ∈ Y (T1 , T2 ), and each .x2 ∈ Y (T2 , T3 ) satisfying .x1 (T2 ) = x2 (T2 ), the function .x3 : [T1 , T3 ] → X defined by © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0_4

51

52

4 Continuous-Time Dynamical Systems

x3 (t) = x1 (t), t ∈ [T1 , T2 ], x3 (t) = x2 (t), t ∈ (T2 , T3 ]

.

belongs to .Y (T1 , T3 ). For each pair of numbers .T1 < T2 , elements of the set .Y (T1 , T2 ) are called trajectories. Let .T1 ∈ R 1 . A function .x : [T1 , ∞) → X is a trajectory if for every .T2 > T1 the restriction of x to .[T1 , T2 ] belongs to .Y (T1 , T2 ). The set of all trajectories .x : [T1 , ∞) → X is denoted by .Y (T1 , ∞). Denote by .Y (R 1 ) = Y (−∞, ∞) the set of all functions .x : R 1 → X such that for each pair of numbers .T1 < T2 the restriction of x to .[T1 , T2 ] belongs to .Y (T1 , T2 ). The elements of .Y (R 1 ) are called trajectories too. θ ∈ X. Fix . Assume that .φ : X → R 1 is a continuous function such that for each pair of numbers .T1 < T2 , each .x ∈ Y (T1 , T2 ), and each .S1 , S2 ∈ [T1 , T2 ] satisfying .S1 < S2 , φ(x(S2 )) ≤ φ(x(S1 )).

.

(4.1)

Assume that .X0 is nonempty, closed subset of X such that for each .r > 0 the set X0 ∩ B( θ , r) is compact

.

(4.2)

and that the following assumptions hold: (A4) For each pair of numbers .T2 > T1 , each .M > 0, and each sequence .{xn }∞ n=1 ⊂ Y (T1 , T2 ) that satisfy .

lim sup{ρ(xn (t), X0 ∩ B( θ , M)) : t ∈ [T1 , T2 ]} = 0,

n→∞

there exists subsequence .{xnk }∞ k=1 that converges uniformly on .[T1 , T2 ] to a trajectory .x ∈ Y (T1 , T2 ) as .k → ∞. (A5) For each bounded, nonempty set B, there exists .M > 0, and for each . > 0, there exists a number .T (B, ) > 0 such that for each .T ≥ T (B, ) and each .x ∈ Y (0, T ) satisfying .x(0) ∈ B, ρ(x(t), X0 ∩ B( θ , M)) ≤ , t ∈ [T (B, ), T ].

.

In Sect. 4.2, we prove the following result. Proposition 4.1 The following properties are equivalent: (1) There exist a trajectory .x ∈ Y (0, ∞) and .M0 > 0 such that for every .t ≥ 0, x(t) ∈ X0 ∩ B( θ , M0 ).

.

(2) There exists .x ∈ Y (0, ∞).

4.1 Preliminaries and Main Results

53

(3) There exists .M > 0, and for each integer .k ≥ 1, there exists .xk ∈ Y (0, Tk ) such that .

lim Tk = ∞ and ρ( θ , xk (0)) ≤ M, k = 1, 2, . . . .

k→∞

In this chapter, in the sequel, we suppose that .Y (0, ∞) = ∅. Define Ω = {z ∈ X : for very  > 0 there exists x ∈ Y (0, ∞)

.

for which lim inf ρ(z, x(t)) ≤ }.

.

t→∞

(4.3)

Clearly, .Ω is a closed subset of the metric space .(X, ρ). In Sect. 4.3, we prove the following result. Proposition 4.2 .Ω = ∅ and .Ω ⊂ X0 . Assumption (A4) implies the following proposition. Proposition 4.3 For every .x ∈ Y (0, ∞), lim ρ(x(t), Ω) = 0.

.

t→∞

It is not difficult to see that the following result holds. Proposition 4.4 Assume that .B ⊂ X be a nonempty, closed set such that for every x ∈ Y (0, ∞),

.

.

lim ρ(x(t), B) = 0.

t→∞

Then .Ω ⊂ B. The following theorem will be proved in Sect. 4.4. Theorem 4.5 Let ., M be positive real numbers. Then there is a positive number T such that for every .x ∈ Y (0, T ) that satisfies ρ(x(0),  θ) ≤ M

.

the inequality .

inf{ρ(x(t), Ω) : t ∈ [0, T ]} ≤ 

holds. The next theorem is proved in Sect. 4.5.

54

4 Continuous-Time Dynamical Systems

Theorem 4.6 The following properties are equivalent: (1) If a bounded function .x ∈ Y (−∞, ∞) satisfies φ(x(t)) = φ(x(0))

.

for all .t ∈ R 1 , then the inclusion .x(t) ∈ Ω holds for all .t ∈ R 1 . (2) If a bounded function .x ∈ Y (−∞, ∞) satisfies x(t) ∈ X0 and φ(x(t)) = φ(x(0))

.

for all .t ∈ R 1 , then the inclusion .x(t) ∈ Ω holds for all real numbers t. (3) For each .M,  > 0, there exist .δ > 0 and .L ≥ 1 such that for each .T > 2L and each .x ∈ Y (0, T ) that satisfy B(x(0),  θ ) ≤ M and φ(x(0)) − φ(x(T )) ≤ δ

.

the inequality ρ(x(t), Ω) ≤ 

.

holds for all .t ∈ [L, T − L]. In Sect. 4.6, we prove the following turnpike result. Theorem 4.7 Assume that properties of Theorem 4.6 hold. Let ., M > 0. Then there exist natural numbers .L, Q such that for every number .T > L and every trajectory .x ∈ Y (0, T ) that satisfy ρ(x(0),  θ) ≤ M

.

q

q

there exist an integer .q ∈ [1, Q] and finite sequences of integers .{ai }i=1 , .{bi }i=1 ⊂ [0, T ] such that ai ≤ bi , i = 1, . . . , q, ai+1 > bi , i ∈ {1, . . . , q} \ {q},

.

ρ(x(t), Ω) ≤ , t ∈ [ai , bi ], i = 1, . . . , q,

.

and q

mes([0, T ] \ ∪i=1 [ai , bi ]) ≤ L.

.

4.2 Proof of Proposition 4.1 Evidently, (1) implies (2) and (2) implies (3). Assume that (3) holds. We show that this implies (1). There exist .M > 0 and .xn ∈ Y (0, n), .n = 1, 2, . . . , such that

4.3 Proof of Proposition 4.2

55

ρ( θ , xn (0)) ≤ M, , n = 1, 2, . . . .

.

(4.4)

By (4.10) and assumption (A5), there exists .M0 > 0 such that for every natural number k there exists an integer .n(k) > k such that for each integer .n ≥ n(k) and each .t ∈ [n(k), n], we have ρ(xn (t), X0 ∩ B( θ , M0 )) ≤ k −1 .

.

(4.5)

For each integer .k ≥ 1, define yk (t) = x2n(k) (t + n(k)), t ∈ [0, n(k)].

.

(4.6)

In view of (4.6), yk ∈ Y (0, n(k)), k = 1, 2, . . . .

.

Equations (4.5) and (4.6) imply that for each integer .k ≥ 1 and each .t ∈ [0, n(k)], ρ(yk (t), X0 ∩ B( θ , M0 )) = ρ(x2n(k) (t + n(k)), X0 ∩ B( θ , M0 )) ≤ k −1 .

.

(4.7)

In view of (4.7), for each integer .m ≥ 1, .

lim sup{ρ(yk (t), X0 ∩ B( θ , M0 )) : t ∈ [0, m]} = 0.

k→∞

(4.8)

By assumption (A4) and (4.8), extracting subsequences and using the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers .{kj }∞ j =1 and .x ∈ Y (0, ∞) such that for each integer .m ≥ 1, ykj (t) → x(t) as j → ∞ uniformly on [0, m].

.

Together with (4.8), this implies that x(t) ∈ X0 ∩ B( θ , M0 ), t ∈ [0, ∞)

.

and (1) holds. This completes the proof of Proposition 4.1.

4.3 Proof of Proposition 4.2 Let .x ∈ Y (0, ∞). Assumption (A5) implies that there exists .M0 > 0 and that for each integer .k ≥ 1 there exists an integer .n(k) > k such that ρ(x(t), X0 ∩ B( θ , M0 )) ≤ k −1 , t ∈ [n(k), ∞).

.

(4.9)

56

4 Continuous-Time Dynamical Systems

Equation (4.9) implies that the sequence .{x(n)}∞ n=1 (here n is a natural number) has θ , M0 ). Hence, a subsequence that converges to a point of .Ω ∩ B( Ω = ∅.

.

Let .z ∈ Ω and . > 0. There exists .x ∈ Y (0, ∞) such that .

lim inf ρ(z, (x(t)) ≤ .. t→∞

(4.10)

In view of (A5), .

lim ρ(x(t), X0 ) = 0.

t→∞

By the equation above and (4.10), ρ(z, X0 ) ≤ .

.

Since the set .X0 is closed and . is an arbitrary positive number, we conclude that z ∈ X0 and .Ω ⊂ X0 . Proposition 4.2 is proved.

.

4.4 Proof of Theorem 4.5 Assume that Theorem 4.5 does not hold. Then for every integer .n ≥ 1 there exists xn ∈ Y (0, n) such that

.

ρ( θ , xn (0)) ≤ M,

(4.11)

ρ(xn (t)), Ω) > , t ∈ [0, n].

(4.12)

.

.

Assumption (A5) implies that there exists .M1 > 0 such that for every integer .k ≥ 1 there exists an integer .n(k) ≥ k such that for each integer .n ≥ n(k) and each .t ∈ [n(k), n], ρ(xn (t), X0 ∩ B(θ, M1 )) ≤ k −1 .

.

(4.13)

For each integer .k ≥ 1, define yk (t) = x2n(k) (t + n(k)), t ∈ [0, n(k)].

.

(4.14)

It follows from (4.12) and (4.14) that for each integer .k ≥ 1 and each .t ∈ [0, n(k)], ρ(yk (t), Ω) > ,

.

(4.15)

4.5 Proof of Theorem 4.6

57

ρ((yk (t), X0 ∩ B( θ , M1 )) ≤ k −1 .

.

(4.16)

By assumption (A4) and (4.16), extracting subsequences and using the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers .{kj }∞ j =1 and .y ∈ Y (0, ∞) such that for each integer .m ≥ 1, ykj (t) → y(t) as j → ∞ uniformly on [0, m].

.

Together with (4.15) and (4.16), this implies that for every .t ∈ [0, ∞), y(t) ∈ X0 ∩ B( θ , M1 ) and ρ(y(t), Ω) ≥ .

.

On the other hand, by Proposition 4.3, .

lim ρ(y(t), Ω) = 0.

t→∞

The contradiction we have reached proves Theorem 4.5.

4.5 Proof of Theorem 4.6 Clearly, (1) implies (2) and (3) implies (1). Therefore, in order to prove the theorem, it is sufficient to show that (2) implies (3). Assume that (2) holds and (3) does not hold. Then there exist ., M > 0 such that for each integer .k ≥ 1 there exist Tk > 2k, xk ∈ Y (0, Tk )

(4.17)

ρ(xk (0),  θ ) ≤ M,

(4.18)

|φ(xk (0)) − φ(xk (Tk ))| ≤ k −1

(4.19)

sup{ρ(xk (t)), Ω) : t ∈ [k, Tk − k]} > .

(4.20)

.

such that .

.

and .

By (A5), (4.17), and (4.18), there exists .M1 > 0, and for each integer .p ≥ 1, there exists an integer .k(p) > p such that for each integer .k ≥ k(p) and each .t ∈ [k(p), Tk ] we have ρ(xk (t), X0 ∩ B(θ, M1 )) ≤ p−1 .

.

(4.21)

58

4 Continuous-Time Dynamical Systems

Let .p ≥ 1 be an integer. In view of (4.20), there exists τp ∈ [2k(p), T2k(p) − 2k(p)]

(4.22)

ρ(x2k(p) (τp ), Ω) > .

(4.23)

yp (t) = x2k(p) (t + τp ), t ∈ [−τp , T2k(p) − τp ].

(4.24)

.

such that .

Define .

Clearly, yp ∈ Y (−τp , T2k(p) − τp ).

.

Equations (4.23) and (4.24) imply that ρ(yp (0), Ω) = ρ(x2k(p) (τp ), Ω) > .

.

(4.25)

By (4.21), (4.22), and (4.24), for all .t ∈ [k(p) − τp , T2k(p) − τp ], ρ(yp (t), Ω ∩ B( θ , M1 )) = ρ(x2k(p) (t + τp ), Ω ∩ B( θ , M1 )) ≤ p−1 .

.

(4.26)

Equations (4.14) and (4.24) imply that φ(yp (−τp )) − φ(yp (T2k(p) − τp )) ≤ (2k(p))−1 ≤ (2p)−1 .

.

(4.27)

In view of (4.22) and (4.26), for each integer .m ≥ 1, .

lim sup{ρ(yp (t), X0 ∩ B( θ , M1 )) : t ∈ [−m, m]} = 0.

p→∞

(4.28)

By assumption (A4) and (4.28), extracting subsequences and using the diagonalization process, we obtain that there exists a strictly increasing sequence of natural 1 numbers .{pi }∞ i=1 and .x ∈ Y (R ) such that for each integer .m ≥ 1, ypi → x as i → ∞ uniformly on [−m, m].

.

(4.29)

It follows from (4.27) and (4.29) that for every .t ∈ R 1 , x(t) = lim ypi (t) ∈ X0 ∩ B( θ , M1 ),

(4.30)

φ(x(t)) = φ(x(0)).

(4.31)

.

i→∞ .

4.6 Proof of Theorem 4.7

59

Property (2) and Eqs. (4.30) and (4.31) imply that x(t) ∈ Ω, t ∈ R 1 .

.

Together with (4.30), this implies that for all sufficiently large natural numbers i, ρ(ypi (0), Ω) ≤ /2.

.

This contradicts (4.25). The contradiction we have reached proves that (3) holds and completes the proof of Theorem 4.6. Since bounded, closed subsets of .X0 are compact, it is not difficult to see that the following result is true. Lemma 4.8 Let .M1 > 0. Then there exist .M0 , r > 0 such that for each .x ∈ X satisfying B(x, r0 ) ∩ B( θ , M1 ) ∩ X0 =→ ∅

.

the inequality .|φ(x)| ≤ M0 holds.

4.6 Proof of Theorem 4.7 We may assume without loss of generality that . < 1 and .M > 1. Assumption (A5) and Lemma 4.8 imply that there exist .M1 > M and an integer .L1 ≥ 1 such that the following property holds: (a) For each .T ≥ L1 and each .x ∈ (0, T ) satisfying .ρ(x(0),  θ ) ≤ M, we have ρ(x(t), X0 ∩ B( θ , M1 )) ≤ 1, |φ(x(t))| ≤ M1

.

for all .t ∈ [L1 , n]. Property (3) of Theorem 4.6 implies there exist .δ ∈ (0, 1) and .L0 > 0 such that the following property holds: (b) For each .T > 2L0 and each .x ∈ Y (0, T ) that satisfy ρ(x(0),  θ ) ≤ M1 + 2 and φ(x(0)) − φ(x(T )) ≤ δ,

.

the inequality ρ(x(t), Ω) ≤ 

.

holds for all .t ∈ [L0 , T − L0 ].

60

4 Continuous-Time Dynamical Systems

Choose integers Q > 2M1 δ −1 + 2

.

(4.32)

and L > (4L0 + 4L1 + 8)(2δ −1 M1 + 2) + M + L1 + 4.

.

(4.33)

Suppose that .T > L and that .x ∈ Y (0, T ) satisfies ρ(x(0),  θ ) ≤ M.

.

(4.34)

Property (a), (4.33), and (4.34) imply that ρ(x(t),  θ )) ≤ M1 + 2,

(4.35)

|φ(x(t))| ≤ M1

(4.36)

.

.

for all .t ∈ [L1 , T ]. By induction, we define a strictly increasing finite sequence .ti ∈ [0, T ], .i = 0, . . . , q. Set t 0 = L1 .

.

If lim φ(x(t)) ≥ lim φ(x(t)) − δ,

.

t→T −

t→t0+

then set .t1 = T and complete to construct the sequence. Assume that lim φ(x(t)) < lim φ(x(t)) − δ.

.

t→T −

t→t0+

Set t1 = sup{t ∈ (t0 , T ] : φ(x(t)) ≥ lim φ(x(t)) − δ}.

.

t→t0+

It is not difficult to see that .t1 is well-defined and .

lim φ(x(t)) ≥ lim φ(x(t)) − δ,

t→t1−

t→t0+

4.6 Proof of Theorem 4.7

61 .

lim φ(x(t)) ≤ lim φ(x(t)) − δ.

t→t1+

t→t0+

If .t1 = T , then we complete to construct the sequence. Assume that .k ≥ 1 is an integer, that we defined a finite strictly increasing sequence .t0 , . . . , tk ∈ [0, T ] such that t 0 = L1 , t k ≤ T ,

.

and that for each .i ∈ {0, . . . , k − 1}, lim φ(x(t)) ≥ lim φ(x(t)) − δ,

.

− t→ti+1

t→ti+

lim φ(x(t)) ≤ lim φ(x(t)) − δ.

.

+ t→ti+1

t→ti+

(It is not difficult to see that the assumption is true with .k = 1.) If .tk = T , then we complete to construct the sequence. Assume that .tk < T . If .

lim φ(x(t)) ≥ lim φ(x(t)) − δ,

t→T −

t→tk+

then we set .tk+1 = T and complete to construct the sequence. Assume that .

lim φ(x(t)) < lim φ(x(t)) − δ.

t→T −

t→tk+

Set tk+1 = sup{t ∈ (tk , T ] : φ(x(t)) ≥ lim φ(x(t)) − δ}.

.

t→tk+

It is not difficult to see that .tk+1 is well-defined, .

.

lim φ(x(t)) ≥ lim φ(x(t)) − δ,

− t→tk+1

t→tk+

lim φ(x(t)) ≤ lim φ(x(t)) − δ.

+ t→tk+1

t→tk+

Thus, the assumption made for k is true for .k + 1 too. Therefore by induction, we constructed the strictly increasing finite sequence .ti ∈ [0, T ], .i = 0, . . . , q, such that t 0 = L1 , t q = T

.

(4.37)

62

4 Continuous-Time Dynamical Systems

and that for every i satisfying .0 ≤ i < q − 1, .

lim φ(x(t)) ≤ lim φ(x(t)) − δ ≤ lim φ(x(t)),

+ t→ti+1

t→ti+

+ t→ti+1

lim φ(x(t)) ≥ lim φ(x(t)) − δ.

.

t→T −

+ t→tq−1

(4.38) (4.39)

By (4.33) and (4.36)–(4.38), 2M1 ≥ M1 + M ≥ lim φ(x(t)) − lim φ(x(t))

.

t→t0+

.

≥



+ t→tq−1

{ lim φ(x(t)) − lim φ(x(t)) : t→ti+

+ t→ti+1

i is an integer, 0 ≤ i ≤ q − 2} ≥ δ(q − 1)

.

and q ≤ 1 + 2δ −1 M1 < Q.

.

(4.40)

Define E = {i ∈ {0, . . . , q − 1} : ti+1 − ti ≥ 2L0 + 4}.

.

(4.41)

Let i ∈ E.

(4.42)

ti+1 − 1 − (ti + 1) ≥ 2L0 + 2.

(4.43)

.

By (4.41) and (4.42), .

Equations (4.37)–(4.39) imply that φ(x(ti + 1)) − φ(x(ti+1 − 1)) ≤ δ.

.

(4.44)

Equations (4.35), (4.43), (4.44) and property (b) applied to the restriction of x to the interval .[ti + 1, ti+1 − 1] imply that ρ(x(t), Ω) ≤ , t ∈ [ti + L0 + 1, ti+1 − 1 − L0 ].

.

(4.45)

Set ai = ti + L0 + 1, bi = ti+1 − L0 − 1.

.

(4.46)

4.6 Proof of Theorem 4.7

63

By (4.37) and (4.46), [0, T ] \ ∪i∈E [ai , bi ]

.

.

.

⊂ ∪{[ti , ti+1 ] : i ∈ {0, . . . , q − 1} \ E} ∪ [0, L1 ]

∪ {[ti , ti + L0 + 1] ∪ [ti+1 − L0 − 1, ti+1 ] : i ∈ E}.

It follows from (4.33), (4.41), (4.44), and (4.47) that mes([0, T ] \ ∪i∈E [ai , bi ])

.

.

≤ q(2L0 + 4) + (2L0 + 2)q + L1 = q(4L0 + 6) + L1 .

≤ (4L0 + 6)(1 + 2M1 δ −1 ) + L1 < L.

Theorem 4.7 is proved.

(4.47)

Chapter 5

General Dynamical Systems with a Lyapunov Function

In this chapter we develop a turnpike theory for a class of general dynamical systems in a metric space with a Lyapunov function. This class of dynamical systems contains as particular cases the classes of dynamical systems studied in Chaps. 3 and 4. We obtain several turnpike results and show that the turnpike phenomenon is stable under small perturbations of objective functions.

5.1 Preliminaries and Two Turnpike Results Assume that (X, ρ) is a metric space. For each x ∈ X and each r > 0, set B(x, r) = {y ∈ X : ρ(x, y) ≤ r} and B 0 (x, r) = {y ∈ X : ρ(x, y) < r}. For each x ∈ X and each pair of nonempty sets A, B ⊂ X, define ρ(x, B) = inf{ρ(x, y) : y ∈ B} and dist(A, B) = sup{ρ(x, B) : x ∈ A}. Let Γ be either R 1 or the set of all integers Z = {0, ±1, ±2, . . . }. Assume that for each pair T1 , T2 ∈ Γ satisfying T1 < T2 , Y (T1 , T2 ) is a nonempty set of functions x : [T1 , T2 ] ∩ Γ → X such that the following properties hold: (A1) For each pair T1 , T2 ∈ Γ , each pair S1 , S2 ∈ Γ satisfying T1 ≤ S1 < S2 ≤ T2 , and each x ∈ Y (T1 , T2 ), the restriction of x to [S1 , S2 ] belongs to Y (S1 , S2 ).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0_5

65

66

5 General Dynamical Systems with a Lyapunov Function

(A2) For each pair T1 , T2 ∈ Γ satisfying T1 < T2 , each S ∈ Γ , and each x ∈ Y (T1 , T2 ), the function x(t +S), t ∈ [T1 , T2 ]∩Γ belongs to Y (T1 +S, T2 +S). (A3) For each triplet T1 , T2 , T3 ∈ Γ satisfying T1 < T2 < T3 , each x1 ∈ Y (T1 , T2 ), and each x2 ∈ Y (T2 , T3 ) satisfying x1 (T2 ) = x2 (T2 ), the function x3 : [T1 , T3 ] ∩ Γ → X defined by x3 (t) = x1 (t), t ∈ [T1 , T2 ] ∩ Γ, x3 (t) = x2 (t), t ∈ (T2 , T3 ] ∩ Γ belongs to Y (T1 , T3 ). If Γ = Z, then we set A = {(y, z) ∈ X × X : there is x ∈ Y (0, 1) such that x(0) = y, x(1) = z}}, define a(x) = {y ∈ X : (x, y) ∈ A}, x ∈ X, and assume that A is closed in the metric space X × X. For each pair of numbers T1 , T2 ∈ Γ satisfying T1 < T2 , elements of the set Y (T1 , T2 ) are called trajectories. Let T1 ∈ Γ . A function x : [T1 , ∞) ∩ Γ → X is a trajectory if for every T2 ∈ (T1 , ∞) ∩ Γ the restriction of x to [T1 , T2 ] ∩ Γ belongs to Y (T1 , T2 ). The set of all trajectories x : [T1 , ∞) ∩ Γ → X is denoted by Y (T1 , ∞). Denote by Y (Γ ) the set of all functions x : Γ → X such that for each pair of numbers T1 , T2 ∈ Γ satisfying T1 < T2 the restriction of x to [T1 , T2 ] ∩ Γ belongs to Y (T1 , T2 ). The elements of Y (Γ ) are called trajectories too. θ ∈ X. Fix  Assume that φ : X → R 1 is a continuous function such that for each pair of numbers T1 ∈ Γ , T2 ∈ (T1 , ∞) ∩ Γ , each x ∈ Y (T1 , T2 ), and each S1 , S2 ∈ [T1 , T2 ] ∩ Γ satisfying S1 < S2 , φ(x(S2 )) ≤ φ(x(S1 )).

(5.1)

Assume that X0 is nonempty, closed subset of X such that for each r > 0 the set X0 ∩ B( θ , r) is compact

(5.2)

and that the following assumptions hold: (A4) For each pair of numbers T1 , T2 ∈ Γ satisfying T2 > T1 , each M > 0, and each sequence {xn }∞ n=1 ⊂ Y (T1 , T2 ) which satisfies lim sup{ρ(xn (t), X0 ∩ B( θ , M)) : t ∈ [T1 , T2 ] ∩ Γ } = 0,

n→∞

5.1 Preliminaries and Two Turnpike Results

67

there exists a subsequence {xnk }∞ k=1 which converges uniformly on [T1 , T2 ] ∩ Γ to a trajectory x ∈ Y (T1 , T2 ) as k → ∞. (A5) For each bounded, nonempty set B, there exists M > 0, and for each  > 0 there exists a positive number T (B, ) ∈ Γ such that for each T ∈ Γ ∩ [T (B, ), ∞) and each x ∈ Y (0, T ) satisfying x(0) ∈ B, ρ(x(t), X0 ∩ B( θ , M)) ≤ , t ∈ [T (B, ), T ] ∩ Γ. (Note that in the case Γ = Z (A4) follows from the compactness of the set X0 ∩ B( θ , M) and the closedness of A.) We suppose that Y (0, ∞) = ∅. Note that if Γ = Z, then all the assumptions made in Chap. 3 hold, and therefore all the results of Chap. 3 are true. If Γ = R 1 , then all the assumptions made in Chap. 4 hold, and therefore all the results of Chap. 4 are true. Here we consider a general and abstract dynamical system and treat discrete-time and continuous-time case simultaneously. Of course, the example of Sect. 2.8 and its discrete-time analog mentioned in Sect. 3.1 are examples of this system. There are also infinite dimensional examples considered in [18, 82]. Define Ω = {z ∈ X : for very  > 0 there exists x ∈ Y (0, ∞) for which lim inf ρ(z, x(t)) ≤ }. t→∞

(5.3)

Clearly, Ω is a closed subset of the metric space (X, ρ). By the results of Chaps. 3 and 4, Ω = ∅, Ω ⊂ X0 ,

(5.4)

lim ρ(x(t), Ω) = 0.

(5.5)

and for each x ∈ Y (0, ∞), t→∞

We also assume that the following assumption holds: (A6) If x ∈ Y (Γ ) is bounded and for all t ∈ Γ , x(t) ∈ X0 and φ(x(t)) = φ(x(0)), then x(t) ∈ Ω for every t ∈ Γ . If Γ = Z and E ⊂ Γ , then we set mes(E) = Card(E). For each x ∈ X, set

68

5 General Dynamical Systems with a Lyapunov Function

π(x) = lim sup{φ(x(T )) : x ∈ Y (0, T ) and x(0) = x}. T →∞

(5.6)

(We assume that the supremum over an empty set is −∞.) It is clear that π(x) ≤ φ(x), x ∈ X.

(5.7)

Evidently, the following result is true: Proposition 5.1 Let T > 0 and x ∈ Y (0, T ). Then π(x(s)) ≤ π(x(0)) for all s ∈ [0, T ] ∩ Γ . The following result is proved in Sect. 5.3. Proposition 5.2 Let x ∈ X. The value π(x) is finite if and only if for every T ∈ Γ ∩ (0, ∞) there exists y ∈ Y (0, T ) satisfying y(0) = x. In Sect. 5.4, we prove the following result. Proposition 5.3 Let M0 > 0. Then there exists M > 0 such that for each x ∈ B( θ , M0 ) satisfying π(x) > −∞, B( θ , M) ∩ Ω ∩ φ −1 (π(x)) = ∅. The next theorem is our first turnpike result. It is proved in Sect. 5.5. Theorem 5.4 Let  > 0 and  x ∈ X satisfy π( x ) > −∞. Then there exist δ > 0 and L > 0 such that for every T ∈ Γ ∩ (2L, ∞) and every x ∈ Y (0, T ) which satisfies x(0) =  x and φ(x(T )) ≥ π( x) − δ the inequality ρ(x(t), Ω) ≤  holds for all t ∈ [L, T − L] ∩ Γ. Clearly, Theorem 5.4 establishes the turnpike property for approximate optimal trajectories which have a fixed starting point with respect to the objective function φ. The turnpike is the set Ω and δ and L depend on  and the initial point  x. For each x ∈ X satisfying π(x) > −∞, set F (x) = Ω ∩ φ −1 (π(x)), which is nonempty in view of Proposition 5.3. The following result which is a generalization of the previous theorem is proved in Sect. 5.7. According to Theorem 5.4, for all t ∈ [L, T − L] ∩ Γ , the distance

5.2 Two Auxiliary Results

69

between x(t) and the turnpike Ω does not exceed . According to our next result, for all t ∈ [L, T − L] ∩ Γ , the distance between x(t) and the set F ( x ) ∩ B( θ , M) does not exceed , where a positive constant M depends on  and the initial point  x and F ( x ) is the set of all points of the turnpike Ω where the value of the Lyapunov function φ is π( x ). Theorem 5.5 Let  > 0 and  x ∈ X satisfy π( x ) > −∞. Then there exist M, L, δ > 0 such that for every T ∈ Γ ∩ (2L, ∞) and every x ∈ Y (0, T ) which satisfies x(0) =  x and φ(x(T )) ≥ π( x) − δ the inequality ρ(x(t), F ( x ) ∩ B( θ , M)) ≤  holds for all t ∈ [L, T − L] ∩ Γ. The next two results are proved in Sect. 5.8. Proposition 5.6 Assume that X0 = X,  x ∈ X satisfies π( x ) > −∞, and that for each S ∈ Γ , the set {z(t) : z ∈ Y (0, S), z(0) =  x , t ∈ [0, S]} is bounded. Then there exists x ∈ Y (0, ∞) such that x(0) =  x and π( x ) = lim φ(x(t)). t→∞

Proposition 5.7 Assume that X0 = X, x ∈ X, and there exists r > 0 such that for each S ∈ Γ ∩ (0, ∞) the set {z(t) : z ∈ Y (0, S), ρ(z(0), x) ≤ r, t ∈ [0, S]} is bounded. Then the function π : X → R 1 ∪ {−∞} is upper semicontinuous at x.

5.2 Two Auxiliary Results Proposition 5.8 Let x ∈ Ω. Then there exists a bounded function y ∈ Y (Γ ) such that y(0) = x, φ(y(t)) = φ(y(0)), t ∈ Γ and y(t) ∈ Ω, t ∈ Γ .

70

5 General Dynamical Systems with a Lyapunov Function

Proof Let k ≥ 1 be an integer. There exists δk ∈ (0, k −1 ) such that for each z ∈ B(x, δk ), we have |φ(x) − φ(z)| ≤ k −1 .

(5.9)

There exists xk ∈ Y (0, ∞) such that lim inf ρ(xk (t), x) < δk . t→∞

(5.10)

In view of (5.9) and (5.10), we may assume without loss of generality that ρ(xk (0), x) < δk ,

(5.11)

|φ(xk (0)) − φ(x)| ≤ k −1 .

(5.12)

By (5.1), the function φ(xk (t)), t ∈ Γ ∩ [0, ∞) is decreasing. Together with (5.9), (5.10), and (5.12), this implies that |φ(x) − φ(xk (t))| ≤ k −1 , t ∈ [0, ∞) ∩ Γ.

(5.13)

In view of (5.10), there exists τk ∈ Γ ∩ (4k, ∞)

(5.14)

ρ(xk (τk ), x) < δk .

(5.15)

yk (t) = xk (t + τk ).

(5.16)

ρ(yk (−τk ), x) < δk , ρ(yk (0), x) < δk .

(5.17)

such that

For each t ∈ [−τk , ∞) ∩ Γ , set

By (5.11) and (5.16),

Equations (5.13) and (5.16) imply that |φ(yk (t)) − φ(x)| ≤ k −1 for all t ∈ Γ ∩ [−τk , ∞).

(5.18)

5.2 Two Auxiliary Results

71

Assumption (A5) implies that there exists M > 0 and for each integer p ≥ 1, there exists an integer k(p) ≥ p such that for each t ∈ Γ ∩ [k(p), ∞), ρ(xk (t), X0 ∩ B( θ , M)) ≤ p−1 for all integers k ≥ 1.

(5.19)

Let p ≥ 1 be an integer. By (5.14), (5.16), and (5. 19), for each t ∈ Γ and each integer k ≥ k(p) + |t|, ρ(yk (t), X0 ∩ B( θ , M)) = ρ(xk (t + τk ), X0 ∩ B( θ , M)) ≤ p−1 .

(5.20)

By (5.20), for each integer m ≥ 1, lim sup{ρ(yk (t), X0 ∩ B( θ , M)) : t ∈ Γ ∩ [−m, m]} = 0.

k→∞

(5.21)

By assumption (A4) and (5.21), extracting subsequences, and using the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers {kj }∞ j =1 and y ∈ Y (Γ ) such that for each integer m ≥ 1, ykj → y as j → ∞ uniformly on [−m, m] ∩ Γ.

(5.22)

It follows from (5.17), (5.18), (5.21), and (5.22) that y(0) = x, y(t) ∈ X0 ∩ B( θ , M), t ∈ Γ, φ(y(t)) = φ(y(0)) = φ(x), t ∈ Γ. Together with assumption (A6), this implies that y(t) ∈ Ω, t ∈ Γ. Proposition 5.8 is proved. Corollary 5.9 Let x ∈ Ω. Then φ(x) = π(x). Proposition 5.10 Let x ∈ Y (0, ∞) satisfy x(0) ∈ Ω and φ(x(t)) = φ(x(0)), t ∈ [0, ∞) ∩ Γ. Then x(t) ∈ Ω for all t ∈ [0, ∞) ∩ Γ. Proof Proposition 5.8 implies that there exists a bounded function y ∈ Y (Γ ) such that y(0) = x(0), φ(y(t)) = φ(x(0)), t ∈ Γ.

72

5 General Dynamical Systems with a Lyapunov Function

Set x(t) = y(t), t ∈ Γ ∩ (−∞, 0). Clearly, x ∈ Y (Γ ) and y(Γ ) is bounded, φ(x(t)) = φ(x(0)), t ∈ Γ. Since y is bounded, it follows from (A5) that x is bounded on Γ too. Proposition 5.8 implies that x(t) ∈ Γ , t ∈ Γ . Proposition 5.10 is proved.

5.3 Proof of Proposition 5.2 Assume that for every integer n ≥ 1, there exists xn ∈ Y (0, n) such that xn (0) = x. It is sufficient to show that π(x) is finite. By (A5), there exists M > 0, and for each integer p ≥ 1, there exists an integer n(p) ≥ p such that for each integer n ≥ n(p) and each t ∈ Γ ∩ [n(p), n], ρ(xn (t), B( θ , M)) ≤ p−1 .

(5.23)

Lemma 4.8 implies that there exist M0 , r0 > 0 such that for each ξ ∈ X satisfying B(ξ, r0 ) ∩ B( θ , M) ∩ X0 = ∅, we have |φ(ξ )| ≤ M0 .

(5.24)

It follows from (5.23) and (5.24) that for each integer p satisfying p−1 < r0 /2, each integer n ≥ n(p), and each t ∈ Γ ∩ [n(p), n], |φ(xn (t))| ≤ M0 and sup{φ(y) : y ∈ Y (0, n), y(0) = x} ≥ −M0 . Thus π(x) ≥ −M0 . On the other hand, π(x) ≤ φ(x). Proposition 5.2 is proved.

5.4 Proof of Proposition 5.3

73

5.4 Proof of Proposition 5.3 Assumption (A5) implies that there exists M1 > M0 , and for each integer k ≥ 1, there exists an integer n(k) > k such that the following property holds: (i) For each integer n ≥ n(k), each y ∈ Y (0, n) satisfying ρ(y(0),  θ ) ≤ M0 , and each t ∈ Γ ∩ [n(k), n], θ , M)) ≤ k −1 . ρ(y(t), X0 ∩ B( Let x ∈ B( θ , M0 ) and π(x) be finite. Proposition 5.2 implies that for each integer n ≥ 1 there exists xn ∈ Y (0, n) such that xn (0) = x,

(5.25)

φ(xn (n)) ≥ sup{φ(ξ(n)) : ξ ∈ Y (0, n), ξ(0) = x} − n−1 ≥ π(x) − n−1 .

(5.26)

For each integer k ≥ 1, define yk ∈ Y (0, n(k)) by yk (t) = x2n(k) (t + n(k)), t ∈ [0, n(k)] ∩ Γ.

(5.27)

Property (i) and (5.27) imply that for each integer k ≥ 1 and each t ∈ [0, n(k)] ∩ Γ , θ , M)) = ρ(x2n(k) (t + n(k)), X0 ∩ B( θ , M)) ≤ k −1 . ρ(yk (t), X0 ∩ B(

(5.28)

In view of (28), for each integer m ≥ 1, θ , M)) : t ∈ Γ ∩ [0, m]} = 0. lim sup{ρ(yk (t), X0 ∩ B(

k→∞

(5.29)

By assumption (A4) and (5.29), extracting subsequences, and using the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers {ki }∞ i=1 and y ∈ Y (0, ∞) such that for each integer m ≥ 1, ykj → y as j → ∞ uniformly on [0, m] ∩ Γ.

(5.30)

It follows from (5.27) and (5.30) that for every t ∈ [0, ∞) ∩ Γ , y(t) = lim ykj (t) = lim x2n(kj ) (t + n(kj )). j →∞

j →∞

(5.31)

74

5 General Dynamical Systems with a Lyapunov Function

Let t ∈ [0, ∞) ∩ Γ . By (5.1) and (5.26), for each integer j ≥ 1 and each t ∈ [0, n(kj )] ∩ Γ , φ(x2n(kj ) (t + n(kj )) ≥ φ(x2n(kj ) (2n(kj ))) ≥ π(x) − (2n(kj ))−1 .

(5.32)

In view of (5.31) and (5.32), φ(y(t)) ≥ π(x).

(5.33)

By (5.6), for each integer j ≥ 1 and each t ∈ [0, n(kj )] ∩ Γ , φ(x2n(kj ) (t + n(kj )) ≤ sup{φ(ξ ) : ξ ∈ Y (0, t + n(kj )), ξ(0) = x} → π(x) as j → ∞ and lim sup φ(x2n(kj ) (t + n(kj )) ≤ π(x). j →∞

Together with (5.31), this implies that φ(y(t)) ≤ π(x). Combined with (5.33), this implies that φ(y(t)) = π(x), t ∈ [0, ∞) ∩ Γ.

(5.34)

In view of (5.29) and (5.30), y(t) ∈ X0 ∩ B( θ , M), t ∈ [0, ∞) ∩ Γ.

(5.35)

In view of (5.35), the sequence {y(n)}∞ n=1 has a limit point ξ ∈ Ω ∩ X0 ∩ B( θ , M). By (5.34), φ(ξ ) = π(x). Proposition 5.3 is proved.

5.5 Proof of Theorem 5.4 Assume that Theorem 5.4 does not hold. Then, for each natural number k, there exist Tk ∈ (2k, ∞) ∩ Γ, xk ∈ Y (0, Tk )

(5.36)

5.5 Proof of Theorem 5.4

75

such that x, xk (0) = 

(5.37)

x ) − k −1 , φ(xk (Tk )) ≥ π(

(5.38)

sup{ρ(xk (t), Ω) : t ∈ [k, Tk − k] ∩ Γ } > .

(5.39)

and

In view of (5.39), for each integer k ≥ 1, there exists τk ∈ [k, Tk − k] ∩ Γ

(5.40)

ρ(xk (τk ), Ω) > .

(5.41)

such that

Equations (5.1), (5.37), and (5.38) imply that for each integer k ≥ 1 and each t ∈ [0, Tk ] ∩ Γ , x ) − 1/k. φ( x ) ≥ φ(xk (t)) ≥ π(

(5.42)

Let k ≥ 1 be an integer. Set yk (t) = xk (t + τk ), t ∈ [−τk , Tk − τk ] ∩ Γ.

(5.43)

In view of (5.36), (5.41), and (5.43), yk ∈ Y (−τk , Tk − τk ), ρ(yk (0), Ω) > .

(5.44)

Assumption (A5) implies that there exists M0 > 0 such that the following property holds: (i) For each integer p ≥ 1, there exists an integer k(p) ≥ p such that for each integer k ≥ k(p) and each t ∈ Γ ∩ [k(p), Tk ], θ , M0 )) ≤ p−1 . ρ(xk (t), X0 ∩ B( Let p ≥ 1 be an integer. Property (i) and (5.43) imply that for each integer k ≥ k(p) and each t ∈ [k(p) − τk , Tk − τk ] ∩ Γ θ , M0 )) = ρ(xk (t + τk ), X0 ∩ B( θ , M0 )) ≤ p −1 . ρ(yk (t), X0 ∩ B(

(5.45)

76

5 General Dynamical Systems with a Lyapunov Function

It follows from (5.40) and (5.45) that for each t ∈ Γ and each integer k ≥ k(p)+|t|, ρ(yk (t), X0 ∩ B( θ , M0 )) ≤ p−1 . This implies that for each integer m ≥ 1, lim sup{ρ(yk (t), X0 ∩ B( θ , M0 )) : t ∈ [−m, m] ∩ Γ } = 0.

k→∞

(5.46)

By assumption (A4) and (5.46), extracting subsequences, and using the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers {kj }∞ j =1 and y ∈ Y (Γ ) such that for each t ∈ Γ , y(t) = lim ykj (t) ∈ X0 ∩ B( θ , M0 ) j →∞

(5.47)

uniformly on [−m, m]∩Γ for every integer m ≥ 1. It follows from (5.44) and (5.47) that ρ(y(0), Ω) ≥ .

(5.48)

Let t ∈ Γ . Equations (5.38), (5.40), (5.43), and (5.47) imply that φ(y(t)) = lim φ(ykj (t)) = lim φ(xkj (t + τkj )) j →∞

j →∞

≥ lim (π( x ) − kj−1 ) = π( x ). j →∞

On the other hand, Eqs. (5.6), (5.43), and (5.47) imply that φ(y(t)) = lim φ(xkj (t + τkj )) j →∞

= lim sup{φ(z(t + kj )) : z ∈ Y (0, t + kj ), z(0) =  x } = π( x ). j →∞

In view of the equation above, φ(x(t)) = π( x ), t ∈ Γ.

(5.49)

Assumption (A6), (5.47), and (5.49) imply that y(t) ∈ Ω, t ∈ Γ , and in particular y(0) ∈ Ω. This contradicts (5.48). The contradiction we have reached completes the proof of Theorem 5.4.

5.6 An Auxiliary Result for Theorem 5.5

77

5.6 An Auxiliary Result for Theorem 5.5 Lemma 5.11 Assume that K ⊂ U ⊂ X, where the set U is open and the set K is nonempty and compact, g : U → R 1 , every point of K is a continuity point of g, and that  > 0. Then there exists δ > 0 such that for each z1 , z2 ∈ X which satisfy ρ(z1 , z2 ) ≤ δ, ρ(zi , K) ≤ δ, i = 1, 2, the inclusion z1 , z2 ∈ U and the inequality |g(z1 ) − g(z2 )| ≤  hold. Proof For each x ∈ K, there exists rx ∈ (0, 1) such that B(x, 4rx ) ⊂ U,

(5.50)

|g(z1 ) − g(z2 )| ≤ /4 for all z1 , z2 ∈ B(x, 4rx ).

(5.51)

In view of (5.50), K ⊂ ∪{B 0 (x, rx ) : x ∈ K}.

(5.52)

Since the set K is compact, (5.52) implies that there exist an integer q ≥ 1 and xi ∈ K, i = 1, . . . , q,

(5.53)

such that q

K ⊂ ∪i=1 B 0 (xi , rxi ).

(5.54)

δ = 2−1 min{rxi : i = 1, . . . , q}.

(5.55)

Set

Assume that z1 , z2 ∈ X satisfy ρ(z1 , z2 ) ≤ δ, ρ(zi , K) ≤ δ, i = 1, 2.

(5.56)

In view of (5.56), there exists y ∈ K such that ρ(z1 , y) ≤ δ, ρ(z2 , y) ≤ 2δ.

(5.57)

By (5.54), there exists j ∈ {1, . . . , q} such that ρ(y, xj ) ≤ rxj .

(5.58)

78

5 General Dynamical Systems with a Lyapunov Function

It follows from (5.55), (5.57), and (5.58) that for i = 1, 2, ρ(zi , xj ) ≤ ρ(zi , y) + ρ(y, xj ) ≤ 2δ + rxj ≤ 2rxj .

(5.59)

By (5.50), (5.51), and (5.59), z1 , z2 ∈ U, |g(z1 ) − g(z2 )| ≤ /4. Lemma 5.11 is proved.

5.7 Proof of Theorem 5.5 We may assume that  < 1. Assumption (A5) implies that there exists M > 2, and for each integer p ≥ 1, there exists an integer Lp > p such that the following property holds: (i) For each T ∈ Γ ∩ [Lp , ∞) and each x ∈ Y (0, T ) satisfying x(0) =  x , we have ρ(x(t), X0 ∩ B( θ , M − 2)) ≤ p−1 , t ∈ Γ ∩ [Lp , T ]. Proposition 5.3 implies that F ( x ) = Ω ∩ φ −1 (π( x )) = ∅.

(5.60)

We show that there exists δ1 > 0 such that the following property holds: (ii) For each z ∈ Ω ∩ B( θ , M) satisfying |φ(z) − π( x )| ≤ 2δ1 , the inequality ρ(z, F ( x ) ∩ B( θ , M)) ≤ /4 is true. Assume the contrary. Then there exists a sequence  {zi }∞ i=1 ⊂ Ω ∩ B(θ , M),

(5.61)

lim φ(zi ) = π( x ),

(5.62)

i→∞

5.7 Proof of Theorem 5.5

79

ρ(zi , F ( x ) ∩ B( θ , M)) > /4.

(5.63)

Since the set Ω ∩ B( θ , M + 1) is compact, we can assume without loss of generality that there exists limi→∞ zi . Equations (5.61) and (5.62) imply that lim zi ∈ Ω ∩ B( θ , M), φ( lim zi ) = π( x ).

i→∞

i→∞

(5.64)

By (5.63), ρ( lim zi , F ( x ) ∩ B( θ , M)) ≥ /4. i→∞

This contradicts (5.64). The contradiction we have reached proves that there exists δ1 > 0 such that property (ii) holds. Lemma 5.11 implies that there exists 1 ∈ (0, /4) such that the following property holds: (iii) |φ(z1 ) − φ(z2 )| ≤ δ1 /4 for each z1 , z2 ∈ X which satisfy ρ(z1 , z2 ) ≤ 41 , ρ(zi , Ω ∩ B( θ , M + 1)) ≤ 41 , i = 1, 2. By definition (5.6), there exists a natural number L0 such that | sup{φ(z(L0 )) : z ∈ Y (0, L0 ) and z(0) =  x } − π( x )| ≤ δ1 /2.

(5.65)

Theorem 5.4 implies that there exist δ ∈ (0, δ1 ) and an integer L > 2L0 + 2L1 such that the following property holds: (iv) For every T ∈ Γ ∩ (2L, ∞) and every x ∈ Y (0, T ) which satisfies x(0) =  x and φ(x(T )) ≥ π( x ) − δ, the inequality ρ(x(t), Ω) ≤ 1 holds for all t ∈ [L, T − L] ∩ Γ. Assume that T ∈ Γ ∩ (2L, ∞), x ∈ Y (0, T ),

(5.66)

x(0) =  x and φ(x(T )) ≥ π( x ) − δ.

(5.67)

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5 General Dynamical Systems with a Lyapunov Function

Property (i) (with p = 1), (5.66), and (5.67) imply that ρ(x(t), X0 ∩ B( θ , M − 2)) ≤ 1, t ∈ Γ ∩ [L1 , T ].

(5.68)

Property (iv), (5.67), and (5.68) imply that ρ(x(t), Ω) ≤ 1 , t ∈ [L, T − L] ∩ Γ.

(5.69)

t ∈ Γ ∩ [L, T − L].

(5.70)

Assume that

In view of (5.69), there exists z∈Ω

(5.71)

ρ(x(t), z) ≤ 21 .

(5.72)

such that

By (5.1), (5.67), and the relation L0 < L ≤ t ≤ T , φ(x(t)) ≥ φ(x(T )) ≥ π( x ) − δ ≥ π( x ) − δ1 .

(5.73)

In view of (5.1), (5.65), and the relation L0 < L ≤ t ≤ T , φ(x(t)) ≤ φ(x(L0 )) ≤ π( x ) + δ1 /2. Together with (5.73), this implies that |φ(x(t)) − π( x )| ≤ δ1 .

(5.74)

It follows from (5.68), (5.70), and (5.72) that x(t) ∈ B( θ , M − 1)

(5.75)

z ∈ Ω ∩ B( θ , M).

(5.76)

and

Property (iii), (5.72), and (5.76) imply that |φ(x(t)) − φ(z)| ≤ δ1 /4.

(5.77)

5.8 Proofs of Propositions 5.6 and 5.7

81

By (5.74) and (5.77), |φ(z) − π( x )| ≤ |φ(z) − φ(x(t))| + |φ(x(t)) − π( x )| ≤ 3δ1 /2.

(5.78)

Property (ii), (5.76), and (5.78) imply that ρ(z, F ( x ) ∩ B( θ , M)) ≤ /4.

(5.79)

It follows from (5.72) and (5.79) that ρ(x(t), F ( x ) ∩ B( θ , M)) ≤ /4 + 21 < /2. Theorem 5.5 is proved.

5.8 Proofs of Propositions 5.6 and 5.7 Proof of Proposition 5.6 For each integer n ≥ 1, there exists xn ∈ Y (0, n) such that xn (0) =  x, φ(xn (n)) ≥ sup{φ(z(n)) : z ∈ Y (0, n), z(0) =  x } − n−1 .

(5.79) (5.80)

Assumption (A5) implies that there exist M0 > 0 and an integer L0 > 0 such that for each integer n ≥ L0 , ρ(xn (t),  θ ) ≤ M0 , t ∈ [L0 , n] ∩ Γ.

(5.81)

By our assumptions, there exists M1 > M0 such that {xn (t) : t ∈ [0, L0 ] ∩ Γ, n = 1, 2, . . . } ⊂ B( θ , M1 ).

(5.82)

By assumption (A4), (5.81), and (5.82), extracting subsequences, and using the diagonalization process, we obtain that there exist a strictly increasing sequence of natural numbers {nj }∞ j =1 and x ∈ Y (0, ∞) such that for each integer m ≥ 1, xnj → x as j → ∞ uniformly on [0, m] ∩ Γ.

(5.83)

It follows from (5.79) and (5.83) that x(0) =  x.

(5.84)

82

5 General Dynamical Systems with a Lyapunov Function

In view of (5.84), lim φ(x(t)) ≤ π(x).

t→∞

(5.85)

It follows from (5.1), (5.80), and (5.83) that for each t ∈ Γ , φ(x(t)) = lim φ(xnj (t)) ≥ lim sup φ(xnj (nj )) j →∞

j →∞

≥ lim (π( x ) − n−1 x ). j ) = π( j →∞

Combined with (5.85), this implies that lim φ(x(t)) = π(x).

t→∞

Proposition 5.6 is proved. Proof of Proposition 5.7 Assume that {xn }∞ n=1 ⊂ X and lim xn = x.

n→∞

(5.86)

We show that π(x) ≥ lim sup π(xn ). n→∞

We may assume without loss of generality that there exists lim π(xn ) > −∞

(5.87)

π(xn ) > −∞ for any integer n ≥ 1.

(5.88)

n→∞

and

Assumption (A5) and (5.86) imply that there exist M0 > 0 and an integer L0 > 0 such that the following property holds: (a) For each integer n ≥ 1, each T ∈ Γ ∩[L0 , ∞), and each y ∈ Y (0, T ) satisfying y(0) = xn , ρ(y(t),  θ ) ≤ M0 , t ∈ [L0 , T ] ∩ Γ. Let r > 0 be as guaranteed by the assumptions of the proposition. In view of (5.86), we may assume without loss of generality that

5.8 Proofs of Propositions 5.6 and 5.7

83

ρ(xn ,  x ) ≤ r, n = 1, 2, . . . .

(5.89)

It follows from (5.88), (5.89), the assumptions of our proposition, the choice of r, and Proposition 5.6 that for each integer n ≥ 1 there exists yn ∈ Y (0, ∞) such that yn (0) = xn , lim φ(yn (t)) = π(xn ). t→∞

(5.90)

Property (a) implies that for each integer n ≥ 1 and each t ∈ [L0 , ∞) ∩ Γ , ρ(yn (t),  θ ) ≤ M0 .

(5.91)

By the assumptions of the proposition and the choice of r, there exists M1 > M0 such that {z(t) : z ∈ Y (0, L0 ), ρ(z(0), x) ≤ r, t ∈ [0, L0 ] ∩ Γ } ⊂ B( θ , M1 ). (5.92) Equations (5.89), (5.91), and (5.92) imply that {yn (t) : t ∈ Γ, n = 1, 2, . . . } ⊂ B( θ , M1 ).

(5.93)

By assumption (A4) and Eq. (5.93), extracting subsequences, using the diagonalization process, and re-indexing, we obtain that there exist a strictly increasing sequence of natural numbers {nj }∞ j =1 and y ∈ Y (0, ∞) such that for each integer m ≥ 1, ynj → y as j → ∞ uniformly on [0, m] ∩ Γ. It follows from (5.86), (5.90), and (5.94) that y(0) = x. By (5.87), (5.90), and (5.94), for each t ∈ Γ ∩ [0, ∞), φ(y(t)) = lim φ(ynj (t)) j →∞

≥ lim sup π(xnj ) = lim π(xn ) j →∞

n→∞

π(y(t)) ≥ lim φ(y(t)) ≥ lim π(xn ). t→∞

Proposition 5.7 is proved.

n→∞

(5.94)

84

5 General Dynamical Systems with a Lyapunov Function

5.9 A Weak Turnpike Result In the sequel we use the following property. (P1) If x1 , x2 ∈ Ω satisfy φ(x1 ) = φ(x2 ), then x1 = x2 . This property holds for models of economic dynamics which are prototype of our dynamical system. When (P1) holds, we obtain the following weak turnpike result. Theorem 5.12 Assume that (P1) holds and that  ∈ (0, 1), M > 0. Then there exist natural numbers L and Q such that for every number T ∈ Γ ∩ (L, ∞) and every trajectory x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M, q

q

there exist an integer q ∈ [1, Q] and finite sequences of integers {ai }i=1 , {bi }i=1 ⊂ Γ ∩ [0, T ] such that ai ≤ bi , i = 1, . . . , q, ai+1 > bi , i ∈ {1, . . . , q} \ {q} and zi ∈ Ω, i = 1, . . . , q such that for each i ∈ {1, . . . , q}, ρ(x(t), zi ) ≤ , t ∈ [ai , bi ] ∩ Γ and q

mes(Γ ∩ [0, T ] \ ∪i=1 [ai , bi ]) ≤ L. For the proof of Theorem 5.12, the following two lemmas are used. The proof of the first one is obvious. Lemma 5.13 Let M,  > 0. Then there exists δ > 0 such that if x1 , x2 ∈ Ω ∩ B( θ , M) satisfy |φ(x1 ) − φ(x2 )| ≤ δ, then ρ(x1 , x2 ) ≤ . Lemma 5.14 Assume that property (P1) holds and that  ∈ (0, 1], M > 0. Then there exist δ > 0 and an integer L ≥ 1 such that for every T ∈ Γ ∩ (2L, ∞) and every x ∈ Y (0, T ) satisfying ρ(x(0),  θ) ≤ M

(5.95)

φ(x(0)) ≤ φ(x(T )) + δ,

(5.96)

and

5.9 A Weak Turnpike Result

85

there exists z ∈ Ω such that the inequality ρ(x(t), z) ≤  holds for all t ∈ [L, T − L] ∩ Γ . Proof Assumption (A5) implies that there exists M0 > M, and for each integer p ≥ 1, there exists an integer k(p) ≥ p such that the following property holds: (a) For each T ∈ Γ ∩ [k(p), ∞) and each z ∈ Y (0, T ) satisfying ρ(z(0),  θ ) ≤ M, we have ρ(z(t), X0 ∩ B( θ , M0 )) ≤ 1/p, t ∈ [k(p), T ] ∩ Γ. Lemma 5.13 implies that there exists 0 ∈ (0, /4) such that the following property holds: (b) For each z1 , z2 ∈ Ω ∩ B( θ , M0 + 1) satisfying |φ(x1 ) − φ(x2 )| ≤ 0 , we have ρ(z1 , z2 ) ≤ /4. Lemma 5.11 implies that there exists 1 ∈ (0, 0 /2) such that the following property holds: (c) For each y1 , y2 ∈ X which satisfy ρ(y1 , y2 ) ≤ 1 , ρ(yi , Ω ∩ B( θ , M0 + 1)) ≤ 1 , i = 1, 2, the inequality |φ(y1 ) − φ(y2 )| ≤ 0 /4. holds. Theorems 3.6 (for Γ = Z) and 4.6 (for Γ = R 1 ) imply that there exist δ ∈ (0, 1 /4) and an integer L ∈ Γ ∩ [1, ∞) such that the following property holds: (d) For every T ∈ Γ ∩ (2L, ∞) and every x ∈ Y (0, T ) which satisfies (5.95) and (5.96), the inequality ρ(x(t), Ω) ≤ 1 holds for all t ∈ [L, T − L] ∩ Γ. We may assume without loss of generality that L > k(2).

(5.97)

T ∈ Γ ∩ (2L, ∞), x ∈ Y (0, T )

(5.98)

Assume that

and that (5.95) and (5.96) hold. Property (d) and Eqs. (5.95)–(5.98) imply that ρ(x(t), Ω) ≤ 1 , t ∈ [L, T − L] ∩ Γ.

(5.99)

86

5 General Dynamical Systems with a Lyapunov Function

Property (a) (with p = 2) and Eqs. (5.95), (5.97), and (5.98) imply that ρ(x(t),  θ ) ≤ M0 + 2−1 , t ∈ [L, T ] ∩ Γ.

(5.100)

t0 ∈ [L, T − L] ∩ Γ

(5.101)

t ∈ [L, T − L] ∩ Γ

(5.102)

z0 , z ∈ Ω

(5.103)

ρ(x(t0 ), z0 ) ≤ 1 ,

(5.104)

ρ(x(t), z) ≤ 1 .

(5.105)

Fix

and

By (5.99)–(5.102), there exist

such that

In view of (5.100), (5.102), (5.104), and (5.105), z, z0 ∈ B( θ , M0 + 1).

(5.106)

Property (c) and Eqs. (5.103)–(5.106) imply that |φ(x(t0 )) − φ(z0 )| ≤ 0 /4,

(5.107)

|φ(x(t)) − φ(z)| ≤ 0 /4.

(5.108)

|φ(x(t0 )) − φ(x(t))| ≤ δ ≤ 0 /4.

(5.109)

By (5.1) and (5.96),

It follows from (5.107)–(5.109) that |φ(z0 ) − φ(z)| ≤ |φ(z0 ) − φ(x(t0 ))| + |φ(x(t0 )) − φ(x(t))| + |φ(x(t)) − φ(z)| ≤ 0 /4 + 0 /4 + 0 /4. The equation above, property (b), and Eqs. (5.103) and (5.106) imply that ρ(z0 , z) ≤ /4.

5.9 A Weak Turnpike Result

87

Together with (5.105), this implies that ρ(z0 , x(t)) ≤ ρ(z0 , z) + ρ(z, x(t)) ≤ /4 + /4 < . Thus z0 ∈ Ω and ρ(z0 , x(t)) < , t ∈ [L, T − L] ∩ Γ. Lemma 5.14 is proved. Proof of Theorem 5.12 Assumption (A5) and Lemma 5.11 imply that there exist M1 > M, M2 > 0, and for each integer p ≥ 1, there exists an integer Lp ≥ p such that the following property holds: (a) For each T ∈ Γ ∩ [Lp , ∞) and each y ∈ Y (0, T ) satisfying ρ(y(0),  θ ) ≤ M, we have ρ(y(t), X0 ∩ B( θ , M1 )) ≤ p−1 , t ∈ Γ ∩ [Lp , T ], |φ(y(t))| ≤ M2 , t ∈ [Lp , T ]. In this section, for a trajectory x, we use the notation limt→S + φ(x(t)) and limt→S − φ(x(t)) in their usual sense if Γ = R 1 and if Γ = Z, then limt→S + φ(x(t)) = φ(x(S + 1)) and limt→S − φ(x(t)) = φ(x(S − 1)). Lemma 5.14 implies that there exist δ ∈ (0, ) and an integer L0 ≥ 1 such that the following property holds: (b) For every T ∈ Γ ∩ (2L0 , ∞) and every x ∈ Y (0, T ) satisfying ρ(x(0),  θ) ≤ M1 and φ(x(0)) ≤ φ(x(T )) + δ, there exists z ∈ Ω such that the inequality ρ(x(t), z) ≤  holds for all t ∈ [L0 , T − L0 ] ∩ Γ . Choose integers Q > 2(M1 + M2 )δ −1 + 1

(5.110)

and L > 2L0 + 2 + (4L0 + 8L1 + 8)(2δ −1 (M1 + M2 ) + 1).

(5.111)

Suppose that T ∈ Γ ∩ [L, ∞) and that x ∈ Y (0, T ) satisfies ρ(x(0),  θ ) ≤ M. Property (a), (5.111), and (5.112) imply that for every t ∈ [L1 , T ] ∩ Γ .

(5.112)

88

5 General Dynamical Systems with a Lyapunov Function

ρ(x(t),  θ )) ≤ M1 + 1, |φ(x(t))| ≤ M2 .

(5.113)

By induction, we define a strictly increasing finite sequence ti ∈ Γ ∩ [0, T ], i = 0, . . . , q. Set t 0 = L1 . If lim φ(x(t)) ≥ lim φ(x(t)) − δ,

t→T −

t→t0+

then set t1 = T and complete to construct the sequence. Assume that lim φ(x(t)) < lim φ(x(t)) − δ.

t→T −

t→t0+

Set t1 = inf{t ∈ (t0 , T ] ∩ Γ : φ(x(t)) < lim φ(x(t)) − δ}. t→L+ 0

It is not difficult to see that t1 is well-defined, in the case Γ = R 1 , lim φ(x(t)) ≤ lim φ(x(t)) − δ ≤ lim φ(x(t))

t→t1+

t→t1 −

t→t0+

and in the case Γ = Z, φ(x(t1 )) < φ(x(t0 )) − δ and for each s ∈ Γ satisfying t0 < s < t1 , φ(x(s)) ≥ φ(x(t0 )) − δ. If t1 = T , then we complete to construct the sequence. Assume that k ≥ 1 is an integer and that we defined a finite strictly increasing sequence t0 , . . . , tk ∈ [0, T ] ∩ Γ such that t 0 = L1 , t k ≤ T 1 , then and that for each i ∈ {0, . . . , k − 1}, if Γ = R+

lim φ(x(t)) ≤ lim φ(x(t)) − δ ≤ lim φ(x(t)),

+ t→ti+1

t→ti+

− t→ti+1

(5.114)

5.9 A Weak Turnpike Result

89

if Γ = Z, then φ(x(ti+1 )) < φ(x(ti )) − δ,

(5.115)

and if s ∈ Γ satisfies ti < s < ti+1 , then φ(x(s)) ≥ φ(x(ti )) − δ.

(5.116)

(It is not difficult to see that the assumption is true with k = 1.) If tk = T , then we complete to construct the sequence. Assume that tk < T . If lim φ(x(t)) ≥ lim φ(x(t)) − δ,

t→T −

t→tk+

then we set tk+1 = T and complete to construct the sequenced. Assume that lim φ(x(t)) < lim φ(x(t)) − δ.

t→T −

t→tk+

Set tk+1 = inf{t ∈ Γ ∩ (tk , T ] : φ(x(t)) < lim φ(x(t)) − δ}. t→tk+

It is not difficult to see that tk+1 is well-defined, tk+1 > tk , if Γ = R 1 , then lim φ(x(t)) ≤ lim φ(x(t)) − δ ≤ lim φ(x(t)),

+ t→tk+1

t→tk+

− t→tk+1

if Γ = Z, then φ(x(tk+1 )) < φ(x(tk )) − δ, and for each s ∈ Γ ∩ (tk , tk+1 ), φ(x(s)) ≥ φ(x(tk )) − δ. Thus, the assumption made for k is true for k+1 too. Therefore, by induction, we constructed the strictly increasing finite sequence ti ∈ [0, T ] ∩ Γ , i = 0, . . . , q such that t 0 = L1 , t q = T , for every integer i satisfying 0 ≤ i < q − 1, if Γ = R 1 , then (5.114) holds, if Γ = Z, then (5.115) holds, and if s ∈ Γ satisfies ti < s < ti+1 , then (5.116) holds. If Γ = R 1 , then

90

5 General Dynamical Systems with a Lyapunov Function

lim φ(x(t)) ≥ lim φ(x(t)) − δ,

t→T −

+ t→tq−1

and if Γ = Z, then for all integers s satisfying tq−1 < s < T , φ(x(s)) > φ(x(tq−1 )) − δ. By the relation t0 = L1 , (5.110), (5.113), and (5.114), 2M2 ≥ lim φ(x(t)) − lim φ(x(t)) t→t0+

=

+ t→tq−1

 { lim φ(x(t)) − lim φ(x(t)) : t→ti+

+ t→ti+1

i is an integer, 0 ≤ i ≤ q − 2} ≥ δ(q − 1) and q ≤ 1 + 2δ −1 M2 < Q.

(5.117)

Define E = {i ∈ {0, . . . , q − 1} : ti+1 − ti ≥ 2L0 + 4}.

(5.118)

Let i ∈ E.

(5.119)

ti+1 − 1 − (ti + 1) ≥ 2L0 + 2.

(5.120)

By (5.118) and (5.119),

Equations (5.114) and (5.120) imply that φ(x(ti+1 − 1)) ≥ φ(x(ti )) − δ.

(5.121)

Equations (5.113), (5.120), and (5.121) and property (b) applied to the restriction of x to the interval [ti + 1, ti+1 − 1] imply that there exists zi ∈ Ω

5.10 Turnpike Results

91

such that ρ(x(t), zi ) ≤ , t ∈ [ti + L0 + 1, ti+1 − 1 − L0 ] ∩ Γ.

(5.122)

Set ai = ti + L0 + 1, bi = ti+1 − L0 − 1.

(5.123)

By (5.111), (5.117), (5.118), and (5.123), mes([0, T ] ∩ Γ \ ∪i∈E [ai , bi ]) ≤ L1 + mes(∪{[ti , ti+1 ] ∩ Γ : i ∈ {0, . . . , q − 1} \ E}) + mes(∪{[ti , ti + L0 + 1] ∩ Γ ∪ [ti+1 − L0 − 1, ti+1 ] ∩ Γ : i ∈ E}) ≤ L1 + (2L0 + 5)(1 + 2M2 δ −1 ) < L. Theorem 5.12 is proved.

5.10 Turnpike Results Theorems 5.4 and 5.5 establish the turnpike property for approximate optimal trajectories which have a fixed starting point with respect to the objective function φ. On the other hand, there are many turnpike results where the turnpike property holds for approximate optimal trajectories with an initial point belonging to a given bounded set [117, 129, 148]. In this chapter we prove the following three turnpike results for approximate optimal trajectories with an initial point belonging to a given bounded set assuming that the function π is continuous at every point of Ω. This assumption holds for models of economic dynamics which are prototype of our dynamical system. Theorem 5.15 Assume that the function π is continuous at every point of Ω. Let  ∈ (0, 1), M > 0. Then there exist δ > 0 and L ∈ Γ ∩ [1, ∞) such that for every T ∈ Γ ∩ (2L, ∞) and every trajectory x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M and φ(x(T )) ≥ π(x(0)) − δ, the value π(x(0)) is finite and ρ(x(t), Ω) ≤  for all t ∈ [L, T − L] ∩ Γ.

92

5 General Dynamical Systems with a Lyapunov Function

The turnpike is the set Ω and δ and L depend on , M. The following result is a generalization of the previous theorem. According to Theorem 5.15, for all t ∈ [L, T − L] ∩ Γ , the distance between ρ(x(t)) and the turnpike Ω does not exceed . According to our next result, for all t ∈ [L, T − L] ∩ Γ , the distance between ρ(x(t)) and the set φ −1 (λ)∩B( θ , M1 )∩Ω does not exceed , where a positive constant λ belongs to φ(B( θ , M1 ) ∩ Ω) and is close to π(x(0)). Theorem 5.16 Assume that the function π is continuous at every point of Ω. Let  ∈ (0, 1), M > 0. Then there exist δ, M1 > 0 and L ∈ Γ ∩ [1, ∞) such that for every T ∈ Γ ∩ (2L, ∞) and every trajectory x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M and φ(x(T )) ≥ π(x(0)) − δ, the value π(x(0)) is finite and there exists λ ∈ φ(B( θ , M1 ) ∩ Ω) ∩ [π(x(0)) − , π(x(0)) + ] such that for all t ∈ [L, T − L] ∩ Γ , ρ(x(t), φ −1 (λ) ∩ B( θ , M1 ) ∩ Ω) ≤  and |φ(x(t)) − λ| ≤ . In our final result, we assume the property (P1) holds and obtain for all t ∈ [L, T − L] ∩ Γ , ρ(x(t), F (x(0))) ≤  for approximate optimal trajectories with an initial point belonging to a given bounded set. Theorem 5.17 Assume that property (P1) holds and that the function π is continuous at every point of Ω. Let  ∈ (0, 1), M > 0. Then there exist δ > 0 and L ∈ Γ ∩ [1, ∞) such that for every T ∈ Γ ∩ (2L, ∞) and every trajectory x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M and φ(x(T )) ≥ π(x(0)) − δ, the value π(x(0)) is finite and for all t ∈ [L, T − L] ∩ Γ , ρ(x(t), F (x(0))) ≤  and |φ(x(t)) − π(x(0))| ≤ .

5.11 Auxiliary Results

93

5.11 Auxiliary Results In the previous section, we stated three turnpike results under the assumption that the function π is continuous at every point of Ω. In this section A and a are defined as in Sect. 5.1. For the discrete-time case when Γ = Z, we show when the continuity of π holds. Proposition 5.18 Let Γ = Z ξ ∈ Ω. Assume that for each  > 0 there exist δ > 0 and ξ1 ∈ Ω such that ρ(ξ, ξ1 ) ≤  and that for each x ∈ B(ξ, δ) the inclusion a(ξ1 ) ⊂ a(x) holds. Then the function π is lower semicontinuous at the point ξ . Proof Let  > 0. There exists 1 ∈ (0, ) such that |φ(z) − φ(ξ )| ≤ /4

(5.124)

for each z ∈ B(ξ, 1 ). There exist δ ∈ (0, 1 ) and ξ1 ∈ Ω such that ρ(ξ, ξ1 ) ≤ 1

(5.125)

and that the following property holds: For each x ∈ B(ξ, δ), a(ξ1 ) ⊂ a(x).

(5.126)

Assume that x ∈ B(ξ, δ). Then the inclusion (5.126) holds. Corollary 5.9, the inclusion ξ, ξ1 ∈ Ω, and (5.126) imply that π(x) ≥ π(ξ1 ) = φ(ξ1 ) ≥ φ(ξ ) − /4 = π(ξ ) − /4. Thus π(x) ≥ π(ξ ) − /4, x ∈ B(ξ, δ). Proposition 5.18 is proved. Proposition 5.19 Let Γ = Z ξ ∈ Ω. Assume that for each  > 0 there exist δ > 0 and ξ1 ∈ Ω such that ρ(ξ, ξ1 ) ≤  and that for each x ∈ B(ξ, δ) the inclusion a(x) ⊂ a(ξ ) holds. Then the function π is upper semicontinuous at the point ξ . Proof Let  > 0. There exists 1 ∈ (0, ) such that |φ(z) − φ(ξ )| ≤ /4

(5.127)

94

5 General Dynamical Systems with a Lyapunov Function

for each z ∈ B(ξ, 1 ). There exist δ ∈ (0, 1 ) and ξ1 ∈ Ω such that ρ(ξ, ξ1 ) ≤ 1

(5.128)

a(x) ⊂ a(ξ1 ).

(5.129)

x ∈ B(ξ, δ).

(5.130)

and that for each x ∈ B(ξ, δ),

Assume that

In view of (5.130), the inclusion (5.129) holds. Corollary 5.9, (5.127)–(5.129), and the inclusion ξ, ξ1 ∈ Ω imply that π(x) ≤ π(ξ1 ) = φ(ξ1 ) ≤ φ(ξ ) +  ≤ π(ξ ) + . Proposition 5.19 is proved.

5.12 Proof of Theorem 5.15 Assumption (A5) implies that there exist M1 > M + 1 and an integer L0 ≥ 1 such that the following property holds: (i) For each T ∈ Γ ∩ [L0 , ∞) and each y ∈ Y (0, T ) satisfying ρ(y(0),  θ ) ≤ M, we have ρ(y(t), X0 ∩ B( θ , M1 − 1)) ≤ 1/2, t ∈ [L0 , T ] ∩ Γ. Assumption (A6) and Theorems 3.6 (in the discrete case) and 4.6 (in the continuous case) imply that there exist 1 ∈ (0, ) and an integer L1 ≥ 1 such that the following property holds: (ii) For every T ∈ Γ ∩ (2L1 , ∞) and every y ∈ Y (0, T ) which satisfies ρ(y(0),  θ ) ≤ M1 and φ(y(0)) − φ(y(T )) ≤ 1 , the inequality ρ(x(t), Ω) ≤  holds for all t ∈ [L1 , T − L1 ] ∩ Γ. In view of (5.2) and (5.4), Ω ∩ B( θ , M1 + 1)

5.12 Proof of Theorem 5.15

95

is a compact set. Corollary 5.9 implies that φ(x) = π(x), x ∈ Ω.

(5.131)

Lemma 5.11 implies that there exists δ ∈ (0, 1 /4) such that the following property holds: (iii) For each z1 , z2 ∈ X which satisfy ρ(zi , Ω ∩ B( θ , M1 + 1)) ≤ δ, i = 1, 2 and ρ(z1 , z2 ) ≤ δ, we have |φ(z1 ) − φ(z2 )| ≤ 1 /4, |π(z1 ) − π(z2 )| ≤ 1 /4. Theorems 3.7 (in the discrete case) and 4.7 (in the continuous case) imply that there exists L3 ≥ 1 such that the following property holds: (iv) For every number T ∈ Γ ∩ [L3 , ∞) and every trajectory y ∈ Y (0, T ) which satisfies ρ(y(0),  θ ) ≤ M1 , we have mes([0, T ] ∩ Γ : ρ(y(t), Ω) > δ}) < L3 . Choose an integer L > 3(L0 + L1 + L3 ).

(5.132)

T ∈ (2L, ∞) ∩ Γ, x ∈ Y (0, T ),

(5.133)

ρ(x(0),  θ ) ≤ M,

(5.134)

φ(x(T )) ≥ π(x(0)) − δ.

(5.135)

Assume that

(Note that here we do not assume that π(x(0)) is finite.)

96

5 General Dynamical Systems with a Lyapunov Function

Property (i) and Eqs. (5.132)–(5.134) imply that for each t ∈ [L0 , T ] ∩ Γ , ρ(x(t), X0 ∩ B( θ , M1 − 1)) ≤ 2−1 .

(5.136)

Property (iv) and Eqs. (5.132) and (5.133) imply that there exists τ ∈ [L0 , L0 + L3 ] ∩ Γ

(5.137)

ρ(x(τ ), Ω) ≤ δ.

(5.138)

z∈Ω

(5.139)

ρ(x(τ ), z) ≤ δ.

(5.140)

ρ(x(τ ),  θ ) ≤ M1 .

(5.141)

such that

In view of (5.138), there exists

such that

By (5.136) and (5.137),

Equations (5.140) and (5.141) imply that ρ(z,  θ ) ≤ M1 + 1.

(5.142)

Property (iii) and Eqs. (5.139), (5.140), and (5.142) imply that |φ(x(τ )) − φ(z)| ≤ 1 /4,

(5.143)

|π(x(τ )) − π(z)| ≤ 1 /4.

(5.144)

In view of (5.131) and (5.139), φ(z) = π(z).

(5.145)

Proposition 5.1 and (5.1) 0imply that π(x(τ )) ≤ π(x(0)) ≤ φ(x(0)). Thus π(x(0)) is finite. By (5.135), (5.143), and (5.144)–(5.146), φ(x(τ )) − φ(x(T )) ≤ φ(z) + 1 /4 − π(x(0)) + δ

(5.146)

5.13 Auxiliary Results for Theorem 5.16

97

≤ φ(z) + 1 /4 − π(x(τ )) + δ ≤ φ(z) + 1 /4 − π(z) + 1 /4 + δ < 1 .

(5.147)

Equations (5.132), (5.133), (5.137), (5.141), and (5.147) and property (ii) applied to the restriction of x to [τ, T ] ∩ Γ imply that ρ(x(t), Ω) ≤ , t ∈ [τ + L1 , T − L1 ] ∩ Γ. Theorem 5.15 is proved.

5.13 Auxiliary Results for Theorem 5.16 Lemma 5.20 Let M,  > 0 and let ξ ∈ Ω satisfy ρ(ξ,  θ ) ≤ M. Then there exists δ > 0 such that for each η ∈ Ω ∩ B( θ , M) satisfying |φ(η) − φ(ξ )| ≤ δ, the inequality θ , M) ∩ Ω, φ −1 (φ(ξ )) ∩ B( θ , M) ∩ Ω) ≤  dist(φ −1 (φ(η)) ∩ B( holds. Proof Assume the contrary. Then, for each integer i ≥ 1, there exists θ , M) ∩ Ω ηi ∈ B( such that |φ(ηi ) − φ(ξ )| ≤ 1/i,

(5.148)

θ , M) ∩ Ω) > . ρ(ηi , φ −1 (φ(ξ )) ∩ B(

(5.149)

Since the set Ω ∩ B( θ M) is compact, we may assume without loss of generality that there exists η = lim ηi . i→∞

Clearly, η ∈ Ω ∩ B( θ , M). By (5.148), φ(η) = lim φ(ηi ) = φ(ξ ). i→∞

98

5 General Dynamical Systems with a Lyapunov Function

This implies that η ∈ φ −1 (φ(ξ )) ∩ B( θ , M) ∩ Ω. Therefore, for all integers i ≥ 1, ρ(ηi , φ −1 (φ(ξ )) ∩ B( θ , M) ∩ Ω) ≤ ρ(ηi , η) → 0 as i → ∞. This contradicts (5.149). The contradiction we have reached completes the proof of Lemma 5.20. Lemma 5.21 Let M,  > 0. Then there exists δ > 0 such that for each c ∈ φ(B( θ , M) ∩ Ω) there exists c0 ∈ φ(B( θ , M) ∩ Ω) such that |c − c0 | ≤ , and for each c˜ ∈ φ(B( θ , M) ∩ Ω) ∩ [c − δ, c + δ], the inequality dist(φ −1 (c) ˜ ∩ B( θ , M) ∩ Ω, φ −1 (c0 ) ∩ B( θ , M) ∩ Ω) ≤  holds. Proof Lemma 5.20 implies that for each ξ ∈ Ω ∩ B( θ , M), there exists δξ ∈ (0, min{, 1}) such that the following property holds: (i) For each η ∈ Ω ∩ B( θ , M) satisfying |φ(η) − φ(ξ )| ≤ δξ , we have dist(φ −1 (φ(η)) ∩ B( θ , M) ∩ Ω, φ −1 (φ(ξ )) ∩ B( θ , M) ∩ Ω) ≤ .

5.13 Auxiliary Results for Theorem 5.16

99

Clearly, φ(B( θ , M) ∩ Ω) ⊂ ∪{(φ(ξ ) − δξ /2, φ(ξ ) + δξ /2) : ξ ∈ B( θ , M) ∩ Ω}, and there exist ξi ∈ B( θ , M) ∩ Ω, i = 1, . . . , q, such that q φ(B( θ , M) ∩ Ω) ⊂ ∪i=1 (φ(ξi ) − δξi /2, φ(ξi ) + δξi /2).

(5.150)

Set δ = min{δξi : i = 1, . . . , q}/4.

(5.151)

c ∈ φ(B( θ , M) ∩ Ω).

(5.152)

Assume that

In view of (5.150) and (5.152), there exists j ∈ {1, . . . , q} such that c ∈ (φ(ξj ) − δξj /2, φ(ξj ) + δξj /2).

(5.153)

cj = φ(ξj ).

(5.154)

Set

By (5.153) and (5.154), |c − cj | < 2−1 δξj < . Assume that c˜ ∈ φ(B( θ , M) ∩ Ω) ∩ [c − δ, c + δ].

(5.155)

It follows from (5.151), (5.153), and (5.154) that |c˜ − φ(ξj )| ≤ 2−1 δξj + δ < δξj . Equations (5.154)–(5.156) and property (i) applied with ξ = ξj imply that dist(φ −1 (c) ˜ ∩ B( θ , M) ∩ Ω, φ −1 (cj ) ∩ B( θ , M) ∩ Ω) ≤ . Lemma 5.21 is proved.

(5.156)

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5 General Dynamical Systems with a Lyapunov Function

5.14 Proofs of Theorems 5.16 and 5.17 Proof of Theorem 5.16 Assumption (A5) implies that there exist M1 > M + 1 and an integer L0 ≥ 1 such that the following property holds: (i) For each T ∈ Γ ∩ [L0 , ∞) and each y ∈ Y (0, T ) satisfying ρ(y(0),  θ ) ≤ M, we have ρ(y(t), X0 ∩ B( θ , M1 − 1)) ≤ 1/2, t ∈ [L0 , T ] ∩ Γ. Lemma 5.21 implies that there exists 0 ∈ (0, /4) such that the following property holds: (ii) For each c ∈ φ(B( θ , M1 ) ∩ Ω), there exists c0 ∈ φ(B( θ , M1 ) ∩ Ω) such that |c − c0 | ≤ /8, and for each c˜ ∈ φ(B( θ , M1 ) ∩ Ω) ∩ [c − 0 , c + 0 ], the inequality dist(φ −1 (c) ˜ ∩ B( θ , M1 ) ∩ Ω, φ −1 (c0 ) ∩ B( θ , M1 ) ∩ Ω) ≤ /4 is true. Lemma 5.11 and the compactness of the set Ω ∩ B( θ , M1 + 1) imply that there exists 1 ∈ (0, 0 /4) such that the following property holds: (iii) For each z1 , z2 ∈ X which satisfy ρ(zi , Ω ∩ B( θ , M1 + 1)) ≤ 21 , i = 1, 2 and ρ(z1 , z2 ) ≤ 21 , we have |φ(z1 ) − φ(z2 )| ≤ 0 /4, |π(z1 ) − π(z2 )| ≤ 0 /4.

5.14 Proofs of Theorems 5.16 and 5.17

101

Assumption (A6) and Theorems 3.6 (in the discrete case) and 4.6 (in the continuous case) imply that there exist 2 ∈ (0, 1 /4) and an integer L1 ≥ 1 such that the following property holds: (iv) For every T ∈ Γ ∩ (2L1 , ∞) and every x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M1 and φ(x(0)) − φ(x(T )) ≤ 2 , the inequality ρ(x(t), Ω) ≤ 1 holds for all t ∈ [L1 , T − L1 ] ∩ Γ. Lemma 5.11 and the compactness of the set Ω ∩ B( θ , M1 + 1) imply that there exists δ ∈ (0, 2 /4) such that the following property holds: (v) For each z1 , z2 ∈ X which satisfy ρ(zi , Ω ∩ B( θ , M1 + 1)) ≤ 2δ, i = 1, 2 and ρ(z1 , z2 ) ≤ 2δ, we have |φ(z1 ) − φ(z2 )| ≤ 2 /4, |π(z1 ) − π(z2 )| ≤ 2 /4. Theorems 3.7 (in the discrete case) and 4.7 (in the continuous case) imply that there exists an integer L2 > 2L1 such that the following property holds: (vi) For every number T ∈ Γ ∩ [L2 , ∞) and every trajectory x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M, we have mes([0, T ] ∩ Γ : ρ(x(t), Ω) > δ}) < L2 . Choose an integer L > 3(L0 + L1 + L2 ).

(5.157)

T ∈ (2L, ∞) ∩ Γ, x ∈ Y (0, T ),

(5.158)

Assume that

102

5 General Dynamical Systems with a Lyapunov Function

ρ(x(0),  θ ) ≤ M,

(5.159)

φ(x(T )) ≥ π(x(0)) − δ.

(5.160)

(Note that here we do not assume that π(x(0)) is finite.) Property (i) and Eqs. (5.157)–(5.159) imply that for each t ∈ [L0 , T ] ∩ Γ , ρ(x(t),  θ ) ≤ M1 − 2−1 .

(5.161)

Property (vi) and Eq. (5.161) imply that there exists τ ∈ [L0 , L0 + L2 ] ∩ Γ

(5.162)

ρ(x(τ ), Ω) ≤ δ.

(5.163)

ξ ∈Ω

(5.164)

ρ(x(τ ), ξ ) ≤ δ.

(5.165)

x(τ ) ∈ B( θ , M1 − 2−1 ), ξ ∈ B( θ , M1 ).

(5.166)

such that

In view of (5.163), there exists

such that

By (5.161), (5.162), and (5.165),

Property (v) and Eqs. (5.164)–(5.166) imply that |φ(x(τ )) − φ(ξ )| ≤ 2 /4,

(5.167)

|π(x(τ )) − π(ξ )| ≤ 2 /4.

(5.168)

Thus π(x(0)) is finite. Proposition 5.1, (5.1), and (5.6) imply that π(x(τ )) ≤ π(x(0)) ≤ φ(x(0)).

(5.169)

Corollary 5.9, (5.1), (5.160), (5.164), and (5.167) imply that π(ξ ) = φ(ξ ) ≥ φ(x(τ )) − 2 /4 ≥ π(x(0)) − δ − 2 /4.

(5.170)

5.14 Proofs of Theorems 5.16 and 5.17

103

It follows from (5.168) and (5.169) that π(ξ ) ≤ 2 /4 + π(x(τ )) ≤ 2 /4 + π(x(0)).

(5.171)

Equations (5.160) and (5.167)–(5.170) imply that φ(x(T )) ≥ π(x(0)) − δ ≥ π(x(τ )) − δ ≥ π(ξ ) − 2 /4 − δ ≥ φ(ξ ) − 2 /4 − 2 /4 ≥ φ(x(τ )) − 32 /4.

(5.172)

In view of (5.157), (5.158), and (5.162), T − τ > L > 3L1 .

(5.173)

Equations (5.166), (5.172), and (5.173) and property (iv) applied to the restriction of x to [τ, T ] ∩ Γ imply that ρ(x(t), Ω) ≤ 1 , t ∈ [τ + L1 , T − L1 ] ∩ Γ.

(5.174)

Let t ∈ [τ + L1 , T − L1 ] ∩ Γ.

(5.175)

In view of (5.174) and (5.175), there exists ξ(t) ∈ Ω

(5.176)

ρ(x(t), ξ(t)) ≤ 1 .

(5.177)

such that

By (5.161), (5.162), (5.174), and (5.177), ρ(ξ(t),  θ ) ≤ M1 .

(5.178)

Property (iii) and Eqs. (5.176)–(5.178) imply that |φ(x(t)) − φ(ξ(t))| ≤ 0 /4,

(5.179)

|π(x(t)) − π(ξ(t))| ≤ 0 /4.

(5.180)

Corollary 5.9 and (5.176) imply that φ(ξ(t)) = π(ξ(t)).

(5.181)

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5 General Dynamical Systems with a Lyapunov Function

It follows from (5.1), (5.172), and (5.175) that φ(x(t)) ≤ φ(x(τ )) ≤ φ(x(T )) + 2 ≤ φ(x(t)) + 2 and |φ(x(t)) − φ(x(τ ))| ≤ 2 .

(5.182)

By (5.167), (5.179), and (5.182), |φ(ξ ) − φ(ξ(t))| ≤ |φ(ξ ) − φ(x(τ ))| + |φ(x(τ )) − φ(x(t))| + |φ(x(t)) − φ(ξ(t))| ≤ 2 /4 + 2 + 0 /4 < 0 .

(5.183)

In view of (5.170) and (5.171), |π(ξ ) − π(x(0))| ≤ 2 /4.

(5.184)

|φ(x(t)) − φ(ξ )| ≤ .

(5.185)

By Eqs. (5.179) and (5.183),

It follows from (5.164), (5.166), (5.176), (5.178), (5.183), and property (ii) (applied ξ )) that there exists with c = φ(ξ ), c˜ = φ(ξ(t)), c0 = φ(  ξ ∈ B( θ , M1 ) ∩ Ω such that |φ( ξ ) − φ(ξ )| ≤ /8, and for each t ∈ Γ ∩ [τ + L1 , T − L1 ], ρ(ξ(t), φ −1 (φ( ξ )) ∩ B( θ , M1 ) ∩ Ω) ≤ /4. Together with (5.177), this implies that ρ(x(t), φ −1 (φ( ξ )) ∩ B( θ , M1 ) ∩ Ω) ≤ . Setting λ = φ( ξ ), we complete the proof of Theorem 5.16. Proof of Theorem 5.17 It is not difficult to see that the following proposition is true. Proposition 5.22 Assume that property (P1) holds and that x ∈ Y (0, ∞). Then there exists limt→∞ x(t) ∈ Ω.

5.15 Extensions of Theorems 5.16 and 5.17

105

Property (P1), assumption (A5), Proposition 5.3, and (5.5) imply that if (P1) holds, x ∈ X, and π(x) is finite, then F (x) ∈ Ω is a singleton and that F : {x ∈ X : π(x) > −∞} → Ω. Since all bounded, closed subsets of Ω are compact, property (P1) implies the next result. Proposition 5.23 Assume that (P1) holds and M,  > 0. Then there exists δ > 0 such that for each z1 , z2 ∈ Ω ∩ B( θ , M) satisfying |φ(z1 ) − φ(z2 )| ≤ δ, the inequality ρ(z1 , z2 ) ≤  holds. Now it is easy to see that Theorem 5.17 follows from Theorem 5.16 and Propositions 5.3 and 5.23.

5.15 Extensions of Theorems 5.16 and 5.17 In the previous sections we showed that approximate optimal trajectories are closed to the turnpike except in regions close to the endpoints. In this chapter we prove the following result which shows under certain additional assumptions that if an initial point of approximate optimal trajectory is close to the turnpike, then the trajectory is close to the turnpike in the region containing the left endpoint. Theorem 5.24 Assume that the following assumptions hold: (B1) For each M > 0, there exists M0 > 0 such that for each T ∈ Γ ∩ (0, ∞) and each y ∈ Y (0, T ) satisfying y(0) ∈ B( θ , M) the inclusion y(t) ∈ B( θ , M0 ) holds for all t ∈ [0, T ] ∩ Γ . (B2) For each , M > 0, there exists δ > 0 such that for each T ∈ (0, ∞) ∩ Γ and each y ∈ Y (0, T ) satisfying θ , M)) ≤ δ, ρ(y(0), X0 ∩ B( the inequality ρ(y(t), X0 ) ≤ , t ∈ [0, T ] ∩ Γ is true.

106

5 General Dynamical Systems with a Lyapunov Function

Assume that the function π is continuous at every point of Ω. Let  ∈ (0, 1), M > 0. Then there exist δ, M1 > 0 and L ∈ Γ ∩ [1, ∞) such that for every T ∈ Γ ∩ (2L, ∞) and every trajectory x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M and φ(x(T )) ≥ π(x(0)) − δ, the value π(x(0)) is finite and there exist λ ∈ φ(B( θ , M1 ) ∩ Ω) ∩ [π(x(0)) − , π(x(0)) + ] and τ ∈ [0, L] ∩ Γ such that for each t ∈ [τ, T − L] ∩ Γ , ρ(x(t), φ −1 (λ) ∩ B( θ , M1 ) ∩ Ω) ≤  and |φ(x(t)) − λ| ≤ . Moreover, if ρ(x(0), Ω) ≤ δ, then τ = 0. The following theorem easily follows from Theorem 5.24 and Propositions 5.3 and 5.23. Theorem 5.25 Assume that assumptions (B1) and (B2) and property (P1) hold and that the function π is continuous at every point of Ω. Let  ∈ (0, 1), M > 0. Then there exist δ, M1 > 0 and L ∈ Γ ∩ [1, ∞) such that for every T ∈ Γ ∩ (2L, ∞) and every trajectory x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M and φ(x(T )) ≥ π(x(0)) − δ, the value π(x(0)) is finite and there exists τ ∈ [0, L] ∩ Γ such that for each t ∈ [τ, T − L] ∩ Γ , ρ(x(t), F (x(0)) ≤  and |φ(x(t)) − π(x(0))| ≤ . Moreover, if ρ(x(0), Ω) ≤ δ, then τ = 0.

5.16 An Auxiliary Result for Theorem 5.24 Lemma 5.26 Assume that (B1) and (B2) hold. Let , M > 0 and L be a natural number. Then there exist δ > 0 and an integer L1 ≥ L such that for every T ∈ [L1 , ∞) ∩ Γ and every x ∈ Y (0, T ) satisfying ρ(x(0),  θ ) ≤ M, ρ(x(0), Ω) ≤ δ, φ(x(T )) ≥ φ(x(0)) − δ the inequality ρ(x(t), Ω) ≤  holds for all t ∈ [0, L] ∩ Γ.

5.16 An Auxiliary Result for Theorem 5.24

107

Proof Assume that the lemma is not true. Then, for each integer k ≥ 1, there exist Tk ∈ [L + k, ∞) ∩ Γ and xk ∈ Y (0, Tk ) such that θ ) ≤ M, ρ(xk (0), 

(5.186)

ρ(xk (0), Ω) ≤ 1/k,

(5.187)

φ(xk (Tk )) ≥ φ(xk (0)) − 1/k,

(5.188)

sup{ρ(xk (t), Ω) : t ∈ [0, L] ∩ Γ } > .

(5.189)

In view of (B1) and (5.186), there exists M0 > M such that θ ) ≤ M0 , t ∈ [0, Tk ] ∩ Γ, k = 1, 2, . . . . φ(xk (t), 

(5.190)

It follows from (B2), (5.187), and (5.190) that lim sup{ρ(xk (t), X0 ) : t ∈ [0, Tk ] ∩ Γ } = 0.

k→∞

(5.191)

By (A4), (5.190) and (5.191), extracting subsequence, using diagonalization process, and re-indexing, we obtain that there exist a subsequence {xkj }∞ j =1 and x ∈ Y (0, ∞) such that for every integer m ≥ 1, xkj converge to x as j → ∞ uniformly on [0, m].

(5.192)

Equations (5.187) and (5.190)–(5.192) imply that θ , M0 ), t ∈ Γ, x(t) ∈ X0 ∩ B( x(0) ∈ Ω.

(5.193)

φ(x(t)) = φ(x(0)), t ∈ [0, ∞) ∩ Γ.

(5.194)

In view of (5.188) and (5.192),

Proposition 5.10, (5.193), and (5.194) imply that x(t) ∈ Ω, t ∈ [0, ∞) ∩ Γ. By the equation above, (5.192), for all sufficiently large natural numbers j , ρ(xkj (t), Ω) < /2, t ∈ [0, L] ∩ Γ. This contradicts (5.189). The contradiction we have reached completes the proof of Lemma 5.26.

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5 General Dynamical Systems with a Lyapunov Function

5.17 Proof of Theorem 5.24 Theorem 5.24 follows from Theorem 5.16 and the following result. Proposition 5.27 Assume that (B1) and (B2) hold and that the function π is continuous at every point of Ω. Let  ∈ (0, 1), M > 0. Then there exist δ, M1 > 0 and L ∈ Γ ∩ [1, ∞) such that for every T ∈ Γ ∩ (2L, ∞) and every trajectory x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M, ρ(x(0), Ω) ≤ δ and φ(x(T )) ≥ π(x(0)) − δ, the value π(x(0)) is finite and there exists λ ∈ φ(B( θ , M1 + 1) ∩ Ω) ∩ [π(x(0)) − , π(x(0)) + ] such that for all t ∈ [0, T − L] ∩ Γ , ρ(x(t), φ −1 (λ) ∩ B( θ , M1 + 1) ∩ Ω) ≤  and |φ(x(t)) − λ| ≤ . Proof Assumption (B1) implies that there exists M1 > M + 1 such that the following property holds: (i) For each T ∈ Γ ∩ (0, ∞) and each y ∈ Y (0, T ) satisfying y(0) ∈ B( θ , M), the inclusion y(t) ∈ B( θ , M1 ) holds for all t ∈ [0, T ] ∩ Γ. Lemma 5.21 implies that there exists 0 ∈ (0, /4) such that the following property holds: (ii) For each c ∈ φ(B( θ , M1 + 1) ∩ Ω) there exists c0 ∈ φ(B( θ , M1 + 1) ∩ Ω) such that |c − c0 | ≤ /8, and for each c˜ ∈ φ(B( θ , M1 + 1) ∩ Ω) ∩ [c − 0 , c + 0 ], the inequality dist(φ −1 (c) ˜ ∩ B( θ , M1 + 1) ∩ Ω, φ −1 (c0 ) ∩ B( θ , M1 + 1) ∩ Ω) ≤ /4

5.17 Proof of Theorem 5.24

109

is true. Lemma 5.11 and the compactness of the set Ω ∩ B( θ , M1 + 1) imply that there exists 1 ∈ (0, 0 /4) such that the following property holds: (iii) For each z1 , z2 ∈ X which satisfy ρ(zi , Ω ∩ B( θ , M1 + 1)) ≤ 21 , i = 1, 2 and ρ(z1 , z2 ) ≤ 21 , we have |φ(z1 ) − φ(z2 )| ≤ 0 /4, |π(z1 ) − π(z2 )| ≤ 0 /4. Assumption (A6) and Theorems 3.6 (in the discrete case) and 4.6 (in the continuous case ) imply that there exist 2 ∈ (0, 1 /4) and an integer L1 ≥ 1 such that the following property holds: (iv) For every T ∈ Γ ∩ (2L1 , ∞) and every x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M1 and φ(x(0)) − φ(x(T )) ≤ 2 , the inequality ρ(x(t), Ω) ≤ 1 holds for all t ∈ [L1 , T − L1 ] ∩ Γ. Lemma 5.26 implies that there exist 3 ∈ (0, 2 /2) and an integer L ≥ 2L1 such that the following property holds: (v) For every T ∈ [L, ∞) ∩ Γ and every x ∈ Y (0, T ) satisfying ρ(x(0),  θ ) ≤ M, ρ(x(0), Ω) ≤ 3 , φ(x(T )) ≥ φ(x(0)) − 3 , the inequality ρ(x(t), Ω) ≤ 1 holds for all t ∈ [0, L1 ] ∩ Γ. Lemma 5.11 and the compactness of the set Ω ∩ B( θ , M1 + 1) imply that there exists δ ∈ (0, 3 /2) such that the following property holds: (vi) For each z1 , z2 ∈ X which satisfy ρ(zi , Ω ∩ B( θ , M1 + 1)) ≤ 2δ, i = 1, 2 and ρ(z1 , z2 ) ≤ 2δ, we have

110

5 General Dynamical Systems with a Lyapunov Function

|φ(z1 ) − φ(z2 )| ≤ 3 /4, |π(z1 ) − π(z2 )| ≤ 3 /4. Assume that T ∈ (L, ∞) ∩ Γ, x ∈ Y (0, T ),

(5.196)

ρ(x(0),  θ ) ≤ M, ρ(x(0), Ω) ≤ δ,

(5.197)

φ(x(T )) ≥ π(x(0)) − δ.

(5.198)

Property (i) and Eq. (5.197) imply that ρ(x(t),  θ ) ≤ M1 , t ∈ [0, T ] ∩ Γ.

(5.199)

In view of (5.197), there exists z∈Ω

(5.200)

ρ(x(0), z) ≤ δ.

(5.201)

ρ( θ , z) ≤ M + 1.

(5.202)

such that

By (5.197) and (5.201),

Property (vi) and Eqs. (5.200)–(5.202) imply that |π(x(0)) − π(z)| ≤ 3 /4.

(5.203)

|φ(x(0)) − φ(z)| ≤ 3 /4.

(5.204)

Corollary 5.9 and (5.200) imply that φ(z) = π(z).

(5.205)

|φ(x(0)) − π(x(0))| ≤ 3 /2.

(5.206)

In view of (5.203)–(5.205),

By (5.1), (5.198), and (5.206), |φ(x(T )) − φ(x(0))| ≤ 3 .

(5.207)

5.17 Proof of Theorem 5.24

111

Property (v), (5.196), (5.197), and (5.207) imply that ρ(x(t), Ω) ≤ 1 , t ∈ [0, L1 ] ∩ Γ.

(5.208)

By property (iv), (5.196), (5.197), and (5.207), ρ(x(t), Ω) ≤ 1 , t ∈ [L1 , T − L1 ] ∩ Γ. It follows from the equation above and (5.208) that ρ(x(t), Ω) ≤ 1 , t ∈ [0, T − L1 ] ∩ Γ.

(5.209)

t ∈ [0, T − L1 ] ∩ Γ.

(5.210)

Let

In view of (5.209) and (5.210), there exists z(t) ∈ Ω

(5.211)

ρ(x(t), z(t)) ≤ 1 .

(5.212)

ρ(z(t),  θ ) ≤ M1 + 1.

(5.213)

such that

By (5.199) and (5.212),

Property (iii) and Eqs. (5.211)–(5.213) imply that |π(z(t)) − π(x(t))| ≤ 0 /4, |φ(z(t)) − φ(x(t))| ≤ 0 /4.

(5.214)

By (5.204), (5.207), and (5.214), |φ(z) − φ(z(t))| ≤ |φ(z) − φ(x(0))| + |φ(x(0)) − φ(x(t))| + |φ(x(t)) − φ(z(t))| ≤ 3 /4 + 3 + 0 /4 < 0 .

(5.215)

It follows from (5.200), (5.210), (5.211), (5.213), (5.215), and property (ii) (applied with c = φ(z), c0 = λ, c˜ = φ(z(t))) that there exists λ ∈ φ(B( θ , M1 + 1) ∩ Ω) such that

112

5 General Dynamical Systems with a Lyapunov Function

|φ(z) − λ| ≤ /8, and for each t ∈ Γ ∩ [0, T − L1 ], ρ(z(t), φ −1 (λ) ∩ B( θ , M1 + 1) ∩ Ω) ≤ /4. Together with (5.206) and (5.212), this implies that ρ(x(t), φ −1 (λ) ∩ B( θ , M1 + 1) ∩ Ω) <  and |λ − π(x(0))| ≤ . Proposition 5.27 is proved.

5.18 Generalizations of Theorems 5.16 and 5.17 In the several turnpike results, we showed that the turnpike property holds when the optimality criterion is defined by the Lyapunov function φ. Now we prove the turnpike results where the optimality criterion is defined by an element of the large class of functions. Let M be a set of functions ψ : X → R 1 such that φ ∈ M and the following assumptions hold: (C1) For each  > 0 and each M > 0, there exists δ > 0 such that for each z ∈ X ∩ B( θ , M) satisfying ρ(z, Ω) ≤ δ there exist ξ1 , ξ2 ∈ Ω such that ρ(z, ξi ) ≤ , i = 1, 2, and for each ψ ∈ M and each T ∈ Γ ∩ (0, ∞), sup{ψ(x(T )) : x ∈ Y (0, T ) and x(0) = ξ1 } −  ≤ sup{ψ(x(T )) : x ∈ Y (0, T ) and x(0) = z} ≤ sup{ψ(x(T )) : x ∈ Y (0, T ) and x(0) = ξ2 } +  (note that by Proposition 5.8 all the values in the equations above are finite). (C2) For each  > 0 and each M > 0, there exists δ > 0 such that for each ξ1 , ξ2 ∈ Ω ∩ B( θ , M) satisfying φ(ξ1 ) ≤ φ(ξ2 ) − , each ψ ∈ M, and each T ∈ Γ ∩ (0, ∞),

5.18 Generalizations of Theorems 5.16 and 5.17

113

sup{ψ(x(T )) : x ∈ Y (0, T ) and x(0) = ξ1 } + δ ≤ sup{ψ(x(T )) : x ∈ Y (0, T ) and x(0) = ξ2 }. (Note that (C2) holds if M = {φ}.) The following result is proved in Sect. 5.19. Theorem 5.28 Assume that the function π is continuous at any point of Ω. Let  ∈ (0, 1), M > 0. Then there exist δ, M1 > 0 and L ∈ Γ ∩ [1, ∞) such that for every T ∈ Γ ∩ (6L, ∞), every ψ ∈ M, and every x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M, ψ(x(T )) ≥ sup{ψ(z(T )) : z ∈ Y (0, T ) and z(0) = x(0)} − δ, the value π(x(0)) is finite and there exist λ ∈ φ(B( θ , M1 ) ∩ Ω) ∩ [π(x(0)) − , π(x(0)) + ] and τ ∈ [0, 3L] ∩ Γ such that for all t ∈ [τ, T − 3L] ∩ Γ , ρ(x(t), φ −1 (λ) ∩ B( θ , M1 ) ∩ Ω) ≤  and |φ(x(t)) − λ| ≤ . Moreover, if (B1) and (B2) hold and ρ(x(0), Ω) ≤ δ, then τ = 0. The next result follows easily from Theorem 5.28 and Propositions 5.3 and 5.23. Theorem 5.29 Assume that the function π is continuous at any point of Ω and (P1) hods. Let  ∈ (0, 1), M > 0. Then there exist δ, M1 > 0 and L ∈ Γ ∩ [1, ∞) such that for every T ∈ Γ ∩ (6L, ∞), every ψ ∈ M, and every x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M, ψ(x(T )) ≥ sup{ψ(z(T )) : z ∈ Y (0, T ) and z(0) = x(0)} − δ, the value π(x(0)) is finite and there exists τ ∈ [0, 3L] ∩ Γ such that for all t ∈ [τ, T − 3L] ∩ Γ , ρ(x(t), F (x(0)) ≤  and |φ(x(t)) − π(x(0))| ≤ . Moreover, if (B1) and (B2) hold and ρ(x(0), Ω) ≤ δ, then τ = 0.

114

5 General Dynamical Systems with a Lyapunov Function

5.19 Proof of Theorem 5.28 Assumption (A5) implies that there exist M1 > M + 1 and an integer L0 ≥ 0 such that the following property holds: (i) For each T ∈ Γ ∩ [L0 , ∞) and each y ∈ Y (0, T ) satisfying ρ(y(0),  θ ) ≤ M, we have ρ(y(t), B( θ , M1 − 1)) ≤ 1/4, t ∈ [L0 , T ] ∩ Γ, and if (B1) holds, then L0 = 0. Lemma 5.21 implies that there exists 0 ∈ (0, /8) such that the following property holds: (ii) For each c ∈ φ(B( θ , M1 ) ∩ Ω), there exists c0 ∈ φ(B( θ , M1 ) ∩ Ω) such that |c − c0 | ≤ /8, and for each c˜ ∈ φ(B( θ , M1 ) ∩ Ω) ∩ [c − 0 , c + 0 ], the inequality dist(φ −1 (c) ˜ ∩ B( θ , M1 ) ∩ Ω, φ −1 (c0 ) ∩ B( θ , M1 ) ∩ Ω) ≤ /4 is true. Lemma 5.11 and the compactness of the set Ω ∩ B( θ , M1 + 1) imply that there exists 1 ∈ (0, 0 /4) such that the following property holds: (iii) For each z1 , z2 ∈ X which satisfy ρ(zi , Ω ∩ B( θ , M1 + 1)) ≤ 21 , i = 1, 2 and ρ(z1 , z2 ) ≤ 21 , we have |φ(z1 ) − φ(z2 )| ≤ 0 /8, |π(z1 ) − π(z2 )| ≤ 0 /8.

5.19 Proof of Theorem 5.28

115

Assumption (A6) and Theorems 5.15 and 5.24 imply that there exist 2 ∈ (0, 1 /4) and an integer L1 ≥ L0 such that the following property holds: (iv) For every T ∈ Γ ∩ (2L1 , ∞) and every x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M1 and φ(x(T )) ≥ π(x(0)) − 2 , the value π(x(0)) is finite and there exists τ ∈ [0, L1 ] ∩ Γ such that the inequality ρ(x(t), Ω) ≤ 1 /4 holds for all t ∈ [τ, T − L1 ] ∩ Γ , and if (B1) and (B2) hold and ρ(x(0), Ω) ≤ 2 , then τ = 0. Fix γ0 ∈ (0, 2 /4). Lemma 5.11 and assumption (C2) imply that there exists δ0 ∈ (0, γ0 /4) such that the following properties holds: (v) For each ξ1 , ξ2 ∈ X satisfying ρ(ξi , X0 ∩ B( θ , M1 + 2)) ≤ 2δ0 , i = 1, 2 and ρ(ξ1 , ξ2 ) ≤ 2δ0 , we have |φ(ξ1 ) − φ(ξ2 )| ≤ γ0 /8; (vi) For each ξ1 , ξ2 ∈ Ω ∩ B( θ , M1 + 2) satisfying φ(ξ1 ) ≤ φ(ξ2 ) − γ0 /4, each T ∈ Γ ∩ (0, ∞), and each ψ ∈ M, sup{ψ(x(T )) : x ∈ Y (0, T ), x(0) = ξ1 } + 2δ0 ≤ sup{ψ(x(T )) : x ∈ Y (0, T ), x(0) = ξ2 }. Assumption (C1) imply that there exists δ1 ∈ (0, δ0 /16) such that the following properties holds: (vii) For each z ∈ X∩B( θ , M1 +2) satisfying ρ(z, Ω) ≤ δ1 , there exist ξ1 , ξ2 ∈ Ω such that

116

5 General Dynamical Systems with a Lyapunov Function

ρ(z, ξi ) ≤ δ0 /16, i = 1, 2 and such that for each T ∈ (0, ∞) ∩ Γ and each ψ ∈ M, sup{ψ(x(T )) : x ∈ Y (0, T ), x(0) = ξ1 } − δ0 /16 ≤ sup{ψ(x(T )) : x ∈ Y (0, T ), x(0) = z} ≤ sup{ψ(x(T )) : x ∈ Y (0, T ), x(0) = ξ2 } + δ0 /16. Theorems 5.15 and 5.24 imply that there exist δ ∈ (0, δ1 /4) and an integer L2 ≥ 1 such that the following property holds: (viii) For every T ∈ Γ ∩ (2L2 , ∞) and every x ∈ Y (0, T ) which satisfies ρ(x(0),  θ ) ≤ M and φ(x(T )) ≥ π(x(0)) − δ, the value π(x(0)) is finite and there exists τ ∈ [0, L2 ] ∩ Γ such that the inequality ρ(x(t), Ω) ≤ δ1 holds for all t ∈ [τ, T − L2 ] ∩ Γ , and if (B1) and (B2) hold and ρ(x(0), Ω) ≤ δ2 , then τ = 0. Theorems 3.7 and 4.7 imply that there exists an integer L ≥ 4L2 + 4L1 + 4L0

(5.216)

such that the following property holds: (ix) For every T ∈ (L, ∞)∩Γ and every x ∈ Y (0, T ) satisfying ρ(x(0),  θ ) ≤ M, we have mes({t ∈ [0, T ] ∩ Γ : ρ(x(t), Ω) > δ}) < L. Assume that T ∈ (6L, ∞) ∩ Γ, ψ ∈ M,

(5.217)

ρ(x(0),  θ ) ≤ M,

(5.218)

x ∈ Y (0, T ) satisfies

ψ(x(T )) ≥ sup{ψ(z(T )) : z ∈ Y (0, T ), z(0) = x(0)} − δ. Property (i), (5.217), and (5.218) imply that for every t ∈ [L0 , ∞) ∩ Γ ,

(5.219)

5.19 Proof of Theorem 5.28

117

ρ(x(t), B( θ , M1 − 1)) ≤ 4−1 and ρ(x(t),  θ ) ≤ M1 − 1/2.

(5.220)

Property (ix) and (5.217)–(5.220) imply that there exist τ0 ∈ [L, 2L] ∩ Γ, τ1 ∈ [T − 2L, T − L] ∩ Γ

(5.221)

such that for i = 0, 1 we have θ ) ≤ M1 − 1/2, ρ(x(τi ), 

(5.222)

ρ(x(τi ), Ω) ≤ δ, i = 0, 1.

(5.223)

There exists y ∈ Y (0, T ) such that y(0) = x(0),

(5.224)

φ(y(T )) ≥ φ(z(T )) − δ/4 for every z ∈ Y (0, T ) satisfying z(0) = y(0). (5.225) Properties (i) and (viii) and Eqs. (5.216)–(5.218), (5.224), and (5.225) imply that there exists τ2 ∈ [0, L0 + L2 ] ∩ Γ

(5.226)

such that ρ(y(t),  θ ) ≤ M1 − 1/2, ρ(y(t), Ω) ≤ δ1 , t ∈ [τ2 , T − L2 ],

(5.227)

and if (B1) and (B2) holds and ρ(x(0), Ω) ≤ δ, then τ2 = 0. By (5.216), (5.217), (5.221), (5.226), and (5.227), τ0 , τ1 ∈ [τ2 , T − L], θ ) ≤ M1 − 1/2, ρ(y(τi ), Ω) ≤ δ1 , i = 0, 1. ρ(y(τi ), 

(5.228) (5.229)

Thus we have shown that there exist τ0 , τ1 ∈ [0, T ] ∩ Γ such that (5.222), (5.223), and (5.229) hold, τ0 ≤ 2L, T − 2L ≤ τ1 ≤ T − L, and if (B1) and (B2) hold and ρ(x(0), Ω) ≤ δ, then τ0 = 0.

(5.230)

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5 General Dynamical Systems with a Lyapunov Function

We will show that for i = 0, 1, |φ(x(τi )) − φ(y(τi ))| ≤ γ0 .

(5.231)

Let i ∈ {0, 1}. Assume that (5.231) does not hold. Then |φ(x(τi )) − φ(y(τi ))| > γ0 .

(5.232)

Property (vii), (5.222), and (5.223) imply that there exist ξ1 , ξ2 ∈ Ω

(5.233)

ρ(x(τi ), ξj ) ≤ δ0 /16, j = 1, 2,

(5.234)

such that

and for each S ∈ (0, ∞) ∩ Γ and each Λ ∈ M, sup{Λ(z(S)) : z ∈ Y (0, S), z(0) = ξ1 } − δ0 /16 ≤ sup{Λ(z(S)) : z ∈ Y (0, S), z(0) = x(τi )} ≤ sup{Λ(z(S)) : z ∈ Y (0, S), z(0) = ξ2 } + δ0 /16.

(5.235)

Property (vii) and (5.229) imply that there exist η1 , η2 ∈ Ω

(5.236)

ρ(y(τi ), ηj ) ≤ δ0 /16, j = 1, 2

(5.237)

such that

and that for each S ∈ (0, ∞) ∩ Γ and each Λ ∈ M, sup{Λ(z(S)) : z ∈ Y (0, S), z(0) = η1 } − δ0 /16 ≤ sup{Λ(z(S)) : z ∈ Y (0, S), z(0) = y(τi )} ≤ sup{Λ(z(S)) : z ∈ Y (0, S), z(0) = η2 } + δ0 /16.

(5.238)

Property (v) and Eqs. (5.222), (5.229), (5.233), (5.234), (5.236), and (5.237) imply that for j = 1, 2, |φ(ξj ) − φ(x(τi ))| ≤ γ0 /8, |φ(y(τi )) − φ(ηj )| ≤ γ0 /8.

(5.239)

5.19 Proof of Theorem 5.28

119

For each pair p, q ∈ {1, 2}, it follows from (5.232) and (5.239) that |φ(ξp ) − φ(ηq )| ≥ |φ(x(τi )) − φ(y(τi ))| − |φ(x(τi )) − φ(ξp )| − |φ(y(τi )) − φ(ηq )| ≥ γ0 − γ0 /4.

(5.240)

Equations (5.239) and (5.240) imply that [min{φ(η1 ), φ(η2 )}, max{φ(η1 ), φ(η2 )}] ∩ [min{φ(ξ1 ), φ(ξ2 )}, max{φ(ξ1 ), φ(ξ2 )}] = ∅.

(5.241)

In view of (5.241), there are two cases: φ(η1 ) > φ(ξ2 );

(5.242)

φ(η2 ) < φ(ξ1 ).

(5.243)

Assume that (5.243) is true. By (5.240) and (5.243), φ(ξ1 ) − φ(η2 ) ≥ 3γ0 /4.

(5.244)

Property (vi) (with ξ1 = η2 , ξ2 = ξ1 ) and Eqs. (5.235) (with S = T − τi , Λ = φ), (5.222), (5.224), (5.225), (5.229), (5.230), (5.233), (5.234), (5.236), (5.237), (5.244), and (5.238) (with Λ = φ, S = T − τi ) imply that φ(y(T )) + δ/4 ≥ sup{φ(z(T )) : z ∈ Y (0, T ), z(0) = x(0)} ≥ sup{φ(z(T − τi )) : z ∈ Y (0, T − τi ), z(0) = x(τi )} ≥ sup{φ(z(T − τi )) : z ∈ Y (0, T − τi ), z(0) = ξ1 } − δ0 /16 ≥ sup{φ(z(T − τi )) : z ∈ Y (0, T − τi ), z(0) = η2 } + 2δ0 − δ0 /16, ≥ sup{φ(z(T − τi )) : z ∈ Y (0, T − τi ), z(0) = y(τi )} + 2δ0 − δ0 /8 ≥ φ(y(T )) + 2δ0 − δ0 /8. This contradicts the choice of δ. The contradiction we have reached proves that (5.242) holds. By (5.240) and (5.242), φ(η1 ) − φ(ξ2 ) > 3γ0 /4.

(5.245)

120

5 General Dynamical Systems with a Lyapunov Function

Property (vi) (with ξ1 = ξ2 , ξ2 = η1 ) and Eqs. (5.235) and (5.238) (with S = T −τi , Λ = ψ), (5.219), (5.222), (5.223), (5.229), (5.230), (5.233), (5.234), (5.236), (5.237), and (5.245) imply that ψ(x(T )) + δ ≥ sup{ψ(z(T )) : z ∈ Y (0, T ), z(0) = x(0)} ≥ sup{ψ(z(T − τi )) : z ∈ Y (0, T − τi ), z(0) = y(τi )} ≥ sup{ψ(z(T − τi )) : z ∈ Y (0, T − τi ), z(0) = η1 } − δ0 /16 ≥ sup{ψ(z(T − τi )) : z ∈ Y (0, T − τi ), z(0) = ξ2 } + 2δ0 − δ0 /16 ≥ sup{ψ(z(T − τi )) : z ∈ Y (0, T − τi ), z(0) = x(τi )} + 2δ0 − δ0 /8 ≥ ψ(x(T )) + 2δ0 − δ0 /8. This contradicts the choice of δ. The contradiction we have reached proves that |φ(x(τi )) − φ(y(τi ))| ≤ γ0 , i = 0, 1.

(5.246)

In view of (5.216), (5.217), and (5.230), τ1 − τ0 ≥ T − 4L ≥ 2L > 4L1 . Proposition 5.1 and Eqs. (5.1), (5.224), (5.225), (5.230), and (5.246) imply that φ(x(τ1 )) ≥ φ(y(τ1 )) − γ0 ≥ φ(y(T )) − γ0 ≥ π(x(0)) − δ/4 − γ0 ≥ π(x(τ0 )) − γ0 − δ/4.

(5.247)

Equations (5.222), (5.247), and (5.247) and property (iv) applied to the restriction of x to [τ0 , τ1 ] ∩ Γ imply that the value π(x(τ0 )) is finite and there exists τ ∈ [0, L1 ] ∩ Γ such that ρ(x(t), Ω) ≤ 1 /4, t ∈ [τ0 + τ, τ1 − L] ∩ Γ,

(5.248)

and if (B1) and (B2) hold and ρ(x(0), Ω) ≤ 2 , then τ = 0. Note that if ρ(x(0), Ω) ≤ δ and (B1) and (B2) hold, then τ0 = 0, τ = 0. Otherwise we assume that τ = L1 .

(5.249)

5.19 Proof of Theorem 5.28

121

By (5.222) and (5.223), there exist ξ˜i ∈ X, i = 0, 1, such that for i = 0, 1, θ ) ≤ M1 . ξ˜i ∈ Ω, ρ(x(τi ), ξ˜i ) ≤ δ, ρ(ξ˜i , 

(5.250)

Equations (5.223), (5.233), (5.234), and (5.250) and property (ii) applied with c = φ(ξ˜0 ) imply that there exists λ ∈ φ(B( θ , M1 ) ∩ Ω) ∩ [φ(ξ˜0 ) − /8, φ(ξ˜0 ) + /8]

(5.251)

such that the following property holds: (x) For each c˜ ∈ φ(B( θ , M1 ) ∩ Ω) ∩ [λ − 0 , λ + 0 ], the inequality ˜ ∩ B( θ , M1 ) ∩ Ω, φ −1 (λ) ∩ B( θ , M1 ) ∩ Ω) ≤ /4 dist(φ −1 (c) is true. Assume that t ∈ [τ0 + τ, τ1 − L] ∩ Γ.

(5.252)

Property (i) and Eqs. (5.220), (5.249), and (5.252) imply that ρ(x(t),  θ ) ≤ M1 − 1/2.

(5.253)

It follows from (5.248) and (5.252) that there exists ξ(t) ∈ Ω

(5.254)

ρ(x(t), ξ(t)) < 1 .

(5.255)

ρ(ξ(t),  θ ) ≤ M1 .

(5.256)

such that

By (5.253) and (5.255),

Proposition 5.1 and (5.252) imply that π(x(0)) ≥ π(x(τ0 )) ≥ π(x(t)) ≥ π(x(τ1 )).

(5.257)

122

5 General Dynamical Systems with a Lyapunov Function

Property (iii) and Eqs. (5.250) and (5.254)–(5.256) imply that |π(x(τi )) − π(ξ˜i )| ≤ 0 /8, i = 0, 1,

(5.258)

|φ(x(τi )) − φ(ξ˜i )| ≤ 0 /8, i = 0, 1,

(5.259)

|π(ξ(t)) − π(x(t))| ≤ 0 /8, |φ(ξ(t)) − φ(x(t))| ≤ 0 /8.

(5.260)

Corollary 5.9, (5.219), (5.250), and (5.257)–(5.259) imply that φ(x(τ0 )) ≤ φ(ξ˜0 ) + 0 /8 = π(ξ˜0 ) + 0 /8 ≤ π(x(τ0 )) + 0 /4 ≤ π(x(0)) + 0 /4 ≤ φ(x(τ )) + 0 /4 + δ ≤ φ(x(τ1 )) + 0 /4 + δ.

(5.261)

In view of (5.1) and (5.261), |φ(x(τ0 )) − π(x0 )| ≤ 0 /4.

(5.262)

|φ(ξ˜0 ) − π(x0 )| ≤ 0 /2.

(5.263)

By (5.259) and (5.262),

Equations (5.1), (5.261), and (5.262) imply that |φ(x(τ0 )) − φ(x(t))| ≤ 0 /4 + δ.

(5.264)

It follows from (5.259), (5.260), and (5.264) that |φ(x(τ0 )) − φ(x(t))| ≤ 0 /4 + δ.

(5.265)

It follows from (5.259), (5.260), and (5.264) that |φ(ξ˜0 ) − φ(ξ(t))| ≤ |φ(ξ˜0 ) − φ(x(τ0 ))| + |φ(x(τ0 )) − φ(x(t))| + |φ(x(t)) − φ(ξ(t))| ≤ 0 /8 + 0 /4 + δ + 0 /8 < 0 .

(5.266)

Property (x) (applied with c˜ = φ(ξ(t)) and Eqs. (5.254), (5.256), and (5.266) imply that ρ(ξ(t), φ −1 (λ) ∩ B( θ , M1 ) ∩ Ω) ≤ /4 and together with (5.255) this implies that

5.20 Continuity of the Function π

123

ρ(x(t), φ −1 (λ) ∩ B( θ , M1 ) ∩ Ω) ≤ .

(5.267)

It follows from (5.251), (5.260), and (5.266) that |φ(x(t)) − λ| ≤ |φ(x(t)) − φ(ξ(t))| + |φ(ξ(t)) − φ(ξ˜0 )| + |φ(ξ˜0 ) − λ| ≤ 0 /8 + 0 + /8 < .

(5.268)

Thus (5.267) and (5.268) hold for all t ∈ [τ0 + τ, τ1 − L1 ] ∩ Γ . By (5.251) and (5.263), |λ − π(x(0))| ≤ . This completes the proof of Theorem 5.28.

5.20 Continuity of the Function π In this section we consider a particular case when X is a vector space equipped with an inner product which induces the norm x = x, x1/2 , x ∈ X and the metric ρ(x, y) = x − y , x, y ∈ X. Assume that a mapping F : X → 2Y is proper (Dom(F ) = ∅). Recall (see Chap. 2) that for each pair of numbers T1 < T2 , Y (T1 , T2 ) is the set of all functions x ∈ W 1,1 (T1 , T2 ; X) such that x  (t) ∈ F (x(t)), t ∈ [T1 , T2 ] almost everywhere (a.e.). We also suppose that the assumptions made in Chap. 2 hold. Assume that X+ is a nonempty closed, convex cone in X such that λX+ ⊂ X+ for each λ ≥ 0. We assume that X is ordered by X+ : for each x, y ∈ X, x ≤ y if and only if y − x ∈ X+ . We suppose that for each x ∈ X, ∅ + x = ∅.

124

5 General Dynamical Systems with a Lyapunov Function

Proposition 5.30 Assume that for each x, y ∈ X satisfying x ≤ y, F (x) ⊂ F (y), φ(x) ≤ φ(y), ξ ∈ Ω, and that for each  > 0 there exists δ > 0 and ξ1 ∈ Ω such that ξ −ξ1 ≤  and B(ξ, δ) ⊂ ξ1 + X+ . Then π is lower semicontinuous at ξ . Proof Let  > 0. There exists 1 ∈ (0, ) such that |φ(z) − φ(ξ )| ≤ /4

(5.269)

for each z ∈ B(ξ, 1 ). There exist δ ∈ (0, 1 ), ξ1 ∈ Ω ∩ B(ξ, 1 )

(5.270)

such that the following property holds: (a) For each x ∈ B(ξ, δ), we have ξ1 ≤ x. Assume that x ∈ B(ξ, δ).

(5.271)

Corollary 5.9 and (5.270) imply that π(ξ1 ) = φ(ξ1 ).

(5.272)

Assume that T > 0 and z ∈ Y (0, T ) satisfies z(0) = ξ1 .

(5.273)

z (t) ∈ F (z(t)).

(5.274)

z1 (t) = z(t) − ξ1 + x, t ∈ [0, T ].

(5.275)

By definition for a.e. t ∈ [0, T ],

Define

Property (a) and Eqs. (5.271), (5.273), and (5.275) imply that z1 (0) = x, z1 (t) ≥ z(t), t ∈ [0, T ]. For a.e. t ∈ [0, T ], z1 (t) = z (t) ∈ F (z(t)) ⊂ F (z1 (t)).

5.20 Continuity of the Function π

125

It is not difficult to see that sup{φ(y(T )) : y ∈ Y (0, T ), y(0) = x} ≥ sup{φ(y(T )) : y ∈ Y (0, T ), y(0) = ξ1 }, φ(z1 (T )) ≥ φ(z(T )), π(x) ≥ π(ξ1 ). In view of (5.269), (5.270), and (5.272), for every x ∈ B(ξ, δ), π(x) ≥ π(ξ1 ) = φ(ξ1 ) ≥ φ(ξ ) − /4 ≥ π(ξ ) − /4. Proposition 5.30 is proved. Proposition 5.31 Assume that for each x, y ∈ X satisfying x ≤ y, F (x) ⊂ F (y), φ(x) ≤ φ(y), ξ ∈ Ω and that for each  > 0 there exist δ > 0 and ξ1 ∈ Ω such that ξ − ξ1 ≤  and B(ξ, δ) ⊂ ξ1 − X+ . Then π is upper semicontinuous at ξ . Proof Let  > 0. There exists 1 ∈ (0, ) such that |φ(z) − φ(ξ )| ≤ /4

(5.276)

for each z ∈ B(ξ, 1 ). There exist δ ∈ (0, 1 ), ξ1 ∈ Ω ∩ B(ξ, 1 ),

(5.277)

and for each x ∈ B(ξ, δ), we have x ≤ ξ1 .

(5.278)

x ∈ B(ξ, δ),

(5.279)

z(0) = x.

(5.280)

Assume that

T > 0, and z ∈ Y (0, T ) satisfies

By definition for a.e. t ∈ [0, T ], z (t) ∈ F (z(t)). Define

126

5 General Dynamical Systems with a Lyapunov Function

z1 (t) = z(t) + ξ1 − x, t ∈ [0, T ].

(5.281)

Equations (5.278), (5.279), and (5.281) imply that z1 (t) ≥ z(t), t ∈ [0, T ].

(5.282)

By (5.282), for a.e. t ∈ [0, T ], z1 (t) = z (t) ∈ F (z(t)) ⊂ F (z1 (t)). Clearly, φ(z1 (T )) ≥ φ(z(T )), sup{φ(y(T )) : y ∈ Y (0, T ), y(0) = ξ1 } ≥ sup{φ(y(T )) : y ∈ Y (0, T ), y(0) = x}, π(z1 ) ≥ π(x). Together with Corollary 5.9, (5.276), and (5.277), this implies that π(x) ≤ π(ξ1 ) = φ(ξ1 ) ≤ φ(ξ ) + /4 = π(ξ ) + /4, π(x) ≤ π(ξ ) + /4 for all x ∈ B(ξ, δ). Proposition 5.31 is proved.

5.21 Examples In this chapter we studied the class of dynamical systems satisfying assumptions (A1)–(A5). The most restrictive assumption is (A4). Here we present two examples of infinite dimensional systems for which (A4) holds. Both of them are discussed in [18]. Example 5.32 (The One-Dimensional Heat Equation) For f ∈ L∞ (0, 1), define S(t)f to be the unique solution u(·, t) of the problem ut = uxx , 0 < x < 1, t > 0, u = 0 at x = 0, 1, u(x, 0) = f (x).

5.21 Examples

127

Set x(0) = f , x(t) = S(t)f , t > 0. It was shown in [18] that for this dynamical system (A4) holds. Example 5.33 (A Semilinear Wave Equation) Let Ω ⊂ R n , n ≥ 3, be a bounded and open set with boundary ∂Ω. Consider the semilinear wave equation utt + βut − Δu + f (u) = 0 in Ω

(5.283)

with boundary condition u|∂Ω = 0

(5.284)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x),

(5.285)

and initial conditions

where β > 0 is a constant and a continuous function f : R 1 → R 1 satisfies the growth condition |f (u)| ≤ c0 (|u|n/n−2 + 1), where c0 > 0 is a constant and a condition lim inf f (u)/u ≥ −λ1 , |u|→∞

where λ1 is the first eigenvalue of the operator −Δ with the boundary conditions (5.284). Let X = H01 (Ω) × L2 (Ω). Then given (u0 , u1 ) ∈ X, there exists a weak solution x(·) = (u, ut ) on [0, ∞) satisfying (5.283)–(5.285). It was shown in [18] that for this dynamical system (A4) holds.

Chapter 6

Discrete-Time Nonautonomous Problems on Half-Axis

In this chapter we study the turnpike phenomenon for discrete-time nonautonomous problems on subintervals of half-axis in metric spaces, which are not necessarily compact. For these optimal control problems the turnpike is not a singleton. We show that the turnpike phenomenon is stable under small perturbations of an objective function. Using the Baire category approach we show that for some classes of problems a typical (generic) problem has a turnpike property.

6.1 Preliminaries and Boundedness Results Let (E, ρE ) and (F, ρF ) be metric spaces. The space E × F is equipped with the metric ρE×F ((x1 , u1 ), (x2 , u2 )) = ρE (x1 , x2 ) + ρF (u1 , u2 ), (xi , ui ) ∈ E × F, i = 1, 2. We suppose that A is a nonempty subset of {0, 1, . . . , } × E, U : A → 2F is a point to set mapping with a graph M = {(t, x, u) : (t, x) ∈ A, u ∈ U (t, x)},

(6.1)

G : M → E and f : M → R 1 . Let 0 ≤ T1 < T2 be integers. We denote by X(T1 , T2 ) the set of all pairs of T2 −1 2 sequences ({xt }Tt=T , {ut }t=T ) such that for each integer t ∈ {T1 , . . . , T2 }, 1 1 (t, xt ) ∈ A,

(6.2)

for each integer t ∈ {T1 , . . . , T2 − 1},

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0_6

129

130

6 Discrete-Time Nonautonomous Problems on Half-Axis

ut ∈ U (t, xt ),

(6.3)

xt+1 = G(t, xt , ut )

(6.4)

and which are called trajectory–control pairs. Let T1 ≥ 0 be an integer. Denote by X(T1 , ∞) the set of all pairs of ∞ sequences {xt }∞ t=T1 ⊂ E, {ut }t=T1 ⊂ F such that for each integer T2 > T1 , T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ). Elements of X(T1 , ∞) are called trajectory– 1 1

2 control pairs. Let 0 ≤ T1 < T2 be integers. A sequence {xt }Tt=T ⊂ E ({xt }∞ t=T1 ⊂ 1

T2 −1 E, respectively) is called a trajectory if there exists a sequence {ut }t=T ⊂ F 1

T2 T2 −1 ({ut }∞ t=T1 ⊂ F , respectively) referred to as a control such that ({xt }t=T1 , {ut }t=T1 ) ∈ ∞ ∞ X(T1 , T2 ) (({xt }t=T1 , {ut }t=T1 ) ∈ X(T1 , ∞), respectively). Let θ0 ∈ E, θ1 ∈ F , a0 > 0 and let ψ : [0, ∞) → [0, ∞) be an increasing function such that

ψ(t) → ∞ as t → ∞.

(6.5)

We suppose that the function f satisfies f (t, x, u) ≥ ψ(ρE (x, θ0 )) − a0 for each (t, x, u) ∈ M.

(6.6)

For each pair of integers T2 > T1 ≥ 0 and each pair of points y, z ∈ E satisfying (T1 , y), (T2 , z) ∈ A we consider the following problems: T 2 −1

T2 −1 2 f (t, xt , ut ) → min, ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), xT1 = y, xT2 = z, 1 1

t=T1 T 2 −1

(P1 ) T2 −1 2 f (t, xt , ut ) → min, ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), xT1 = y, 1 1

(P2 )

t=T1 T 2 −1

T2 −1 2 f (t, xt , ut ) → min, ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ). 1 1

(P3 )

t=T1

For each pair of integers T2 > T1 ≥ 0 and each pair of points y, z ∈ E satisfying (T1 , y), (T2 , z) ∈ A we define U (T1 , T2 , y, z) = inf{ f

T 2 −1

T2 −1 2 f (t, xt , ut ) : ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), 1 1

t=T1

xT1 = y, xT2 = z},

(6.7)

6.1 Preliminaries and Boundedness Results

131

U f (T1 , T2 , y) = inf{U f (T1 , T2 , y, h) : h ∈ E, (T2 , h) ∈ A},

(6.8)

f (T1 , T2 , z) = inf{U f (T1 , T2 , h, z) : h ∈ E, (T1 , h) ∈ A}, U

(6.9)

U f (T1 , T2 ) = inf{U f (T1 , T2 , h, ξ ) : h, ξ ∈ E, (T1 , h), (T2 , ξ ) ∈ A}.

(6.10)

∞ We suppose that bf > 0 is an integer, ({xt }∞ t=0 , {ut }t=0 ) ∈ X(0, ∞), f

f

f

{xt : t = 0, 1, . . . } is bounded , f

(6.11)

f

Δf := sup{|f (t, xt , ut )| : t = 0, 1, . . . } < ∞

(6.12)

and that the following assumptions hold. (A1) For each S1 > 0 there exist S2 > 0 and an integer c > 0 such that T 2 −1

f

f

f (t, xt , ut ) ≤

t=T1

T 2 −1

f (t, xt , ut ) + S2

t=T1

T2 −1 2 for each pair of integers T1 ≥ 0, T2 ≥ T1 + c and each ({xt }Tt=T , {ut }t=T )∈ 1 1 X(T1 , T2 ) satisfying ρE (θ0 , xj ) ≤ S1 , j = T1 , T2 . (A2) For each  > 0 there exists δ > 0 such that for each (Ti , zi ) ∈ A, i = 1, 2 f satisfying ρE (zi , xTi ) ≤ δ, i = 1, 2 and T2 ≥ bf there exist integers τ1 , τ2 ∈ (0, bf ], T1 +τ1 −1 1 +τ1 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T1 + τ1 ), 1 1 (1)

(1)

T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T2 − τ2 , T2 ) 2 −τ2 2 −τ2 (2)

(2)

such that (1)

f

(1)

xT1 = z1 , xT1 +τ1 = xT1 +τ1 , T1 +τ 1 −1 

(1)

(1)

f (t, xt , ut ) ≤

T1 +τ 1 −1 

t=T1

f

f

f

f

f (t, xt , ut ) + ,

t=T1 (2)

f

(2)

xT2 = z2 , xT2 −τ2 = xT2 −τ2 , T 2 −1 t=T2 −τ2

f (t, xt(2) , u(2) t )

≤

T 2 −1 t=T2 −τ2

f (t, xt , ut ) + .

132

6 Discrete-Time Nonautonomous Problems on Half-Axis

Section 6.6 contains examples of optimal control problems satisfying assumptions (A1) and (A2). Many examples can also be found in [119–121, 131, 137, 138, 147]. The following result is proved in Section 3.8 of [148]. Theorem 6.1 1. There exists S > 0 such that for each pair of integers T2 > T1 ≥ 0 and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), 1 1 T 2 −1

f (t, xt , ut ) + S ≥

T 2 −1

t=T1

f

f

f (t, xt , ut ).

t=T1

∞ 2. For each ({xt }∞ t=0 , {ut }t=0 ) ∈ X(0, ∞) either T −1

f (t, xt , ut ) −

t=0

T −1

f

f

f (t, xt , ut ) → ∞ as T → ∞

t=0

or sup{|

T −1

f (t, xt , ut ) −

t=0

T −1

f

f

f (t, xt , ut )| : T ∈ {1, 2, . . . }} < ∞.

(6.13)

t=0

Moreover, if (6.13) holds, then sup{ρE (xt , θ0 ) : t = 0, 1, . . . } < ∞. ∞ We say that ({xt }∞ t=0 , {ut }t=0 ) ∈ X(0, ∞) is (f )-goof if (6.13) holds [31, 117, 129, 133, 142, 147]. The next boundedness result is proved in Section 3.8 [148]. It has a prototype in [117].

Theorem 6.2 Let c > 0 be an integer and M0 > 0. Then there exists M1 > 0 such T2 −1 2 , {ut }t=T )∈ that for each pair of integers T1 ≥ 0, T2 ≥ T1 + c and each ({xt }Tt=T 1 1 X(T1 , T2 ) satisfying T 2 −1 t=T1

f (t, xt , ut ) ≤

T 2 −1

f

f

f (t, xt , ut ) + M0

t=T1

the inequality ρE (θ0 , xt ) ≤ M1 holds for all t = T1 , . . . , T2 − 1. Let L > 0 be an integer. Denote by AL the set of all (S, z) ∈ A for which there S+τ −1 ) ∈ X(S, S + τ ) such that exists an integer τ ∈ (0, L] and ({xt }S+τ t=S , {ut }t=S

6.1 Preliminaries and Boundedness Results

f

xS = z, xS+τ = xS+τ ,

133 S+τ −1

f (t, xt , ut ) ≤ L.

t=S

L the set of all (S, z) ∈ A such that S ≥ L and there exists an integer Denote by A τ ∈ (0, L] and ({xt }St=S−τ , {ut }S−1 t=S−τ ) ∈ X(S − τ, S) satisfying f

xS−τ = xS−τ , xS = z,

S−1 

f (t, xt , ut ) ≤ L.

t=S−τ

The following Theorems 6.3–6.5 are also boundedness results. They are proved in Section 3.8 of [148]. Theorem 6.3 Let L > 0 be an integer and M0 > 0. Then there exists M1 > 0 such that for each integer T1 ≥ 0, each integer T2 ≥ T1 + 2L and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfying 1 1 L , (T1 , xT1 ) ∈ AL , (T2 , xT2 ) ∈ A T 2 −1

f (t, xt , ut ) ≤ U f (T1 , T2 , xT1 , xT2 ) + M0

t=T1

the inequality ρE (θ0 , xt ) ≤ M1 holds for all t = T1 , . . . , T2 − 1. Theorem 6.4 Let L > 0 be an integer and M0 > 0. Then there exists M1 > 0 such that for each integer T1 ≥ 0, each integer T2 ≥ T1 + L and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfying 1 1 (T1 , xT1 ) ∈ AL ,

T 2 −1

f (t, xt , ut ) ≤ U f (T1 , T2 , xT1 ) + M0

t=T1

the inequality ρE (θ0 , xt ) ≤ M1 holds for all t = T1 , . . . , T2 − 1. Theorem 6.5 Let L > 0 be an integer and M0 > 0. Then there exists M1 > 0 such that for each integer T1 ≥ 0, each integer T2 ≥ T1 + L and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfying 1 1 L , (T2 , xT2 ) ∈ A

T 2 −1

f (T1 , T2 , xT2 ) + M0 f (t, xt , ut ) ≤ U

t=T1

the inequality ρE (θ0 , xt ) ≤ M1 holds for all t = T1 , . . . , T2 − 1.

134

6 Discrete-Time Nonautonomous Problems on Half-Axis

6.2 Turnpike Properties We say that f possesses the turnpike property (or TP for short) if for each  > 0 and each M > 0 there exist δ > 0 and an integer L > 0 such that for each integer T2 −1 2 T1 ≥ 0, each integer T2 ≥ T1 +2L and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which 1 1 satisfies T 2 −1

f (t, xt , ut ) ≤ min{U f (T1 , T2 ) + M, U f (T1 , T2 , xT1 , xT2 ) + δ}

t=T1

there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 . . . , τ2 . f

f

Moreover, if ρE (xT2 , xT2 ) ≤ δ, then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ, T1 ≥ L, then τ1 = T1 . We say that f possesses the strong turnpike property (or STP for short) if for each  > 0 and each M > 0 there exist δ > 0 and an integer L > 0 such that for T2 −1 2 each integer T1 ≥ 0, each integer T2 ≥ T1 + 2L and each ({xt }Tt=T , {ut }t=T ) ∈ 1 1 X(T1 , T2 ) which satisfies T 2 −1

f (t, xt , ut ) ≤ min{U f (T1 , T2 ) + M, U f (T1 , T2 , xT1 , xT2 ) + δ}

t=T1

there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 . . . , τ2 . f

f

Moreover, if ρE (xT2 , xT2 ) ≤ δ, then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ, then τ1 = T1 . Theorem 6.1 implies the following two results. Theorem 6.6 Assume that f has TP and that , M > 0. Then there exist δ > 0 and an integer L > 0 such that for each integer T1 ≥ 0, each integer T2 ≥ T1 + 2L and T2 −1 2 each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 T 2 −1 t=T1

f (t, xt , ut ) ≤ min{

T 2 −1

f

f

f (t, xt , ut ) + M, U f (T1 , T2 , xT1 , xT2 ) + δ}

t=T1

there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 , . . . , τ2 .

6.2 Turnpike Properties

135 f

f

Moreover, if ρE (xT2 , xT2 ) ≤ δ, then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ, T1 ≥ L, then τ1 = T1 . Theorem 6.7 Assume that f has STP and that , M > 0. Then there exist δ > 0 and an integer L > 0 such that for each integer T1 ≥ 0, each integer T2 ≥ T1 + 2L T2 −1 2 and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 T 2 −1

f (t, xt , ut ) ≤ min{

t=T1

T 2 −1

f

f

f (t, xt , ut ) + M, U f (T1 , T2 , xT1 , xT2 ) + δ}

t=T1

there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 , . . . , τ2 . f

f

Moreover, if ρE (xT2 , xT2 ) ≤ δ, then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ, then τ1 = T1 . It is not difficult to show (see Section 3.3 of [148]) that the following result is true. Proposition 6.8 Let L > 0 be an integer, T1 ≥ 0, T2 ≥ T1 + 2L be integers, L . Then (T1 , z1 ) ∈ AL , (T2 , z2 ) ∈ A U f (T1 , T2 , z1 , z2 ) ≤

T 2 −1

f

f

f (t, xt , ut ) + 2L(1 + a0 ).

t=T1

Proposition 6.8 and Theorems 6.6 and 6.7 imply the following two results. Theorem 6.9 Assume that f has TP, L0 > 0 is an integer and that  > 0. Then there exist δ > 0 and an integer L > L0 such that for each integer T1 ≥ 0, each T2 −1 2 integer T2 ≥ T1 + 2L and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 L0 , (T1 , xT1 ) ∈ AL0 , (T2 , xT2 ) ∈ A T 2 −1

f (t, xt , ut ) ≤ U f (T1 , T2 , xT1 , xT2 ) + δ

t=T1

there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 , . . . , τ2 . f

f

Moreover, if ρE (xT2 , xT2 ) ≤ δ then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ, T1 ≥ L, then τ1 = T1 .

136

6 Discrete-Time Nonautonomous Problems on Half-Axis

Theorem 6.10 Assume that f has STP, L0 > 0 is an integer and that  > 0. Then there exist δ > 0 and an integer L > L0 such that for each integer T1 ≥ 0, each T2 −1 2 , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies integer T2 ≥ T1 + 2L and each ({xt }Tt=T 1 1 L0 , (T1 , xT1 ) ∈ AL0 , (T2 , xT2 ) ∈ A T 2 −1

f (t, xt , ut ) ≤ U f (T1 , T2 , xT1 , xT2 ) + δ

t=T1

there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 , . . . , τ2 . f

f

Moreover, if ρE (xT2 , xT2 ) ≤ δ then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ, then τ1 = T1 . It is not difficult to show (see Section 3.3 of [148]) that the following result is true. Proposition 6.11 Let L > 0 be an integer, (T1 , z) ∈ AL , T2 ≥ T1 +L be an integer. Then U (T1 , T2 , z) ≤ f

T 2 −1

f

f

f (t, xt , ut ) + L(1 + a0 ).

t=T1

Proposition 6.11 and Theorems 6.6, 6.7 imply the following two results. Theorem 6.12 Assume that f has TP, L0 > 0 is an integer and that  > 0. Then there exist δ > 0 and an integer L > L0 such that for each integer T1 ≥ 0, each T2 −1 2 integer T2 ≥ T1 + 2L and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 (T1 , xT1 ) ∈ AL0 ,

T 2 −1

f (t, xt , ut ) ≤ U f (T1 , T2 , xT1 ) + δ

t=T1

there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 , . . . , τ2 . f

f

Moreover, if ρE (xT2 , xT2 ) ≤ δ then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ, T1 ≥ L, then τ1 = T1 . Theorem 6.13 Assume that f has STP, L0 > 0 is an integer and that  > 0. Then there exist δ > 0 and an integer L > L0 such that for each integer T1 ≥ 0, each T2 −1 2 , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies integer T2 ≥ T1 + 2L and each ({xt }Tt=T 1 1

6.2 Turnpike Properties

137

(T1 , xT1 ) ∈ AL ,

T 2 −1

f (t, xt , ut ) ≤ U f (T1 , T2 , xT1 ) + δ

t=T1

there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 , . . . , τ2 . f

f

Moreover, if ρE (xT2 , xT2 ) ≤ δ then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ, then τ1 = T1 . The next theorem is proved in Section 3.10 of [148]. Theorem 6.14 f has TP if and only if the following properties hold: ∞ (P1) for each (f )-good pair of sequences ({xt }∞ t=0 , {ut }t=0 ) ∈ X(0, ∞), f

lim ρE (xt , xt ) = 0;

t→∞

(P2) for each  > 0 and each M > 0 there exist δ > 0 and an integer L > 0 such T +L T +L−1 that for each integer T ≥ 0 and each ({xt }t=T , {ut }t=T ) ∈ X(T , T + L) which satisfies T +L−1 

f (t, xt , ut ) ≤ min{U f (T , T + L, xT , xT +L ) + δ,

t=T T +L−1 

f

f

f (t, xt , ut ) + M}

t=T f

there exists an integer s ∈ [T , T + L] such that ρE (xs , xs ) ≤ . We say that f possesses the weak turnpike property (or WTP for short) if for each  > 0 and each M > 0 there exist natural numbers Q, l such that for each integer T2 −1 2 T1 ≥ 0, each integer T2 ≥ T1 +lQ and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which 1 1 satisfies T 2 −1 t=T1

f (t, xt , ut ) ≤

T 2 −1

f

f

f (t, xt , ut ) + M

t=T1 q

q

there exist finite sequences of integers {ai }i=1 , {bi }i=1 ⊂ {T1 , . . . , T2 } such that an integer q ≤ Q, 0 ≤ bi − ai ≤ l, i = 1, . . . , q, bi ≤ ai+1 for all integers i satisfying 1 ≤ i < q, f

q

ρE (xt , xt ) ≤  for all integers t ∈ [T1 , T2 ] \ ∪i=1 [ai , bi ].

138

6 Discrete-Time Nonautonomous Problems on Half-Axis

WTP was studied in [108, 109] for discrete-time unconstrained problems, in [110] for variational problems and in [127] for discrete-time constrained problems. The next result is proved in Section 3.12 of [148]. Theorem 6.15 f has WTP if and only if f has (P1) and (P2). The proof of Theorem 6.14 is based on the following result (see Lemma 3.35 of [148]). Lemma 6.16 Assume that (P1) holds and that  > 0. Then there exists δ > 0 and an integer L > 0 such that for each integer T1 ≥ L, each integer T2 ≥ T1 + 2bf T2 −1 2 and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 f

ρE (xTi , xTi ) ≤ δ, i = 1, 2, T 2 −1

f (t, xt , ut ) ≤ U f (T1 , T2 , xT1 , xT2 ) + δ

t=T1 f

the inequality ρE (xt , xt ) ≤  holds for all t = T1 , . . . , T2 .

6.3 Perturbed Problems In this section we suppose that the following assumption holds. (A3) For each  > 0 there exists δ > 0 such that for each (Ti , zi ) ∈ A, i = 1, 2 f satisfying ρE (zi , xTi ) ≤ δ, i = 1, 2 and T2 ≥ bf there exist integers τ1 , τ2 ∈ (0, bf ], T1 +τ1 −1 1 +τ1 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T1 + τ1 ), 1 1 (1)

(1)

T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T2 − τ2 , T2 ) 2 −τ2 2 −τ2 (2)

(2)

such that (1)

f

(1)

xT1 = z1 , xT1 +τ1 = xT1 +τ1 , T1 +τ 1 −1 

(1)

(1)

f (t, xt , ut ) ≤

t=T1

T1 +τ 1 −1 

f

f

f (t, xt , ut ) + ,

t=T1 (1)

f

ρ(xt , xt ) ≤ , t = T1 , . . . , T1 + τ1 ,

6.4 Examples

139 f

xT(2) = z2 , xT(2) = xT2 −τ2 , 2 2 −τ2 T 2 −1

f (t, xt(2) , u(2) t )≤

t=T2 −τ2

T 2 −1

f

f

f (t, xt , ut ) + ,

t=T2 −τ2 f

ρ(xt(2) , xt ) ≤ , t = T2 − τ2 , . . . , T2 . Clearly, (A3) implies (A2). Assume that φ : E×E → [0, 1] is a continuous function satisfying φ(x, x) = 0 for all x ∈ E and such that the following properties hold: (i) for each  > 0 there exists δ > 0 such that if x, y ∈ E and φ(x, y) ≤ δ, then ρE (x, y) ≤ . (ii) for each  > 0 there exists δ > 0 such that if x, y ∈ E and ρE (x, y) ≤ δ, then φ(x, y) ≤ . For each r ∈ (0, 1) set f

fr (t, x, u) = f (t, x, u) + rφ(x, xt ), (t, x, u) ∈ M. f

f

f

f

Clearly, for any r ∈ (0, 1), (A1), (A3) hold for fr with (xt r , ut r ) = (xt , ut ), t = 0, 1, . . . . The next result is proved in Section 3.17 of [148]. Theorem 6.17 Let r ∈ (0, 1). Then fr has TP, (P1), and (P2).

6.4 Examples Example 6.18 We consider an example which is a particular case of the problem introduced in Sect. 6.1. Let a1 > 0, ψ1 : [0, ∞) → [0, ∞) be an increasing function such that ψ1 (t) → ∞ as t → ∞, μ : {0, 1, . . . } → R 1 , π : {0, 1, . . . } × E → R 1 , L : M → [0, ∞), π(t, x) ≥ −a1 for all (t, x) ∈ {0, 1, . . . } × E, π be bounded on bounded subsets of {0, 1, . . . } × E, μ(t) ≥ −a1 for all t ∈ {0, 1, . . . }, and for all (t, x, u) ∈ M, L(t, x, u) − π(t, G(t, x, u)) ≥ −a1 + ψ1 (ρE (θ0 , x)), f

f

L(t, x, u) = 0 if and only if x = xt , u = ut ,

(6.14) (6.15)

140

6 Discrete-Time Nonautonomous Problems on Half-Axis

f (t, x, u) = μ(t) + L(t, x, u) + π(t, x) − π(t + 1, G(t, x, u)).

(6.16)

By the relation above, for all (t, x, u) ∈ M. f (t, x, u) ≥ −3a1 + ψ1 (ρE (θ0 , x)) and (6.6) holds with a0 = 3a1 , ψ0 = ψ1 . T2 −1 2 , {ut }t=T ) ∈ X(T1 , T2 ). It is not Let T2 > T1 ≥ 0 be integers and ({xt }Tt=T 1 1 difficult to see that T 2 −1

f (t, xt , ut ) =

t=T1

T 2 −1

μ(t) +

t=T1

=

T 2 −1

L(t, xt , ut ) +

t=T1

μ(t) +

t=T1 T 2 −1

T 2 −1

T 2 −1

T 2 −1

(π(t, xt ) − π(t + 1, xt+1 ))

t=T1

L(t, xt , ut ) + π(T1 , xT1 ) − π(T2 , xT2 ),

t=T1

f

f

f (t, xt , ut ) =

t=T1

T 2 −1

f

f

μ(t) + π(T1 , xT1 ) − π(T2 , xT2 )

t=T1

and T 2 −1

f (t, xt , ut ) −

t=T1

T 2 −1

f

f

f (t, xt , ut )

t=T1 f

f

≥ π(T1 , xT1 ) − π(T2 , xT2 ) − π(T1 , xT1 ) + π(T2 , xT2 ). Since π is bounded on bounded subsets of {0, 1, . . . , } × E we conclude that (A1) holds. We suppose that the following assumptions hold: (B1) for each  > 0 there exists δ > 0 such that for each integer T ≥ 0, each pair f f (T , z1 ), (T +1, z2 ) ∈ A satisfying ρE (z1 , xT ), ρE (z2 , xT +1 ) ≤ δ, there exists u ∈ U (T , z1 ) such that f

ρF (u, uT ) ≤ , z2 = G(T , z1 , u); (B2) for each  > 0 there exists δ > 0 such that for each (T , z, ξ ) ∈ M satisfying f f f f ρE (z, xT ) ≤ δ, ρF (ξ, uT ) ≤ δ, we have f (T , z, ξ ) ≤ f (T , xT , uT ) + . It is clear that (B1) and (B2) imply (A2) and (A3). Example 6.19 We consider an example which is a particular case of the problem introduced in Sect. 6.1. Let a1 > 0, ψ1 : [0, ∞) → [0, ∞) be an increasing function

6.5 The Space of Objective Functions and the Stability Result

141

such that ψ1 (t) → ∞ as t → ∞, μ : {0, 1, . . . } → R 1 , π : {0, 1, . . . } × E → R 1 , L : M → [0, ∞), π(t, x) ≥ −a1 for all (t, x) ∈ {0, 1, . . . } × E, π be bounded on bounded subsets of {0, 1, . . . } × E, μ(t) ≥ −a1 for all t ∈ {0, 1, . . . }, and for all (t, x, u) ∈ M, (6.14)–(6.16) hold. Then as it was shown in Example 6.18, (A1) and (6.6) hold with a0 = 3a1 , ψ0 = ψ1 . We suppose that there exists a natural number df such that the following assumptions hold: (B3) for each  > 0 there exists δ > 0 such that for each integer T ≥ 0, each pair f f (T , z1 ), (T + df , z2 ) ∈ A satisfying ρE (z1 , xT ), ρE (z2 , xT +df ) ≤ δ, there T +d

T +d −1

exists ({xt }t=T f , {ut }t=T f

) ∈ X(T , T + df ) such that f

xT = z1 , xT +df = z2 , ρ(ut , ut ) ≤ , t = T , . . . , T + df − 1; (B4) for each  > 0 there exists δ > 0 such that for each (T , z) ∈ A satisfying f f ρE (z, xT ) ≤ δ and each u ∈ U (T , z) satisfying, ρF (u, uT ) ≤ δ, we have f

f

f

ρE (xT +1 , G(T , z, u)) ≤ , f (T , z, u) ≤ f (T , xT , uT ) + . (B3) and (B4) imply (A2) and (A3). Assume now that for any nonempty bounded set Ω ⊂ E the function π is bounded on {0, 1, . . . , } × Ω, the function μ is bounded and that the following property holds: for each M,  > 0 there exists δ > 0 such that for each (t, x, u) ∈ M which satisfies ρE (θ0 , x) ≤ M and L(t, x, u) ≤ δ f

we have ρE (x, xt ) ≤ . It was shown in Section 3.6 of [148] that f has TP.

6.5 The Space of Objective Functions and the Stability Result Denote by Mψ the set of all functions g : M → R 1 , which satisfy g(t, x, u) ≥ ψ(ρE (x, θ0 )) − a0 for each (t, x, u) ∈ M.

(6.17)

142

6 Discrete-Time Nonautonomous Problems on Half-Axis

For each g ∈ Mψ and each pair of integers T2 > T1 ≥ 0 and each pair of points y, z ∈ E satisfying (T1 , y), (T2 , z) ∈ A we define U (T1 , T2 , y, z) = inf{ g

T 2 −1

T2 −1 2 g(t, xt , ut ) : ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), 1 1

t=T1

xT1 = y, xT2 = z},

U g (T1 , T2 , y) = inf{

T 2 −1

g(t, xt , ut ) :

t=T1 T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), xT1 = y}, 1 1

U g (T1 , T2 , z) = inf{

T 2 −1

g(t, xt , ut ) :

t=T1 T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), xT2 = z}, 1 1

U f (T1 , T2 ) = inf{

T 2 −1

T2 −1 2 g(t, xt , ut ) : ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 )}. 1 1

(6.18)

t=T1

We assume that f : M → R 1 satisfies all the assumptions introduced in Sect. 6.1 (see (A1) and (A2)). We equip the space Mψ with the uniformity determined by the base E(N, ) = {(g1 , g2 ) ∈ Mψ × Mψ : |g1 (t, x, u) − g2 (t, x.u)| ≤  for each (t, x, u) ∈ M satisfying ρE (x, θ0 ), ρF (u, θ1 ) ≤ N} ∩ {(g1 , g2 ) ∈ Mψ × Mψ : |g1 (t, x, u) − g2 (t, x, u)| ≤  for each (t, x, u) ∈ M satisfying min{g1 (t, x, u), g2 (t, x, u)} ≤ N},

(6.19)

where N,  > 0. Clearly, the uniform space Mψ is Hausdorff and has a countable base. Therefore Mψ is metrizable (by a metric dψ ). It is not difficult to see that the uniform space Mψ is complete. It is equipped with a topology induced by the uniformity which is called sometimes as a weak topology. The following proposition is easily deduced from growth condition (6.17).

6.5 The Space of Objective Functions and the Stability Result

143

Proposition 6.20 For each integer M0 > 0 there exists M1 > 0 such that for each integer T1 ≥ 0, each integer T2 ∈ [T1 , T1 + M0 ], each g ∈ Mψ and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 T 2 −1

g(t, xt , ut ) ≤ M0

t=T1

the inequality ρE (xt , θ0 ) ≤ M1 holds for every t ∈ {T1 , . . . , T2 − 1}. Proposition 6.21 Let g ∈ Mψ and D, c,  > 0. Then there exists a neighborhood V of g in Mψ such that for each pair of integers T1 ≥ 0, T2 ∈ (T1 , T1 + c], each T2 −1 2 h ∈ V and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 min{

T 2 −1

g(t, xt , ut ),

t=T1

T 2 −1

h(t, xt , ut )} ≤ D

(6.20)

t=T1

the inequality |

T 2 −1 t=T1

g(t, xt , ut ) −

T 2 −1

h(t, xt , ut )| ≤ 

t=T1

holds. Proof Fix N > D + a0 c, δ ∈ (0, min{1, /c}).

(6.21)

V = {h ∈ Mψ : (g, h) ∈ E(N, δ)}.

(6.22)

Define

Assume that h ∈ V , T1 ≥ 0 and T2 ∈ (T1 , T1 + c] are integers and that T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfies (6.20). Let 1 1 j ∈ {T1 , . . . , T2 − 1}. By (6.17) and (6.20), min{g(j, xj , uj ), h(j, xj , uj )} ≤ D + a0 c. Together with (6.19), (6.21), (6.22) and the inclusion h ∈ V this implies that |g(j, xj , uj ) − h(j, xj , uj )| ≤ δ.

144

6 Discrete-Time Nonautonomous Problems on Half-Axis

Combined with (6.21) this implies that |

T 2 −1

g(t, xt , ut ) −

T 2 −1

t=T1

h(t, xt , ut )| ≤ δc < .

t=T1

Proposition 6.21 is proved. In Sect. 6.7 we prove the following result which shows the stability of the turnpike phenomenon under small perturbations of the objective function. Theorem 6.22 Assume that f ∈ Mψ has TP, L is a natural number and  ∈ (0, 1). Then there exists an integer L1 > L, δ ∈ (0, ) and a neighborhood V of f in Mψ such that for each g ∈ V and each pair of integers T1 ≥ 0, T2 ≥ T1 + 4L1 , U g (T1 , T2 , z) is finite for each z ∈ E satisfying (T1 , z) ∈ AL , U g (T1 , T2 , z1 , z2 ) is L and that for finite for each z1 , z2 ∈ E satisfying (T1 , z1 ) ∈ AL and (T2 , z2 ) ∈ A T2 T2 −1 each ({xt }t=T1 , {ut }t=T1 ) ∈ X(T1 , T2 ) which satisfies at least one of the following conditions: (i) T 2 −1

g(t, xt , ut ) ≤ min{U g (T1 , T2 ) + L, U g (T1 , T2 , xT1 , xT2 ) + δ};

t=T1

(ii) T 2 −1

g(t, xt , ut ) ≤ U g (T1 , T2 , xT1 ), (T1 , xT1 ) ∈ AL ;

t=T1

(iii) T 2 −1

g(t, xt , ut ) ≤ U g (T1 , T2 , xT1 , xT2 ) + δ,

t=T1

L (T1 , xT1 ) ∈ AL , (T2 , xT2 ) ∈ A there exist integers τ1 ∈ [T1 , T1 + L1 ], τ2 ∈ [T2 − L1 , T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 . . . , τ2 . f

Moreover, if ρE (xT2 , xT2 ) ≤ δ, then τ2 = T2 and if T1 ≥ L1 and f

ρE (xT1 , xT1 ) ≤ δ, then τ1 = T1 .

6.6 Auxiliary Results

145

It is clear that U g (T1 , T2 , z) is finite for each (T1 , z) ∈ AL if T2 − T1 ≥ L L if and U g (T1 , T2 , z1 , z2 ) is finite for each (T1 , z1 ) ∈ AL and each (T2 , z2 ) ∈ A T2 − T1 ≥ 2L.

6.6 Auxiliary Results Lemma 6.23 Assume that f ∈ Mψ has TP, M > 0 and  ∈ (0, 1). Then there exists an integer L0 ≥ 4 such that for each integer L1 > L0 there exists a neighborhood V of f in Mψ such that for each g ∈ V and each pair of integers T2 −1 2 , {ut }t=T ) ∈ X(T1 , T2 ) which T1 ≥ 0, T2 ∈ (T1 , T1 + L1 ] and for each ({xt }Tt=T 1 1 satisfies T 2 −1

g(t, xt , ut ) ≤

t=T1

T 2 −1

f

f

f (t, xt , ut ) + M

(6.23)

t=T1

the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) > }) ≤ L0 /2 holds. Proof Theorems 6.14 and 6.15 imply that there exists an integer L0 ≥ 4 such that the following property holds: (a) for each pair of integers T1 ≥ 0, T2 ≥ T1 + L0 /2 and for each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 T 2 −1 t=T1

f (t, xt , ut ) ≤

T 2 −1

f

f

f (t, xt , ut ) + M + 4

t=T1

the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) > }) ≤ L0 /4 holds. Let L1 > L0 be an integer. Proposition 6.21 implies that there exists a neighborhood V of f in Mψ such that the following property holds: (b) for each pair of integers T1 ≥ 0, T2 ∈ (T1 , T1 + L1 ], each g ∈ V and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1

146

6 Discrete-Time Nonautonomous Problems on Half-Axis

min{

T 2 −1

f (t, xt , ut ),

t=T1

T 2 −1

g(t, xt , ut )} ≤ M + L1 Δf

t=T1

the inequality |

T 2 −1

f (t, xt , ut ) −

t=T1

T 2 −1

g(t, xt , ut )| ≤ 1

(6.24)

t=T1

holds. Assume that T1 ≥ 0, T2 ∈ (T1 , T1 + L1 ] are integers, g ∈ V and that T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfies (6.23). By (6.12) and (6.23), 1 1 T 2 −1

g(t, xt , ut ) ≤ M + Δf L1 .

t=T1

Together with property (b) this implies (6.24). By (6.23) and (6.24), T 2 −1

f (t, xt , ut ) ≤

t=T1

T 2 −1

g(t, xt , ut ) + 1 ≤

t=T1

T 2 −1

f

f

f (t, xt , ut ) + M + 1.

t=T1

Combined with property (a) this implies that f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) > }) ≤ L0 /2. Lemma 6.23 is proved. Lemma 6.24 Assume that f ∈ Mψ has TP,  > 0, M ≥ 8 and L is a natural number. Then there exist integers L0 ≥ L + 4, L1 > 8L0 + 2L0 L(a0 + 1) and a neighborhood V of f in Mψ such that for each g ∈ V , each pair of integers T2 −1 2 , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies at T1 ≥ 0, T2 ≥ T1 + L1 and each ({xt }Tt=T 1 1 least one of the following conditions: (i) T 2 −1 t=T1

g(t, xt , ut ) ≤ U g (T1 , T2 ) + M;

6.6 Auxiliary Results

147

(ii) T 2 −1

g(t, xt , ut ) ≤ U g (T1 , T2 , xT1 ) + M, (T1 , xT1 ) ∈ AL ;

t=T1

(iii) T 2 −1

g(t, xt , ut ) ≤ U g (T1 , T2 , xT1 , xT2 ) + M,

t=T1

L ; (T1 , xT1 ) ∈ AL , (T2 , xT2 ) ∈ A (iv) T 2 −1

L g (T1 , T2 , xT2 ) + M, (T2 , xT2 ) ∈ A g(t, xt , ut ) ≤ U

t=T1

the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) ≤ }) ≥ L0 /2 holds. Proof Lemma 6.23 implies that there exists an integer L0 ≥ 4 + L and a neighborhood V0 of f in Mψ such that the following property holds: (a) for each g ∈ V0 and each pair of integers T1 ≥ 0, T2 ∈ (T1 , T1 + 8(L0 + 1)] T2 −1 2 and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 T 2 −1 t=T1

g(t, xt , ut ) ≤

T 2 −1

f

f

f (t, xt , ut ) + M

t=T1

the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) > }) ≤ L0 /2 is true. Proposition 6.21 implies that there exists a neighborhood V ⊂ V0 of f in Mψ such that the following property holds: (b) for each pair of integers T1 ≥ 0, T2 ∈ (T1 , T1 + 8L0 ], each g ∈ V and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1

148

6 Discrete-Time Nonautonomous Problems on Half-Axis

min{

T 2 −1

f (t, xt , ut ),

t=T1

T 2 −1

g(t, xt , ut )} ≤ 2L0 (Δf + 1) + L(1 + a0 )

t=T1

the inequality |

T 2 −1

f (t, xt , ut ) −

t=T1

T 2 −1

g(t, xt , ut )| ≤ 8−1

t=T1

is true. Choose an integer L1 > L0 (8 + 2L(a0 + 1)).

(6.25)

g ∈ V , T1 ≥ 0, T2 ≥ T1 + L1

(6.26)

Assume that

T2 −1 2 are integers and that ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfies at least one of 1 1 T2 −1 2 conditions (i)–(iv). We construct ({ xt }Tt=T , { ut }t=T ) ∈ X(T1 , T2 ). If condition 1 1 (i) holds we set f

f

 xt = xt , t = T1 , . . . , T2 ,  ut = ut , t = T1 , . . . , T2 − 1. T2 −1 2 If condition (ii) holds, then there exist ({ xt }Tt=T , { ut }t=T ) ∈ X(T1 , T2 ) and an 1 1 integer τ1 ∈ (0, L] such that f

 xT1 = xT1 ,

T1 +τ 1 −1 

f (t,  xt ,  ut ) ≤ L,

t=T1 f

 xt = xt , t = T + τ1 , . . . , T2 , f

 ut = ut , t = T + τ1 , . . . , T2 − 1. T2 −1 2 If condition (iii) holds, then there exist ({ xt }Tt=T , { ut }t=T ) ∈ X(T1 , T2 ) and 1 1 integers τ1 , τ2 ∈ (0, L] such that

 xT1 = xT1 ,  xT2 = xT2 , f

f

 xt = xt , t = T + τ1 , . . . , T2 − τ2 ,  ut = ut , t = T + τ1 , . . . , T2 − τ2 − 1, T1 +τ 1 −1  t=T1

f (t,  xt ,  ut ) ≤ L,

T 2 −1 t=T2 −τ2

f (t,  xt ,  ut ) ≤ L.

6.6 Auxiliary Results

149

T2 −1 2 If condition (iv) holds, then there exist ({ xt }Tt=T , { ut }t=T ) ∈ X(T1 , T2 ) and an 1 1 integer τ1 ∈ (0, L] such that

 xT2 = xT2 , f

f

 xt = xt , t = T1 , . . . , T2 − τ1 ,  ut = ut , t = T1 , . . . , T2 − τ1 − 1, T 2 −1

f (t,  xt ,  ut ) ≤ L.

t=T2 −τ1

There exists a natural number q such that qL0 ≤ T2 − T1 < (q + 1)L0 .

(6.27)

−1 q > (T2 − T1 )L−1 0 − 1 ≥ L1 L0 − 1 ≥ 8.

(6.28)

By (6.25)–(6.27),

It follows from property (b), (6.28), the relation L0 ≥ L+4 and the construction T2 −1 2 of ({ xt }Tt=T , { ut }t=T ) that for all i = 1, . . . , q − 2, 1 1 |

T1 +(i+1)L  0 −1

g(t,  xt ,  ut ) −

t=T1 +iL0

T1 +(i+1)L  0 −1

f (t, xt ,  ut )| ≤ 8−1 . f

f

(6.29)

t=T1 +iL0

By (6.17), (6.27), the relation L0 ≥ L + 4 and the construction of the trajectory T2 −1 2 ({ xt }Tt=T , { ut }t=T ), 1 1 T 2 −1

T 2 −1

f (t,  xt ,  ut ) ≤

t=T1 +(q−1)L0

f

f

f (t, xt ,  ut ) + L(a0 + 1).

(6.30)

t=T1 +(q−1)L0

Property (b) and Eqs. (6.12), (6.26), (6.27), and (6.30) imply that T 2 −1 t=T1 +(q−1)L0

g(t,  xt ,  ut ) ≤

T 2 −1

f

f

f (t, xt , ut ) + L(a0 + 1) + 1.

t=T1 +(q−1)L0

(6.31) T2 −1 2 By (6.6), the relation L0 ≥ L + 4 and the construction of ({ xt }Tt=T , { ut }t=T ), 1 1 T1 +L 0 −1 t=T1

f (t,  xt ,  ut )

150

6 Discrete-Time Nonautonomous Problems on Half-Axis T1 +L 0 −1

≤

f

f

f (t, xt ,  ut ) + L(a0 + 1).

(6.32)

t=T1

Property (b) and Eqs. (6.12), (6.26), and (6.32) imply that T1 +L 0 −1

g(t,  xt ,  ut ) ≤

T1 +L 0 −1

t=T1

≤

f (t,  xt ,  ut ) + 1

t=T1

T1 +L 0 −1

f

f

f (t, xt ,  ut ) + L(a0 + 1) + 1.

(6.33)

t=T1

It follows from (6.27), (6.29), and (6.31) that in the all cases T 2 −1

g(t,  xt ,  ut ) ≤

t=T1

T 2 −1

f

f

f (t, xt ,  ut ) + q + 1 + 2L(a0 + 1).

(6.34)

t=T1

In order to complete the proof of our lemma we need only to show that f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) ≤ }) ≥ L0 /2. Assume the contrary. Then f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) ≤ }) < L0 /2.

(6.35)

Property (a), (6.26), (6.27), and (6.35) imply that for all i = 0, . . . , q − 2, T1 +(i+1)L  0 −1

g(t, xt , ut ) >

T1 +(i+1)L  0 −1

t=T1 +iL0 T 2 −1

f

f

f

f

f (t, xt ,  ut ) + M,

t=T1 +iL0 T 2 −1

g(t, xt , ut ) >

t=T1 +(q−1)L0

f (t, xt , ut ) + M.

t=T1 +(q−1)L0

By the equations above T 2 −1 t=T1

g(t, xt , ut ) ≥

T 2 −1

f

f

f (t, xt , ut ) + L(a0 + 1) + Mq.

(6.36)

t=T1

It follows from (6.34), (6.36), the inequality M ≥ 8, conditions (i)–(iv) and the T2 −1 2 construction of ({ xt }Tt=T , { ut }t=T ) that 1 1

6.7 Proof of Theorem 6.22 T 2 −1

f

151

f

f (t, xt , ut ) ≤

t=T1

T 2 −1

g(t, xt , ut ) − Mq ≤

t=T1

≤ −(q − 1)M +

T 2 −1

g(t,  xt ,  ut ) − qM + M

t=T1 T 2 −1

f

f

f (t, xt , ut ) + q + 1 + 2L(1 + a0 )

t=T1

and q ≤ 3 + 2L(1 + a0 ).

(6.37)

In view of (6.26), (6.27) and (6.37), L1 ≤ L0 (q + 1) ≤ L0 (4 + 2L(1 + a0 )). This contradicts (6.25). The contradiction we have reached completes the proof of Lemma 6.24.

6.7 Proof of Theorem 6.22 We may assume without loss of generality that L > 4 + 4bf + (bf + 1)(Δf + 1)

(6.38)

and that the following property holds: f

(a) if (T , z) ∈ A and ρE (z, xt ) ≤ , then z ∈ AL and if in addition T ≥ L, the L . z∈A Lemma 6.16 and (A2) imply that there exists δ ∈ (0, ) and an integer L¯ 0 > 4bf such that the following properties hold: (b) for each integer T1 ≥ L¯ 0 , each integer T2 ≥ T1 + 2bf and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 f

ρE (xTi , xTi ) ≤ 4δ, i = 1, 2, T 2 −1

f (t, xt , ut ) ≤ U f (T1 , T2 , xT1 , xT2 ) + 4δ

t=T1 f

the inequality ρE (xt , xt ) ≤  holds for all t = T1 , . . . , T2 ;

152

6 Discrete-Time Nonautonomous Problems on Half-Axis f

(c) for each (Ti , zi ) ∈ A, i = 1, 2 satisfying ρE (zi , xTi ) ≤ δ, i = 1, 2 and T2 ≥ bf there exist T +b

T +b −1

f f 1 1 ({xt(1) }t=T , {u(1) t }t=T1 1

) ∈ X(T1 , T1 + bf ),

T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T2 − bf , T2 ) 2 −bf 2 −bf (2)

(2)

such that f

xT(1) = z1 , xT(1) = xT1 +bf , 1 1 +bf T1 +bf −1



T1 +bf −1 (1) (1) f (t, xt , ut )



≤

t=T1

f

f

f

f

f (t, xt , ut ) + 1,

t=T1 (2)

f

(2)

xT2 = z2 , xT2 −bf = xT2 −bf , T 2 −1

(2)

(2)

f (t, xt , ut ) ≤

t=T2 −bf

T 2 −1

f (t, xt , ut ) + 1.

t=T2 −bf

Lemma 6.24 implies that there exists integers L0 > L + 2L¯ 0 + 8, L1 > 8L0 + 2L0 L(a0 + 1)

(6.39)

a neighborhood V0 of f in Mψ such that the following property holds: (d) for each g ∈ V0 , each pair of integers T1 ≥ 0, T2 ≥ T1 + L1 and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies at least one of the following 1 1 conditions: T 2 −1

g(t, xt , ut ) ≤ U g (T1 , T2 ) + L;

t=T1 T 2 −1

g(t, xt , ut ) ≤ U g (T1 , T2 , xT1 ) + L, (T1 , xT1 ) ∈ AL ;

t=T1 T 2 −1

g(t, xt , ut ) ≤ U g (T1 , T2 , xT1 , xT2 ) + L,

t=T1

L ; (T1 , xT1 ) ∈ AL , (T2 , xT2 ) ∈ A

6.7 Proof of Theorem 6.22 T 2 −1

153

L g (T1 , T2 , xT2 ) + L, (T2 , xT2 ) ∈ A g(t, xt , ut ) ≤ U

t=T1

the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) ≤ δ}) ≥ L0 /2 is true. Proposition 6.21 implies that there exists a neighborhood V ⊂ V0 of f in Mψ such that the following property holds: (e) for each pair of integers S1 ≥ 0, S2 ∈ (S1 , S1 + L1 ], each g ∈ V and each S2 −1 2 ({xt }St=S , {ut }t=S ) ∈ X(S1 , S2 ) which satisfies 1 1 min{

S 2 −1

f (t, xt , ut ),

t=S1

S 2 −1

g(t, xt , ut )} ≤ (L1 + 2)Δf + 8 + a0

t=S1

the inequality |

S 2 −1

f (t, xt , ut ) −

t=S1

S 2 −1

g(t, xt , ut )| ≤ δ

t=S1

is true. Assume that g ∈ V and pair of integers T1 ≥ 0, T2 ≥ T1 + 4L1

(6.40)

T2 −1 2 and that ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfies at least one of the following 1 1 conditions:

(i) T 2 −1

g(t, xt , ut ) ≤ min{U g (T1 , T2 ) + L, U g (T1 , T2 , xT1 , xT2 ) + δ};

t=T1

(ii) T 2 −1 t=T1

g(t, xt , ut ) ≤ U g (T1 , T2 , xT1 ) + δ, (T1 , xT1 ) ∈ AL ;

154

6 Discrete-Time Nonautonomous Problems on Half-Axis

(iii) T 2 −1

g(t, xt , ut ) ≤ U g (T1 , T2 , xT1 , xT2 ) + δ,

t=T1

L . (T1 , xT1 ) ∈ AL , (T2 , xT2 ) ∈ A Property (d), conditions (i)–(iii), and (6.40) imply that f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) ≤ δ}) ≥ L0 /2.

(6.41)

S1 = min{t ∈ {max{T1 , L¯ 0 }, . . . , T2 } : ρE (xt , xt ) ≤ δ}.

(6.42)

Set f

f (Note that if T1 ≥ L¯ 0 and ρE (xT1 , xT1 ) ≤ δ, then S1 = T1 .) Clearly, S1 is welldefined. If S1 ≥ T1 + L1 , then it follows from properties (a), (d) and conditions (i)–(iii) and (6.40) that f

Card({t ∈ {T1 , . . . , S1 } : ρE (xt , xt ) ≤ δ}) ≥ L0 /2. This contradicts (6.42) (see (6.39)). The contradiction we have reached proves that S1 < T1 + L1 .

(6.43)

Set f

S2 = max{t ∈ {S1 , . . . , T2 } : ρE (xt , xt ) ≤ δ}.

(6.44)

If S2 ≤ T2 − L1 , then it follows from properties (a), (d) and conditions (i)–(iii) and Eqs. (6.39), (6.40) and (6.44) that f

Card({t ∈ {S2 , . . . , T2 } : ρE (xt , xt ) ≤ δ}) ≤ L0 /2. This contradicts (6.44). Thus S2 > T2 − L1 . In order to complete the proof of Theorem 6.22 it is sufficient to show that f

ρE (xt , xt ) ≤ , t ∈ {S1 , . . . , S2 }. Assume the contrary. Then there exists an integer

(6.45)

6.7 Proof of Theorem 6.22

155

τ0 ∈ {S1 , . . . , S2 }

(6.46)

ρE (xτ0 , xτf0 ) > .

(6.47)

such that

In view of (6.40) and (6.43)–(6.45), S2 − S1 ≥ T2 − T1 − 2L1 ≥ 2L1 .

(6.48)

By (6.48), there exists an integer S3 such that τ0 ∈ [S3 , S3 + L0 /4 + 1] ⊂ [S1 , S2 ].

(6.49)

S4 = S3 + L0 /4 + 1.

(6.50)

Set

Set f

(6.51)

f

(6.52)

S5 = max{t ∈ {S1 , . . . , S3 } : ρE (xt , xt ) ≤ δ}. S6 = min{t ∈ {S4 , . . . , S2 } : ρE (xt , xt ) ≤ δ}. It follows from (6.49)–(6.52) that S6 − S5 ≥ S4 − S3 ≥ L0 /4 + 1.

(6.53)

Assume that S6 − S5 ≥ L1 . Properties (a) and (d), conditions (i)–(iii) and Eqs. (6.40), (6.50) and (6.51) imply that f

Card({t ∈ {S5 , . . . , S6 } : ρE (xt , xt ) ≤ δ}) ≥ L0 /2. On the other hand, by (6.50)–(6.52), f

Card({t ∈ {S5 , . . . , S6 } : ρE (xt , xt ) ≤ δ}) ≤ S4 − S3 + 2 ≤ L0 /4 + 3. The contradiction we have reached proves that S6 − S5 ≤ L1 .

156

6 Discrete-Time Nonautonomous Problems on Half-Axis

Together with (6.53) this implies that

L0 /4 + 1 ≤ S6 − S5 ≤ L1 .

(6.54)

Property (c) and Eqs. (6.39), (6.51), (6.52), and (6.54) imply that there exist S6 −1 6 ({yt }St=S , {vt }t=S ) ∈ X(S5 , S6 ) 5 5

such that yS5 = xS5 , yS6 = yS6 , f

f

yt = xt , t ∈ {S5 + bf , . . . , S6 − bf }, vt = ut , t ∈ {S5 + bf , . . . , S6 − bf − 1}, S5 +bf −1



S5 +bf −1



f (t, yt , vt ) ≤

t=S5

f

f

f

f

f (t, xt , ut ) + 1,

t=S5

S 6 −1

S6 

f (t, yt , vt ) ≤

t=S6 −bf

f (t, xt , ut ) + 1.

(6.55)

t=S6 −bf

By (6.12), (6.54), and (6.55), S 6 −1

f (t, yt , vt ) ≤

t=S5

S 6 −1

f

f

f (t, xt , ut ) + 2 ≤ 2 + Δf (L1 + 2).

(6.56)

t=S5

In view of conditions (i)–(iii), S 6 −1

g(t, xt , ut ) ≤ U g (S5 , S6 , xS5 , xS6 ) + δ.

t=S5

Property (e) and Eqs. (6.40) and (6.54)–(6.56) imply that U g (S5 , S6 , xS5 , xS6 ) ≤ inf{

S 6 −1

S6 −1 6 g(t, ξt , ηt ) : ({ξt }St=S , {ηt }t=S ) ∈ X(S5 , S6 ), 5 5

t=S5

ξSi = xSi , i = 5, 6,

S 6 −1 t=S5

f (t, ξt , ηt ) ≤ Δf (L1 + 2) + 3}

(6.57)

6.7 Proof of Theorem 6.22

157

≤ δ + inf{

S 6 −1

f (t, ξt , ηt ) :

t=S5 6 6 −1 ({ξt }St=S , {ηt }St=S ) ∈ X(S5 , S6 ), ξSi = xSi , i = 5, 6, 5 5

S 6 −1

f (t, ξt , ηt ) ≤ Δf (L1 + 2) + 3} ≤ δ + U f (S5 , S6 , xS5 , xS6 ).

(6.58)

t=S5

By (6.57) and (6.58), S 6 −1

g(t, xt , ut ) ≤ Δf (L1 + 2) + 4.

(6.59)

t=S5

Property (e) and (6.54) and (6.59) imply that |

S 6 −1

f (t, xt , ut ) −

t=S5

S 6 −1

g(t, xt , ut )| ≤ δ.

t=S5

Together with (6.57) and (6.58) this implies that S 6 −1

f (t, xt , ut ) ≤

t=S5

S 6 −1

g(t, xt , ut ) + δ

t=S5

≤ U g (S5 , S6 , xS5 , xS6 ) + 2δ ≤ U f (S5 , S6 , xS5 , xS6 ) + 3δ. By the equation above, property (b) and Eqs. (6.39), (6.42), (6.51), (6.52) and (6.54), f

ρE (xt , xt ) ≤ , t ∈ {S5 , . . . , S6 }. Together with (6.49) and (6.50) this implies that ρE (xτ0 , xτf0 ) ≤ . This contradicts (6.47). The contradiction we have reached competes the proof of Theorem 6.22.

158

6 Discrete-Time Nonautonomous Problems on Half-Axis

6.8 Problems with Discount We use the notation, definitions and assumptions introduced in Sects. 6.1–6.3 and 6.5. Let g ∈ Mψ and α = {αt }∞ t=0 ⊂ (0, 1]. For each pair of integers T2 > T1 ≥ 0 and each pair of points y, z ∈ E satisfying (T1 , y), (T2 , z) ∈ A we define U

αg

(T1 , T2 , y, z) = inf{

T 2 −1

αt g(t, xt , ut ) :

t=T1 T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), 1 1

xT1 = y, xT2 = z}, U αg (T1 , T2 , y) = inf{

T 2 −1

αt g(t, xt , ut ) :

t=T1 T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), xT1 = y}, 1 1

αg (T1 , T2 , y) = inf{ U

T 2 −1

αt g(t, xt , ut ) :

t=T1 T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ), xT2 = y}, 1 1

U αg (T1 , T2 ) = inf{

T 2 −1

T2 −1 2 αt g(t, xt , ut ) : ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 )}. 1 1

t=T1

(6.60) We assume that all the assumptions of f posed in Sects. 6.1–6.3 and 6.5 hold and prove the following stability result for problems with discount. Theorem 6.25 Assume that f ∈ Mψ has TP, L is a natural number and  ∈ (0, 1). Then there exists an integer L1 > L, δ ∈ (0, ) λ ∈ (0, 1) and a neighborhood V of f in Mψ such that for each g ∈ V and each pair of integers T1 ≥ 0, T2 ≥ T1 + 4L1 , each α = {αt }∞ t=0 ⊂ (0, 1] which satisfies αt1 αt−1 ≥ λ for each t1 , t2 ∈ {T1 , . . . , T2 } satisfying |t2 − t1 | ≤ L1 2 T2 −1 2 and that for each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies at least one of the 1 1 following conditions:

6.9 Auxiliary Results for Theorem 6.25

159

(i) T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 ) + δ min{αt : t = T1 , . . . , T2 − 1};

t=T1

(ii) (T1 , xT1 ) ∈ AL ,

T 2 −1

αt g(t, xt , ut )

t=T1

≤ U αg (T1 , T2 , xT1 ) + δ min{αt : t = T1 , . . . , T2 − 1}; (iii) L (T1 , xT1 ) ∈ AL , (T2 , xT2 ) ∈ A and T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 , xT1 , xT2 ) + δ min{αt : t = T1 , . . . , T2 − 1}

t=T1

there exist integers τ1 ∈ [T1 , T1 + L1 ], τ2 ∈ [T2 − L1 , T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 . . . , τ2 . f

Moreover, if ρE (xT2 , xT2 ) ≤ δ, then τ2 = T2 and if T1 ≥ L1 and f

ρE (xT1 , xT1 ) ≤ δ, then τ1 = T1 .

6.9 Auxiliary Results for Theorem 6.25 We assume in this section that f has TP. Lemma 6.26 Assume that M ≥ 2 and  ∈ (0, 1). Then there exists an integer L0 ≥ 4 such that for each integer L1 > L0 there exist λ ∈ (0, 1) and a neighborhood V of f in Mψ such that for each g ∈ V and each pair of integers T1 ≥ 0, T2 ∈ (T1 , T1 + L1 ], each α = {αt }∞ t=0 ⊂ (0, 1] which satisfies ≥ λ for each t1 , t2 ∈ {T1 , . . . , T2 − 1} αt1 αt−1 2 T2 −1 2 and for each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1

160

6 Discrete-Time Nonautonomous Problems on Half-Axis

T 2 −1

αt g(t, xt , ut ) ≤

t=T1

T 2 −1

αt f (t, xt , ut ) + 4−1 M min{αt : t = T1 , . . . , T2 − 1} f

f

t=T1

(6.61) the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) > }) ≤ L0 /2

(6.62)

holds. Proof Lemma 6.23 implies that there exists an integer L0 ≥ 4 such that for each integer L1 > L0 the following property holds: (a) there exists a neighborhood V of f in Mψ such that for each g ∈ V and each T2 −1 2 , {ut }t=T ) ∈ pair of integers T1 ≥ 0, T2 ∈ (T1 , T1 + L1 ] and each ({xt }Tt=T 1 1 X(T1 , T2 ) which satisfies T 2 −1

g(t, xt , ut ) ≤

t=T1

T 2 −1

f

f

f (t, xt , ut ) + M

t=T1

equation (6.62) is true. Let L1 > L0 be an integer and let V be a neighborhood of f in Mψ such that property (a) holds. Fix λ ∈ (2−1 , 1) such that 4(1 − λ)(Δf + 1)(2L1 + 2L1 a0 + 2) < 1.

(6.63)

g ∈ V , T1 ≥ 0, T2 ∈ (T1 , T1 + L1 ]

(6.64)

Assume that

are integers, α = {αt }∞ t=0 ⊂ (0, 1] satisfies αt1 αt−1 ≥ λ for each t1 , t2 ∈ {T1 , . . . , T2 − 1} satisfying |t1 − t2 | ≤ L1 2 (6.65) T2 −1 2 and ({xt }Tt=T , {u } ) ∈ X(T , T ) satisfies (6.61). We show that (6.62) holds. t 1 2 t=T 1 1 Assume the contrary. Then by property (a) and (6.64), T 2 −1 t=T1

g(t, xt , ut ) >

T 2 −1 t=T1

f

f

f (t, xt , ut ) + M.

(6.66)

6.9 Auxiliary Results for Theorem 6.25

161

It follows from (6.17), (6.64), and (6.66) that T 2 −1

αt g(t, xt , ut ) = −a0

t=T1

T 2 −1

αt +

T 2 −1

t=T1

≥ −a0

αt (g(t, xt , ut ) + a0 )

t=T1 T 2 −1

αt

t=T1

+ min{αt : t = T1 , . . . , T2 − 1}{

T 2 −1

(g(t, xt , ut ) + a0 )

t=T1

≥ −a0

T 2 −1

αt + min{αt : t = T1 , . . . , T2 − 1}(

T 2 −1

f

f

(f (t, xt , ut ) + a0 ) + M).

t=T1

t=T1

(6.67) Equations (6.6), (6.12) and (6.63)–(6.65) imply that max{αt : t = T1 , . . . , T2 − 1}

T 2 −1

f

f

(f (t, xt , ut ) + a0 )

t=T1

− min{αt : t = T1 , . . . , T2 − 1}

T 2 −1

f

f

(f (t, xt , ut ) + a0 )

t=T1

≤ max{αt : t = T1 , . . . , T2 − 1}(1 − λ)

T 2 −1

f

f

(f (t, xt , ut ) + a0 )

t=T1

≤ max{αt : t = T1 , . . . , T2 − 1}(1 − λ)(Δf + a0 )L1 ≤ max{αt : t = T1 , . . . , T2 − 1}.

(6.68)

By (6.6), (6.67), and (6.68), T 2 −1

αt g(t, xt , ut ) ≥ −a0

t=T1

+ max{αt : t = T1 , . . . , T2 − 1}

T 2 −1

αt

t=T1 T 2 −1 t=T1

f

f

(f (t, xt , ut ) + a0 )

162

6 Discrete-Time Nonautonomous Problems on Half-Axis

− max{αt : t = T1 , . . . , T2 − 1} + M min{αt : t = T1 , . . . , T2 − 1} ≥

T 2 −1

f

f

αt f (t, xt , ut ) − max{αt : t = T1 , . . . , T2 − 1}

t=T1

+ M min{αt : t = T1 , . . . , T2 − 1} ≥

T 2 −1

αt f (t, xt , ut ) + 2−1 M min{αt : t = T1 , . . . , T2 − 1}. f

f

t=T1

This contradicts (6.61). The contradiction we have reached proves (6.62) and Lemma 6.26. Lemma 6.27 Assume that  ∈ (0, 1) and L is a natural number. Then there exist integers L0 > L + 4, L1 > 8L0 + 2L0 L(a0 + 1) + L0 (2L0 + 4), λ ∈ (0, 1) and a neighborhood V of f in Mψ such that for each g ∈ V , each pair of integers T1 ≥ 0, T2 ≥ T1 + L1 , each α = {αt }∞ t=0 ⊂ (0, 1] which satisfies αt1 αt−1 ≥ λ for each t1 , t2 ∈ {T1 , . . . , T2 − 1} satisfying |t1 − t2 | ≤ L1 2

(6.69)

the following assertion holds. T2 −1 2 Assume that ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfies at least one of the 1 1 following conditions: (i) T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 ) + 4 min{αt : t = T1 , . . . , T2 − 1};

t=T1

(ii) T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 , xT1 ) + 4 min{αt : t = T1 , . . . , T2 − 1},

t=T1

(T1 , xT1 ) ∈ AL ; (iii) T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 , xT1 , xT2 ) + 4 min{αt : t = T1 , . . . , T2 − 1},

t=T1

L ; (T1 , xT1 ) ∈ AL , (T2 , xT2 ) ∈ A

6.9 Auxiliary Results for Theorem 6.25

163

(iv) T 2 −1

αg (T1 , T2 , xT2 ) + 4 min{αt : t = T1 , . . . , T2 − 1}, αt g(t, xt , ut ) ≤ U

t=T1

L (T2 , xT2 ) ∈ A Then f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) ≤ }) ≥ L0 /2.

(6.70)

Proof Lemma 6.26 implies that there exists an integer L0 > 4 + L, λ0 ∈ (0, 1) and a neighborhood V0 of f in Mψ such that the following property holds: (a) for each g ∈ V0 , each pair of integers T1 ≥ 0, T2 ∈ (T1 , T1 + 2L0 ], each α = {αt }∞ t=0 ⊂ (0, 1] which satisfies αt1 αt−1 ≥ λ0 for each t1 , t2 ∈ {T1 , . . . , T2 − 1} 2 T2 −1 2 and for each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 T 2 −1

αt g(t, xt , ut ) ≤

t=T1

T 2 −1

f

f

αt f (t, xt , ut ) + 8 min{αt : t = T1 , . . . , T2 − 1}

t=T1

the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) > }) ≤ L0 /2 is true. Choose an integer q0 > 8L + 16(L0 + 1)(a0 + 1)

(6.71)

and λ ∈ (λ0 , 1) such that λq0 +1 > 2−1 , (1 − λ)λ−1 (L0 + 1)(Δf + a0 + 1) < 8−1 .

(6.72)

164

6 Discrete-Time Nonautonomous Problems on Half-Axis

Proposition 6.21 implies that there exists a neighborhood V ⊂ V0 of f in Mψ such that the following property holds: (b) for each pair of integers T1 ≥ 0, T2 ∈ (T1 , T1 + 8L0 ], each g ∈ V and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 min{

T 2 −1

f (t, xt , ut ),

t=T1

T 2 −1

g(t, xt , ut )} ≤ 2L0 (Δf + 1) + L(1 + 2a0 )

t=T1

the inequality |

T 2 −1

f (t, xt , ut ) −

t=T1

T 2 −1

g(t, xt , ut )| ≤ 8−1

t=T1

is true. Choose an integer L1 > L0 (8 + 2L(a0 + 1)) + L0 (2a0 + 2q0 + 4).

(6.73)

Assume that T1 ≥ 0, T2 ∈ (T1 , T1 + L1 ]

(6.74)

are integers, α = {αt }∞ t=0 ⊂ (0, 1] satisfies (6.69), g∈V

(6.75)

T2 −1 2 and that ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfies at least one of the conditions 1 1 (i)–(iv). In order to complete the proof of the lemma it is sufficient to show that (6.70) holds. T2 −1 2 xt }Tt=T , { ut }t=T ) ∈ X(T1 , T2 ). If condition (i) holds, then We construct ({ 1 1 we set f

f

 xt = xt , t = T1 , . . . , T2 ,  ut = ut , t = T1 , . . . , T2 − 1. T2 −1 2 If condition (ii) holds, then there exist ({ xt }Tt=T , { ut }t=T ) ∈ X(T1 , T2 ) and an 1 1 integer τ1 ∈ [0, L] such that

 xT1 = xT1 ,

T1 +τ 1 −1 

f (t,  xt ,  ut ) ≤ L,

t=T1 f

 xt = xt , t = T + τ1 , . . . , T2 ,

6.9 Auxiliary Results for Theorem 6.25

165

f

 ut = ut , t = T + τ1 , . . . , T2 − 1. T2 −1 2 If condition (iii) holds, then there exist ({ xt }Tt=T , { ut }t=T ) ∈ X(T1 , T2 ) and 1 1 integers τ1 , τ2 ∈ [0, L] such that

 xT1 = xT1 ,  xT2 = xT2 , f

f

 xt = xt , t = T + τ1 , . . . , T2 − τ2 ,  ut = ut , t = T + τ1 , . . . , T2 − τ2 − 1, T1 +τ 1 −1 

T 2 −1

f (t,  xt ,  ut ) ≤ L,

f (t,  xt ,  ut ) ≤ L.

t=T2 −τ2

t=T1

T2 −1 2 If condition (iv) holds, then there exist ({ xt }Tt=T , { ut }t=T ) ∈ X(T1 , T2 ) and an 1 1 integer τ1 ∈ [0, L] such that

 xT2 = xT2 , f

f

 xt = xt , t = T1 , . . . , T2 − τ1 ,  ut = ut , t = T1 , . . . , T2 − τ1 − 1, T 2 −1

f (t,  xt ,  ut ) ≤ L.

t=T2 −τ1

There exists a natural number q such that qL0 ≤ T2 − T1 < (q + 1)L0 .

(6.76)

By (6.73), (6.74), and (6.76), −1 q ≥ (T2 − T1 )L−1 0 − 1 ≥ L1 L0 − 1 ≥ 8.

(6.77)

It follows from property (b), (6.12), (6.75), the relation L0 > L + 4 and the T2 −1 2 construction of ({ xt }Tt=T , { ut }t=T ) that for all i = 1, . . . , q − 2, 1 1 |

T1 +(i+1)L  0 −1

g(t,  xt ,  ut ) −

t=T1 +iL0

T1 +(i+1)L  0 −1

f (t, xt ,  ut )| ≤ 8−1 . f

f

(6.78)

t=T1 +iL0

T2 −1 2 By (6.6), (6.76) and the construction of ({ xt }Tt=T , { ut }t=T ), 1 1 T 2 −1 t=T1 +(q−1)L0

f (t,  xt ,  ut ) ≤

T 2 −1 t=T1 +(q−1)L0

f

f

f (t, xt , ut ) + L(a0 + 1).

(6.79)

166

6 Discrete-Time Nonautonomous Problems on Half-Axis

Property (b) and Eqs. (6.12), (6.75), (6.76), and (6.79) imply that T 2 −1 t=T1 +(q−1)L0

≤

T 2 −1

g(t,  xt ,  ut ) ≤

f (t,  xt ,  ut ) + 1

t=T1 +(q−1)L0

T 2 −1

f

f

f (t, xt , ut ) + L(1 + a0 ) + 1.

(6.80)

t=T1 +(q−1)L0

By (6.6), (6.76), the relation L0 > L + 4 and the construction of the trajectory– T2 −1 2 control pair ({ xt }Tt=T , { ut }t=T ), 1 1 T1 +L 0 −1

f (t,  xt ,  ut ) ≤

T1 +L 0 −1

t=T1

f

f

f (t, xt , ut ) + L(a0 + 1).

(6.81)

t=T1

Property (b) and Eqs. (6.12), (6.75), and (6.81) imply that T1 +L 0 −1

g(t,  xt ,  ut ) ≤

T1 +L 0 −1

t=T1

≤

f (t,  xt ,  ut ) + 1

t=T1

T1 +L 0 −1

f

f

f (t, xt , ut ) + L(a0 + 1) + 1.

(6.82)

t=T1

Let {S1 , S2 } ∈ {(T1 + iL0 , T1 + (i + 1)L0 : i = 1, . . . , q − 2}. Equations (6.12), (6.17), (6.69), and (6.73) imply that S 2 −1 t=S1

αt g(t,  xt ,  ut ) =

S 2 −1

αt (g(t,  xt ,  ut ) + a0 ) − a0

t=S1

≤ max{αt : t ∈ {S1 , . . . , S2 − 1}}

S 2 −1

αt

t=S1 S 2 −1

S 2 −1

t=S1

t=S1

(g(t,  xt ,  ut ) + a0 ) − a0

≤ max{αt : t ∈ {S1 , . . . , S2 − 1}}

S 2 −1 t=S1

f

f

(f (t, xt , ut ) + a0 )

αt

6.9 Auxiliary Results for Theorem 6.25

167

+ 8−1 max{αt : t ∈ {S1 , . . . , S2 − 1}} − a0

S 2 −1

αt

t=S1

≤

S 2 −1

f

f

α(f (t, xt , ut ) + a0 )

t=S1

+ (1 − λ) max{αt : t ∈ {S1 , . . . , S2 − 1}}

S 2 −1

f

f

(f (t, xt , ut ) + a0 )

t=S1

+ 8−1 max{αt : t ∈ {S1 , . . . , S2 − 1}} − a0

S 2 −1

αt

t=S1

≤

S 2 −1

f

f

αt f (t, xt , ut ) + (1 − λ) max{αt : t ∈ {S1 , . . . , S2 − 1}}L0 (Δf + a0 )

t=S1

+ 8−1 max{αt : t ∈ {S1 , . . . , S2 − 1}} ≤

S 2 −1

αt f (t, xt , ut )+λ−1 (1−λ) min{αt : t ∈ {S1 , . . . , S2 −1}}L0 (Δf +a0 ) f

f

t=S1

+ 8−1 λ−1 min{αt : t ∈ {S1 , . . . , S2 − 1}} ≤

S 2 −1

αt f (t, xt , ut ) + 2−1 min{αt : t ∈ {S1 , . . . , S2 − 1}}. f

f

(6.83)

t=S1

By (6.12), (6.17), (6.69), (6.72), (6.73), (6.76), (6.80), and (6.82), for every {S1 , S2 } ∈ {(T1 , T1 + L0 ), (T1 + (q − 1)L0 , T2 )} we have S 2 −1 t=S1

αt g(t,  xt ,  ut ) ≤

S 2 −1

αt (g(t,  xt ,  ut ) + a0 ) − a0

t=S1

≤ max{αt : t ∈ {S1 , . . . , S2 − 1}}

S 2 −1

αt

t=S1 S 2 −1

S 2 −1

t=S1

t=S1

(g(t,  xt ,  ut ) + a0 ) − a0

αt

168

6 Discrete-Time Nonautonomous Problems on Half-Axis S 2 −1

≤ max{αt : t ∈ {S1 , . . . , S2 − 1}}(

f

f

(f (t, xt , ut ) + a0 ) + L(a0 + 1) + 1)

t=S1

− a0

S 2 −1

αt

t=S1

≤ max{αt : t ∈ {S1 , . . . , S2 − 1}}(L(a0 + 1) + 1) − a0

S 2 −1

αt

t=S1

+

S 2 −1

f

f

αt (f (t, xt , ut ) + a0 )

t=S1

+ (1 − λ) max{αt : t ∈ {S1 , . . . , S2 − 1}}

S 2 −1

f

f

α(f (t, xt , ut ) + a0 )

t=S1

≤

S 2 −1

αt f (t, xt , ut ) + min{αt : t ∈ {S1 , . . . , S2 − 1}}λ−1 (L(a0 + 1) + 1) f

f

t=S1

+ min{αt : t ∈ {S1 , . . . , S2 − 1}}λ−1 (1 − λ)(2L0 Δf + a0 ) ≤

S 2 −1

αt f (t, xt , ut ) + 2−1 min{αt : t ∈ {S1 , . . . , S2 − 1}} f

f

t=S1

+ 2 min{αt : t ∈ {S1 , . . . , S2 − 1}}(L(a0 + 1) + 1).

(6.84)

In order to complete the proof of the lemma it is sufficient to show that (6.70) holds. Assume the contrary. Then f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) ≤ }) < L0 /2.

(6.85)

Property (a) and Eqs. (6.69), (6.75), (6.76) and (6.85) imply that for all i = 0, . . . , q − 2, T1 +(i+1)L  0 −1 t=T1 +iL0

αt g(t, xt , tut )

6.9 Auxiliary Results for Theorem 6.25

>

T1 +(i+1)L  0 −1

f

169

f

αt f (t, xt , ut ) + 8 min{αt : t ∈ {T1 + iL0 , . . . , Ti + (i + 1)L0 − 1}}

t=T1 +iL0

(6.86) and T 2 −1

αt g(t, xt , tut )

t=T1 +(q−1)L0

>

T 2 −1

f

f

αt f (t, xt , ut )+8 min{αt : t ∈ {T1 +(q −1)L0 , . . . , T2 −1}}.

t=T1 +(q−1)L0

(6.87) It follows from (6.76), (6.86), and (6.87) that T 2 −1

αt g(t, xt , ut ) >

t=T1

+ 8(

q−2 

T 2 −1

f

f

αt f (t, xt , ut )

t=T1

min{αt : t ∈ {T1 + iL0 , . . . , Ti + (i + 1)L0 − 1}}

i=0

+ min{αt : t ∈ {T1 + (q − 1)L0 , . . . , T2 − 1}}).

(6.88)

In view of (6.76), (6.83), and (6.84), T 2 −1 t=T1

+ 2−1

q−2 

αt g(t,  xt ,  ut ) ≤

T 2 −1

f

f

αt f (t, xt , ut )

t=T1

min{αt : t ∈ {T1 + iL0 , . . . , Ti + (i + 1)L0 − 1}}

i=0

+ 2−1 min{αt : t ∈ {T1 + (q − 1)L0 , . . . , T2 − 1}} + 2(L(a0 + 1) + 1) min{αt : t ∈ {T1 , . . . , Ti + L0 − 1}} + min{αt : t ∈ {T1 + (q − 1)L0 , . . . , T2 − 1}}.

(6.89)

By (6.88), (6.89), conditions (i)–(iv) and the construction of the trajectory– T2 −1 2 control pair ({ xt }Tt=T , { ut }t=T ), 1 1

170

6 Discrete-Time Nonautonomous Problems on Half-Axis T 2 −1

f

f

αt f (t, xt , ut ) + 8(

t=T1

q−2 

min{αt : t ∈ {T1 + iL0 , . . . , T1 + (i + 1)L0 − 1}}

i=0

+ min{αt : t ∈ {T1 + (q − 1)L0 , . . . , T2 − 1}})
2(

q−2 

min{αt : t ∈ {T1 + iL0 , . . . , T1 + (i + 1)L0 − 1}}

i=0

+ min{αt : t ∈ {T1 + (q − 1)L0 , . . . , T2 − 1}}). There exists j ∈ {0, . . . , q − 2} such that

(6.91)

6.10 Proof of Theorem 6.25

171

min{αt : t ∈ {T1 + iL0 , . . . , Ti + (i + 1)L0 − 1}} ≤ min{αt : t ∈ {T1 + j L0 , . . . , Ti + (j + 1)L0 − 1}}

(6.92)

for every i ∈ {0, . . . , q − 2}. In view of (6.73) and (6.77), q − 2 ≥ L1 L−1 0 − 3 ≥ 2q0 + 2.

(6.93)

By (6.93), there exists an integer j0 ≥ 0 such that j0 + q0 ≤ q − 2, j0 ≤ j ≤ j0 + q0 .

(6.94)

Equations (6.69), (6.72), (6.73), (6.91), (6.92), and (6.94) imply that (2L(a0 + 1) + 1) min{αt : t ∈ {T1 + j L0 , . . . , T1 + (j + 1)L0 − 1}}λ−1 ≥

q−2 

min{αt : t ∈ {T1 + iL0 , . . . , T1 + (i + 1)L0 − 1}}

i=0

≥

j0 +q 0 −1 

min{αt : t ∈ {T1 + iL0 , . . . , T1 + (i + 1)L0 − 1}}

i=j0

≥ min{αt : t ∈ {T1 + j L0 , . . . , T1 + (j + 1)L0 − 1}}λq0 q0 and 2L(a0 + 1) + 1 ≥ q0 λq0 +1 ≥ 2−1 q0 . This contradicts (6.71). The contradiction we have reached proves Lemma 6.27.

6.10 Proof of Theorem 6.25 In view of (A2), we may assume without loss of generality that L > 4 + 4bf + (bf + 1)(Δf + 1) and that the following property holds: f if (T , z) ∈ A and ρE (z, xT ) ≤ , then z ∈ AL and if in addition T ≥ L, then L . z∈A Lemma 6.16 and (A2) imply that there exists δ ∈ (0, ) and an integer L¯ 0 > 0 such that the following properties hold:

172

6 Discrete-Time Nonautonomous Problems on Half-Axis

(a) for each integer T1 ≥ L¯ 0 , each integer T2 T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1

≥ T1 + 2bf and each

f

ρE (xTi , xTi ) ≤ 4δ, i = 1, 2, T 2 −1

f (t, xt , ut ) ≤ U f (T1 , T2 , xT1 , xT2 ) + 8δ

t=T1 f

the inequality ρE (xt , xt ) ≤  holds for all t = T1 , . . . , T2 ; f (b) for each (Ti , zi ) ∈ A, i = 1, 2 satisfying ρE (zi , xTi ) ≤ δ, i = 1, 2 and T2 > bf there exist T +b

T +b −1

f f 1 1 ({xt(1) }t=T , {u(1) t }t=T1 1

) ∈ X(T1 , T1 + bf ),

T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T2 − bf , T2 ) 2 −bf 2 −bf (2)

(2)

such that f

xT(1) = z1 , xT(1) = xT1 +bf , 1 1 +bf T1 +bf −1



T1 +bf −1 (1) (1) f (t, xt , ut )



≤

t=T1

f

f

f

f

f (t, xt , ut ) + 1,

t=T1 (2)

f

(2)

xT2 = z2 , xT2 −bf = xT2 −bf , T 2 −1

(2) (2) f (t, xt , ut )

≤

t=T2 −bf

T 2 −1

f (t, xt , ut ) + 1.

t=T2 −bf

Lemma 6.27 implies that there exist integers L0 > L + 8 + 2L¯ 0 , L1 > 8L0 + 2L0 L(a0 + 1) + L0 (2L0 + 4)

(6.95)

a neighborhood V0 of f in Mψ and a number λ satisfying 1 > λ > 2−1 , λ−1 (1 − λ)a0 L1 < δ, (λ−1 − 1)(2 + (L1 + 2)(Δf + a0 + 2)) < δ such that the following property holds:

(6.96)

6.10 Proof of Theorem 6.25

173

(c) for each g ∈ V0 , each pair of integers T1 ≥ 0, T2 > T1 + L1 , each α = {αt }∞ t=0 ⊂ (0, 1] which satisfies αt1 αt−1 ≥ λ for each t1 , t2 ∈ {T1 , . . . , T2 − 1} satisfying |t1 − t2 | ≤ L1 2 T2 −1 2 and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies at least one of the 1 1 following conditions: T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 ) + 4 min{αt : t = T1 , . . . , T2 − 1};

t=T1 T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 , xT1 )

t=T1

+ 4 min{αt : t = T1 , . . . , T2 − 1}, (T1 , xT1 ) ∈ AL ; T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 , xT1 , xT2 ) + 4 min{αt : t = T1 , . . . , T2 − 1},

t=T1

L ; (T1 , xT1 ) ∈ AL , (T2 , xT2 ) ∈ A T 2 −1

αg (T1 , T2 , xT2 ) αt g(t, xt , ut ) ≤ U

t=T1

L + 4 min{αt : t = T1 , . . . , T2 − 1}, (T2 , xT2 ) ∈ A the inequality f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) ≤ δ}) ≥ L0 /2 is true. Proposition 6.21 implies that there exists a neighborhood V ⊂ V0 of f in Mψ such that the following property holds: (d) for each pair of integers S1 ≥ 0, S2 ∈ (S1 , S1 + L1 ], each g ∈ V and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 min{

T 2 −1 t=T1

the inequality

f (t, xt , ut ),

T 2 −1 t=T1

g(t, xt , ut )} ≤ (L1 + 2)Δf + 8 + a0

174

6 Discrete-Time Nonautonomous Problems on Half-Axis

|

T 2 −1 t=T1

f (t, xt , ut ) −

T 2 −1

g(t, xt , ut )| ≤ δ

t=T1

is true. Assume that g ∈ V , T1 ≥ 0, T2 ≥ T1 + 4L1

(6.97)

are integers, α = {αt }∞ t=0 ⊂ (0, 1] satisfies αt1 αt−1 ≥ λ for each t1 , t2 ∈ {T1 , . . . , T2 − 1} satisfying |t2 − t1 | ≤ L1 2 (6.98) T2 −1 2 and that ({xt }Tt=T , {u } ) ∈ X(T , T ) satisfies at least one of the following t 1 2 t=T1 1 conditions: (i) T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 ) + δ min{αt : t = T1 , . . . , T2 − 1};

t=T1

(ii) (T1 , xT1 ) ∈ AL ,

T 2 −1

αt g(t, xt , ut )

t=T1

≤ U αg (T1 , T2 , xT1 ) + δ min{αt : t = T1 , . . . , T2 − 1}; (iii) L (T1 , xT1 ) ∈ AL , (T2 , xT2 ) ∈ A and T 2 −1

αt g(t, xt , ut ) ≤ U αg (T1 , T2 , xT1 , xT2 ) + δ min{αt : t = T1 , . . . , T2 − 1}.

t=T1

Property (c) and Eqs. (6.97) and (6.98) imply that f

Card({t ∈ {T1 , . . . , T2 } : ρE (xt , xt ) ≤ δ}) ≥ L0 /2.

(6.99)

6.10 Proof of Theorem 6.25

175

Set f S1 = min{t ∈ {max{T1 , L¯ 0 }, . . . , T2 } : ρE (xt , xt ) ≤ δ}.

(6.100)

f (Note that if T1 ≥ L¯ 0 and ρE (xT1 , xT1 ) ≤ δ, then S1 = T1 .) Clearly, in view of (6.95), (6.97) and (6.100), S1 is well-defined. If S1 > T1 + L1 , then it follows from properties (a), (c) and conditions (i)–(iii) and Eqs. (6.97), (6.98), (6.100) that f

Card({t ∈ {T1 , . . . , S1 } : ρE (xt , xt ) ≤ δ}) ≥ L0 /2. This contradicts (6.100). The contradiction we have reached proves that S1 ≤ T1 + L1 .

(6.101)

Set f

S2 = max{t ∈ {S1 , . . . , T2 } : ρE (xt , xt ) ≤ δ}.

(6.102)

If S2 ≤ T2 − L1 , then it follows from properties (b), (c), conditions (i)–(iii) and Eqs. (6.97), (6.98) and (6.102) that f

Card({t ∈ {S2 , . . . , T2 } : ρE (xt , xt ) ≤ δ}) ≥ L0 /2. This contradicts (6.102). Thus S2 ≥ T2 − L1 .

(6.103)

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

ρE (xt , xt ) ≤ , t ∈ {S1 , . . . , S2 }. Assume the contrary. Then there exists an integer τ0 ∈ {S1 , . . . , S2 } such that ρE (xτ0 , xτf0 ) > .

(6.104)

S2 − S1 ≥ T2 − T1 − 2L1 ≥ 2L1 .

(6.105)

In view of (6.97) and (6.103),

176

6 Discrete-Time Nonautonomous Problems on Half-Axis

By (6.95), there exists an integer S3 such that τ0 ∈ [S3 , S3 + L0 /4+1] ⊂ [S1 , S2 ] and set S4 = S3 + L0 /4+1.

(6.106)

Set f

S5 = max{t ∈ {S1 , . . . , S3 } : ρE (xt , xt ) ≤ δ}, f

S6 = min{t ∈ {S4 , . . . , S2 } : ρE (xt , xt ) ≤ δ}.

(6.107)

It follows from (6.106) and (6.107) that S6 − S5 ≥ S4 − S3 ≥ L0 /4 + 1.

(6.108)

Assume that S6 − S5 ≥ L1 . Properties (b) and (c), conditions (i)–(iii) and Eqs. (6.97), (6.107) and (6.108) imply that f

Card({t ∈ {S5 , . . . , S6 } : ρE (xt , xt ) ≤ δ}) ≥ L0 /2. On the other hand, by (6.106)–(6.108), f

Card({t ∈ {S5 , . . . , S6 } : ρE (xt , xt ) ≤ δ}) ≤ S4 − S3 + 2 ≤ L0 /4 + 3. The contradiction we have reached proves that S6 − S5 ≤ L1 . Together with (6.108) this implies that

L0 /4 + 1 ≤ S6 − S5 ≤ L1 . Property (b) and Eqs. (6.95) and (6.107)–(6.109) imply that there exist S6 −1 6 ({yt }St=S , {vt }t=S ) ∈ X(S5 , S6 ) 5 5

such that yS5 = xS5 , yS6 = yS6 ,

(6.109)

6.10 Proof of Theorem 6.25

177

f

f

yt = xt , t ∈ {S5 +bf , . . . , S6 −bf }, vt = ut , t ∈ {S5 +bf , . . . , S6 −bf −1}, (6.110) S5 +bf −1 S5 +bf −1   f f f (t, yt , vt ) ≤ f (t, xt , ut ) + 1, t=S5 S 6 −1

t=S5

f (t, yt , vt ) ≤

t=S6 −bf

S 6 −1

f

f

f (t, xt , ut ) + 1.

(6.111)

t=S6 −bf

By (6.12) and (6.109)–(6.111), S 6 −1

f (t, yt , vt ) ≤ 2 + Δf (L1 + 2).

(6.112)

t=S5

Property (d) and Eqs. (6.97), (6.109), and (6.112) imply that S 6 −1

g(t, yt , vt ) ≤ 3 + Δf (L1 + 2).

(6.113)

t=S5

By (6.17), (6.98), and (6.109), S 6 −1

αt g(t, xt , ut ) =

t=S5

S 6 −1

αt (g(t, xt , ut ) + a0 ) − a0

t=S5

αt

t=S5 S 6 −1

S 6 −1

t=S5

t=S5

S 6 −1

S 6 −1

≥ min{αt : t ∈ {S5 , . . . , S6 − 1}}

≥ min{αt : t ∈ {S5 , . . . , S6 − 1}}

S 6 −1

(g(t, xt , ut ) + a0 ) − a0

g(t, xt , ut ) − a0 (1 − λ)

t=S5

αt

αt .

t=S5

(6.114) In view of conditions (i)–(iii) and (6.114), min{αt : t ∈ {S5 , . . . , S6 − 1}}

S 6 −1

g(t, xt , ut )

t=S5

≤

S 6 −1 t=S5

αt g(t, xt , ut ) + (1 − λ)a0

S 6 −1 t=S5

αt

178

6 Discrete-Time Nonautonomous Problems on Half-Axis

≤ U αg (S5 , S6 , xS5 , xS6 ) + δ min{αt : t ∈ {S5 , . . . , S6 − 1}} + (1 − λ)a0

S 6 −1

(6.115)

αt .

t=S5

Property (d) and Eqs. (6.17), (6.96)–(6.98), (6.109), (6.110) and (6.112) imply that U αg (S5 , S6 , xS5 , xS6 ) ≤ inf{

S 6 −1

αt g(t, ξt , ηt ) :

t=S5 S6 −1 6 ({ξt }St=S , {ηt }t=S ) ∈ X(S5 , S6 ), 5 5 S 6 −1

f (t, ξt , ηt ) ≤

t=S5

≤ inf{λ

−1

S 6 −1

f (t, yt , vt ), ξSi = xSi , i = 5, 6}.

t=S5

min{αt : t ∈ {S5 , . . . , S6 − 1}}

S 6 −1

(g(t, ξt , ηt ) + a0 ) − a0

t=S5

S 6 −1 t=S5

S6 −1 6 ({ξt }St=S , {ηt }t=S ) ∈ X(S5 , S6 ), 5 5 S 6 −1

f (t, ξt , ηt ) ≤

t=S5

S 6 −1

f (t, yt , vt ), ξSi = xSi , i = 5, 6}

t=S5

≤ λ−1 min{αt : t ∈ {S5 , . . . , S6 − 1}} inf{

S 6 −1

g(t, ξt , ηt ) :

t=S5 S6 −1 6 ({ξt }St=S , {ηt }t=S ) ∈ X(S5 , S6 ), 5 5 S 6 −1 t=S5

f (t, ξt , ηt ) ≤

S 6 −1

f (t, yt , vt ), ξSi = xSi , i = 5, 6}

t=S5

+ a0 (λ−1 − 1) min{αt : t ∈ {S5 , . . . , S6 − 1}}(S6 − S5 ) ≤ λ−1 min{αt : t ∈ {S5 , . . . , S6 − 1}} inf{

S 6 −1 t=S5

f (t, ξt , ηt ) + δ :

αt :

6.10 Proof of Theorem 6.25

179 S6 −1 6 ({ξt }St=S , {ηt }t=S ) ∈ X(S5 , S6 ), 5 5

S 6 −1

f (t, ξt , ηt ) ≤

t=S5

S 6 −1

f (t, yt , vt ), ξSi = xSi , i = 5, 6}

t=S5

+ a0 (λ−1 − 1) min{αt : t ∈ {S5 , . . . , S6 − 1}}L1 ≤ λ−1 min{αt : t ∈ {S5 , . . . , S6 − 1}}U f (S5 , S6 , xS5 , xS6 ) + δλ−1 min{αt : t ∈ {S5 , . . . , S6 − 1}} + δ min{αt : t ∈ {S5 , . . . , S6 − 1}}. (6.116) In view of (6.110) and (6.112), U f (S5 , S6 , xS5 , xS6 ) ≤ 2 + (Lf + 2)Δf .

(6.117)

By (6.115) and (6.116),

(

S 6 −1

g(t, xt , ut )) min{αt : t ∈ {S5 , . . . , S6 − 1}}

t=S5

≤ δ min{αt : t ∈ {S5 , . . . , S6 − 1}} + (1 − λ)a0

S 6 −1

αt

t=S5

+ λ−1 min{αt : t ∈ {S5 , . . . , S6 − 1}}U f (S5 , S6 , xS5 , xS6 ) + δλ−1 min{αt : t ∈ {S5 , . . . , S6 − 1}} + δ min{αt : t ∈ {S5 , . . . , S6 − 1}}. (6.118) It follows from (6.96), (6.98), (6.109), (6.117), and (6.118) that S 6 −1

g(t, xt , ut ) ≤ δ + (1 − λ)a0 λ−1 L1 + λ−1 U f (S5 , S6 , xS5 , xS6 ) + δλ−1 + δ

t=S5

≤ U f (S5 , S6 , xS5 , xS6 ) + 4δ + (λ−1 − 1)(L1 + 2)Δf ≤ U f (S5 , S6 , xS5 , xS6 ) + 5δ. Property (d) and Eqs. (6.97), (6.109), (6.117), and (6.119) imply that

(6.119)

180

6 Discrete-Time Nonautonomous Problems on Half-Axis

|

S 6 −1

g(t, xt , ut ) −

t=S5

S 6 −1

f (t, xt , ut )| ≤ δ.

t=S5

Combined with (6.119) this implies that S 6 −1

f (t, xt , ut ) ≤ U f (S5 , S6 , xS5 , xS6 ) + 6δ.

t=S5

Together with property (a) and Eqs. (6.107)–(6.110) this implies that f

ρE (xt , xt ) ≤ , t ∈ {S5 , . . . , S6 }. Combined with (6.106) this implies that ρE (xτ , xτf ) ≤ . This contradicts (6.104). The contradiction we have reached completes the proof of Theorem 6.25.

6.11 Genericity Results for Discrete-Time Problems Assume that a set M ⊂ Mψ is equipped with a metric ds which induces a uniformity which is stronger than the relative uniformity determined by the base E(N, ), N,  > 0 (see (6.19)) and is metrizable by the metric dψ . Recall that the topology induced by the metric dψ in Mψ is called weak. The space M ⊂ Mψ is equipped with the relative weak topology (which is called the weak topology) and with the topology induced by the metric ds which is called the strong topology. Assume that Mreg is the set of all f ∈ M such that there exists an integer bf > f f ∞ 0, ({xt }∞ t=0 , {ut }t=0 ) ∈ X(0, ∞), Δf ≥ 0 such that (A1), (A3) and Eqs. (6.11) and (6.12) hold. Assume that φ : E × E → [0, 1] is a continuous function satisfying φ(x, x) = 0 for all x ∈ E and such that the following properties hold: (i) for each  > 0 there exists δ > 0 such that if x, y ∈ E and φ(x, y) ≤ δ, then ρE (x, y) ≤ . (ii) for each  > 0 there exists δ > 0 such that if x, y ∈ E and ρE (x, y) ≤ δ, then φ(x, y) ≤ . Let f ∈ Mreg and let r ∈ (0, 1). Define f

fr (t, x, u) = f (t, x, u) + rφ(x, xt ), (t, x, u) ∈ M.

6.11 Genericity Results for Discrete-Time Problems

181 f

f

f

f

It was shown in Sect. 6.3 that (A1) and (A3) hold for fr with (xt r , ut r ) = (xt , ut ), t = 0, 1, . . . , bfr = bf and Δfr = Δf . Assume that for every f ∈ Mreg and every r ∈ (0, 1), fr ∈ M and fr → f as r → 0+ in the strong topology. Theorem 6.17 implies that for every f ∈ Mreg and every r ∈ (0, 1), fr has properties TP, WTP, ¯ reg the closure of Mreg in the metric space (M, ds ). P1, P2. Denote by M We prove the following result. Theorem 6.28 Let M > 0 and  ∈ (0, 1). Then there exists an open (in the weak ¯ reg such that topology) and everywhere dense (in the strong topology) set F ⊂ M {fr : f ∈ Mreg , r ∈ (0, 1)} ⊂ F and that for each g ∈ F there exists an open set in the weak topology V ⊂ Mψ ∗ ∞ containing g, δ ∈ (0, 1), L > 0 and ({xt∗ }∞ t=0 , {ut }t=0 ) ∈ X(0, ∞) and the following assertion holds. Assume that h ∈ V , T1 ≥ 0, T2 ≥ T1 + 2L are integers and a trajectory–control T2 −1 2 pair ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfies 1 1 T 2 −1

h(t, xt , ut ) ≤ min{U h (T1 , T2 ) + M, U h (T1 , T2 , xT1 , xT2 ) + δ}.

t=T1

Then there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 . . . , τ2 . f

f

Moreover, if ρE (xT2 , xT2 ) ≤ δ, then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ, T1 ≥ L, then τ1 = T1 . Proof Clearly, the set {fr : f ∈ Mreg , r ∈ (0, 1)} ¯ reg equipped with the relative strong topology. Let f ∈ is everywhere dense in M Mreg and r ∈ (0, 1). Then fr has TP. Theorem 6.22 implies that there exists an open neighborhood V (f, r) of fr in Mψ in the weak topology and L(f, r), δ(f, r) > 0 such that the following property holds: for each h ∈ V (f, r), each pair of integers T1 ≥ 0, T2 ≥ T1 + 2L(f, r) and each T2 −1 2 ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 T 2 −1

h(t, xt , ut ) ≤ min{U h (T1 , T2 ) + M, U h (T1 , T2 , xT1 , xT2 ) + δ(f, r)}

t=T1

there exist integers τ1 ∈ [T1 , T1 + L(f, r)], τ2 ∈ [T2 − L(f, r), T2 ] such that

182

6 Discrete-Time Nonautonomous Problems on Half-Axis f

ρE (xt , xt ) ≤ , t = τ1 . . . , τ2 ; f

f

moreover, if ρE (xT2 , xT2 ) ≤ δ(f, r), then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ(f, r), T1 ≥ L(f, r), then τ1 = T1 . Define ¯ reg . F = ∪{V (f, r) : f ∈ Mreg , r ∈ (0, 1)} ∩ M It is not difficult to see that the set F is open in the weak topology, everywhere dense in the strong topology and that our assertion holds. Now we assume that the metric space (M, ds ) is complete and prove the following generic result. Theorem 6.29 Then there exists an everywhere dense (in the strong topology) set ¯ reg which is a countable intersection of open (in the weak topology) subsets F ⊂M ¯ of Mreg such that {fr : f ∈ Mreg , r ∈ (0, 1)} ⊂ F ∗ ∞ and that for each f ∈ F there exist ({xt∗ }∞ t=0 , {ut }t=0 ) ∈ X(0, ∞) such that the following assertion holds. For each M,  > 0 there exists a neighborhood V of f in Mψ in the weak topology and δ, L > 0 such that for each g ∈ V , each pair of integers T1 ≥ 0, T2 −1 2 T2 ≥ T1 + 2L and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfying 1 1 T 2 −1

g(t, xt , ut ) ≤ min{U g (T1 , T2 ) + M, U g (T1 , T2 , xT1 , xT2 ) + δ}

t=T1

there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that f

ρE (xt , xt ) ≤ , t = τ1 . . . , τ2 .

6.12 Proof of Theorem 6.29 The set {fr : f ∈ Mreg , r ∈ (0, 1)} ¯ reg with the strong topology. Let f ∈ Mreg , r ∈ (0, 1)} is everywhere dense in M and n ≥ 1 be an integer. By (A3), there exists Δ(f, r, n) ∈ (0, 2−n )

6.12 Proof of Theorem 6.29

183

such that the following property holds: f

(a) for each (Ti , zi ) ∈ A, i = 1, 2 satisfying ρE (zi , xTi ) ≤ Δ(f, r, n), i = 1, 2 and T2 ≥ bf there exist (1) T +b

(1) T +b −1

f f 1 1 ({xt }t=T , {ut }t=T 1 1

) ∈ X(T1 , T1 + bf ),

T2 −1 2 ({xt(2) }Tt=T , {u(2) t }t=T2 −bf ) ∈ X(T2 − bf , T2 ) 2 −bf

such that (1)

(1)

f

(2)

(2)

f

xT1 = z1 , xT1 +bf = xT1 +bf , xT2 = z2 , xT2 −bf = xT2 −bf , T1 +bf −1



T1 +bf −1 (1) (1) fr (t, xt , ut )

≤

t=T1



fr (t, xt , ut ) + 2−n , f

f

t=T1

T 2 −1

(2)

(2)

fr (t, xt , ut ) ≤

t=T2 −bf

T 2 −1

fr (t, xt , ut ) + 2−n , f

f

t=T2 −bf

ρ(xt , xt ) ≤ 2−n , t = T1 , . . . , T1 + bf , f

(1)

ρ(xt , xt ) ≤ 2−n , t = T2 − bf , . . . , T2 . f

(2)

Theorem 6.22 and Proposition 6.21 imply that there exists an open neighborhood U (f, r, n) of fr in Mψ in the weak topology and an integer L(f, r, n) > n, δ(f, r) ∈ (0, 2−n ) such that the following properties hold: (b) For each h ∈ U (f, r, n), each pair of integers T1 ≥ 0, T2 ≥ T1 + 2L(f, r, n) T2 −1 2 , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies and each ({xt }Tt=T 1 1 T 2 −1

h(t, xt , ut ) ≤ min{U h (T1 , T2 ) + n, U h (T1 , T2 , xT1 , xT2 ) + δ(f, r, n)}

t=T1

there exist integers τ1 ∈ [T1 , T1 + L(f, r)], τ2 ∈ [T2 − L(f, r), T2 ] such that ρE (xt , xt ) ≤ 2−n Δ(f, r, n), t = τ1 . . . , τ2 ; f

f

f

moreover, if ρE (xT2 , xT2 ) ≤ δ(f, r, n), then τ2 = T2 and if ρE (xT1 , xT1 ) ≤ δ(f, r, n), T1 ≥ L(f, r, n), then τ1 = T1 .

184

6 Discrete-Time Nonautonomous Problems on Half-Axis

(c) For each h ∈ U (f, r, n), each integers T ≥ 0 and each trajectory–control pair T +b T +b −1 ({xt }t=T f , {ut }t=T f ) ∈ X(T , T + bf ) which satisfies T +bf −1



T+ bf −1

fr (t, xt , ut ) ≤

t=T



f

f

f (t, xt , ut ) + 4

t=T

the inequality T +bf −1

|



T +bf −1

h(t, xt , ut ) −

t=T



fr (t, xt , ut )| ≤ 2−n Δ(f, r, n)

t=T

is true. Define ¯ F = (∩∞ n=1 ∪ {U (f, r, n) : f ∈ Mreg , r ∈ (0, 1)}) ∩ Mreg .

(6.120)

It is not difficult to see that the set F is a countable intersection of open in ¯ reg . the weak topology, everywhere dense in the strong topology subsets of M Assume that f ∈ F.

(6.121)

By (6.120) and (6.121), for each integer n ≥ 1 there exists f (n) ∈ Mreg , r (n) ∈ (0, 1) such that f ∈ U (f (n) , r (n) , n).

(6.122)

Property (b) and (6.122) imply that the following property holds: (d) for each integer n ≥ 1, each pair of integers T1 ≥ 0, T2 ≥ T1 +2L(f (n) , r (n) , n) T2 −1 2 and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 T 2 −1

f (t, xt , ut ) ≤ min{U f (T1 , T2 )+n, U f (T1 , T2 , xT1 , xT2 )+δ(f (n) , r (n) , n)}

t=T1

we have f (n)

ρE (xt , xt

) ≤ 2−n Δ(f (n) , r (n) , n),

t = T1 + L(f (n) , r (n) , n), . . . , T2 − L(f (n) , r (n) , n).

6.12 Proof of Theorem 6.29

185

Property (d) implies that the following property holds: (e) for each pair of integers n, m ≥ 1, each pair of integers T1 ≥ 0, T2 ≥ T1 + 2 max{L(f (n) , r (n) , n), L(f (m) , r (m) , m)} T2 −1 2 and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) which satisfies 1 1 T 2 −1

f (t, xt , ut )

t=T1

≤ min{U f (T1 , T2 )+1, U h (T1 , T2 , xT1 , xT2 )+min{δ(f (i) , r (i) , i) : i = n, m}} we have for i = n, m, f (i)

ρE (xt , xt

) ≤ 2−i Δ(f (i) , r (i) , i), t = T1 + L(f (i) , r (i) , i), . . . , T2 − L(f (i) , r (i) , i)

and for all integers t ∈ [T1 + max{L(f (n) , r (n) , n), L(f (m) , r (m) , m)}, T2 − max{L(f (n) , r (n) , n), L(f (m) , r (m) , m)}] we have f (m)

ρE (xt

f (n)

, xt

) ≤ 2−n Δ(f (n) , r (n) , n) + 2−m Δ(f (m) , r (m) , m).

Property (e) implies that the following property holds: (f) for each pair of integers n, m ≥ 1 and each integer t ≥ max{L(f (n) , r (n) , n), L(f (m) , r (m) , m)} we have f (m)

ρE (xt

f (n)

, xt

) ≤ 2−n Δ(f (n) , r (n) , n) + 2−m Δ(f (m) , r (m) , m).

Construct a strictly increasing sequence of natural numbers {nk }∞ k=1 such that n1 = 1

(6.123)

and that for each integer k ≥ 1, nk+1 > L(f (nk ) , r (nk ) , nk ) + nk (bf (nk ) + 1),

(6.124)

186

6 Discrete-Time Nonautonomous Problems on Half-Axis

2−nk+1 Δ(f (nk+1 ) , r (nk+1 ) , nk+1 ) < 2−nk Δ(f (nk ) , r (nk ) , nk ).

(6.125)

Set f (n1 )

xt∗ = xt

f (n1 )

, t = 0, . . . , L(f (n2 ) , r (n2 ) , n2 ), u∗t = ut

,

t = 0, . . . , L(f (n2 ) , r (n2 ) , n2 ) − 1.

(6.126)

Assume that k ≥ 1 is an integer and that we defined L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )

({xt∗ }t=0

L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )−1

, {u∗t }t=0

)

∈ X(0, L(f (nk+1 ) , r (nk+1 ) , nk+1 )) such that x∗

L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )

f (nk ) . L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )

=x

(6.127)

(Clearly, our assumption holds for k = 1.) Property (f), Eqs. (6.124), (6.125) and the inequality L(f (nk+1 ) , r (nk+1 ) , nk+1 ) > nk+1 imply that for each integer t ≥ L(f (nk+1 ) , r (nk+1 ) , nk+1 ), f (nk )

ρE (xt

f (nk+1 )

, xt

)

≤ 2−nk Δ(f (nk ) , r (nk ) , nk ) + 2−nk+1 Δ(f (nk+1 ) , r (nk+1 ) , nk+1 ) ≤ 2−nk +1 Δ(f (nk ) , r (nk ) , nk ) ≤ Δ(f (nk ) , r (nk ) , nk ).

(6.128)

Property (a) and (6.128) imply that there exists ({xt∗ }

L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )+b t=L(f

(nk+1 )

,r

(nk+1 )

f (nk )

,nk+1 )

, {u∗t }

L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )+b t=L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )

f (nk )

−1

)

∈ X(L(f (nk+1 ) , r (nk+1 ) , nk+1 ), L(f (nk+1 ) , r (nk+1 ) , nk+1 ) + bf (nk ) ) such that x∗

L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )

f (nk ) , L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )

=x

(6.129)

6.12 Proof of Theorem 6.29

187

x∗

L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )+b

f (nk )

f (nk+1 ) L(f (nk+1 ) ,r (nk+1 ) ,nk+1 )+b

=x

(6.130)

, f (nk )

 (n ) {f (nkk ) (t, xt∗ , u∗t ) : r

t = L(f (nk+1 ) , r (nk+1 ) , nk+1 ), . . . , L(f (nk+1 ) , r (nk+1 ) , nk+1 ) + bf (nk ) − 1} ≤

 (n ) f (nk ) f (nk ) {f (nkk ) (t, xt , ut ): r

t = L(f (nk+1 ) , r (nk+1 ) , nk+1 ), . . . , L(f (nk+1 ) , r (nk+1 ) , nk+1 )+bf (nk ) −1}+2−nk , (6.131) (n ) ∗ f k −nk ρE (xt , xt )≤2 , t ∈ L(f (nk+1 ) , r (nk+1 ) , nk+1 ), . . . , L(f (nk+1 ) , r (nk+1 ) , nk+1 ) + bf (nk ) . (6.132) Set f (nk+1 )

xt∗ = xt

,

t ∈ {L(f (nk+1 ) , r (nk+1 ) , nk+1 ) + bf (nk ) , . . . , L(f (nk+2 ) , r (nk+2 ) , nk+2 )}, (6.133) f (nk+1 )

u∗t = ut

,

t ∈ {L(f (nk+1 ) , r (nk+1 ) , nk+1 ) + bf (nk ) , . . . , L(f (nk+2 ) , r (nk+2 ) , nk+2 ) − 1}. (6.134) It is not difficult to see that L(f (nk+2 ) ,r (nk+2 ) ,nk+2 )

({xt∗ }t=0

L(f (nk+2 ) ,r (nk+2 ) ,nk+2 −1)

, {u∗t }t=0

)

∈ X(0, L(f (nk+2 ) , r (nk+2 ) , nk+2 )). Therefore by induction we defined ∗ ({xt∗ }∞ t=0 , {ut }∞ ) ∈ X(0, ∞)

such that (6.126) holds, for each integer k ≥ 1 Eqs. (6.127) and (6.129)–(6.134) are true. Assume that k ≥ 2 and t ≥ L(f (nk ) , r (nk ) , nk ) are integers. By (6.124), there exists an integer q≥k

(6.135)

188

6 Discrete-Time Nonautonomous Problems on Half-Axis

such that t ∈ {L(f (nq ) , r (nq ) , nq ), . . . , L(f (nq+1 ) , r (nq+1 ) , nq+1 )}.

(6.136)

It follows from (6.124), (6.125), (6.135), and (6.136) that f (nk )

ρE (xt

f (nq )

, xt

) ≤ 2−nk +1 Δ(f (nk ) , r (nk ) , nk ).

(6.137)

If t ∈ {L(f (nq ) , r (nq ) , nq ), . . . , L(f (nq ) , r (nq ) , nq ) + bf (nq−1 ) },

(6.138)

then it follows from Eq. (6.132) applied with k = q − 1, property (f) and Eqs. (6.124) and (6.138) that f (nq )

ρE (xt∗ , xt

(nq−1 )

) ≤ ρE (xt∗ , xt

f

f

) + ρE (xt

(nq−1 )

f (nq )

, xt

)

≤ 2−nq −1 + 2−nq−1 +1 Δ(f (nq−1 ) , r (nq−1 ) , nq−1 ) ≤ 2−nq−1 +1 . By (6.133) applied with k = q − 1, for each t ∈ {L(f (nq ) , r (nq ) , nq ) + bf (nq ) , . . . , L(f (nq+1 ) , r (nq+1 ) , nq+1 )} we have f (nq )

xt∗ = xt

.

Thus in all the cases f (nq )

ρE (xt∗ , xt

) ≤ 2−nq−1 +1 .

(6.139)

In view of (6.137) and (6.139), f (nk )

ρE (xt∗ , xt

f (nk )

) ≤ ρE (xt

f (nq )

, xt

f (nq )

) + ρE (xt

, xt∗ )

≤ 2−nk +1 Δ(f (nk ) , r (nk ), nk ) + 2−nq−1 +1 ≤ 2−nk +1 Δ(f (nk ) , r (nk ), nk ) + 2−nk−1 +1 ≤ 3 · 2−nk−1 . Thus we have shown that the following property holds: (g) for each pair of integers k ≥ 2 and t ≥ L(f (nk ) , r (nk ) , nk ) we have f (nk )

ρE (xt∗ , xt

) ≤ 3 · 2−nk−1 .

6.12 Proof of Theorem 6.29

189

Assume that M > 1 and  ∈ (0, 1). Choose an integer k > 4 such that nk > M, 4 · 2−nk−1 < .

(6.140)

g ∈ U (f (nk ) , r (nk ) , nk ),

(6.141)

Assume that

δ = δ(f (nk ) , r (nk ) , nk ), T1 ≥ 0, T2 ≥ T1 + 2L(f (nk ) , r (nk ) , nk ) T2 −1 2 are integers and that ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfies 1 1 T 2 −1

g(t, xt , ut )

t=T1

≤ min{U g (T1 , T2 )+M, U g (T1 , T2 , xT1 , xT2 )+δ(f (nk ) , r (nk ) , nk )}.

(6.142)

Property (b) and Eqs. (6.140) and (6.141) imply that for each integer t ∈ [T1 + L(f (nk ) , r (nk ) , nk ), T2 − L(f (nk ) , r (nk ) , nk )] we have f (nk )

ρE (xt , xt

) ≤ 2−nk Δ(f (nk ) , r (nk ) , nk ).

(6.143)

Property (g) and Eqs. (6.140) and (6.143) imply that for each integer t ∈ [T1 + L(f (nk ) , r (nk ) , nk ), T2 − L(f (nk ) , r (nk ) , nk )] we have f (nk )

ρE (xt , xt∗ ) ≤ ρE (xt , xt

f (nk )

) + ρE (xt

, xt∗ )

≤ 2−nk Δ(f (nk ) , r (nk ) , nk ) + 3 · 2−nk−1 ≤ 4 · 2−nk−1 < . Theorem 6.29 is proved.

190

6 Discrete-Time Nonautonomous Problems on Half-Axis

6.13 Examples of the Space M Define φ : E × E → [0, 1] by φ(x, y) = min{1, ρE (x, y)}, x, y ∈ E. Clearly, the function φ is continuous and properties (i), (ii) hold. Set φ0 (x) = min{1, x}, x ∈ R 1 . Then φ0 is Lipschitz with a Lipschitz constant 1, φ = φ0 ◦ ρE and φ is also a Lipschitz function on E × E with a Lipschitz constant 1. We equip the space Mψ with the uniformity determined by the base Es () = {(h, g) ∈ Mψ × Mψ : |h(t, x, u) − g(t, x.u)| ≤  for each (t, x, u) ∈ M} ∩ {(h, g) ∈ Mψ × Mψ : |(h − g)(t, x1 , u) − (h − g)(t, x2 , u)| ≤ ρE (x1 , x2 ) for each (t, x1 , u), (t, x2 , u) ∈ M} ∩ {(h, g) ∈ Mψ × Mψ : |(h − g)(t, x, u1 ) = (h − g)(t, x, u2 ) for each (t, x, ui ) ∈ M, i = 1, 2}, where  > 0. Clearly, the uniform space Mψ is Hausdorff and has a countable base. Therefore Mψ is metrizable (by a metric ds ). It is not difficult to see that the metric space (Mψ , ds ) is complete and the uniformity introduced above is stronger than the uniformity defined by (6.19). Denote by Ml the set of all f ∈ Mψ such that for each integer t ≥ 0, the function f (t, ·, ·) is lower semicontinuous on the set {(x.u) ∈ E × F : (t, x, u) ∈ M}, by Mc the set of all f ∈ Mψ such that for each integer t ≥ 0, the function f (t, ·, ·) is continuous on the set {(x.u) ∈ E × F : (t, x, u) ∈ M}, by ML the set of all f ∈ Mψ such that for each integer t ≥ 0, the function f (t, ·, ·) is Lipschitz on the set {(x.u) ∈ E × F : (t, x, u) ∈ M}, by MlL the set of all f ∈ Mψ such that for each integer t ≥ 0, the function f (t, ·, ·) is Lipschitz on all bounded subsets of {(x.u) ∈ E × F : (t, x, u) ∈ M}. Clearly, all these sets are closed in (Mψ , ds ) and M can be any of them.

6.14 Extensions of the Generic Results

191

6.14 Extensions of the Generic Results ˜ ψ the set of all functions g : {0, 1, . . . , } × E × F → R 1 , which satisfy Denote by M g(t, x, u) ≥ ψ(ρE (x, θ0 )) − a0 for each (t, x, u) ∈ {0, 1, . . . , } × E × F. ˜ ψ denote by R(g) the restriction of g to M. Clearly, R(g) ∈ Mψ For each g ∈ M ˜ ψ. for every g ∈ M ˜ ψ and each pair of integers T2 > T1 ≥ 0 and each pair of points For each g ∈ M y, z ∈ E satisfying (T1 , y), (T2 , z) ∈ A we define U g (T1 , T2 , y, z) = U R(g) (T1 , T2 , y, z), U g (T1 , T2 , y) = U R(g) (T1 , T2 , y), R(g) (T1 , T2 , z), g (T1 , T2 , z) = U U U g (T1 , T2 ) = U R(g) (T1 , T2 ). ˜ ψ with the uniformity determine by the base We equip the space M ˜ ψ ×M ˜ ψ : |g1 (t, x, u) − g2 (t, x.u)| ≤  ˜ E(N, ) = {(g1 , g2 ) ∈ M for each (t, x, u) ∈ {0, 1, . . . , } × E × F satisfying ρE (x, θ0 ), ρ(u, θ1 ) ≤ N } ˜ ψ ×M ˜ ψ : |g1 (t, x, u) − g2 (t, x, u)| ≤  ∩ {(g1 , g2 ) ∈ M for each (t, x, u) ∈ {0, 1, . . . } × E × F satisfying min{g1 (t, x, u), g2 (t, x, u)} ≤ N }, ˜ ψ is Hausdorff and has a countable where N,  > 0. Clearly, the uniform space M ˜ ψ is metrizable (by a metric d˜ψ ). It is not difficult to see that base. Therefore M ˜ ψ is complete. It is equipped with a topology induced by the the uniform space M ˜ψ→ uniformity which is called a weak topology. It is clear that the mapping R : M Mψ is uniformly continuous. ˜ ⊂ M ˜ ψ is equipped with a metric d˜s which induces Assume that a set M a uniformity which is stronger than the uniformity introduced above. The space ˜ ⊂ Mψ is equipped with the topology induced by the metric ds which is called M the strong topology. ˜ such that there exists an integer bf > 0, ˜ reg the set of all f ∈ M Denote by M f ∞ f ∞ Δf > 0 and ({xt }t=0 , {ut }t=0 ) ∈ X(0, ∞) such that (A1), (A3) and Eqs. (6.11) and (6.12) hold.

192

6 Discrete-Time Nonautonomous Problems on Half-Axis

Assume that φ : E × E → [0, 1] is a continuous function satisfying φ(x, x) = 0 for all x ∈ E and such that properties (i), (ii) of Sect. 6.11 hold. ˜ reg and let r ∈ (0, 1). Define Let f ∈ M f

fr (t, x, u) = f (t, x, u) + rφ(x, xt ), (t, x, u) ∈ {0, 1, . . . , } × E × F. f

f

f

f

It was shown in Sect. 6.3 that (A1) and (A3) hold for fr with (xt r , ut r ) = (xt , ut ), t = 0, 1, . . . , bfr = bf and Δfr = Δf . Theorem 6.17 implies that fr has properties ˜ reg and every r ∈ (0, 1), fr ∈ M ˜ TP, WTP, P1, P2. Assume that for every f ∈ M and fr → f as r → 0+ in the strong topology. ˜ reg ) the closure of M ˜ ds ). The ˜ reg in the metric space (M, Denote by cl(M following analog of Theorem 6.28 is true. Theorem 6.30 Let M > 0 and  ∈ (0, 1). Then there exists an open (in the weak ˜ reg ) such topology) and everywhere dense (in the strong topology) set F˜ ⊂ cl(M that ˜ reg , r ∈ (0, 1)} ⊂ F˜ {fr : f ∈ M ˜ψ such that for each g ∈ F˜ there exists an open set in the weak topology V ⊂ M ∞ ∞ ∗ ∗ containing g, δ ∈ (0, 1), L > 0 and ({xt }t=0 , {ut }t=0 ) ∈ X(0, ∞) and the following assertion holds. Assume that h ∈ V , T1 ≥ 0, T2 ≥ T1 + 2L are integers and a trajectory–control T2 −1 2 , {ut }t=T ) ∈ X(T1 , T2 ) satisfies pair ({xt }Tt=T 1 1 T 2 −1

h(t, xt , ut ) ≤ min{U h (T1 , T2 ) + M, U h (T1 , T2 , xT1 , xT2 ) + δ}.

t=T1

Then there exist integers τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that ρE (xt , xt∗ ) ≤ , t = τ1 . . . , τ2 . Moreover, if ρE (xT2 , xT∗2 ) ≤ δ, then τ2 = T2 and if ρE (xT1 , xT∗1 ) ≤ δ, T1 ≥ L, then τ1 = T1 . ˜ Proof Our result follows from Theorem 6.28 applied for the space M = R(M) equipped with the metric ds = dψ . Let F be as guaranteed by Theorem 6.28 with ˜ Set M = R(M). ˜ reg ). F˜ = R−1 (F) ∩ cl(M Clearly, ˜ reg , r ∈ (0, 1)} ⊂ F˜ {fr : f ∈ M

6.14 Extensions of the Generic Results

193

˜ ψ → Mψ is which is dense in the strong topology. Note that the mapping R : M continuous in the weak topologies. This implies that ˜ ⊂ M, ¯ R(cl(M ˜ reg )) ⊂ M ¯ reg . R(cl(M)) Thus ˜ reg )) F˜ = R−1 (F) ∩ (cl(M ˜ reg ) in the weak topology. Now the assertion of Theorem 6.30 is an open set in cl(M follows from the assertion of Theorem 6.28. The following analog of Theorem 6.29 is true. Theorem 6.31 Then there exists an everywhere dense (in the strong topology) set ˜ reg ) which is a countable intersection of open (in the weak topology) F˜ ⊂ cl(M ˜ reg ) such that subsets of cl(M ˜ reg , r ∈ (0, 1)} ⊂ F˜ {fr : f ∈ M ∗ ∞ and that for each f ∈ F˜ there exist ({xt∗ }∞ t=0 , {ut }t=0 ) ∈ X(0, ∞) such that the following assertion holds. ˜ ψ in the weak For each M,  > 0 there exists a neighborhood V of f in M topology and δ, L > 0 such that for each g ∈ V , each pair of integers T1 ≥ 0, T2 −1 2 T2 ≥ T1 + 2L and each ({xt }Tt=T , {ut }t=T ) ∈ X(T1 , T2 ) satisfying 1 1 T 2 −1

g(t, xt , ut ) ≤ min{U g (T1 , T2 ) + M, U g (T1 , T2 , xT1 , xT2 ) + δ}

t=T1

the inequality ρE (xt , xt∗ ) ≤ , t = T1 + L, . . . , T2 − L is true. ˜ Proof Our result follows from Theorem 6.29 applied for the space M = R(M) equipped with the metric ds = dψ . Let F be as guaranteed by Theorem 6.29 with ˜ Set M = R(M). F˜ = R−1 (F). Clearly, ˜ reg , r ∈ (0, 1)} ⊂ F˜ {fr : f ∈ M

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6 Discrete-Time Nonautonomous Problems on Half-Axis

˜ ψ → Mψ is which is dense in the strong topology. Note that the mapping R : M continuous in the weak topologies and ˜ ⊂ cl(R(M)) ˜ = M, ¯ R(cl(M ˜ reg )) ⊂ M ¯ reg . R(cl(M)) Let F be as guaranteed by Theorem 6.29. Define ˜ reg )). F˜ = R−1 (F)(cl(M It is not difficult to see that ˜ reg , r ∈ (0, 1)} ⊂ F, ˜ {fr : f ∈ M F˜ is dense in the strong topology and is a countable intersection of open sets in ˜ reg )) in the weak topology. Now the assertion of Theorem 6.31 follows from cl(M the assertion of Theorem 6.29.

6.15 Smooth Integrands We use the convention that ∞/∞ = 1 and ∞ − ∞ = 0. Let (X,  · ) and (Y,  · ) be Banach spaces. Denote by C(X, Y ) the set of all continuous mappings f : X → Y . Let k ≥ 1 be an integer. For each mapping f : X → Y we denote by f (k) (x) its Frechet derivative of the order k at a point x ∈ X if it exists. Denote by C k (X, Y ) the set of all mappings f : X → Y such that f (k) (x) exists at any point x ∈ X and the mapping x → f (k) (x), x ∈ X is continuous. We set C 0 (X, Y ) = C(X, Y ), f (0) = f , f ∈ C(X, Y ). Let k ≥ 0 be an integer. For each f, g ∈ C (k) (X, Y ) define (k) d˜X,Y (f, g) = sup{f (j ) (x) − g (j ) (x) : j = 0, . . . , k, x ∈ X}, (k) (k) (k) dX,Y (f, g) = d˜X,Y (f, g)(1 + d˜X,Y (f, g))−1 . (k) It is known that (C (k) (X, Y ), dX,Y ) is a complete metric space [41, 89]. Let k ≥ 1 be an integer. We assume that there exists a function λ∗ ∈ C k (X, R 1 ) such that

0 ≤ λ∗ (x) ≤ 1, x ∈ X, {x ∈ X : λ∗ (x) > 0} ⊂ {x ∈ X : x ≤ 1},

6.15 Smooth Integrands

195

λ∗ (z) > 0 for some z ∈ X, (j )

sup{λ∗ (z) : z ∈ X, j = 0, . . . , k} < ∞.

(6.144)

(Note that the function λ∗ exists if the space X is Hilbert. It also exists if k = 1 and X has an equivalent Frechet differentiable norm. For more details concerning the existence of the function λ∗ see [40].) In view of Deville-Godefroy-Zizler variational principle [40] (see also Theorem 5.9 of [122]), there exists φ ∈ C k (X; R 1 ) such that φ(0) = 0, φ(z) ∈ [0, 1], z ∈ X, sup{φ (j ) (z) : z ∈ X, j = 0, . . . , k} < ∞ and that the following property holds: for each  > 0 there exists δ > 0 such that if x ∈ X and φ(x) ≤ δ, then x ≤ . Assume that (E,  · ) and (F,  · ) are Banach spaces, ρE (x, y) = x − y, x, y ∈ E, ρF (x, y) = x − y, x, y ∈ F . We consider the Banach space E × F equipped with the norm (x, y) = x + y, x ∈ E, y ∈ F ˜ ψ introduced in Sect. 6.14. and the space M Assume that k ≥ 1 is an integer and that there exists a function λ∗ ∈ C k (E, R 1 ) satisfying (6.144) with X = E. Then there exists a function φ0 ∈ C k (E; R 1 ) such that φ0 (0) = 0, φ0 (z) ∈ [0, 1], z ∈ E, (j )

sup{φ0 (z) : z ∈ E, j = 0, . . . , k} < ∞ and that the following property holds: for each  > 0 there exists δ > 0 such that if x ∈ E and φ(x) ≤ δ, then x ≤ . ˜ ψ such that for each integer t ≥ 0, f (t, ·, ·) ∈ ˜ the set of all f ∈ M Denote by M C k (E × F ). This set is equipped with the uniformity determined by the base ˜ ×M ˜ : for each integer t ≥ 0, E() = {(g1 , g2 ) ∈ M (k)

dE×F,R 1 (g1 (t, ·, ·), g2 (t, ·, ·)) ≤ }, where  > 0. This uniformity is metrizable and complete and induces a topology which is called strong. Set φ(x, y) = φ0 (x − y), x, y ∈ E. It is not difficult to see ˜ that Theorems 6.30 and 6.31 hold for the space M.

Chapter 7

Infinite Dimensional Control

The study of infinite dimensional optimal control has been a rapidly growing area of research [1–4, 15–17, 19–21, 37, 53, 61, 65, 67, 68, 73, 86, 87, 92, 101–104, 113]. In this chapter we present preliminaries which we need in order to study turnpike properties of infinite dimensional optimal control problems. We discuss unbounded operators, C0 semigroups, evolution equations and admissible control operators.

7.1 Unbounded Operators Let (X,  · ) and (Y,  · ) be Banach spaces. Denote by L(X, Y ) the set of all linear continuous operators from X to Y . For every A ∈ L(X, Y ), define A = sup{Ax : x ∈ X, x ≤ 1}. Set L(X) = L(X, X), X∗ = L(X, R 1 ), and Y ∗ = L(Y, R 1 ). For all x ∈ X, set I x = x. In the space X, we always consider the norm convergence. Let D(A) be a linear subspace of X (not necessarily closed), and let A : D(A) → Y be a linear operator. We say that A is densely defined if D(A) is dense in X. We say that A is closed if the graph G(A) = {(x, y) ∈ X × Y : x ∈ D(A), y = Ax} of A is closed in X × Y . We say that A is closable if there exists a closed operator ¯ → Y such that A¯ : D(A) ¯ ⊂ X, Ax ¯ = Ax, x ∈ D(A). D(A) ⊂ D(A) Let the symbol ·, · be referred to as the duality pairing between X ∗ and X. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0_7

197

198

7 Infinite Dimensional Control

Similar to a bounded operator, for any linear operator (not necessarily bounded) A : D(A) ⊂ X → X, we still define the resolvent ρ(A) = {λ ∈ C 1 : (λI − A)−1 ∈ L(X)}, the spectrum σ (A) = C 1 \ ρ(A), the point spectrum (or the set of eigenvalues) σp (A) of A, and we set Ran(A) = {Ax : x ∈ D(A)}. Let A : D(A) ⊂ X → X be densely defined. Clearly, the map x → Ax, y , x ∈ D(A), y ∈ X∗ is well-defined. Suppose that y ∈ X∗ and |Ax, y | ≤ cy x for all x ∈ D(A). Then the functional fy (x) = Ax, y , x ∈ D(A), can be extended linearly and continuously to the whole X which is the closure of D(A). Such an extension (still denoted by itself) fy is in X∗ and unique. Hence we obtain Ax, y = fy (x) = x, fy for all x ∈ D(A). Define D(A∗ ) = {y ∈ X∗ : ∃cy ≥ 0 such that |Ax, y | ≤ cy x ∀x ∈ D(A)},

(7.1)

A∗ y = fy , y ∈ D(A∗ ). Clearly, A∗ : D(A∗ ) ⊂ X∗ → X∗ is a linear operator satisfying Ax, y = x, A∗ y for all x ∈ D(A) and all y ∈ D(A∗ ). The mapping A∗ is called the adjoint operator of A.

7.2 C0 Semigroup Let (X,  · ) be a Banach space, and let {T (t) : t ∈ [0, ∞)} ⊂ L(X). We call T (·) a C0 semigroup (or a strongly continuous semigroup of operators) on X if the following properties hold: T (0) = I,

(7.2)

T (t + s) = T (s)T (t) for all s, t ≥ 0,

(7.3)

lim T (s)x − x = 0, x ∈ X.

(7.4)

s→0

7.2 C0 Semigroup

199

In the case when T (t) is defined for all t ∈ R 1 and (7.3) holds for all t, s ∈ R 1 , we call T (·) as C0 group or a strongly continuous group of operators. Equations (7.3) and (7.4) are usually referred to as the semigroup property and the strong continuity, respectively. The following result is true (see Proposition 4.7, Chapter 2 of [73]). Proposition 7.1 Let T (·) be a C0 semigroup on X. Then there exist constants M ≥ 1 and ω ∈ R 1 such that T (t) ≤ Meωt , t ∈ [0, ∞). Let T (·) be a C0 semigroup on X, and let D(A) = {x ∈ X : lim t −1 (T (t) − I )x exists}, t→0

Ax = lim t −1 (T (t) − I )x, x ∈ D(A). t→0

The operator A : D(A) → X is called the generator of the semigroup T (·). We also say that A generates the C0 semigroup T (·). In general, the operator A is not bounded. The following two results hold. Proposition 7.2 (Proposition 4.10, Chapter 2 of [73]) Let T (·) be a C0 semigroup on X. Then the generator A of T (·) is a densely defined closed operator and n ∩∞ n=1 D(A ) is dense in X. Furthermore, if S(·) is another C0 semigroup on X with the same generator A as T (·), then S(·) = T (·). Theorem 7.3 ([57, 106] and Theorem 4.11, Chapter 2 of [73]) Let A : D(A) ⊂ X → X be a linear operator. Then the following properties are equivalent: (i) A generates a C0 semigroup T (·) on X such that T (t) ≤ Meωt for all t ≥ 0 with some M ≥ 1 and ω ∈ R 1 . (ii) A is a densely defined and closed operator such that for the above M ≥ 1 and ω ∈ R 1 , {λ ∈ C 1 : Re λ > ω} ⊂ ρ(A), and (λI − A)−n  ≤ M(Re λ − ω)−n for all integers n ≥ 0 and all λ ∈ C 1 with Re λ > ω. (iii) A is a densely defined and closed operator such that for the above M ≥ 1 and ω ∈ R 1 , there exists a sequence of positive numbers λk → ∞ as k → ∞ such that for all integers k ≥ 1, λk ∈ ρ(A), and (λk I − A)−n  ≤ M(λk − ω)−n for all integers n ≥ 0 and all integers k ≥ 1. Because the generator A determines the C0 semigroup T (·) uniquely and in the case A ∈ L(X) the C0 semigroup has an explicit expression eAt =

200

∞

7 Infinite Dimensional Control −1 t n An

(Proposition 4.9, Chapter 2 of [73]), we denote by eAt the C0 semigroup generated by A. n=0 (n!)

7.3 Evolution Equations Let (X,  · ) be a Banach space, A : D(A) ⊂ X → X generate a C0 semigroup eAt on X, T > 0, f : [0, T ] → X be a Bochner integrable function, and y0 ∈ X. We consider the following evolution equation: y  (t) = Ay(t) + f (t), t ∈ [0, T ],

(P0 )

y0 = y(0). A continuous function y : [0, T ] → X is called a (mild) solution of (P0 ) if 

t

y(t) = e y0 + At

eA(t−s) f (s)ds, t ∈ [0, T ].

0

A continuous function y : [0, T ] → X is called a weak solution of (P0 ) if for any x ∗ ∈ D(A∗ ), y(·), x ∗ is an absolutely continuous function on [0, T ] and that for all t ∈ [0, T ], y(t), x ∗ = y0 , x ∗ +



t

[y(s), A∗ x ∗ + f (s), x ∗ ]ds.

0

We have the following result: Proposition 7.4 ([17], Proposition 5.2, Chapter 2 of [73]) A continuous function y : [0, T ] → X is a solution of (P0 ) if and only if it is a weak solution of (P0 ). The following useful result is also valid (see Proposition 4.14, Chapter 2 of [73]). Proposition 7.5 For any x ∈ D(A), eAt x ∈ D(A) for all t ≥ 0 and (d/dt)(eAt x) = AeAt x = eAt Ax for all t ≥ 0, 

t

eAt x − eAs x =

eAr Axdr for all t > s ≥ 0.

s

7.4 C0 Groups Let (H, ·, · ) be a Hilbert space equipped with an inner product ·, · which induces the norm  · . A one-parametric family S(t), t ∈ R 1 , of continuous linear operators

7.5 Admissible Control Operators

201

from H onto H is a strongly continuous group of continuous linear operators on H or C0 group on H if S(0)x = x for all x ∈ H, S(t1 + t2 ) = S(t1 )S(t2 ) for all t1 , t2 ∈ R 1 , lim S(t)x = x, x ∈ H.

t→0

Let S(t), t ∈ R 1 , be a C0 group on H . Then the generator of S is the linear operator A : D(A) ⊂ H → H defined by D(A) = {x ∈ H : lim t −1 (S(t) − I )x exists}, t→0

Ax = lim t −1 (S(t) − I )x, x ∈ D(A). t→0

The following result is true. Theorem 7.6 ([37]) Let S(·) be a C0 group on H . Then the following assertions hold: (i) D(A) is dense in H and A is a closed linear operator. (ii) For every x 0 ∈ D(A), there exists a unique function x ∈ C 1 (R 1 ; H ) ∩ C 0 (R 1 ; D(A)) such that x(0) = x 0 , x(t) ∈ D(A) for all t ∈ R 1 and (dx/dt)(t) = Ax(t) for all t ∈ R 1 ; moreover, this solution satisfies x(t) = S(t)x 0 for all t ∈ R 1 . (iii) S(t)∗ , t ∈ R 1 is a C0 group on H and its generator is the adjoint operator A∗ of A. It is clear that if S(·) is a C0 group with the generator A : D(A) → H , then S(−t), t ∈ R 1 , is also a C0 group with the generator −A : D(A) → H . Let A : D(A) ⊂ H → H . It is known that A and −A are both generators of C0 semigroups if and only if A is a generator of a C0 group [104]. Note that a C0 group is determined by its generator A uniquely and is denoted by eAt , t ∈ R 1 . Clearly, for all t ∈ R 1 , e−At = eA(−t) .

7.5 Admissible Control Operators Let J be an open interval and U be a Hilbert space. Denote by H 1 (J ; U ) the Sobolev space of all locally absolutely continuous functions z : J → U for which dz/dt ∈ L2 (J ; U ). The space H 2 (J ; U ) is the set of all locally absolutely continuous functions z : J → U for which dz/dt ∈ H 1 (J ; U ). The space H01 (J ; U ) is the set of all functions in H 1 (J ; U ) which have limits equal to zero at the endpoints of J . Define

202

7 Infinite Dimensional Control

H02 (J ; U ) = {h ∈ H 2 (J ; U ) ∩ H01 (J ; U ) : dh/dx ∈ H01 (J ; U )}. In this section we collect several useful results of [104]. Proposition 7.7 (Proposition 2.1.2 of [104]) Let eAt , t ≥ 0, be a C0 semigroup on a Hilbert space E. Then the mapping (t, z) → eAt z, t ∈ [0, ∞), z ∈ E, is continuous on [0, ∞) × E in the product topology. Let X and Z be Hilbert spaces and D(A) be a linear subspace of X. A linear operator A : D(A) → Z is called closed if its graph G(A) = {(f, Af ) : f ∈ D(A)} is closed in X × Z. If A is closed, then D(A) is a Hilbert space with the graph norm  · gr : z2gr = z2X + Az2Z , z ∈ D(A). Proposition 7.8 (Proposition 2.10.1 of [104]) Let X be a Hilbert space and A : D(A) → X be a densely defined linear operator such that ρ(A) = ∅. Then, for every β ∈ ρ(A) ∩ R 1 , the space D(A) with the norm z1 = (βI − A)(z), z ∈ D(A), is a Hilbert space denoted by X1 . The norms generated as above for different β ∈ ρ(A) ∩ R 1 are equivalent to the graph norm and the embedding X1 ⊂ X is continuous. If L ∈ L(X) satisfies LD(A) ⊂ D(A), then L ∈ L(X1 ). Note that the relation ρ(A) = ∅ implies that the operator A is closed. Let A be as in Proposition 7.8. It is clear that A∗ has the same property. We define a Hilbert space X1d = D(A∗ ) equipped with the norm zd1 = (βI − A∗ )z, z ∈ D(A∗ ), where β ∈ ρ(A) ∩ R 1 , or equivalently, β ∈ ρ(A∗ ) ∩ R 1 . Proposition 7.9 (Proposition 2.10.2 of [104]) Let A be as in Proposition 7.8, and let β ∈ ρ(A) ∩ R 1 .We denote by X−1 the completion of X with respect to the norm z−1 = (βI − A)−1 z, z ∈ X.

(7.5)

The norms generated as above for different β ∈ ρ(A) ∩ R 1 are equivalent (in particular, X−1 is independent of the choice of β). Moreover, X−1 is the dual of X1d with respect to the pivot space X. If L ∈ L(X) satisfies L∗ D(A∗ ) ⊂ D∗ (A), then L has a unique extension to an operator L˜ ∈ L(X−1 ). Proposition 7.10 (Proposition 2.10.3 of [104]) Let A : D(A) → X be a densely defined linear operator such that ρ(A) = ∅, β ∈ ρ(A) ∩ R 1 , X1 be as in Proposition 7.8, and X−1 be as in Proposition 7.9. Then A ∈ L(X1 , X) and A has a unique extension A˜ ∈ L(X, X−1 ). Moreover, the operators (βI − A)−1 ∈ L(X, X1 ) ˜ −1 ∈ L(X−1 , X) are unitary. and (βI − A) Proposition 7.11 (Proposition 2.10.4 of [104]) We use the notation from Proposition 7.10 and assume that A generates a C0 semigroup S(·) on X. Then, for every

7.5 Admissible Control Operators

203

˜ ˜ t ≥ 0, S(t) has a unique extension S(t) ∈ L(X−1 ), S(t), t ≥ 0 is a C0 semigroup on X−1 , and A˜ is its generator. Let U and X be Hilbert spaces, S(·) be a C0 semigroup on X, and A : D(A) → X be its generator. We use the notation above and denote by A and S(·) also the extension of the generator to X and the extension of the semigroup to X−1 , respectively. Consider the differential equation z (t) = Az(t) + f (t),

(7.6)

where f ∈ L1loc ([0, ∞); X−1 ). A solution of (7.6) in X−1 is a function z ∈ L1loc ([0, ∞); X) ∩ C([0, ∞); X−1 ) which satisfies the following equations in X−1 : 

t

z(t) − z(0) =

[Az(s) + f (s)]ds for all t ∈ [0, ∞).

(7.7)

0

It is also called a “strong” solution of (7.6) in X−1 . Equation (7.7) implies that z is an a.c. function with values in X−1 and (7.6) holds for almost every t ≥ 0, with the derivative computed with respect to the norm of X−1 . We can also define the concept of a “weak” solution of (7.6) in X−1 by regarding instead of (7.7) that for every φ ∈ X1d and every t ≥ 0,  z(t) − z(0), φ X−1 ,Xd = 1

t

[z(s), A∗ φ X + f (s), φ X−1 ,Xd ]ds. 1

0

These two concepts are equivalent. Proposition 7.12 (Proposition 4.1.4 of [104]) Suppose that z is a solution of (7.6) in X−1 , and let z0 = z(0). Then z is given by  z(t) = S(t)z0 +

t

S(t − σ )f (σ )dσ, t ≥ 0.

(7.8)

0

In particular, for every z0 ∈ X, there exists at most one solution in X−1 of (7.6) which satisfies initial condition z(0) = z0 . Note that z satisfying (7.8) is called the mild solution of (7.6) with the initial state z0 ∈ X. It is not difficult to see that the following result is valid. Proposition 7.13 Let f ∈ L1loc ([0, ∞); X−1 ), T > 0, and z ∈ C 0 ([0, T ]; X−1 ). Then z satisfies (7.8) for all t ∈ [0, T ] in X−1 if and only if for all ξ ∈ X1d = D(A∗ ) and all t ∈ [0, T ],

204

7 Infinite Dimensional Control

 z(t), ξ X−1 ,Xd = S(t)z0 , ξ X−1 ,Xd + 1

1

 = S(t)z0 , ξ X−1 ,Xd + 1

t

0

0

t

[S(t − σ )f (σ ), ξ X−1 ,Xd ]dσ 1

[f (σ ), S ∗ (t − σ )ξ X−1 ,Xd ]dσ. 1

Proposition 7.14 (Proposition 12.1.2 of [104]) Assume that Z1 , Z2 , and Z3 are Hilbert spaces, F ∈ L(Z1 , Z3 ), G ∈ L(Z2 , Z3 ), and Ran(F ) = {F (z) : z ∈ Z1 } ⊂ Ran(G) = {G(z) : z ∈ Z2 }. Then there exists an operator L ∈ L(Z1 , Z2 ) such that F = GL. Corollary 7.15 Assume that Z1 and Z2 are Hilbert spaces and G ∈ L(Z1 , Z2 ) satisfies Ran(G) = Z2 . Then there exists an operator L ∈ L(Z2 , Z1 ) such that GL = I —the identity operator in Z2 . Let B ∈ L(U, X−1 ) and τ ≥ 0. Define Φτ ∈ L(L2 (0, ∞; U ), X−1 ) by  Φτ u =

τ

S(τ − σ )Bu(σ )dσ.

(7.9)

0

The operator B ∈ L(U, X−1 ) is called an admissible control operator for S(·) if for some τ > 0, Ran(Φτ ) = {Φτ u : u ∈ L2 (0, ∞; U )} ⊂ X. Proposition 7.16 (Proposition 4.2.2 and (4.2.5) of [104]) Assume that B ∈ L(U, X−1 ) is an admissible control operator for S(·). Then, for every t ≥ 0, Φt ∈ L(L2 (0, ∞; U ), X), and for all T > t > 0, Φt  ≤ ΦT . Proposition 7.17 (Proposition 4.2.5 of [104]) Assume that B ∈ L(U, X−1 ) is an admissible control operator for S(·). Then, for every z0 ∈ X and every u ∈ L2loc (0, ∞; U ), the initial value problem z (t) = Az(t) + Bu(t), z(0) = z0 , has a unique solution in X−1 . This solution is given by  z(t) = S(t)z0 + Φt u = S(t)z0 + 0

t

S(t − σ )Bu(σ )dσ,

7.5 Admissible Control Operators

205

and it satisfies 1 z ∈ C([0, ∞); X) ∩ Hloc ((0, ∞); X−1 ).

Proposition 7.18 (Proposition 4.2.6 of [104]) Assume that B ∈ L(U, X−1 ) is an admissible control operator for S(·). Then, for every z0 ∈ X and every u ∈ L2loc (0, ∞; U ), there exists a unique function z ∈ C([0, ∞); X) such that for every t ≥ 0 and every ψ ∈ D(A∗ ), 

t

z(t) − z0 , ψ X =

[z(σ ), A∗ ψ X + u(σ ), B ∗ ψ U ]dσ.

0

Propositions 7.16 and 7.17 and Theorem 7.3 imply the following result. Proposition 7.19 Assume that B ∈ L(U, X−1 ) is an admissible control operator for S(·) and T > 0. Then there exists a constant cT > 0 such that for every z0 ∈ X and every u ∈ L2 (0, T ; U ), the unique solution z of the initial value problem z (t) = Az(t) + Bu(t), t ∈ (0, T ) a.e., z(0) = z0 , satisfies z ∈ C([0, T ]; X) and that for all t ∈ [0, T ], z(t) ≤ cT (z0  + u). Proposition 7.16 and Corollary 7.15 imply the following result. Proposition 7.20 Assume that B ∈ L(U, X−1 ) is an admissible control operator for S(·), T > 0, and Ran(ΦT ) = X. Then there exists L ∈ L(X, L2 (0, T ; U )) such that ΦT Lx = x for all x ∈ X. The following result was proved in [148], Theorem 1.23. Theorem 7.21 Assume that B ∈ L(U, X−1 ) is an admissible control operator for a C0 semigroup S(·), T > 0, Ran(ΦT ) = X, xf ∈ X, uf ∈ U , x ∗ (t) = xf , u∗ (t) = uf , t ∈ [0, ∞), and x ∗ (·) is a unique solution in X−1 of the initial value problem z (t) = Az(t) + Bu∗ (t), z(0) = xf . Then there exists a constant c > 0 such that for each z0 , z1 ∈ X, there exist u ∈ L2 (0, T ; U ) and z ∈ C 0 ([0, T ]; X) which is a solution of the problem z (t) = Az(t) + Bu(t), t ∈ [0, T ] a.e., z(0) = z0 in X−1 and satisfy z(T ) = z1 ,

206

7 Infinite Dimensional Control

u(t) − uf  ≤ c(z1 − xf  + z0 − xf ), t ∈ [0, T ], z(t) − xf  ≤ c(z1 − xf  + z0 − xf ), t ∈ [0, T ].

7.6 Examples Let U and X be Hilbert spaces, S(·) be a C0 semigroup on X, A : D(A) → X be its generator, and B ∈ L(U, X−1 ) be an admissible control operator for S(·). We say that the pair (A, B) is exactly controllable in a time τ > 0 if Ran(Φτ ) = X [37, 104]. It is known that the exact controllability in time τ > 0 is equivalent to the following property: For each pair z0 , z1 ∈ X, there exists u ∈ L2 (0, τ ; U ) such that the solution z of the initial value problem z = Az + Bu, z(0) = z0

(7.10)

satisfies z(τ ) = z1 . In this section we consider examples of the pairs (A, B) which are exactly controllable in some time τ > 0. All of these examples were discussed in [104] and [37]. It should be mentioned that in Chap. 8 our turnpike results will be established for large and general classes of infinite dimensional optimal control problems. One of these classes contains problems which are defined by an integrand and a pair of operators (A, B) introduced above, which is exactly controllable in some time τ > 0. Therefore, each example of pairs (A, B) considered below gives us a subclass of infinite dimensional optimal control problems, for which the results of Chap. 8 hold. Example 7.22 (Example 11.2.2 of [104]) We consider the problem of controlling the vibrations of an elastic membrane by a force field acting on a part of this membrane. More precisely, let n be a natural number, and let Ω ⊂ R n be a bounded open set with ∂Ω of class C 2 or Ω be a rectangular domain. The physical problem described above can be modeled by the equations ∂ 2w − Δw = u in Ω × (0, ∞), ∂t 2 w = 0 on ∂Ω × (0, ∞), w(x, 0) = f (x),

∂w (x, 0) = g(x) for x ∈ Ω, ∂t

where f is the initial displacement and g is the initial velocity. Let O be a nonempty open subset of Ω and u ∈ L2 (0, ∞; L2 (O)) be the input function. For any such u,

7.6 Examples

207

we assume that u(x, t) = 0 for all x ∈ Ω \ O. The equation above can be written in the form (7.10) using the following spaces and operators: X = H01 (Ω) × L2 (Ω), D(A) = (H 2 (Ω) ∩ H01 (Ω)) × H01 (Ω), U = L2 (O) ⊂ L2 (Ω), A

        f g f 0 = for all ∈ D(A), Bu = for all u ∈ U. g Δf g u

Here B ∈ L(U, X). It was shown in Example 11.2.2 of [104] that B is an admissible control operator for the C0 semigroup eAt and that the pair (A, B) is exactly controllable in time τ > 0 if Γ and O satisfy the assumptions of Theorem 7.4.1 of [104], where Γ is a relatively open subset of ∂Ω. In particular, (A, B) is exactly controllable in time τ > 0 if there exist x0 ∈ R n and  > 0 such that N ({x ∈ ∂Ω : (x − x0 ) · ν(x) > 0}) ⊂ closO, τ > 2r(x0 ), where r(x0 ) = sup{x − x0  : x ∈ Ω}, | · | is the Euclidean norm in R n , ν is the unit outward normal vector field in ∂Ω, and N (D) = {x ∈ Ω : d(x, D) < } for any D ⊂ Ω with d(x, D) = inf{|x − y| : y ∈ D}. Example 7.23 (Example 11.2.4 of [104]) Let n be a natural number, let Ω ⊂ R n be a bounded open set with ∂Ω of class C 2 or let Ω be a rectangular domain, and let O be a nonempty open subset of Ω. We consider the problem of controlling the vibrations of an elastic plate occupying the domain Ω by a force field acting on O. More precisely, we consider the following initial and boundary initial value problem: ∂ 2w + Δ2 w = u in Ω × (0, ∞), ∂t 2 w = Δw = 0 on ∂Ω × (0, ∞), w(x, 0) = 0,

∂w (x, 0) = 0 for x ∈ Ω, ∂t

where u ∈ L2 (0, ∞; L2 (O)) is the input function. As usual we assume that u(x, t) = 0 for all x ∈ Ω \O. The equations above determine a system with the state space X = (H 2 (Ω) ∩ H01 (Ω)) × L2 (Ω) and the input space U = L2 (Ω) which is exactly controllable in any time τ > 0, as it was shown in Example 11.2.4 of [104], if the pair (Ω, O) satisfies one of the assumptions (A1) or (A2) in Example 11.2.3 of [104]. More precisely, let H = L2 (Ω), D(A0 ) = H1 be the Sobolev space H 2 (Ω) ∩ H01 (Ω), A0 : D(A0 ) → H be defined by A0 φ = −Δφ, φ ∈ D(A0 ), and let H2 = D(A20 ) be endowed with the graph norm. Let χ be the Hilbert space H1 × H , consider the dense subset of χ defined by D(A) = H2 × H1 , and let the linear operator A : D(A) → χ be defined by

208

7 Infinite Dimensional Control

 0 I . A= −A20 0 

Then the equations above can be written in the form z = Az + Bu, z(0) = 0,   0 for all u ∈ U . It was shown in u Example 11.2.4 of [104] that the pair (A, B) is exactly controllable in any time τ > 0 if the pair (Ω, O) satisfies one of the assumptions (A1) or (A2) in Example 11.2.3 of [104].

where B ∈ L(U, χ ) is defined by Bu =

Example 7.24 (Example 11.2.6 of [104]) We consider the problem of controlling the vibrations of a string occupying the interval [0, π ] by means of a force u(t) acting at its left end. The equations describing this problem are formulated as a well-posed boundary control system in subsection 10.2.2 of [104]: ∂ 2w ∂ 2w (x, t) = (x, t), 0 < x < π, t ≥ 0, ∂t 2 ∂x 2 ∂w (0, t) = u(t), t ≥ 0, ∂x

w(π, t) = 0, w(x, 0) = f (x),

∂w (x, 0) = g(x), 0 < x < π. ∂t

For this problem, X = HR1 (0, π ) × L2 (0, π ), A : D(A) → X is defined by D(A) = 1 {f ∈ H 2 (0, π ) ∩ HR1 (0, π ) : ∂f ∂x (0) = 0} × HR (0, π ),       g f f A = ∂ 2 f for all ∈ D(A). g g 2 ∂x

    f f = −g(0) for all ∈ D(A). It was g g shown in Example 10.2.6 of [104] that the pair (A, B) is exactly controllable in any time τ ≥ 2π.

The control operator B satisfies B ∗

Example 7.25 (Example 11.2.7 of [104]) We consider the boundary control of the non-homogeneous elastic string. The model is described by the equation ∂w ∂ 2w ∂ (a(x) (x, t)) − b(x)w(x, t), 0 < x < π, t > 0, (x, t) = 2 ∂x ∂x ∂t

7.6 Examples

209

w(0, t) = u(t), w(π, t) = 0, w(·, 0) = f,

∂w (·, 0) = g. ∂t

Here a ∈ C 2 [0, π ], b ∈ L∞ (0, π ), a(x) ≥ m > 0, b(x) ≥ 0 for all x ∈ [0, π ]. These equations correspond to a well-posed boundary control system with state space X = L2 (0, π ) × H −1 (0, π ). The generator A is defined by A

      f f g for all ∈ D(A) = H01 (0, π ) × L2 (0, π ), = g g −A0 f

where A0 ∈ L(H01 (0, π ), H −1 (0, π )) is defined by A0 f =

∂f ∂ (a ) + bf for all f ∈ H01 (0, π ). ∂x ∂x

The control operator B of this system is determined by B∗

    ∂ φ φ ψ)| for all = a(0) (A−1 ∈ D(A∗ ) = D(A). x=0 ψ ψ ∂x 0

It was shown in Example 11.2.7 of [104] that the pair (A, B) is exactly controllable π in any time τ ≥ 2 0 (a(x))−1/2 dx. Example 7.26 (Example 11.2.8 of [104]) We consider the problem of controlling the vibrations of a beam occupying the interval [0, π ] by means of a torque u(t) acting at its left end. The model is described by the initial and boundary value problem ∂ 2w ∂ 4w (x, t) = − (x, t), 0 < x < π, t > 0, ∂t 2 ∂x 4 w(0, t) = 0, w(π, t) = 0,

∂ 2w ∂ 2w (0, t) = u(t), (π, t) = 0, ∂x 2 ∂x 2

w(·, 0) = f,

∂w (·, 0) = g. ∂t

These equations correspond to a control system where H = L2 (0, π ), H1 = 2 H 2 (0, π ) ∩ H01 (0, π ), A0 : H1 → H is defined by A0 f = − ∂∂xf2 for all f ∈ H1 , H1/2 = H01 (0, π ), H−1/2 = H0−1 (0, π ). The unique extensions of A0 to unitary operators from H1/2 onto H−1/2 and from H onto H−1 are still denoted by A0 . The space H3/2 = A−1 0 H1/2 is H3/2 = {g ∈ H 3 (0, π ) ∩ H01 (0, π ) :

∂ 2g ∂ 2g (0) = (π ) = 0}. ∂x 2 ∂x 2

210

7 Infinite Dimensional Control

We set X = H1/2 × H−1/2 , D(A) = H3/2 × H1/2 ,       f f g for all A ∈ D(A). = g g −A20 f The control operator B of this system is determined by     d f f −1 B = − (A0 g)|x=0 for all ∈ D(A∗ ) = D(A). g g dx ∗

It was shown in Example 11.2.8 of [104] that the pair (A, B) is exactly controllable in any time τ > 0. Example 7.27 (Example 11.2.9 of [104]) We consider the problem of controlling the vibrations of a beam occupying the interval [0, 1] by means of an angular velocity u(t) applied at its left end. The equations describing this problem have been formulated as a well-posed boundary control system in Section 10.5 of [104] as follows: ∂ 2w ∂ 4w (x, t) = − (x, t), 0 < x < 1, t > 0, ∂t 2 ∂x 4 w(0, t) = 0, w(1, t) = 0,

∂w ∂w (0, t) = u(t), (1, t) = 0, ∂x ∂x

w(·, 0) = f,

∂f (·, 0) = g. ∂t

We denote X = V ×L2 (0, 1), where V = {h ∈ H 2 (0, 1) : h(0) = h(1) = 0}. The norm on X is defined by  z2 = z1 2V + z2 2L2 , where z1 2V =

1 0

|

∂h ∂x (1)

=

∂ 2 z1 2 | dx. ∂x 2

Let Z ⊂ X be defined by Z = (V ∩ H 4 (0, 1)) × V , L : Z → X, G : Z → R 1 be defined by  L

0 4

∂ − ∂x 4

   I ∂z2 z (0), , G 1 = z2 ∂x 0

KerG = {z ∈ Z : G(z) = 0} = (V ∩ H 4 (0, 1)) × H02 (0, 1), A = L|KerG and a control operator B be defined by

7.6 Examples

B

211

∗



ψ1 ψ2



∂ 2 ψ1 =− (0), for all ∂x 2



ψ1 ψ2



∈ D(A∗ ) = D(A).

It was shown in Example 11.2.9 of [104] that the pair (A, B) is exactly controllable in any time τ > 0. The next two examples are discussed in [37]. Example 7.28 The transport equation. Let L > 0. We consider the linear control system yt + yx = 0, t ∈ (0, T ), x ∈ (0, L), y(t, 0) = u(t), t ∈ (0, T ), where u(t) ∈ R 1 , y(t, ·) : (0, L) → R 1 , with X = L2 (0, L), D(A) = {f ∈ H 1 (0, L) : f (0) = 0}, Af = −fx , f ∈ D(A), U = R 1 and B : R 1 → D(A∗ ) defined by (Bu)z = uz(0) for all u ∈ R 1 , and z ∈ D(A∗ ). It was shown in [37] that the pair (A, B) is controllable. Example 7.29 The Korteweg–de Vries equation. Let L > 0. We consider the linear control system yt + yx + yxxx = 0, t ∈ (0, T ), x ∈ (0, L), y(t, 0) = y(t, L) = 0, yx (t, L) = u(t), t ∈ (0, T ), where u(t) ∈ R 1 , y(t, ·) : (0, L) → R 1 , with X = L2 (0, L), D(A) = {f ∈ H 3 (0, L) : f (0) = f (L) = fx (L) = 0}, Af = −fx − fxxx , f ∈ D(A), U = R 1 , and B : R 1 → D(A∗ ) is defined by (Bu)z = uzx (L) for all u ∈ R 1 and z ∈ D(A∗ ). It was shown in [37] that the pair (A, B) is controllable.

Chapter 8

Continuous-Time Nonautonomous Problems on Half-Axis

In this chapter, we establish sufficient and necessary conditions for the turnpike phenomenon for continuous-time optimal control problems on subintervals of halfaxis in metric spaces. For these optimal control problems, the turnpike is not a singleton. The results of this chapter will be obtained for three large classes of problems that will be treated simultaneously.

8.1 Preliminaries Let (E, ρE ) be a complete metric space and (F, ρF ) be a metric space. We suppose that A is a nonempty subset of [0, ∞) × E and U : A → 2F is a point to set mapping with a graph M = {(t, x, u) : (t, x) ∈ A, u ∈ U(t, x)}. We suppose that M is a Borel measurable subset of [0, ∞) × E × F . Assume that for each pair of numbers T1 , T2 satisfying 0 ≤ T1 < T2 < ∞ we are given a set X(T1 , T2 ) of pairs (x, u) (called as trajectory–control pairs) such that x : [T1 , T2 ] → E is a continuous function and u : [T1 , T2 ] → F is a Lebesgue function satisfying (t, x(t)) ∈ A, t ∈ [T1 , T2 ], u(t) ∈ U (t, x(t)), t ∈ [T1 , T2 ] almost everywhere (a.e.).

(8.1) (8.2)

We suppose that the following property holds: if 0 ≤ T1 ≤ S1 < S2 ≤ T2 < ∞, (x, u) ∈ X(T1 , T2 ), x˜ is the restriction of x to [S1 , S2 ], and u˜ is the restriction of u to [S1 , S2 ], then (x, ˜ u) ˜ ∈ X(S1 , S2 );

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0_8

213

214

8 Continuous-Time Nonautonomous Problems on Half-Axis

for each triplet of nonnegative numbers T1 < T2 < T3 , each (x1 , u1 ) ∈ X(T1 , T2 ), and each (x2 , u2 ) ∈ X(T2 , T3 ) satisfying x1 (T2 ) = x2 (T2 ), a pair (x3 , u3 ) ∈ X(T1 , T3 ), where x3 (t) = x1 (t), t ∈ [T1 , T2 ], x3 (t) = x2 (t), t ∈ (T2 , T3 ], u3 (t) = u1 (t), t ∈ [T1 , T2 ], u3 (t) = u2 (t), t ∈ (T2 , T3 ]. Let a0 > 0, θ0 ∈ E, θ1 ∈ F , let ψ : [0, ∞) → [0, ∞) be an increasing function such that ψ(t) → ∞ as t → ∞,

(8.3)

and let ψ1 : M → [0, ∞) satisfy ψ1 (t, x, u) ≥ ψ(ρE (x, θ0 )), (t, x, u) ∈ M.

(8.4)

Denote by Mψ the set of all Borelian functions g : M → R 1 such that for each (t, x, u) ∈ M, g(t, x, u) ≥ ψ1 (t, x, u) − a0 , (t, x, u) ∈ M.

(8.5)

For each pair of numbers T2 > T1 ≥ 0, each (x, u) ∈ X(T1 , T2 ), and each g ∈ Mψ , define  I (T1 , T2 , x, u) = g

T2

g(t, x(t), u(t))dt ∈ (−∞, ∞].

T1

We consider functionals of the form I g (T1 , T2 , x, u), where 0 ≤ T1 < T2 , (x, u) ∈ X(T1 , T2 ), and g ∈ Mψ . Let g ∈ Mψ . For each pair of numbers T2 > T1 ≥ 0 and each pair of points (T1 , y), (T2 , z) ∈ A, we define U g (T1 , T2 , y, z) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 ), x(T1 ) = y, x(T2 ) = z},

(8.6)

U g (T1 , T2 , y) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 ), x(T1 ) = y},

(8.7)

g (T1 , T2 , z) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 ), x(T2 ) = z}, U

(8.8)

U g (T1 , T2 ) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 )}.

(8.9)

8.1 Preliminaries

215

We equip the space Mψ with the uniformity determined by the base E(N, , λ) = {(f, g) ∈ Mψ × Mψ : |f (t, x, u) − g(t, x.u)| ≤  for each (t, x, u) ∈ M satisfying ρE (x, θ0 ), ρ(u, θ1 ) ≤ N } ∩ {(f, g) ∈ Mψ × Mψ : (|f (t, x, u)| + 1)(|g(t, x, u)| + 1)−1 ∈ [λ−1 , λ] for each (t, x, u) ∈ M satisfying ρE (x, θ0 ) ≤ N},

(8.10)

where N,  > 0, λ > 1. Clearly, the uniform space Mψ is Hausdorff and has a countable base. Therefore, Mψ is metrizable (by a metric dψ ). It is not difficult to see that the uniform space Mψ is complete. It is equipped with a topology induced by the uniformity. In this chapter, we assume that the following assumption holds: (A0) For each L0 > 0, there exists L1 > 0 such that for each T1 ≥ 0, each T2 ∈ (T1 , T1 + L0 ], each (x, u) ∈ X(T1 , T2 ), and each g ∈ Mψ that satisfy ρE (x(T1 ), θ0 ) ≤ L0 , I g (T1 , T2 , x, u) ≤ L0 the inequality ρE (θ0 , x(t)) ≤ L1 is true for all t ∈ [T1 , T2 ]. Let f ∈ Mψ . In some cases, we will use the following assumption: (A0)’ For each L0 > 0, there exists L1 > 0 such that for each T1 ≥ 0, each T2 ∈ (T1 , T1 + L0 ], and each (x, u) ∈ X(T1 , T2 ) that satisfy I f (T1 , T2 , x, u) ≤ L0 and min{ρE (θ0 , x(t)) : t ∈ [T1 , T2 ]} ≤ L0 the inequality ρE (θ0 , x(t)) ≤ L1 is true for all t ∈ [T1 , T2 ]. Assumption (A0) and Eqs. (8.3)–(8.5) easily imply the following result. Proposition 8.1 Assume that M1 > 0, 0 < τ < τ0 < τ1 . Then there exists M2 > 0 such that for each T1 ≥ 0, each T2 ∈ [T1 + τ0 , T1 + τ1 ], each g ∈ Mψ , and each (x, u) ∈ X(T1 , T2 ) that satisfy

216

8 Continuous-Time Nonautonomous Problems on Half-Axis

I g (T1 , T2 , x, u) ≤ M1 the inequality ρE (θ0 , x(t)) ≤ M2 is true for all t ∈ [T1 + τ, T2 ].

8.2 The First Class of Problems We begin with the description of the first class of problems. Let (E,  · ) be a Banach space and E ∗ be its dual. Let {A(t) : t ∈ [0, ∞)} be the family of closed densely defined linear operators with the domain and range in the Banach space E. Let (F, ρF ) be a metric space, θ0 = 0, θ1 ∈ F, ρE (x, y) = x − y, x, y ∈ E. We suppose that A is a nonempty subset of [0, ∞) × E and U : A → 2F is a point to set mapping with a graph M = {(t, x, u) : (t, x) ∈ A, u ∈ U (t, x)}. We suppose that M is a Borel measurable subset of [0, ∞)×E×F and G : M → E is a Borelian function. We consider the homogeneous Cauchy problem x (t) = A(t)x(t), t ∈ [0, ∞).

(8.11)

We assume that there exists a function U : {(t, s) ∈ R 2 : 0 ≤ s ≤ t < ∞} → L(E) that has the following properties [21]: (i) For each x0 ∈ E, the function (t, s) → U (t, s)x0 is continuous on the set {(t, s) ∈ R 2 : 0 ≤ s ≤ t < ∞}. (ii) U (s, s) = I d for all s ∈ [0, ∞), where I d is the identity operator. (iii) U (t, s)U (s, τ ) = U (t, τ ) for all numbers t ≥ s ≥ τ ≥ 0. (iv) For each s ≥ 0, there exists a densely linear subspace Es of E such that for each x0 ∈ Es the function t → U (t, s)x0 is continuously differentiable on [s, ∞) and (∂/∂t)U (t, s)x0 = A(t)U (t, s)x0 , t ∈ [s, ∞).

(8.12)

(v) There exists an increasing function τ → Δτ > 0, τ > 0 such that for each τ > 0, each s ≥ 0, and each t ∈ [s, s + τ ], U (t, s) ≤ Δτ .

8.2 The First Class of Problems

217

In this case problem, (8.11) is called well-posed [21]. Let 0 ≤ T1 < T2 , and consider the following equation: x (t) = A(t)x(t) + f (t), t ∈ [T1 , T2 ], x(0) = xT1 ,

(8.13)

where xT1 ∈ E and f ∈ L1 (T1 , T2 ; E). A continuous function x : [T1 , T2 ] → E is a solution of (8.13) if  x(t) = U (t, T1 )x(T1 ) +

t

U (t, s)f (s)ds, t ∈ [T1 , T2 ].

(8.14)

T1

Assume that (8.14) holds and τ ∈ [T1 , T2 ]. It is not difficult to see [148, p. 386] that x : [τ, T2 ] → E is a solution of the equation y (t) = A(t)y(t) + f (t), t ∈ [τ, T2 ], y(τ ) = x(τ ). Let 0 ≤ T1 < T2 < T3 , z0 ∈ E, f ∈ L1 (T1 , T3 ; E), a continuous function x1 : [T1 , T2 ] → E is a solution of the equation x (t) = A(t)x(t) + f (t), t ∈ [T1 , T2 ], x(T1 ) = z0 , and a continuous function x2 : [T2 , T3 ] → E is a solution of the equation x (t) = A(t)x(t) + f (t), t ∈ [T2 , T3 ], x(T2 ) = x1 (T2 ). Set x(t) = x1 (t), t ∈ [T1 , T2 ], x(t) = x2 (t), t ∈ [T2 , T3 ]. Clearly, the function x : [T1 , T3 ] → E is continuous. It is not difficult to see [148, p. 387] that x(·) is a solution of the equation x (t) = A(t)x(t) + f (t), t ∈ [T1 , T3 ], x(T1 ) = z0 . Let 0 ≤ T1 < T2 . We consider the following equation: x (t) = A(t)x(t) + G(t, x(t), u(t)), t ∈ [T1 , T2 ].

(8.15)

A pair of functions x : [T1 , T2 ] → E, u : [T1 , T2 ] → F is called a (mild) solution of (8.15) if x : [T1 , T2 ] → E is a continuous function, u : [T1 , T2 ] → F is a Lebesgue measurable function, (t, x(t)) ∈ A, t ∈ [T1 , T2 ], u(t) ∈ U (t, x(t)), t ∈ [T1 , T2 ] almost everywhere (a.e.),

(8.16) (8.17)

218

8 Continuous-Time Nonautonomous Problems on Half-Axis

G(s, x(s), u(s)), s ∈ [T1 , T2 ] is Bochner integrable, and for every t ∈ [T1 , T2 ],  x(t) = U (t, T1 )x(T1 ) +

t

U (t, s)G(s, x(s), u(s))ds.

(8.18)

T1

The set of all pairs (x, u), which are solutions of (8.15), is denoted by X(T1 , T2 ). Let T1 ≥ 0. A pair of functions x : [T1 , ∞) → E, u : [T1 , ∞) → F is called a (mild) solution of the system x (t) = A(t)x(t) + G(t, x(t), u(t)), t ∈ [T1 , ∞)

(8.19)

if for every T2 > T1 , x : [T1 , T2 ] → E, u : [T1 , T2 ] → F is a solution of (8.15). The set of all such pairs (x, u), which are solutions of the equation above, is denoted by X(T1 , ∞). A function x : I → E, where I is either [T1 , T2 ] or [T1 , ∞) (0 ≤ T1 < T2 ), is called a trajectory if there exists a Lebesgue measurable function u : I → F (referred to as a control) such that (x, u) ∈ X(T1 , T2 ) or (x, u) ∈ X(T1 , ∞), respectively. For this class of problem for each (t, x, u) ∈ M, set ψ1 (t, x, u) = max{ψ(x), ψ(u), ψ((G(t, x, u) − a0 x)+ )(G(t, x, u) − a0 x)+ }.

(8.20)

In Sect. 8.4, we show that assumption (A0) holds for this class of problems.

8.3 The Second Class of Problems Let (E, ·, · )E be a Hilbert space equipped with an inner product ·, · E that induces the norm ·E , and let (F, ·, · F ) be a Hilbert space equipped with an inner product ·, · F that induces the norm ·F . For simplicity, we set ·, · E = ·, · , ·E = ·, ·, · F = ·, · ,  · F =  · , if E, F are understood. Set θ0 = 0, θ1 = 0, ρE (x, y) = x − y, x, y ∈ E, and ρF (x, y) = x − y, x, y ∈ F . We suppose that A0 is a nonempty subset of E and U0 : A0 → 2F is a point to set mapping with a graph M0 = {(x, u) : x ∈ A0 , u ∈ U0 (x)}. We suppose that M0 is a Borel measurable subset of E × F . Define

8.3 The Second Class of Problems

219

A = [0, ∞) × A0 , U : A → 2F by U (t, x) = U0 (x), (t, x) ∈ A, M = [0, ∞) × M0 . Let a linear operator A : D(A) → E generate a C0 semigroup S(t) = eAt , t ∈ [0, ∞) on E. As usual, we denote by S(t)∗ the adjoint of S(t). Then S(t)∗ , t ∈ [0, ∞), is a C0 semigroup, and its generator is the adjoint A∗ of A. The domain D(A∗ ) is a Hilbert space equipped with the graph norm  · D(A∗ ) : z2D(A∗ ) = z2E + A∗ z2E , z ∈ D(A∗ ). Let D(A∗ ) be the dual of D(A∗ ) with the pivot space E. In particular, E1d := D(A∗ ) ⊂ E ⊂ D(A∗ ) = E−1 . (Here we use the notation of Chap. 7.) Let G : M → D(A∗ ) = E−1 , B ∈ L(F, E−1 ) be an admissible control operator for eAt , t ≥ 0, and for all (t, x, u) ∈ M, G(t, x, u) = Bu. Let 0 ≤ T1 < T2 . We consider the following equation: x (t) = Ax(t) + Bu(t), t ∈ [T1 , T2 ] a.e.

(8.21)

A pair of functions x : [T1 , T2 ] → E, u : [T1 , T2 ] → F is called a (mild) solution of (8.21) if x : [T1 , T2 ] → E is a continuous function, u : [T1 , T2 ] → F is a Lebesgue measurable function, u ∈ L2 (T1 , T2 ; F ), (t, x(t)) ∈ A, t ∈ [T1 , T2 ],

(8.22)

u(t) ∈ U (t, x(t)), t ∈ [T1 , T2 ] a.e.,

(8.23)

and for each t ∈ [T1 , T2 ],  x(t) = e

A(t−T1 )

x(T1 ) +

t

eA(t−s) Bu(s)ds

(8.24)

T1

in E−1 . The set of all pairs (x, u), which are solutions of (8.21), is denoted by X(T1 , T2 , A, G). In the sequel for simplicity, we use the notation X(T1 , T2 ) = X(T1 , T2 , A, G) if the pair (A, G) is understood.

220

8 Continuous-Time Nonautonomous Problems on Half-Axis

Let T1 ≥ 0. A pair of functions x : [T1 , ∞) → E, u : [T1 , ∞) → F is called a (mild) solution of the system x (t) = Ax(t) + Bu(t), t ∈ [T1 , ∞)

(8.25)

if for every T2 > T1 , x : [T1 , T2 ] → E, and u : [T1 , T2 ] → F is a solution of (8.21). The set of all such pairs (x, u), which are solutions of (8.25), is denoted by X(T1 , ∞). A function x : I → E, where I is either [T1 , T2 ] or [T1 , ∞) (0 ≤ T1 < T2 ), is called a trajectory if there exists a Lebesgue measurable function u : I → F (referred to as a control) such that (x, u) ∈ X(T1 , T2 ) or (x, u) ∈ X(T1 , ∞), respectively. For this class of problems, for each (t, x, u) ∈ M, set ψ1 (t, x, u) = max{ψ(x), K0 u2 } − a0 ,

(8.26)

where K, a0 are positive constants. In Sect. 8.4, we show that assumption (A0) holds for this class of problems.

8.4 A0 and the First and Second Classes of Problems In this section, we prove that assumption (A0) holds for the first and second classes of problems. Namely, we prove the following result where the two classes are treated simultaneously. Proposition 8.2 Let M0 , M1 , τ0 > 0. Then there exists M2 > M1 such that for each g ∈ Mψ , each T1 ≥ 0, each T2 ∈ (T1 , T1 + τ0 ], and each (x, u) ∈ X(T1 , T2 ) satisfying x(T1 ) ≤ M1 ,

(8.27)

I g (T1 , T2 , x, u) ≤ M0 ,

(8.28)

the following inequality holds: x(t) ≤ M2 for all t ∈ [T1 , T2 ].

(8.29)

Proof Let us consider the first class of problems. Fix a positive number δ < min{16−1 τ0 , 8−1 (a0 + 1)−1 , 8−1 (a0 + 1)−1 Δ−1 δ }.

(8.30)

By (8.5) and (8.20), there exist h0 > M1 + 1 and γ0 > 0 such that for each g ∈ Mψ ,

8.4 A0 and the First and Second Classes of Problems

221

g(t, x, u) ≥ 4(M0 + a0 τ0 )δ −1 for each (t, x, u) ∈ M satisfying x ≥ h0 , (8.31) g(t, x, u) ≥ 8(G(t, x, u) − a0 x)+ for each (t, x, u) ∈ M satisfying G(t, x, u) − a0 x ≥ γ0 .

(8.32)

Choose a number M2 > (8M1 + 8 + 2h0 + 2γ0 δ + M0 + a0 τ0 )(Δδ + 1).

(8.33)

Let g ∈ Mψ , T1 ≥ 0, T2 ∈ (T1 , T1 + τ0 ], (x, u) ∈ X(T1 , T2 ), and Eqs. (8.27) and (8.28) hold. We show that (8.29) holds. Assume the contrary. Then there exists t0 ∈ [T1 , T2 ] such that x(t0 ) > M2 .

(8.34)

By the choice of h0 (see (8.31)) and (8.27), (8.28), there exists t1 ∈ [T1 , t0 ] such that x(t1 ) ≤ h0 , |t1 − t0 | ≤ δ.

(8.35)

There exists t2 ∈ [t1 , t0 ] such that x(t2 ) ≥ x(t) for all t ∈ [t1 , t0 ].

(8.36)

In view of (8.33)–(8.36), (8.37)

t2 > t1 . By (8.18), (8.36), and (8.37),  x(t2 ) − U (t2 , t1 )x(t1 ) = 

t2

U (t2 , s)G(x(s), u(s))ds.

(8.38)

t1

It follows from property (v) from Sect. 8.2 and (8.35) and (8.38) that  x(t2 )−Δδ h0 ≤ x(t2 )−Δδ x(t1 ) ≤ Δδ

t2

t1

Define

G(s, x(s), u(s))ds.

(8.39)

222

8 Continuous-Time Nonautonomous Problems on Half-Axis

E1 = {t ∈ [t1 , t2 ] : G(t, x(t), u(t)) ≥ a0 x(t) + γ0 }, E2 = [t1 , t2 ] \ E1 .

(8.40) (8.41)

By (8.28), (8.32), (8.35), (8.36), and (8.39)–(8.41), Δ−1 δ x(t2 ) ≤ h0 +  ≤ h0 + a0

t2



t2

x(t)|dt +

t1

t1



t2

G(s, x(s), u(s))ds

t1

(G(s, x(s), u(s)) − a0 x(s))+ ds

 ≤ h0 + a0 δx(t2 ) + E1

(G(t, x(t), u(t)) − a0 x(t))+ dt

 + E2

(G(t, x(t), u(t)) − a0 x(t))+ dt 

≤ h0 + a0 x(t2 )δ + γ0 δ + E1

(G(t, x(t), u(t)) − a0 x(t))+ dt

≤ h0 + a0 x(t2 )δ + γ0 δ + 8−1

 g(t, x(t), u(t))dt E1

≤ h0 + a0 x(t2 )δ + γ0 δ + 8−1 (M0 + a0 τ0 ).

(8.42)

It follows from (8.30), (8.34), (8.36), and (8.42) that −1 −1 −1 2−1 M2 Δ−1 δ ≤ 2 Δδ x(t2 ) ≤ h0 + γ0 δ + 8 (M0 + a0 τ0 )

and M2 ≤ (2h0 + 2γ0 δ + 4−1 (M0 + a0 τ0 ))Δδ . This contradicts (8.33). The contradiction we have reached proves Proposition 8.2 for the first class of problems. Consider the second class of problems. Recall (see Proposition 7.1) that there exist M∗ ≥ 1 and ω∗ ∈ R 1 such that eAt  ≤ M∗ eω∗ t , t ∈ [0, ∞) and that for every τ ≥ 0,

(8.43)

8.4 A0 and the First and Second Classes of Problems



τ

Φτ u =

223

eA(τ −s) Bu(s)ds, u ∈ L2 (0, τ ; F ),

(8.44)

0

where Φτ ∈ L(L2 (0, τ ; F ), E) (see Proposition 7.16). Choose a number δ ∈ (0, 1). By (8.5) and (8.26), there exist h 0 > M1 + 1

(8.45)

and γ > 1 such that g(t, x, u) ≥ 4(M0 + a0 τ0 )δ −1 for each (t, x, u) ∈ M satisfying x ≥ h0 , (8.46) g(t, x, u) ≥ K0 u2 /2 for each u ∈ F satisfying u ≥ γ . (8.47) Choose M2 > M1 + M∗ e|ω∗ |δ h0 + Φ1 γ δ 1/2 + Φ1 (2K0−1 (M0 + a0 τ0 ))1/2 .

(8.48)

Let g ∈ Mψ , T1 ≥ 0, T2 ∈ (T1 , T1 + τ0 ], (x, u) ∈ X(T1 , T2 ), and (8.27) and (8.28) hold. We show that (8.29) holds. Assume the contrary. Then there exists t0 ∈ [T1 , T2 ] such that (8.34) holds. By the choice of h0 , (8.45), (8.46), there exists t1 ∈ [T1 , t0 ] satisfying (8.35). There exists t2 ∈ [t1 , t0 ] such that x(t2 ) ≥ x(t) for all t ∈ [t1 , t0 ].

(8.49)

Clearly, (x, u) ∈ X(t1 , t2 ), x(t2 ) = e

A(t2 −t1 )

 x(t1 ) +

t2

eA(t2 −s) Bu(s)ds

(8.50)

t1

in E−1 = D(A∗ ) . Proposition 7.16, (8.35), (8.43), (8.44), and (8.50) imply that x(t2 ) ≤ x(t1 )M∗ e ≤ M∗ e|ω∗ |δ h0 + 



|ω∗ |δ

t2 −t1 0

 +

t2

eA(t2 −s) Bu(s)ds

t1

eA(t2 −t1 −s) Bu(t1 + s)ds

224

8 Continuous-Time Nonautonomous Problems on Half-Axis

≤ M∗ e|ω∗ |δ h0 + Φt2 −t1 u(t1 + ·)L2 (0,t2 −t1 ;F ) ≤ M∗ e|ω∗ |δ h0 + Φ1 (



t2

u(s)2 ds)1/2 .

(8.51)

t1

Define Ω1 = {t ∈ [t1 , t2 ] : u(t) ≥ γ }, Ω2 = [t1 , t2 ] \ Ω1 .

(8.52)

By (8.26), (8.28), (8.34), (8.47)–(8.49), (8.51), and (8.52), M2 < x(t0 ) ≤ x(t2 ) ≤ M∗ e|ω∗ |δ h0   2 1/2 + Φ1 [( u(s) ds) + ( u(s)2 ds)1/2 ] Ω1

Ω2

≤ M∗ e|ω∗ |δ h0 + Φ1 (2K0−1

 g(s, x(s), u(s))ds)1/2 + Φ1 γ δ 1/2 Ω1

≤ M∗ e|ω∗ |δ h0 + Φ1 γ δ 1/2 + Φ1 (2K0−1 (M0 + a0 τ0 ))1/2 < M2 . The contradiction we have reached proves Proposition 8.2 for the second class of problems.

8.5 The Third Class of Problems Let (E, ρE ) be a complete metric space, (F, ρF ) be a metric space, and θ0 ∈ E, θ1 ∈ F. We suppose that A is a nonempty subset of [0, ∞) × E and U : A → 2F is a point to set mapping with a graph M = {(t, x, u) : (t, x) ∈ A, u ∈ U (t, x)}. We suppose that M is a Borel measurable subset of [0, ∞) × E × F and G : M → [0, ∞) is a Borelian function. Let 0 ≤ T1 < T2 < ∞. Denote by X(T1 , T2 ) the set of all pairs of functions x : [T1 , T2 ] → E and u : [T1 , T2 ] → F such that x is continuous on [T1 , T2 ], u is Lebesgue measurable on function [T1 , T2 ], (t, x(t)) ∈ A, t ∈ [T1 , T2 ],

(8.53)

8.5 The Third Class of Problems

225

u(t) ∈ U (t, x(t)), t ∈ [T1 , T2 ] almost everywhere (a.e.),

(8.54)

G(s, x(s), u(s)), s ∈ [T1 , T2 ] is integrable, and for every pair S1 , S2 ∈ [T1 , T2 ] satisfying S1 < S2 ,  ρE (x(S1 ), x(S2 )) ≤

S2

G(t, x(t), u(t))dt.

(8.55)

S1

(See Section 1.1 of [5].) Pairs (x, u) ∈ X(T1 , T2 ) are called trajectory–control pairs. Let T1 ≥ 0. Denoted by X(T1 , ∞) the set of all pairs of functions x : [T1 , ∞) → E, u : [T1 , ∞) → F such that for every T2 > T1 , the pair x : [T1 , T2 ] → E, u : [T1 , T2 ] → F belongs to X(T1 , T2 ). Pairs (x, u) ∈ X(T1 , ∞) are called trajectory– control pairs. A function x : I → E, where I is either [T1 , T2 ] or [T1 , ∞) (0 ≤ T1 < T2 ), is called a trajectory if there exists a Lebesgue measurable function u : I → F (referred to as a control) such that (x, u) ∈ X(T1 , T2 ) or (x, u) ∈ X(T1 , ∞), respectively. For this class of problems for each (t, x, u) ∈ M, set ψ1 (t, x, u) = max{ψ(ρE (x, θ0 )), ψ(ρF (u, θ1 )), ψ((G(t, x, u) − a0 ρE (x, θ0 ))+ )(G(t, x, u) − a0 ρE (x, θ0 ))+ }.

(8.56)

For each pair of numbers T2 > T1 ≥ 0, each (x, u) ∈ X(T1 , T2 ), and each g ∈ Mψ , define  I (T1 , T2 , x, u) = g

T2

g(t, x(t), u(t))dt ∈ (−∞, ∞].

T1

We show now that assumption (A0) holds for this class of problems. Proposition 8.3 Assumption (A0) holds for the third class of problems. Moreover, for each L0 > 0, there exists L1 > 0 such that for each g ∈ Mψ , each T1 ≥ 0, each T2 ∈ (T1 , T1 + L0 ], and each (x, u) ∈ X(T1 , T2 ) satisfying I g (T1 , T2 , x, u) ≤ L0 and min{ρE (x(t), θ0 ) : t ∈ [T1 , T2 ]} ≤ L0 the following inequality holds: ρE (x(t), θ0 ) ≤ L1 for all t ∈ [T1 , T2 ].

226

8 Continuous-Time Nonautonomous Problems on Half-Axis

Proof In view of (8.5) and (8.56), for each g ∈ Mψ and each (t, x, u) ∈ M, g(t, x, u) ≥ max{ψ(ρE (x, θ0 )), ψ(ρF (u, θ1 )), ψ((G(t, x, u) − a0 ρE (x, θ0 ))+ )(G(t, x, u) − a0 ρE (x, θ0 ))+ } − a0 .

(8.57)

Let L0 > 0. Fix a positive number δ < min{16−1 L0 , 8−1 (a0 + 1)−1 }.

(8.58)

By (8.3) and (8.57), there exist h0 > L0 + 1 and γ0 > 0 such that for each g ∈ Mψ , g(t, x, u) ≥ 4(L0 + a0 τ0 )δ −1 for each (t, x, u) ∈ M satisfying ρE (x, θ0 ) ≥ h0 , (8.59) g(t, x, u) ≥ 8(G(t, x, u) − a0 ρE (x, θ0 ))+ for each (t, x, u) ∈ M satisfying G(t, x, u) − a0 ρE (x, θ0 ) ≥ γ0 .

(8.60)

Choose a number L1 > 8L0 + 8 + 2h0 + 8γ0 δ + 4L0 (1 + a0 ) + 4.

(8.61)

Let g ∈ Mψ , T1 ≥ 0, T2 ∈ (T1 , T1 + L0 ], (x, u) ∈ X(T1 , T2 ), I g (T1 , T2 , x, u) ≤ L0 ,

(8.62)

min{ρE (x(t), θ0 ) : t ∈ [T1 , T2 ]} ≤ L0 .

(8.63)

ρE (x(t), θ0 ) ≤ L1 for all t ∈ [T1 , T2 ].

(8.64)

and

We show that

Assume the contrary. Then there exists t0 ∈ [T1 , T2 ] such that ρE (x(t0 ), θ0 ) > L1 .

(8.65)

By the choice of h0 (see (8.59)) and (8.62), (8.63), there exists t1 ∈ [T1 , T2 ] such that ρE (x(t1 ), θ0 ) ≤ h0 , |t1 − t0 | ≤ δ.

(8.66)

8.5 The Third Class of Problems

227

There exists t2 ∈ [min{t1 , t0 }, max{t1 , t0 }]

(8.67)

such that ρE (x(t2 ), θ0 ) ≥ ρE (x(t), θ0 ), t ∈ [min{t1 , t0 }, max{t1 , t0 }].

(8.68)

In view of (8.61), (8.65), (8.66), and (8.68), ρE (x(t2 ), x(t1 )) ≥ ρE (x(t2 ), θ0 ) − ρE (x(t1 ), θ0 ) ≥ ρE (x(t2 ), θ0 ) − h0 ≥ 2−1 ρE (x(t2 ), θ0 ).

(8.69)

By (8.55) and (8.69), 2−1 ρE (x(t2 ), θ0 ) ≤ ρE (x(t2 ), x(t1 )) ≤



max{t1 ,t2 } min{t1 ,t2 }

G(t, x(t), u(t))dt.

(8.70)

Define E1 = {t ∈ [min{t1 , t2 }, max{t1 , t2 }] : G(t, x(t), u(t)) ≥ a0 ρE (x(t), θ0 ) + γ0 }, E2 = [min{t1 , t2 }, max{t1 , t2 }] \ E1 .

(8.71)

By (8.60), (8.62), (8.66), (8.67), (8.70), and (8.71), 2−1 ρE (x(t2 ), θ0 ) ≤



 G(t, x(t), u(t))dt + E1

G(t, x(t), u(t))dt E2

 ≤ E1

(G(t, x(t), u(t)) − a0 ρE (x(t), θ0 ))+ dt

 + E2

(G(t, x(t), u(t)) − a0 ρE (x(t), θ0 ))+ dt

+ a0 ρE (x(t2 ), θ0 )δ  ≤ a0 ρE (x(t2 ), θ0 )δ + γ0 δ + 8−1 g(t, x(t), u(t))dt E1

≤ a0 δρE (x(t2 ), θ0 ) + γ0 δ + 8−1 (L0 + a0 L0 ). It follows from (8.58), (8.65), (8.68), and (8.72) that

(8.72)

228

8 Continuous-Time Nonautonomous Problems on Half-Axis

4−1 L1 ≤ 4−1 ρE (x(t2 ), θ0 ) ≤ γ0 δ + 8−1 (L0 + a0 L0 ), L1 ≤ 4γ0 δ + L0 (1 + a0 ). This contradicts (8.61). The contradiction we have reached proves (8.64) and Proposition 8.3.

8.6 Boundedness Results We consider the general class of problems introduced in Sect. 8.1 and use the notation, definitions, and assumptions introduced there. We suppose that (xf , uf ) ∈ X(0, ∞) satisfies sup{ρE (xf (t), θ0 ) : t ∈ [0, ∞)} < ∞, Δf := sup{|I f (j, j + 1, xf , uf )| : j = 0, 1, . . . } < ∞.

(8.73) (8.74)

We suppose that there exists a number bf > 0 and the following assumptions hold: (A1) For each S1 > 0, there exist S2 > 0 and c > 0 such that I f (T1 , T2 , xf , uf ) ≤ I f (T1 , T2 , x, u) + S2 for each T1 ≥ 0, each T2 ≥ T1 + c, and each (x, u) ∈ X(T1 , T2 ) satisfying ρE (x(Tj ), θ0 ) ≤ S1 , j = 1, 2. (A2) For each  > 0, there exists δ > 0 such that for each (Ti , zi ) ∈ A, i = 1, 2, satisfying ρE (zi , xf (Ti )) ≤ δ, i = 1, 2, and T2 ≥ bf , there exist τ1 , τ2 ∈ (0, bf ] and (x1 , u1 ) ∈ X(T1 , T1 + τ1 ), (x2 , u2 ) ∈ X(T2 − τ2 , T2 ), which satisfies x1 (T1 ) = z1 , x1 (T1 + τ1 ) = xf (T1 + τ1 ), I f (T1 , T1 + τ1 , x1 , u1 ) ≤ I f (T1 , T1 + τ1 , xf , uf ) + , x2 (T2 ) = z2 , x2 (T2 − τ2 ) = xf (T2 − τ2 ), I f (T2 − τ2 , T2 , x2 , u2 ) ≤ I f (T2 − τ2 , T2 , xf , uf ) + . Relations (8.5) and (8.74) imply the following result. Lemma 8.4 Let c > 0. Then Δf (c) = sup{|I f (T1 , T2 , xf , uf )| : T1 ≥ 0, T2 ∈ (T1 , T1 + c]} < ∞. The following result is proved in Sect. 8.10.

8.6 Boundedness Results

229

Theorem 8.5 1. There exists S > 0 such that for each pair of numbers T2 > T1 ≥ 0 and each (x, u) ∈ X(T1 , T2 ), I f (T1 , T2 , x, u) + S ≥ I f (T1 , T2 , xf , uf ). 2. For each (x, u) ∈ X(0, ∞), either I f (0, T , x, u) − I f (0, T , xf , uf ) → ∞ as T → ∞ or sup{|I f (0, T , x, u) − I f (0, T , xf , uf )| : T ∈ (0, ∞)} < ∞.

(8.75)

Moreover, if (8.75) holds, then sup{ρE (x(t), θ0 )) : t ∈ (0, ∞)} < ∞. We say that (x, u) ∈ X(0, ∞) is (f )-good if (8.75) holds. The next boundedness result is proved in Sect. 8.11. Theorem 8.6 Let M0 > 0, c > 0, c0 ∈ (0, c). Then there exists M1 > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + c, and each (x, u) ∈ X(T1 , T2 ) satisfying I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M0 the inequality ρE (x(t), θ0 ) ≤ M1 holds for all t ∈ [T1 + c0 , T2 ] and ρE (x(t), θ0 ) ≤ M1 for all t ∈ [T1 , T2 ] if assumption (A0)’ holds. Let L > 0. Denote by AL the set of all (s, z) ∈ A for which there exist τ ∈ (0, L] and (x, u) ∈ X(s, s + τ ) such that x(s) = z, x(s + τ ) = xf (s + τ ), I f (s, s + τ, x, u) ≤ L. L the set of all (s, z) ∈ A for which s ≥ L and there exist τ ∈ (0, L] Denote by A and (x, u) ∈ X(s − τ, s) such that x(s − τ ) = xf (s − τ ), x(s) = z, I f (s − τ, s, x, u) ≤ L. Theorems 8.7–8.9 stated below are also boundedness results. They follow easily from Theorem 8.6. For details, see Chapter 6 of [148]. Theorem 8.7 Let L > 0, M0 > 0, and c ∈ (0, L). Then there exists M1 > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) satisfying

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L , (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + M0 the inequality ρE (x(t), θ0 ) ≤ M1 holds for all t ∈ [T1 + c, T2 ] and ρE (x(t), θ0 ) ≤ M1 for all t ∈ [T1 , T2 ] if assumption (A0)’ holds. Theorem 8.8 Let L > 0, M0 > 0, and c ∈ (0, L). Then there exists M1 > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + L, and each (x, u) ∈ X(T1 , T2 ) satisfying (T1 , x(T1 )) ∈ AL , I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 )) + M0 the inequality ρE (x(t), θ0 ) ≤ M1 holds for all t ∈ [T1 + c, T2 ] and ρE (x(t), θ0 ) ≤ M1 for all t ∈ [T1 , T2 ] if assumption (A0)’ holds. Theorem 8.9 Let L > 0, M0 > 0, and c ∈ (0, L). Then there exists M1 > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + L, and each (x, u) ∈ X(T1 , T2 ) satisfying L , I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T2 )) + M0 (T2 , x(T2 )) ∈ A the inequality ρE (x(t), θ0 ) ≤ M1 holds for all t ∈ [T1 + c, T2 ] and ρE (x(T1 ), θ0 ) ≤ M1 for all t ∈ [T1 , T2 ] if assumption (A0)’ holds. The next result is proved in Sect. 8.12. Theorem 8.10 Let M0 > 0 and c > 0. Then there exists M1 > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + c, and each (x, u) ∈ X(T1 , T2 ) satisfying ρE (x(T1 ), θ0 ) ≤ M0 , I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M0 the inequality ρE (x(t), θ0 ) ≤ M1 holds for all t ∈ [T1 , T2 ]. The next three theorems follow from Theorem 8.10. For details, see Chapter 6 of [148]. Theorem 8.11 Let L > 0, M0 > 0. Then there exists M1 > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) satisfying L , ρE (x(T1 ), θ0 ) ≤ M0 , (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A

8.7 Turnpike Results

231

I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + M0 the inequality ρE (x(t), θ0 ) ≤ M1 holds for all t ∈ [T1 , T2 ]. Theorem 8.12 Let L > 0, M0 > 0. Then there exists M1 > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + L, and each (x, u) ∈ X(T1 , T2 ) satisfying ρE (x(T1 ), θ0 ) ≤ M0 , (T1 , x(T1 )) ∈ AL , I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 )) + M0 the inequality ρE (x(T1 ), θ0 ) ≤ M1 holds for all t ∈ [T1 , T2 ]. Theorem 8.13 Let L > 0, M0 > 0. Then there exists M1 > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + L, and each (x, u) ∈ X(T1 , T2 ) satisfying L , ρE (x(T1 ), θ0 ) ≤ M0 , (T2 , x(T2 )) ∈ A f (T1 , T2 , x(T2 )) + M0 I f (T1 , T2 , x, u) ≤ U the inequality ρE (x(T1 ), θ0 ) ≤ M1 holds for all t ∈ [T1 , T2 ].

8.7 Turnpike Results We say that the integrand f possesses the turnpike property (or TP for short) if for each  > 0 and each M > 0 there exist δ > 0 and L > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy I f (T1 , T2 , x, u) ≤ min{U f (T1 , T2 ) + M, U f (T1 , T2 , x(T1 ), x(T2 )) + δ} there exist τ1 , τ2 ∈ [0, L] such that ρE (x(t), xf (t)) ≤  for all t ∈ [T1 + τ1 , T2 − τ2 ]. Moreover, if ρE (x(T2 ), xf (T2 )) ≤ δ, then τ2 = 0, and if T1 ≥ L and ρE (x(T1 ), xf (T1 )) ≤ δ, then τ1 = 0. We say that the integrand f possesses the strong turnpike property (or STP for short) if for each  > 0 and each M > 0 there exist δ > 0 and L > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy I f (T1 , T2 , x, u) ≤ min{U f (T1 , T2 ) + M, U f (T1 , T2 , x(T1 ), x(T2 )) + δ} there exist τ1 , τ2 ∈ [0, L] such that

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ρE (x(t), xf (t)) ≤  for all t ∈ [T1 + τ1 , T2 − τ2 ]. Moreover, if ρE (x(T1 ), xf (T1 )) ≤ δ, then τ1 = 0, and if ρE (x(T2 ), xf (T2 )) ≤ δ then τ2 = 0. Theorem 8.5 implies the following two results. Theorem 8.14 Assume that f has TP and that , M > 0. Then there exist δ > 0 and L > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy I f (T1 , T2 , x, u) ≤ min{I f (T1 , T2 , xf , uf ) + M, U f (T1 , T2 , x(T1 ), x(T2 )) + δ} there exist τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that ρE (x(t), xf (t)) ≤  for all t ∈ [τ1 , τ2 ]. Moreover, if ρE (x(T2 ), xf (T2 )) ≤ δ, then τ2 = T2 , and if T1 ≥ L and ρE (x(T1 ), xf (T1 )) ≤ δ, then τ1 = T1 . Theorem 8.15 Assume that f has STP and that , M > 0. Then there exist δ > 0 and L > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy I f (T1 , T2 , x, u) ≤ min{I f (T1 , T2 , xf , uf ) + M, U f (T1 , T2 , x(T1 ), x(T2 )) + δ} there exist τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that ρE (x(T1 ), xf (T1 )) ≤  for all t ∈ [τ1 , τ2 ]. Moreover, for i = 1, 2, if ρE (x(Ti ), xf (Ti )) ≤ δ, then τi = Ti . It is easy to see that the following result holds (for details, see [148]). Proposition 8.16 Let L > 0. Then for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each L , pair of points (T1 , z1 ) ∈ AL , (T2 , z2 ) ∈ A U f (T1 , T2 , z1 , z2 ) ≤ I f (T1 , T2 , xf , uf ) + 2L(1 + a0 ). Proposition 8.16 and Theorems 8.14 and 8.15 imply the following two results. Theorem 8.17 Assume that f has TP and that , L0 > 0. Then there exist δ > 0 and L > L0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy L0 , (T1 , x(T1 )) ∈ AL0 , (T2 , x(T2 )) ∈ A I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ

8.7 Turnpike Results

233

there exist τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that ρE (x(t), xf (t)) ≤  for all t ∈ [τ1 , τ2 ]. Moreover, if ρE (x(T2 ), xf (T2 )) ≤ δ, then τ2 = T2 , and if T1 ≥ L and ρE (x(T1 ), xf (T1 )) ≤ δ, then τ1 = T1 . Theorem 8.18 Assume that f has STP and that , L0 > 0. Then there exist δ > 0 and L > L0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy L0 , (T1 , x(T1 )) ∈ AL0 , (T2 , x(T2 )) ∈ A I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ there exist τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that ρE (x(t), xf (t)) ≤  for all t ∈ [τ1 , τ2 ]. Moreover, if i ∈ {1, 2} and ρE (x(Ti ), xf (Ti )) ≤ δ, then τi = Ti . It is easy to see that the following result holds. Proposition 8.19 Let L > 0. Then for each (T1 , z) ∈ AL , each T2 ≥ T1 + L, U f (T1 , T2 , z) ≤ I f (T1 , T2 , xf , uf ) + L(1 + a0 ). Proposition 8.19 implies the following two results. Theorem 8.20 Assume that f has TP and that , L0 > 0. Then there exist δ > 0 and L > L0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy (T1 , x(T1 )) ∈ AL0 , I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 )) + δ there exist τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that ρE (x(t), xf (t)) ≤  for all t ∈ [τ1 , τ2 ]. Moreover, if ρE (x(T2 ), xf (T2 )) ≤ δ, then τ2 = T2 , and if T1 ≥ L and ρE (x(T1 ), xf (T1 )) ≤ δ, then τ1 = T1 . Theorem 8.21 Assume that f has STP and that , L0 > 0. Then there exist δ > 0 and L > L0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy

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(T1 , x(T1 )) ∈ AL0 , I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 )) + δ there exist τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that ρE (x(t), xf (t)) ≤  for all t ∈ [τ1 , τ2 ]. Moreover, if i ∈ {1, 2} and ρE (x(T2 ), xf (T2 )) ≤ δ, then τi = Ti . It is easy to see that the following result holds. L Proposition 8.22 Let L > 0. Then for each T1 ≥ 0 and each (T2 , z) ∈ A satisfying T2 ≥ T1 + L, f (T1 , T2 , z) ≤ I f (T1 , T2 , xf , uf ) + L(1 + a0 ). U Proposition 8.22 implies the following two results. Theorem 8.23 Assume that f has TP and that , L0 > 0. Then there exist δ > 0 and L > L0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy L0 , (T2 , x(T2 )) ∈ A f (T1 , T2 , x(T2 )) + δ I f (T1 , T2 , x, u) ≤ U there exist τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that ρE (x(t), xf (t)) ≤  for all t ∈ [τ1 , τ2 ]. Moreover, if ρE (x(T2 ), xf (T2 )) ≤ δ, then τ2 = T2 , and if T1 ≥ L and ρE (x(T1 ), xf (T1 )) ≤ δ, then τ1 = T1 . Theorem 8.24 Assume that f has STP and that , L0 > 0. Then there exist δ > 0 and L > L0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2L, and each (x, u) ∈ X(T1 , T2 ) that satisfy L0 , (T2 , x(T2 )) ∈ A f (T1 , T2 , x(T2 )) + δ I f (T1 , T2 , x, u) ≤ U there exist τ1 ∈ [T1 , T1 + L], τ2 ∈ [T2 − L, T2 ] such that ρE (x(t), xf (t)) ≤  for all t ∈ [τ1 , τ2 ]. Moreover, if i ∈ {1, 2} and ρE (x(T2 ), xf (T2 )) ≤ δ, then τi = Ti .

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235

The next theorem is the main result of this chapter. It is proved in Sect. 8.13. Theorem 8.25 f has TP if and only if the following properties hold: (P1)

For each (f )-good pair (x, u) ∈ X(0, ∞), lim ρE (x(t), xf (t)) = 0.

t→∞

(P2)

For each  > 0 and each M > 0, there exist δ > 0 and L > 0 such that for each T ≥ 0 and each (x, u) ∈ X(T , T + L) that satisfy I f (T , T + L, x, u) ≤ min{U f (T , T + L, x(T ), x(T + L)) + δ, I f (T , T + L, xf , uf ) + M} there exists s ∈ [T , T + L] such that ρE (x(s), xf (s)) ≤ .

We say that f possesses the weak turnpike property (or WTP for short) if for each  > 0 and each M > 0 there exist a natural number Q and l > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + lQ, and each (x, u) ∈ X(T1 , T2 ) that satisfy I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M q

q

there exist finite sequences {ai }i=1 , {bi }i=1 ⊂ [T1 , T2 ] such that an integer q ≤ Q, 0 ≤ bi − ai ≤ l, i = 1, . . . , q, bi ≤ ai+1 for all integers i satisfying 1 ≤ i < q, q

ρE (x(t), xf (t)) ≤  for all t ∈ [T1 , T2 ] \ ∪i=1 [ai , bi ]. The next result is proved in Sect. 8.14. Theorem 8.26 f has WTP if and only if f has (P1) and (P2). Theorem 8.27 Consider the three classes of problems. For the second class of problems, we assume that the following assumption holds: (A3) For each  > 0, there exist δ, L > 0 such that for each S1 ≥ L and each S2 ∈ [S1 , S1 + δ] the inequality I f (S1 , S2 , xf , uf ) ≤ . The integrand f has (P1) and (P2) if and only if the following property holds: (P3) For each  > 0 and each M > 0, there exists L > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + L, and each (x, u) ∈ X(T1 , T2 ) that satisfy I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M the inequality

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mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) ≤ L is valid. In view of Theorem 8.26, (P1) and (P2) imply (P3). Clearly, (P3) implies (P2). Therefore, Theorem 8.27 follows from the next result that is proved in Sect. 8.16. Proposition 8.28 We consider three classes of problems. For the second class of problems, we assume that (A3) holds. Let f have (P3). Then f has (P1).

8.8 Lower Semicontinuity Property We continue to consider the general class of problems studied in Sect. 8.7. We say that f possesses lower semicontinuity property (or LSC property for short) if for each T2 > T1 ≥ 0 and each sequence (xj , uj ) ∈ X(T1 , T2 ), j = 1, 2, . . . , which satisfy sup{I f (T1 , T2 , xj , uj ) : j = 1, 2, . . . } < ∞, there exist a subsequence {(xjk , ujk )}∞ k=1 and (x, u) ∈ X(T1 , T2 ) such that xjk (t) → x(t) as k → ∞ for every t ∈ [T1 , T2 ], I f (T1 , T2 , x, u) ≤ lim inf I f (T1 , T2 , xj , uj ). j →∞

LSC property plays an important role in the calculus of variations and optimal control theory [35]. Let S ≥ 0. A pair (x, u) ∈ X(S, ∞) is called (f )-overtaking optimal if for every (y, v) ∈ X(S, ∞) satisfying x(S) = y(S), lim sup[I f (S, T , x, u) − I f (S, T , y, v)] ≤ 0. T →∞

A pair (x, u) ∈ X(S, ∞) is called f -weakly optimal if for every (y, v) ∈ X(S, ∞) satisfying x(S) = y(S), lim inf[I f (S, T , x, u) − I f (S, T , y, v)] ≤ 0. T →∞

A pair (x, u) ∈ X(S, ∞) is called (f )-minimal if for every T > S I f (S, T , x, u) = U f (S, T , x(S), x(T )). The next result is proved in Sect. 8.18.

8.8 Lower Semicontinuity Property

237

Theorem 8.29 Assume that f has (P1), (P2), and (LSC) property and S ≥ 0. Then the following assertions hold: 1. Assume that (x, u) ∈ X(S, ∞) is (f )-good. Then there exists an (f )-overtaking optimal pair (x∗ , u∗ ) ∈ X(S, ∞) such that x∗ (S) = x(S). 2. Assume that (x, ˜ u) ˜ ∈ X(S, ∞) is (f )-good and that (x∗ , u∗ ) ∈ X(S, ∞) satisfies x∗ (S) = x(S). ˜ Then the following conditions are equivalent: (i) (ii) (iii) (iv) (v)

(x∗ , u∗ ) is (f )-overtaking optimal. (x∗ , u∗ ) is (f )-weakly optimal. (x∗ , u∗ ) is (f )-minimal and (f )-good. (x∗ , u∗ ) is (f )-minimal and satisfies limt→∞ ρE (x∗ (t), xf (t)) = 0. (x∗ , u∗ ) is (f )-minimal and satisfies lim inft→∞ ρE (x∗ (t), xf (t)) = 0.

The next result easily follows from Theorems 8.25 and 8.26. Theorem 8.30 Assume that f has (P1), (P2) and LSC property and , L > 0. Then there exists τ0 > 0 such that for every T0 ≥ 0 and every (f )-overtaking optimal pair (x, u) ∈ X(T0 , ∞) that satisfy (T0 , x(T0 )) ∈ AL the inequality ρE (x(t), xf (t)) ≤  holds for all t ≥ T0 + τ0 . In the sequel, we use the following result that can be easily proved. Proposition 8.31 Let f have (P1) and (T0 , z0 ) ∈ A. There exists an (f )-good pair (x, u) ∈ X(T0 , ∞) satisfying x(T0 ) = z0 if and only if (T0 , z0 ) ∈ ∪{AL : L > 0}. LSC property implies the following result. Proposition 8.32 Let f have LSC property, T2 > T2 ≥ 0, L > 0. Then the following assertions hold: 1. There exists (x, u) ∈ X(T1 , T2 ) such that I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , y, v) for all (y, v) ∈ X(T1 , T2 ). L , then there exists (x, u) ∈ 2. If T2 − T1 ≥ 2L, (T1 , z1 ) ∈ AL , (T2 , z2 ) ∈ A X(T1 , T2 ) such that x(Ti ) = zi , i = 1, 2 and I f (T1 , T2 , x, u) = U f (T1 , T2 , z1 , z2 ). 3. If T2 − T1 ≥ L, (T1 , z) ∈ AL , then there exists (x, u) ∈ X(T1 , T2 ) such that x(T1 ) = z and I f (T1 , T2 , x, u) = U f (T1 , T2 , z). L , then there exists (x, u) ∈ X(T1 , T2 ) such that 4. If T2 − T1 ≥ L, (T2 , z) ∈ A x(T2 ) = z and f (T1 , T2 , z). I f (T1 , T2 , x, u) = U

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8.9 Perturbed Problems In this section, we use the following assumption: (A4) For each  > 0, there exists δ > 0 such that for each (Ti , zi ) ∈ A, i = 1, 2, satisfying ρE (zi , xf (t)) ≤ δ, i = 1, 2, and T2 ≥ bf , there exist τ1 , τ2 ∈ (0, bf ] and (x1 , u1 ) ∈ X(T1 , T1 + τ1 ), (x2 , u2 ) ∈ X(T2 − τ2 , T2 ) that satisfy x1 (T1 ) = z1 , x1 (T1 + τ1 ) = xf (T1 + τ1 ), I f (T1 , T1 + τ1 , x1 , u1 ) ≤ I f (T1 , T1 + τ1 , xf , uf ) + , ρE (x1 (t), xf (t)) ≤ , t ∈ [T1 , T1 + τ1 ], x2 (T2 ) = z2 , x2 (T2 − τ2 ) = xf (T2 − τ2 ), I f (T2 − τ2 , T2 , x2 , u2 ) ≤ I f (T2 − τ2 , T2 , xf , uf ) + , ρE (x2 (t), xf (t)) ≤ , t ∈ [T2 − τ2 , T2 ]. Clearly, (A4) implies (A2). Assume that φ : E × E → [0, 1] is a continuous function satisfying φ(0, 0) = 0 and such that the following property holds: (i) For each  > 0, there exists δ > 0 such that for each x, y ∈ E satisfying φ(x, y) ≤ δ, we have ρE (x, y) ≤ . For each r ∈ (0, 1), set fr (t, x, u) = f (t, x, u) + rφ(x, xf (t)), (t, x, u) ∈ M. Clearly, for any r ∈ (0, 1), fr is a Borelian function; if (A4) holds, then (A1) and (A4) hold for fr with (xfr , ufr ) = (xf , uf ). In the next result, we consider each of the three classes of problems. For the second class, we assume that (A3) holds. Theorem 8.33 Let (A4) hold and r ∈ (0, 1). Then fr has (P1) and (P2). Moreover, if (xf , uf ) is (f )-minimal, then (xf , uf ) is a unique (fr )-overtaking optimal with an initial point xf (0).

8.10 Auxiliary Results for Theorems 8.5 and Its Proof The next result follows easily from (A2). Proposition 8.34 Let γ > 0. Then there exists δ > 0 such that for each (T , z1 ), (T + 2bf , z2 ) ∈ A satisfying

8.10 Auxiliary Results for Theorems 8.5 and Its Proof

239

ρE (z1 , xf (T )) ≤ δ, ρE (z2 , xf (T + 2bf )) ≤ δ, there exists (x, u) ∈ X(T , T + 2bf ), which satisfies x(T ) = z1 , x(T + 2bf ) = z2 , I f (T , T + 2bf , x, u) ≤ I f (T , T + 2bf , xf , uf ) + γ . Lemma 8.35 There exist numbers S > 0, c0 > 1 such that for each T1 ≥ 0, each T2 ≥ T1 + c0 , and each (x, u) ∈ X(T1 , T2 ), I f (T1 , T2 , xf , uf ) ≤ I f (T1 , T2 , x, u) + S.

(8.76)

Proof In view of (8.3), (8.5), and (8.74), there exists S1 > 0 such that ψ(S1 ) > a0 + 1 + sup{|I f (j, j + 1, xf , uf )| : j = 0, 1, . . . }.

(8.77)

By (A1), there exist S2 > 0, c0 > 1 such that I f (T1 , T2 , xf , uf ) ≤ I f (T1 , T2 , x, u) + S2

(8.78)

for each T1 ≥ 0, each T2 ≥ T1 + c0 , and each (x, u) ∈ X(T1 , T2 ) satisfying ρE (x(T1 ), θ0 ), ρE (x(T2 ), θ0 ) ≤ S1 . Fix a number S ≥ S2 + 2 + 2a0 (4 + 2c0 ) + 4(c0 + 4) × sup{|I f (j, j + 1, xf , uf )| : j = 0, 1, . . . }. Assume that T1 ≥ 0, T2 ≥ T1 + c0 , (x, u) ∈ X(T1 , T2 ). We show that (8.76) is true. If ρE (x(t), θ0 ) ≥ S1 , t ∈ [T1 , T2 ], then by (8.4), (8.5), (8.77), and (8.79) for all t ∈ [T1 , T2 ], f (t, x(t), u(t)) ≥ −a0 + ψ(S1 ) and

(8.79)

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8 Continuous-Time Nonautonomous Problems on Half-Axis

I f (T1 , T2 , x, u) ≥ (T2 − T1 )(ψ(S1 ) − a0 ) ≥ I f (T1 , T2 , xf , uf ) − S and (8.76) holds. Assume that inf{ρE (x(t), θ0 ) : t ∈ [T1 , T2 ]} < S1 .

(8.80)

τ1 = inf{t ∈ [T1 , T2 ] : ρE (x(t), θ0 ) ≤ S1 },

(8.81)

τ2 = sup{t ∈ [T1 , T2 ] : ρE (x(t), θ0 ) ≤ S1 }.

(8.82)

Set

Clearly, τ1 , τ2 are well-defined and ρE (x(τi ), θ0 ) ≤ S1 , i = 1, 2.

(8.83)

By (8.4), (8.5), (8.81), and (8.82) for each t ∈ [T1 , τ1 ] ∪ [T2 , τ2 ], ρE (x(t), θ0 ) ≥ S1 , f (t, x(t), u(t)) ≥ −a0 + ψ(S1 ).

(8.84)

It follows from (8.74), (8.77), and (8.84) that I f (T1 , τ1 , x, u) ≥ I f (T1 , τ1 , xf , uf ) − 2Δf − 2a0 , I f (τ2 , T2 , x, u) ≥ I f (τ2 , T2 , xf , uf ) − 2Δf − 2a0 .

(8.85)

If τ2 − τ1 < c0 , then by (8.5), (8.74), and (8.85), I f (T1 , T2 , x, u) ≥ I f (T1 , τ1 , xf , uf ) + I f (τ2 , T2 , xf , uf ) − 4Δf − 4a0 − a0 c0 ≥ I f (T1 , T2 , xf , uf ) − 4a0 − a0 c0 − Δf c0 − 5Δf − 2a0 ≥ I f (T1 , T2 , xf , uf ) − S. Assume that τ2 − τ1 ≥ c0 . It follows from (8.78) and (8.83) that I f (τ1 , τ2 , xf , uf ) ≤ I f (τ1 , τ2 , x, u) + S2 . Together with (8.79) and (8.85), this implies (8.76). Lemma 8.35 is proved.

8.11 Proof of Theorem 8.6

241

Proof of Theorem 8.5 Assertion 1 follows from Lemma 8.35. Let (x, u) ∈ X(0, ∞). Assumption 1 implies that either I f (0, T , x, u) − I f (0, T , xf , uf ) → ∞ as T → ∞ or sup{|I f (0, T , x, u) − I f (0, T , xf , uf )| : T ∈ (0, ∞)} < ∞.

(8.86)

Assume that (8.86) holds. Then the boundedness of x follows from Proposition 8.1.

8.11 Proof of Theorem 8.6 We may assume that c < 1/2 and c0 < c/4. By Theorem 8.5, there exists S0 > 0 such that the following property holds: (i) For each T2 > T1 ≥ 0 and each (x, u) ∈ X(T1 , T2 ), I f (T1 , T2 , x, u) + S0 ≥ I f (T1 , T2 , xf , uf ). Proposition 8.1 implies that there exists M1 > 0 such that the following property holds: (ii) For each T ≥ 0 and each (x, u) ∈ X(T , T + c/2) satisfying I f (T , T + c/2, x, u) ≤ M0 + 2S0 + 2 + 2a0 + 2Δf , we have ρE (x(t), θ0 ) ≤ M1 for all t ∈ [c0 + T , T + c/2], and if (A0)’ holds, then ρE (x(t), θ0 ) ≤ M1 for all t ∈ [T , T + c/2]. Assume that T1 ≥ 0, T2 ≥ T1 + c, (x, u) ∈ X(T1 , T2 )

(8.87)

I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M0 .

(8.88)

and

242

8 Continuous-Time Nonautonomous Problems on Half-Axis

Property (i) and (8.88) imply that for each a ∈ R 1 satisfying [a, a + 2−1 c] ⊂ [T1 , T2 ], we have I f (a, a + c/2, x, u) = I f (T1 , T2 , x, u) − I f (T1 , a, x, u) − I f (a + c/2, T2 , x, u) ≤ I f (T1 , T2 , x, u) + M0 − I f (T1 , a, xf , uf ) + S0 − I f (a + c/2, T2 , xf , uf ) + S0 = I f (a, a + c/2, xf , uf ) + 2S0 + M0 ≤ 2Δf + 2a0 + 2S0 + M0 , and combined with property (ii), this implies that ρE (x(t), θ0 ) ≤ M1 for all t ∈ [c0 + a, a + c/2], and if (A0)’ holds, then ρE (x(t), θ0 ) ≤ M1 for all t ∈ [a, a + c/2]. Since a is any number satisfying (8.87), this completes the proof of Theorem 8.6.

8.12 Proof of Theorem 8.10 We may assume that c < 1. Theorem 8.5 implies that there exists S0 > 0 such that the following property holds: (i) For each T2 > T1 ≥ 0 and each (x, u) ∈ X(T1 , T2 ), we have I f (T1 , T2 , x, u) + S0 ≥ I f (T1 , T2 , xf , uf ). By Theorem 8.6, there exists M2 > 0 such that the following property holds: (ii) For each T1 ≥ 0, each T2 ≥ T1 + c, and each (x, u) ∈ X(T1 , T2 ), satisfying I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M0 , the inequality ρE (x(t), θ0 ) ≤ M2 holds for all t ∈ [T1 + c/4, T2 ]. Assumption (A0) implies that there exists M1 > M2 such that the following property holds: (iii) For each T ≥ 0 and each (x, u) ∈ X(T , T + c/4) satisfying

8.13 Proof of Theorem 8.25

243

ρE (x(T ), θ0 ) ≤ M0 , I f (T , T + c/4, x, u) ≤ M0 + S0 + a0 c + 2|Δf | + 2a0 , we have ρE (x(t), θ0 ) ≤ M1 for all t ∈ [T , T + c/4]. Assume that T1 ≥ 0, T2 ≥ T1 + c, (x, u) ∈ X(T1 , T2 ), ρE (x(T1 ), θ0 ) ≤ M0 ,

(8.89)

I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M0 .

(8.90)

Property (ii) and (8.90) imply that ρE (x(t), θ0 ) ≤ M2 < M1 , t ∈ [T1 + c/4, T2 ].

(8.91)

It follows from (8.90) and property (i) that I f (T1 , T1 + c/4, x, u) = I f (T1 , T2 , x, u) − I f (T1 + c/4, T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M0 − I f (T1 + c/4, T2 , x, u) + S0 ≤ S0 + M0 + I f (T1 , T1 + c/4, xf , uf ) ≤ M0 + S0 + 2Δf + 2a0 .

(8.92)

Property (iii) and Eqs. (8.89) and (8.92) imply ρE (x(t), θ0 ) ≤ M1 , t ∈ [T1 , T1 + c/4]. This completes the proof of Theorem 8.10.

8.13 Proof of Theorem 8.25 First we show that TP implies (P1) and (P2). In view of Theorem 8.5, TP implies (P2). We show that TP implies (P1). Assume that TP holds and (x, u) ∈ X(0, ∞) is (f )-good. Let  > 0. There exists S > 0 such that for each T > 0, |I f (0, T , x, u) − I f (0, T , xf , uf )| < S. This implies that for each pair of numbers T2 > T1 ≥ 0, |I f (T1 , T2 , x, u) − I f (T1 , T2 , xf , uf )| < 2S.

(8.93)

244

8 Continuous-Time Nonautonomous Problems on Half-Axis

Let δ > 0. We show that there exists Tδ > 0 such that for each T > Tδ , I f (Tδ , T , x, u) ≤ U f (Tδ , T , x(Tδ ), x(T )) + δ. Assume the contrary. Then for each T ≥ 0, there exists S > T such that I f (T , S, x, u) > U f (T , S, x(T ), x(S)) + δ. This implies that there exists a strictly increasing sequence of numbers {Ti }∞ i=0 such that T0 = 0, and for every integer i ≥ 0, I f (Ti , Ti+1 , x, u) > U f (Ti , Ti+1 , x(Ti ), x(Ti+1 )) + δ. By the relation above, there exists (y, v) ∈ X(0, ∞) such that for every integer i ≥ 0, y(Ti ) = x(Ti ), I f (Ti , Ti+1 , x, u) > I f (Ti , Ti+1 , y, v) + δ. Combined with (8.93), this implies that for each integer k ≥ 1, I f (0, Tk , y, v) − I f (0, Tk , xf , uf ) = I f (0, Tk , y, v) − I f (0, Tk , x, u) + I f (0, Tk , x, u) − I f (0, Tk , xf , uf ) ≤ −kδ + 2S → −∞ as k → ∞. This contradicts Theorem 8.5. The contradiction we have reached proves that the following property holds: (i) For each δ > 0, there exists Tδ > 0 such that for each T > Tδ , I f (Tδ , T , x, u) ≤ U f (Tδ , T , x(Tδ ), x(T )) + δ. Theorem 8.14 implies that there exist δ > 0, L > 0 such that the following property holds: (ii) For each S1 ≥ 0, each S2 ≥ S1 + 2L, and each (z, ξ ) ∈ X(S1 , S2 ) that satisfy I f (S1 , S2 , z, ξ ) ≤ min{I f (S1 , S2 , xf , uf )+2S, U f (S1 , S2 , z(S1 ), z(S2 ))+δ},

8.13 Proof of Theorem 8.25

245

we have ρE (z(t), xf (t)) ≤ , t ∈ [S1 + L, S2 − L]. Let Tδ > 0 be as guaranteed by (i). Properties (i) and (ii), (8.93), and the choice of Tδ imply that for each T ≥ Tδ + 2L, ρE (x(t), xf (t)) ≤ , t ∈ [Tδ , T − L] and (P1) holds. Thus TP implies (P1) and (P2). Now we show that (P1) and (P2) imply TP. We begin with the following auxiliary result. Lemma 8.36 Assume that (P1) holds and that  > 0. Then there exists δ, L > 0 such that for each T1 ≥ L, each T2 ≥ T1 + 2bf , and each (x, u) ∈ X(T1 , T2 ) that satisfy ρE (x(Ti ), xf (Ti )) ≤ δ, i = 1, 2, I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ the inequality ρE (x(t), xf (t)) ≤  holds for all t ∈ [T1 , T2 ]. Proof By (A2), for each integer k ≥ 1, there exists δk ∈ (0, 2−k ) such that the following property holds: (iii) For each (Ti , zi ) ∈ A, i = 1, 2, satisfying ρE (zi , xf (Ti )) ≤ δk , i = 1, 2, and T2 ≥ bf , there exist τ1 , τ2 ∈ (0, bf ] and (x1 , u1 ) ∈ X(T1 , T1 + τ1 ), (x2 , u2 ) ∈ X(T2 − τ2 , T2 ) that satisfy x1 (T1 ) = z1 , x1 (T1 + τ1 ) = xf (T1 + τ1 ), x2 (T2 ) = z2 , x2 (T2 − τ2 ) = xf (T2 − τ2 ), I f (T1 , T1 + τ1 , x1 , u1 ) ≤ I f (T1 , T1 + τ1 , xf , uf ) + 2−k , I f (T2 − τ2 , T2 , x2 , u2 ) ≤ I f (T2 − τ2 , T2 , xf , uf ) + 2−k . We may assume without loss of generality that {δk }∞ k=1 is a decreasing sequence. Assume that the lemma does not hold. Then for each integer k ≥ 1, there exist Tk,1 ≥ k + bf , Tk,2 ≥ Tk,1 + 2bf , and (xk , uk ) ∈ X(Tk,1 , Tk,2 ) such that

(8.94)

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8 Continuous-Time Nonautonomous Problems on Half-Axis

ρE (xk (Tk,i ), xf (Tk,i )) ≤ δk , i = 1, 2, I f (Tk,1 , Tk,2 , xk , uk ) ≤ U f (Tk,1 , Tk,2 , xk (Tk,1 ), xk (Tk,2 )) + δk , sup{ρE (xk (t), xf (t)) : t ∈ [Tk,1 , Tk,2 ]} > .

(8.95) (8.96) (8.97)

Extracting a subsequence and re-indexing, we may assume without loss of generality that for each integer k ≥ 1, Tk+1,1 ≥ Tk,2 + 4bf .

(8.98)

Let k ≥ 1 be an integer. Property (iii), (8.94) and (8.95) imply that there exist τk,1 , τk,2 ∈ (0, bf ] and (x˜k , u˜ k ) ∈ X(Tk,1 − τk,1 , Tk,2 + τk,2 ) such that (x˜k (t), u˜ k (t)) = (xk (t), uk (t)), t ∈ [Tk,1 , Tk,2 ],

(8.99)

x˜k (Tk,1 − τk,1 ) = xf (Tk,1 − τk,1 ), x˜k (Tk,2 + τk,2 ) = xf (Tk,2 + τk,2 ),

(8.100)

I f (Tk,1 − τk,1 , Tk,1 , x˜k , u˜ k ) ≤ I f (Tk,1 − τk,1 , Tk,1 , xf , uf ) + 2−k ,

(8.101)

I f (Tk,2 , Tk,2 + τk,2 , x˜k , u˜ k ) ≤ I f (Tk,2 , Tk,2 + τk,2 , xf , uf ) + 2−k .

(8.102)

Property (iii) and (8.94) and (8.95) imply that there exist τk,3 , τk,4 ∈ (0, bf ] and ( xk ,  uk ) ∈ X(Tk,1 , Tk,2 ) such that  xk (Tk,1 ) = xk (Tk,1 ),  xk (Tk,2 ) = xk (Tk,2 ),

(8.103)

( xk (t),  uk (t)) = (xf (t), uf (t)), t ∈ [Tk,1 + τk,3 , Tk,2 − τk,4 ],

(8.104)

I f (Tk,1 , Tk,1 + τk,3 ,  xk ,  uk ) ≤ I f (Tk,1 , Tk,1 + τk,3 , xf , uf ) + 2−k ,

(8.105)

I f (Tk,2 − τk,4 , Tk,2 ,  xk ,  uk ) ≤ I f (Tk,2 − τk,4 , Tk,2 , xf , uf ) + 2−k .

(8.106)

In view of (8.103)–(8.106), U f (Tk,1 , Tk,2 , xk (Tk,1 ), xk (Tk,2 )) ≤ I f (Tk,1 , Tk,2 ,  xk ,  uk ) ≤ I f (Tk,1 , Tk,2 , xf , uf ) + 2−k+1 .

(8.107)

By (8.96), (8.107) and the inequality δk < 2−k , I f (Tk,1 , Tk,2 , xk , uk ) ≤ I f (Tk,1 , Tk,2 , xf , uf ) + 2−k+2 . It follows from (8.99)–(8.102) and (8.108) that

(8.108)

8.13 Proof of Theorem 8.25

247

I f (Tk,1 − τk,1 , Tk,2 + τk,2 , x˜k , u˜ k ) ≤ I f (Tk,1 − τk,1 , Tk,1 , xf , uf ) + I f (Tk,1 , Tk,2 , xk , uk ) + I f (Tk,2 , Tk,2 + τk,2 , xf , uf ) + 2−k+1 ≤ I f (Tk,1 − τk,1 , Tk,2 + τk,2 , xf , uf ) + 2−k+3 .

(8.109)

By (8.98) and (8.100), there exists (x, u) ∈ X(0, ∞) such that for each integer k ≥ 1, (x(t), u(t)) = (x˜k (t), u˜ k (t)), t ∈ [Tk,1 − τk,1 , Tk,2 + τk,2 ],

(8.110)

(x(t), u(t)) = (xf (t), uf (t)), t ∈ [0, ∞) \ ∪∞ k=1 [Tk,1 − τk,1 , Tk,2 + τk,2 ]. (8.111) It follows from (8.111) that for each integer k ≥ 1, I f (0, Tk,2 + τk,2 , x, u) − I f (0, Tk,2 + τk,2 , xf , uf ) k  = (I f (Ti,1 − τi,1 , Ti,2 + τi,2 , x˜i , u˜ i ) − I f (Ti,1 − τi,1 , Ti,2 + τi,2 , xf , uf )) i=1

≤

k 

2−i+3 ≤ 8.

i=1

Combined with Theorem 8.5 implies that (x, u) is (f )-good. Together with (P1), this implies that lim ρE (x(t), xf (t)) = 0.

t→∞

On the other hand, in view of (8.97), (8.99), and (8.110), lim sup ρE (x(t), xf (t)) ≥ . t→∞

The contradiction we have reached completes the proof of Lemma 8.36. Completion of the Proof of Theorem 8.25 Assume that properties (P1) and (P2) hold. Let , M > 0. By Lemma 8.36, there exist δ0 , L0 > 0 such that the following property holds: (iv) For each T1 ≥ L0 , each T2 ≥ T1 + 2bf , and each (x, u) ∈ X(T1 , T2 ) that satisfy

248

8 Continuous-Time Nonautonomous Problems on Half-Axis

ρE (x(Ti ), xf (Ti )) ≤ δ0 , i = 1, 2, I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ0 the inequality ρE (x(t), xf (t)) ≤  holds for all t ∈ [T1 , T2 ]. By Theorem 8.5, there exists S0 > 0 such that the following property holds: (v) For each T2 > T1 ≥ 0 and each (x, u) ∈ X(T1 , T2 ), we have I f (T1 , T2 , x, u) + S0 > I f (T1 , T2 , xf , uf ). In view of (P2), there exist δ ∈ (0, δ0 ) and L1 > 0 such that the following property holds: (vi) For each T ≥ 0 and each (x, u) ∈ X(T , T + L1 ) that satisfy I f (T , T + L1 , x, u) ≤ min{U f (T , T + L1 , x(T ), x(T + L1 )) + δ, I f (T , T + L1 , xf , uf ) + 2S0 + M}, there exists S ∈ [T , T + L1 ] such that ρE (x(S), xf (S)) ≤ δ0 . Set L = L0 + L1 + bf .

(8.112)

Assume that T1 ≥ 0, T2 ≥ T1 + 2L and that (x, u) ∈ X(T1 , T2 ) satisfies I f (T1 , T2 , x, u) ≤ U f (T1 , T2 ) + M,

(8.113)

I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ.

(8.114)

I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M.

(8.115)

By (8.113),

Property (v) and (8.115) imply that for each pair of numbers Q1 , Q2 ∈ [T1 , T2 ] satisfying Q1 < Q2 , I f (Q1 , Q2 , x, u) = I f (T1 , T2 , x, u) − I f (T1 , Q1 , x, u) − I f (Q2 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M − I f (T1 , Q1 , xf , uf ) + S0 − I f (Q2 , T2 , xf , uf ) + S0 = I f (Q1 , Q2 , xf , uf ) + M + 2S0 .

(8.116)

8.14 Proof of Theorem 8.26

249

It follows from (8.112) and (8.116) that I f (max{T1 , L0 }, max{T1 , L0 } + L1 , x, u) ≤ I f (max{T1 , L0 }, max{T1 , L0 } + L1 , xf , uf ) + 2S0 + M,

(8.117)

I f (T2 − L1 , T2 , x, u) ≤ I f (T2 − L1 , T2 , xf , uf ) + 2S0 + M.

(8.118)

By (8.114), (8.117), (8.118), and property (vi), there exist τ1 ∈ [max{T1 , L0 }, max{T1 , L0 } + L1 ], τ2 ∈ [T2 − L1 , T2 ]

(8.119)

such that ρE (x(τi ), xf (τi )) ≤ δ0 , i = 1, 2.

(8.120)

If ρE (x(T2 ), xf (T2 )) ≤ δ, then we may assume that τ2 = T2 , and if T1 ≥ L0 and ρE (x(T1 ), xf (T1 )) ≤ δ, then we may assume that τ1 = T1 . In view of (8.112) and (8.119), τ2 − τ1 ≥ T2 − T1 − L1 − L1 − L0 ≥ L0 + 2bf .

(8.121)

Property (iv), (8.114), (8.120), and (8.121) imply that ρE (x(t), xf (t)) ≤ , t ∈ [τ1 , τ2 ]. Thus TP holds and Theorem 8.25 is proved.

8.14 Proof of Theorem 8.26 Assume that f has properties (P1) and (P2). Theorem 8.25 implies that f has TP. Let , M > 0. By Theorem 8.5, there exists S0 > 0 such that the following property holds: (i) For each τ2 > τ1 ≥ 0 and each (y, v) ∈ X(τ1 , τ2 ), I f (τ1 , τ2 , y, v) + S0 ≥ I f (τ1 , τ2 , xf , uf ). TP implies that there exist δ ∈ (0, ) and L0 > 0 such that the following property holds: (ii) For each τ1 ≥ 0, each τ2 ≥ τ1 + 2L0 , and each (y, v) ∈ X(τ1 , τ2 ) that satisfy

250

8 Continuous-Time Nonautonomous Problems on Half-Axis

I f (τ1 , τ2 , y, v) ≤ min{U f (τ1 , τ2 ) + M + 3S0 , U f (τ1 , τ2 , y(τ1 ), y(τ2 )) + δ}, we have ρE (y(t), xf (t)) ≤ , t ∈ [τ1 + L0 , τ2 − L0 ]. Set l = 2L0 + 1.

(8.122)

Q ≥ 2 + 2(M + S0 )δ −1 .

(8.123)

Choose a natural number

Assume that T1 ≥ 0, T2 ≥ T1 + lQ and (x, u) ∈ X(T1 , T2 ) satisfies I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M.

(8.124)

Property (i) and (8.124) imply that for each pair of numbers τ1 , τ2 ∈ [T1 , T2 ] satisfying τ1 < τ2 , I f (τ1 , τ2 , x, u) = I f (T1 , T2 , x, u) − I f (T1 , τ1 , x, u) − I f (τ2 , T2 , x, u) ≤ M + I f (T1 , T2 , xf , uf ) − I f (T1 , τ1 , xf , uf ) + S0 − I f (τ2 , T2 , xf , uf ) + S0 ≤ I f (τ1 , τ2 , xf , uf ) + M + 2S0 .

(8.125)

t 0 = T1 .

(8.126)

Set

If I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ, then we set t 1 = T2 , s 1 = T2 .

(8.127)

Assume that I f (T1 , T2 , x, u) > U f (T1 , T2 , x(T1 ), x(T2 )) + δ.

(8.128)

8.14 Proof of Theorem 8.26

251

It is easy to see that for each t ∈ (T1 , T2 ) such that t − T1 is sufficiently small, we have I f (T1 , t, x, u) − U f (T1 , t, x(T1 ), x(t)) < δ. Set t˜1 = inf{t ∈ (T1 , T2 ] : I f (T1 , t, x, u) − U f (T1 , t, x(T1 ), x(t)) > δ}. Clearly, t˜1 ∈ (T1 , T2 ] is well-defined. There exist s1 , t1 ∈ [T1 , T2 ] such that t0 < s1 < t˜1 , s1 ≥ t˜1 − 1/4, I f (t0 , s1 , x, u) − U f (t0 , s1 , x(t0 ), x(s1 )) ≤ δ, t˜1 ≤ t1 < t˜1 + 1/4, t1 ≤ T2 , I f (t0 , t1 , x, u) − U f (t0 , t1 , x(t0 ), x(t1 )) > δ.

(8.129) (8.130) (8.131) (8.132)

Assume that k ≥ 1 is an integer, and we defined finite sequences of numbers {ti }ki=0 ⊂ [T1 , T2 ], {si }ki=1 ⊂ [T1 , T2 ] such that T1 = t0 < t 1 · · · < t k ,

(8.133)

ti−1 < si ≤ ti , ti − si ≤ 2−1 ,

(8.134)

for each integer i = 1, . . . , k,

I f (ti−1 , si , x, u) − U f (ti−1 , si , x(ti−1 ), x(si )) ≤ δ,

(8.135)

and if ti < T2 , then I f (ti−1 , ti , x, u) − U f (ti−1 , ti , x(ti−1 ), x(ti )) > δ.

(8.136)

(Note that in view of (8.126), (8.127), (8.129), (8.132), our assumption holds for k = 1.) By (8.136), there exists (y, v) ∈ X(T1 , T2 ) such that y(ti ) = x(ti ), i = 1, . . . , k, I f (ti−1 , ti , x, u)−I f (ti−1 , ti , y, v) > δ for all intergers i ∈ [1, k]\{k}, (y(t), v(t)) = (x(t), u(t)), t ∈ [tk−1 , T2 ]. Property (i), (8.124), (8.137), and (8.138) imply that

(8.137) (8.138)

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8 Continuous-Time Nonautonomous Problems on Half-Axis

I f (T1 , T2 , xf , uf ) + M ≥ I f (T1 , T2 , x, u) ≥ I f (T1 , T2 , y, v) + δ(k − 1) ≥ I f (T1 , T2 , xf , uf ) − S0 + δ(k − 1), k ≤ 1 + δ −1 (M + S0 ).

(8.139)

If tk = T2 , then the construction of the sequence is completed. Assume that tk < T2 . If I f (tk , T2 , x, u) ≤ U f (tk , T2 , x(tk ), x(T2 )) + δ, then we set tk+1 = T2 , sk+1 = T2 , and the construction of the sequence is completed. Assume that I f (tk , T2 , x, u) > U f (tk , T2 , x(tk ), x(T2 )) + δ. It is easy to see that for each t ∈ (tk , T2 ) such that t − tk is sufficiently small, we have I f (tk , t, x, u) − U f (tk , t, x(tk ), x(t)) < δ. Set t˜ = inf{t ∈ (tk , T2 ] : I f (tk , t, x, u) − U f (tk , t, x(tk ), x(t)) > δ}. Clearly, t˜ is well-defined and t˜ ∈ (tk , T2 ]. There exist sk+1 , tk+1 ∈ [T1 , T2 ] such that tk < sk+1 < t˜, sk+1 ≥ t˜ − 1/4, t˜ + 4−1 > tk+1 ≥ t˜, I f (tk , sk+1 , x, u) − U f (tk , sk+1 , x(tk ), x(sk+1 )) ≤ δ, I f (tk , tk+1 , x, u) − U f (tk , tk+1 , x(tk ), x(tk+1 )) > δ. It is not difficult to see that the assumption made for k also holds for k + 1. In q view of (8.139), by induction, we constructed finite sequences {ti }i=0 ⊂ [T1 , T2 ], q {si }i=1 ⊂ [T1 , T2 ] such that q ≤ 1 + δ −1 (M + S0 ),

(8.140)

T1 = t0 < t1 · · · < tq = T2 , for each integer i = 1, . . . , q, ti−1 < si ≤ ti , ti − si ≤ 2−1 ,

(8.141)

8.14 Proof of Theorem 8.26

253

I f (ti−1 , si , x, u) − U f (ti−1 , si , x(ti−1 ), x(si )) ≤ δ,

(8.142)

and if ti < T2 , then I f (ti−1 , ti , x, u) − U f (ti−1 , ti , x(ti−1 ), x(ti )) > δ.

(8.143)

Assume that i ∈ {0, . . . , q − 1}, ti+1 − ti ≥ 2L0 + 1.

(8.144)

By (8.141) and (8.144), si+1 − ti ≥ 2L0 .

(8.145)

In view of (8.142), I f (ti , si+1 , x, u) − U f (ti , si+1 , x(ti ), x(si+1 )) ≤ δ.

(8.146)

Property (i) and (8.124) imply that I f (ti , si+1 , x, u) = I f (T1 , T2 , x, u) − I f (T1 , ti , x, u) − I f (si+1 , T2 , x, u) ≤ M + I f (T1 , T2 , xf , uf ) − I f (T1 , ti , xf , uf ) + S0 − I f (si+1 , T2 , xf , uf ) + S0 ≤ I f (ti , si+1 , xf , uf ) + M + 2S0 ≤ M + 3S0 + U f (ti , si+1 ).

(8.147)

It follows from (8.145)–(8.147) and property (ii) that ρE (x(t), xf (t)) ≤ , t ∈ [ti + L0 , si+1 − L0 ], and together with (8.141), this implies that ρE (x(t), xf (t)) ≤ , t ∈ [ti + L0 , ti+1 − L0 − 1].

(8.148)

By (8.148), {t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > } ⊂ ∪{[ti , ti +L0 ]∪[ti+1 −1−L0 , ti+1 ] : i ∈ {0, . . . , q −1} and ti+1 −ti ≥ 2L0 +1} ∪ {[ti , ti+1 ] : i ∈ {0, . . . , q − 1} and ti+1 − ti < 2L0 + 1}. In view of (8.123) and (8.140), the maximal length of intervals in the right-hand side of the relation above does not exceed 2L0 + 1 = l, and their number does not exceed 2q ≤ Q. Thus WTP holds. It is not difficult to see that WTP implies (P1) Theorem 8.26 is proved.

254

8 Continuous-Time Nonautonomous Problems on Half-Axis

8.15 An Auxiliary Result Lemma 8.37 Let  > 0. Then there exist δ, L > 0 such that for each T1 ≥ L, each T2 > T1 , and each (x, u) ∈ X(T1 , T2 ) that satisfy ρE (x(Ti ), xf (Ti )) ≤ δ, i = 1, 2, the following inequality holds: I f (T1 , T2 , T , x, u) ≥ I f (T1 , T2 , T , xf , uf ) − . Proof By (A2), there exists δ > 0 such that the following property holds: (i) For each (T , z) ∈ A satisfying ρE (z, xf (T )) ≤ δ, there exist τ1 ∈ (0, bf ], (x˜1 , u˜ 1 ) ∈ X(T , T + τ1 ) such that x˜1 (T ) = z , x˜1 (T + τ1 ) = xf (T + τ1 ), I f (T , T + τ1 , x˜1 , u˜ 1 ) ≤ I f (T , T + τ1 , xf , uf ) + /4, and if T ≥ bf , then there exist τ2 ∈ (0, bf ] and (x˜2 , u˜ 2 ) ∈ X(T − τ2 , T ) such that x˜2 (T − τ2 ) = xf (T − τ2 ) , x˜2 (T ) = z, I f (T − τ2 , T , x˜2 , u˜ 2 ) ≤ I f (T − τ2 , T , xf , uf ) + /4. Theorem 8.5 implies that there exists L > 2bf such that for each T1 ≥ L − bf and each T2 > T1 I f (T1 , T2 , xf , uf ) < U f (T1 , T2 , xf (T1 ), xf (T2 )) + /4.

(8.149)

Assume that T1 ≥ L, T2 > T1 , (x, u) ∈ X(T1 , T2 ), and ρE (x(T1 ), xf (T1 )) ≤ δ, ρE (x(T2 ), xf (T2 )) ≤ δ.

(8.150)

By (8.150) and property (i), there exist τ1 , τ2 ∈ (0, bf ] and (y, v) ∈ X(T1 −τ1 , T2 + τ2 ) such that y(T1 − τ1 ) = xf (T1 − τ1 ),

(8.151)

(y(t), v(t)) = (x(t), u(t)) for all t ∈ [T1 , T2 ],

(8.152)

I f (T1 − τ1 , T1 , y, v) ≤ I f (T1 − τ1 , T1 , xf , uf ) + /4,

(8.153)

8.16 Proof of Proposition 8.28

255

y(T2 + τ2 ) = xf (T2 + τ2 ), I f (T2 , T2 + τ2 , y, v) ≤ I f (T2 , T2 + τ2 , xf , uf ) + /4.

(8.154) (8.155)

It follows from (8.149), (8.151), and (8.154) that I f (T1 − τ1 , T2 + τ2 , xf , uf ) < I f (T1 − τ1 , T2 + τ2 , y, v) + /4.

(8.156)

Relations (8.152), (8.153), (8.155), and (8.156) imply that I f (T1 , T2 , x, u) = I f (T1 − τ1 , T2 + τ2 , y, v) − I f (T1 − τ1 , T1 , y, v) − I f (T2 , T2 + τ2 , y, v) ≥ I f (T1 − τ1 , T2 + τ2 , xf , uf ) − /4 − I f (T1 − τ1 , T1 , xf , uf ) − /4 − I f (T2 , T2 + τ2 , xf , uf ) − /4 = I f (T1 , T2 , xf , uf ) − 3/4. Lemma 8.37 is proved.

8.16 Proof of Proposition 8.28 Proposition 7.1 implies that there exist M∗ ≥ 1, ω∗ ∈ R 1 such that eAt  ≤ M∗ eω∗ t , t ∈ [0, ∞), in the case of the second class of problems. We show that (P1) holds. Assume that (x, u) ∈ X(0, ∞) is (f )-good. We show that lim ρE (x(t), xf (t)) = 0.

t→∞

Assume the contrary. Then there exist  > 0 and a sequence of numbers {tk }∞ k=1 such that t1 ≥ 8, tk+1 ≥ tk + 8 for all integers k ≥ 1,

(8.157)

ρE (x(tk ), xf (tk )) >  for all integers k ≥ 1.

(8.158)

256

8 Continuous-Time Nonautonomous Problems on Half-Axis

Theorem 8.5 implies that there exists M1 > 0 such that ρE (xf (t), θ0 ) ≤ M1 , ρE (x(t), θ0 ) ≤ M1 , t ∈ [0, ∞),

(8.159)

|I f (0, t, x, u) − I f (0, t, xf , uf )| < M1 , t ∈ (0, ∞). By the relation above, for all S > T ≥ 0, |I f (T , S, x, u) − I f (T , S, xf , uf )| ≤ 2M1 .

(8.160)

By (8.16), there exists M2 > 0 such that I f (S1 , S2 , xf , uf ) ≤ M2 for all S1 ≥ 0 and all S2 ∈ (S1 , S1 + 1].

(8.161)

From now, we consider the three classes of problems separately and first consider the first class of problems. Recall that in this case θ0 = 0, θ1 = 0, and ρE (z1 , z2 ) = z1 − z2 , z1 , z2 ∈ E. Fix 1 = 4−1 (2Δ1 + 1)−1 .

(8.162)

By (8.5) and (8.20), there exists h0 > 0 such that f (t, x, u) ≥ 41−1 (M1 + M2 + a0 + 8)(G(t, x, u) − a0 x+ )

(8.163)

for each (t, x, u) ∈ M satisfying G(t, x, u) − a0 x ≥ h0 . There exists δ ∈ (0, 1) such that δ < /8, Δ1 δ < 1 /32, δ(a0 M1 + h0 ) < 1 /4.

(8.164)

Let k ≥ 1 be an integer. We show that for all t ∈ [tk − δ, tk ], x(t) − xf (t) ≥ δ.

(8.165)

Assume the contrary. Then there exists τ ∈ [tk − δ, tk ]

(8.166)

x(τ ) − xf (τ ) < δ.

(8.167)

such that

By (8.18) and (8.166),

8.16 Proof of Proposition 8.28

257



tk

x(tk ) = U (tk , τ )x(τ ) +

U (tk , s)G(s, x(s), u(s))ds,

(8.168)

U (tk , s)G(s, xf (s), uf (s))ds.

(8.169)

τ



tk

xf (tk ) = U (tk , τ )xf (τ ) + τ

Property (v) (from Sect. 8.2), (8.166), (8.167), (8.169) imply that x(tk ) − xf (tk ) = U (tk , τ )xf (τ ) − x(τ ) 

tk

+

U (tk , s)G(s, x(s), u(s))ds

τ



tk

+

U (tk , s)G(s, xf (s), uf (s))ds

τ



tk

≤ Δδ xf (τ ) − x(τ ) + Δδ

G(s, x(s), u(s))ds

τ



tk

+ Δδ

G(s, xf (s), uf (s))ds

τ



tk

≤ Δ1 δ + Δδ



tk

G(s, x(s), u(s))ds + Δδ

τ

G(s, xf (s), uf (s))ds.

τ

(8.170)

Let (y, v) ∈ {(x, u), (xf , uf )}.

(8.171)

Set Ω1 = {t ∈ [τ, tk ] : G(t, y(t), v(t)) − a0 y(t) ≥ h0 }, Ω2 = [τ, tk ] \ Ω1 . (8.172) By (8.159)–(8.161), (8.163), (8.164), (8.166), (8.171), and (8.172), 

tk

 G(s, y(s), v(s))ds ≤ a0

τ

tk

y(s)ds

τ



tk

+ τ

(G(s, y(s), v(s)) − a0 y(s))+ ds 

≤ a0 δM1 + h0 δ + Ω1

(G(s, y(s), v(s)) − a0 y(s))+ ds

258

8 Continuous-Time Nonautonomous Problems on Half-Axis

≤ a0 δM1 + h0 δ + 4−1 1 (M1 + M2 + a0 + 8)−1

 f (s, y(s), v(s))ds Ω1

≤ a0 δM1 + h0 δ + (4−1 1 (M1 + M2 + a0 + 8)−1 )(I f (τ, tk , y, v) + a0 δ) ≤ δ(a0 M1 + h0 ) + (4−1 1 (M1 + M2 + a0 + 8)−1 )(I f (τ, tk , xf , uf ) + 2M1 + a0 δ) ≤ δ(a0 M1 + h0 ) + (4−1 1 (M1 + M2 + a0 + 8)−1 )(M2 + 2M1 + a0 δ) < 1 /4 + 1 /2. In view of (8.158), (8.164), (8.170) and the equation above,  ≤ x(tk ) − xf (tk ) ≤ Δ1 δ + 2Δδ (1 /4 + 1 /2) ≤ 1 /32 + 2Δ1 1 ≤ 1 (2Δ1 + 1) = 4−1 , a contradiction. The contradiction we have reached proves that (8.165) holds for all t ∈ [tk − δ, tk ]. Together with (8.157) and (8.158), this implies that mes({t ∈ [0, T ] : x(t) − xf (t) ≥ δ}) → ∞ as T → ∞. On the other hand, property (P3) and (8.160) imply that there exists L > 0 such that for all T > 0, mes({t ∈ [0, T ] : x(t) − xf (t) ≥ δ}) ≤ L. The contradiction we have reached proves that limt→∞ (x(t) − xf (t)) = 0, and (P1) holds for the first class of problems. Consider the second class of problems. Recall (see Proposition 7.16 and (7.9)) that for each τ ≥ 0, Φτ ∈ L(L2 (0, τ ; , F ), E) is defined by  Φτ u =

τ

eA(τ −s) Bu(s)ds, u ∈ L2 (0, τ ; F ).

(8.173)

0

Lemma 8.37 and (A2) and (A3) imply that there exist δ ∈ (0, 1), L¯ > 0 such that M∗ e|ω∗ | δ < /8, δ < bf /16, 64a0 δ < K0  2 (Φ1  + 1)−2

(8.174)

and the following properties hold: (i) For each (Ti , zi ) ∈ A, i = 1, 2, satisfying zi − xf (Ti ) ≤ δ, i = 1, 2 and T2 ≥ bf , there exist τ1 , τ2 ∈ (0, bf ] and (x1 , u1 ) ∈ X(T1 , T1 + τ1 ), (x2 , u2 ) ∈ X(T2 − τ2 , T2 ) satisfying

8.16 Proof of Proposition 8.28

259

x1 (T1 ) = z1 , x1 (T1 + τ1 ) = xf (T1 + τ1 ), x2 (T2 ) = z2 , x2 (T2 − τ2 ) = xf (T2 − τ2 ),

I f (T1 , T1 + τ1 , x1 , u1 ) ≤ I f (T1 , T1 + τ1 , xf , uf ) + 32−1  2 K0 (Φ1  + 1)−2 , I f (T2 − τ2 , T2 , x2 , u2 ) ≤ I f (T2 − τ2 , T2 , xf , uf ) + 32−1  2 K0 (Φ1  + 1)−2 . ¯ each t2 ∈ (t1 , t1 + 4δ], (ii) For each t1 ≥ L, I f (t1 , t2 , xf , uf ) ≤ 32−1  2 K0 (Φ1  + 1)−2 . ¯ each S2 > S1 , and each (ξ, η) ∈ X(S1 , S2 ) that satisfy (iii) For each S1 ≥ L, ξ(Si ) − xf (Si ) ≤ δ, i = 1, 2, we have I f (S1 , S2 , ξ, η) ≥ I f (S1 , S2 , xf , uf ) − 32−1  2 K0 (Φ1  + 1)−2 . Property (P3) and (8.160) imply that there exists L > 0 such that for all T > 0, mes({t ∈ [0, T ] : x(t) − xf (t) ≥ δ}) ≤ L. Therefore, there exists T0 > 0 such that for each S > T0 , mes({t ∈ [T0 , S] : x(t) − xf (t) > δ}) ≤ δ/2.

(8.175)

Since the pair (x, u) is (f )-good, it follows from Theorem 8.6 that there exists T1 > 0 such that for each S > T1 , I f (T1 , S, x, u) ≤ U f (T1 , S, x(T1 ), x(S)) + 32−1  2 K0 (Φ1  + 1)−2 ,

(8.176)

I f (T1 , S, xf , uf ) ≤ U f (T1 , S, xf (T1 ), xf (S)) + 32−1  2 K0 (Φ1  + 1)−2 . (8.177) Let k ≥ 1 be an integer such that ¯ tk > T0 + T1 + 1 + L.

(8.178)

inf{x(t) − xf (t) : t ∈ [tk − δ, tk ]} < δ.

(8.179)

Assume that

Therefore, there exists τ ∈ [tk − δ, tk ]

(8.180)

260

8 Continuous-Time Nonautonomous Problems on Half-Axis

such that x(τ ) − xf (τ ) < δ.

(8.181)

It is clear that (x, u), (xf , uf ) ∈ X(τ, tk ). In view of (8.24), (8.173), and (8.180), 

x(tk ) = eA(tk −τ ) x(τ ) +

tk τ

xf (tk ) = e

eA(tk −s) Bu(s)ds = eA(tk −τ ) x(τ ) + Φtk −τ u(τ + ·),

A(tk −τ )

 xf (τ ) +

(8.182)

tk

e

A(tk −s)

Buf (s)ds

τ

= eA(tk −τ ) xf (τ ) + Φtk −τ uf (τ + ·).

(8.183)

Proposition 7.16, (8.158), (8.173), (8.174), (8.180)–(8.183) imply that  ≤ x(tk ) − xf (tk ) = eA(tk −τ ) (x(τ ) − xf (τ )) + Φtk −τ u(τ + ·) + Φtk −τ uf (τ + ·) ≤ M∗ e|ω∗ |δ x(τ ) − xf (τ ) 

tk

+ Φ1 (



tk

u(s)2 ds)1/2 + Φ1 (

τ

uf (s)2 ds)1/2

τ



tk

< /8 + Φ1 (



tk

u(s)2 ds)1/2 + Φ1 (

τ

uf (s)2 ds)1/2

τ

and −1

(4

(Φ1  + 1)



−1 2

) ≤ max{

tk

 u(s) ds, 2

τ

tk

uf (s)2 ds}.

(8.184)

τ

Let (y, v) ∈ {(x, u), (xf , uf )}, 

tk

 v(s)2 ds ≥ max{

τ

τ

tk



tk

u(s)2 ds,

uf (s)2 ds}.

(8.185) (8.186)

τ

In view of (8.184)–(8.186), 

tk τ

v(s)2 ds ≥ (4−1 )2 (Φ1  + 1)−2 .

(8.187)

8.16 Proof of Proposition 8.28

261

By (8.26) and (8.196), 

tk

 f (s, y(s), v(s))ds + a0 (tk − τ ) ≥ K0

τ

tk

v(s)2 ds

τ −1 2

≥4

 K0 (Φ1  + 1)−2 .

(8.188)

It follows from (8.180) and (8.188) that 

tk

f (s, y(s), v(s))ds ≥ 4−1 K0  2 (Φ1  + 1)−2 − a0 δ.

(8.189)

τ

In view of (8.175) and (8.178), for each S > tk , mes({t ∈ [tk , S] : y(t) − xf (t) > δ}) ≤ δ/2.

(8.190)

Inequality (8.190) implies that there exist S1 ∈ [tk , tk + δ]

(8.191)

y(S1 ) − xf (S1 ) ≤ δ

(8.192)

S2 ∈ [S1 + 2bf , S1 + 2bf + δ]

(8.193)

y(S2 ) − xf (S2 ) ≤ δ.

(8.194)

such that

and

such that

It follows from (8.180), (8.189), and (8.191) that I f (τ, S1 , y, v) = I f (τ, tk , y, v) + I f (tk , S1 , y, v) ≥ 4−1 K0  2 (Φ1  + 1)−2 − a0 δ − a0 δ.

(8.195)

Property (iii) and (8.178) and (8.191)–(8.194) imply that I f (S1 , S2 , y, v) ≥ I f (S1 , S2 , xf , uf ) − 32−1 K0  2 (Φ1  + 1)−2 . By (8.178)–(8.180), (8.191), (8.195), (8.196), and property (ii), I f (τ, S2 , y, v) = I f (τ, S1 , y, v) + I f (S1 , S2 , y, v)

(8.196)

262

8 Continuous-Time Nonautonomous Problems on Half-Axis

≥ 4−1 K0  2 (Φ1  + 1)−2 − 2a0 δ + I f (S1 , S2 , xf , uf ) − 32−1 K0  2 (Φ1  + 1)−2 ≥ K0  2 (Φ1  + 1)−2 (4−1 − 32−1 ) − 2a0 δ + I f (τ, S2 , xf , uf ) − I f (τ, S1 , xf , uf ) ≥ I f (τ, S2 , xf , uf ) + K0  2 (Φ1  + 1)−2 (4−1 − 32−1 ) − 2a0 δ − 16−1 K0  2 (Φ1  + 1)−2 .

(8.197)

Property (i), (8.180), (8.181), and (8.191)–(8.194) imply that there exists (x, ˜ u) ˜ ∈ X(τ, S2 ) such that x(τ ˜ ) = y(τ ), x(τ ˜ + bf ) = xf (τ + bf ),

(8.198)

I f (τ, τ +bf , x, ˜ u) ˜ ≤ I f (τ, τ +bf , xf , uf )+32−1  2 K0 (Φ1 +1)−2 ,

(8.199)

x(S ˜ 2 ) = y(S2 ), x(S ˜ 2 − bf ) = xf (S2 − bf ),

(8.200)

I f (S2 − bf , S2 , x, ˜ u) ˜ ≤ I f (S2 − bf , S2 , xf , uf ) + 32−1  2 K0 (Φ1  + 1)−2 , (8.201) (x(t), ˜ u(t)) ˜ = (xf (t), uf (t)), t ∈ [τ + bf , S2 − bf ]. (8.202) In view of (8.199), (8.201), and (8.202), I f (τ, S2 , x, ˜ u) ˜ ≤ I f (τ, S2 , xf , uf ) + 16−1  2 K0 (Φ1  + 1)−2 .

(8.203)

It follows from (8.174), (8.197), and (8.203) that I f (τ, S2 , y, v) − I f (τ, S2 , x, ˜ u) ˜ ≥  2 K0 (Φ1  + 1)−2 (4−1 − 32−1 − 8−1 ) − 2δa0 ≥  2 K0 (Φ1  + 1)−2 16−1 .

(8.204)

By (8.198), (8.200), and (8.204), I f (τ, S2 , y, v) ≥ U f (τ, S2 , y(τ ), y(S2 )) +  2 K0 (Φ1  + 1)−2 /16.

(8.205)

It follows from (8.176)–(8.178), (8.180), and (8.125) that τ > T1 and I f (τ, S2 , y, v) ≤ U f (τ, S2 , y(τ ), y(S2 )) + 32−1  2 K0 (Φ1  + 1)−2 .

8.16 Proof of Proposition 8.28

263

This contradicts (8.205). The contradiction we have reached proves that (8.179) is not true and x(t) − xf (t) ≥ δ, t ∈ [tk − δ, tk ] ¯ Together with (8.157), for each natural number k satisfying tk ≥ T0 + T1 + 1 + L. this implies that mes({t ∈ [0, T ] : x(t) − xf (t) ≥ δ}) → ∞ as T → ∞. This contradicts (8.175) holding for all S > T0 . The contradiction we have reached proves that limt→∞ (x(t) − xf (t) = 0 and (P1) holds for the second class of problems too. Consider the third class of problems. Fix 1 ∈ (0, 4−1 ).

(8.206)

By (8.3), (8.5), and (8.56), there exists h0 > 0 such that f (t, x, u) ≥ 41−1 (M1 + M2 + a0 + 8)(G(t, x, u) − a0 ρE (x, θ0 )+ )

(8.207)

for each (t, x, u) ∈ M satisfying G(t, x, u) − a0 ρE (x, θ0 ) ≥ h0 . There exists δ ∈ (0, 1) such that δ < /8, δ(a0 M1 + h0 ) < 1 /4.

(8.208)

Let k ≥ 1 be an integer. We show that for all t ∈ [tk − δ, tk ], ρE (|x(t), xf (t)) ≥ δ.

(8.209)

Assume the contrary. Then there exists τ ∈ [tk − δ, tk ]

(8.210)

ρE (x(τ ), xf (τ )) < δ.

(8.211)

such that

By (8.55) and (8.210),  ρE (x(tk ), x(τ )) ≤

tk

G(s, x(s), u(s))ds,

(8.212)

τ

 ρE (xf (tk ), xf (τ )) ≤

tk

G(s, xf (s), uf (s))ds. τ

(8.213)

264

8 Continuous-Time Nonautonomous Problems on Half-Axis

Let (y, v) ∈ {(x, u), (xf , uf )}.

(8.214)

Set Ω1 = {t ∈ [τ, tk ] : G(t, y(t), v(t)) − a0 ρE (y(t), θ0 ) ≥ h0 }, Ω2 = [τ, tk ] \ Ω1 . (8.215) By (8.5), (8.159), (8.161), (8.207), (8.208), (8.210), (8.214), and (8.215), 

tk

 G(s, y(s), v(s))ds ≤ a0

τ

tk

ρE (y(s), θ0 )ds τ



tk

+ τ

(G(s, y(s), v(s)) − a0 ρE y(s), θ0 ))+ ds 

≤ a0 δM1 + h0 δ + Ω1 −1

≤ a0 δM1 + h0 δ + 4

(G(s, y(s), v(s)) − a0 ρE y(s), θ0 ))+ ds

1 (M1 + M2 + a0 + 8)

−1

 f (s, y(s), v(s))ds Ω1

≤ a0 δM1 + h0 δ + (4−1 1 (M1 + M2 + a0 + 8)−1 )(I f (τ, tk , y, v) + a0 δ) ≤ δ(a0 M1 + h0 ) + (4−1 1 (M1 + M2 + a0 + 8)−1 )(I f (τ, tk , xf , uf ) + 2M1 + a0 δ) ≤ δ(a0 M1 + h0 ) + (4−1 1 (M1 + M2 + a0 + 8)−1 )(M2 + 2M1 + a0 δ) < 1 /4 + 1 /2.

(8.216)

In view of (8.158), (8.206), (8.208), (8.211)–(8.214), and (8.216),  ≤ ρE (x(tk ), xf (tk )) ≤ ρE (x(tk ), x(τ )) + ρE (x(τ ), xf (τ )) + ρE (xf (τ ), x(tk )) ≤ δ + 31 /2 < , a contradiction. The contradiction we have reached proves that (8.209) holds for all t ∈ [tk − δ, tk ]. Together with (8.157), this implies that mes({t ∈ [0, T ] : ρE (x(tk ), xf (tk )) ≥ δ}) → ∞ as T → ∞.

8.17 Auxiliary Results for Theorem 8.29

265

On the other hand, property (P3) and (8.160) imply that there exists L > 0 such that for all T > 0, mes({t ∈ [0, T ] : ρE (x(tk ), xf (tk )) ≥ δ}) ≤ L. The contradiction we have reached proves that limt→∞ ρE (x(t), xf (t)) = 0 and (P1) holds for the third class of problems. Proposition 8.28 is proved.

8.17 Auxiliary Results for Theorem 8.29 Recall that for each z ∈ R 1 , z = max{i ∈ R 1 : i is an integer, i ≤ z}. Proposition 8.38 Assume that f has (P1), (P2) and LSC property, T0 ≥ 0, z0 ∈ E, (T0 , z0 ) ∈ ∪{AL : L ∈ (0, ∞)}. Then there exists an (f )-good and (f )-minimal pair (x∗ , u∗ ) ∈ X(T0 , ∞) such that x∗ (T0 ) = z0 . Proof There exists L0 > 0 such that (T0 , z0 ) ∈ AL0 .

(8.217)

It follows from Theorem 8.5 that there exists S0 > 0 such that for each T2 > T1 ≥ 0 and each (x, u) ∈ X(T1 , T2 ), I f (T1 , T2 , x, u) + S0 ≥ I f (T1 , T2 , xf , uf ).

(8.218)

Fix an integer k0 ≥ L0 . LSC property and (8.217) imply that for each integer k ≥ k0 there exists (xk , uk ) ∈ X(T0 , T0 + k) satisfying xk (T0 ) = z0 , I f (T0 , T0 + k, xk , uk ) = U f (T0 , T0 + k, z0 ).

(8.219) (8.220)

In view of (8.4), (8.49), and (8.217), for each integer k ≥ k0 , U f (T0 , T0 + k, z0 ) ≤ L0 + I f (T0 , T0 + k, xf , uf ) + a0 L0 .

(8.221)

By (8.218), (8.220), and (8.221), for each integer k ≥ k0 and each pair of numbers T1 , T2 ∈ [T0 , T0 + k] satisfying T1 < T2 , I f (T1 , T2 , xk , uk ) = I f (T0 , T0 + k, xk , uk ) − I f (T0 , T1 , xk , uk ) − I f (T0 + k, T2 , xk , uk ) ≤ I f (T0 , T0 + k, xf , uf ) + L0 (1 + a0 ) − I f (T0 , T1 , xf , uf )

266

8 Continuous-Time Nonautonomous Problems on Half-Axis

+ S0 − I f (T0 + k, T2 , xk , uk ) + S0 = I f (T1 , T2 , xf , uf ) + 2S0 + L0 (1 + a0 ).

(8.222)

By (8.222) and LSC property, extracting subsequences, using the diagonalization process, and re-indexing, we obtain that there exists a strictly increasing sequence of natural numbers {kp }∞ p=1 such that k1 ≥ k0 , and for each integer i ≥ 0, there exists f limp→∞ I (T0 +i, T0 +i+1, xkp , ukp ) and there exists (yi , vi ) ∈ X(T0 +i, T0 +i+1) such that xkp (t) → yi (t) as p → ∞ for all t ∈ [T0 + i, T0 + i + 1],

(8.223)

I f (T0 + i, T0 + i + 1, yi , vi ) ≤ lim I f (T0 + i, T0 + i + 1, xkp , ukp ).

(8.224)

p→∞

In view of (8.223), there exists (x∗ , u∗ ) ∈ X(T0 , ∞) such that for each integer i ≥ 0, (x∗ (t), u∗ (t)) = (yi (t), vi (t)), t ∈ [T0 + i, T0 + i + 1].

(8.225)

It follows from (8.222), (8.224), and (8.225) that for every integer q ≥ 1, I f (T0 , T0 + q, x∗ , u∗ ) ≤ lim I f (T0 , T0 + q, xkp , ukp ) p→∞

≤ I f (T0 , T0 + q, xf , uf ) + 2S0 + L0 (1 + a0 ).

(8.226)

Theorem 8.5, (8.219), (8.223), (8.225), and (8.226) imply that (x∗ , u∗ ) is (f )-good and x∗ (T0 ) = z0 .

(8.227)

In order to complete the proof of the proposition, it is sufficient to show that (x∗ , u∗ ) is (f )-minimal. Assume the contrary. Then there exist Δ > 0, an integer τ0 ≥ 1, and (y, v) ∈ X(T0 , T0 + τ0 ) such that y(T0 ) = x∗ (T0 ), y(T0 + τ0 ) = x∗ (T0 + τ0 ),

(8.228)

I f (T0 , T0 + τ0 , x∗ , u∗ ) > I f (T0 , T0 + τ0 , y, v) + 2Δ.

(8.229)

By (A2) and Lemma 8.37, there exist L1 , δ > 0 such that the following properties hold: (i) For each (T , ξ ) ∈ A satisfying ρE (ξ, xf (T )) ≤ δ, there exist τ1 ∈ (0, bf ] and (x˜1 , u˜ 1 ) ∈ X(T , T + τ1 ) such that

8.17 Auxiliary Results for Theorem 8.29

267

x˜1 (T ) = ξ , x˜1 (T + τ1 ) = xf (T + τ1 ), I f (T , T + τ1 , x˜1 , u˜ 1 ) ≤ I f (T , T + τ1 , xf , uf ) + Δ/8, and if T ≥ bf , then there exist τ2 ∈ (0, bf ] and (x˜2 , u˜ 2 ) ∈ X(T − τ2 , T ) such that x˜2 (T − τ2 ) = xf (T − τ2 ) , x˜2 (T ) = ξ, I f (T − τ2 , T , x˜2 , u˜ 2 ) ≤ I f (T − τ2 , T , xf , uf ) + Δ/8. (ii) If T2 > T1 ≥ L1 , (x, u) ∈ X(T1 , T2 ), ρE (x(Ti ), xf (Ti )) ≤ δ, i = 1, 2, we have I f (T1 , T2 , x, u) ≥ I f (T1 , T2 , xf , uf ) − Δ/8. Theorems 8.20 and 8.25 and (8.217), (8.219), and (8.220) imply that there exists an integer L2 > L0 + L1 such that for each integer k ≥ k0 + 2L2 , ρE (xk (t), xf (t)) ≤ δ, t ∈ [T0 + L2 , T0 + k − L2 ].

(8.230)

In view of (8.223), (8.225), and (8.230), ρE (x∗ (t), xf (t)) ≤ δ for all t ≥ T0 + L2 .

(8.231)

By (8.223)–(8.225), there exists a natural number p0 such that kp0 > k0 + 2L1 + 2L2 + 2τ0 + 2 + 2bf + 2T0 ,

(8.232)

I f (T0 , T0 + τ0 + L2 , x∗ , u∗ ) ≤ I f (T0 , T0 + τ0 + L2 , xkp0 , ukp0 ) + Δ/2.

(8.233)

Property (i) and (8.228), (8.230)–(8.234) imply that there is (x, u) ∈ X(T0 , T0 +kp0 ) such that (x(t), u(t)) = (y(t), v(t)), t ∈ [T0 , T0 + τ0 ],

(8.234)

(x(t), u(t)) = (x∗ (t), u∗ (t)), t ∈ (T0 + τ0 , T0 + τ0 + L2 ],

(8.235)

x(T0 + τ0 + L2 + bf ) = xf (T0 + τ0 + L2 + bf ),

(8.236)

I f (T0 + τ0 + L2 , T0 + τ0 + L2 + bf , x, u) ≤ I f (T0 + τ0 + L2 , T0 + τ0 + L2 + bf , xf , uf ) + Δ/8,

(8.237)

(x(t), u(t)) = (xkp0 (t), ukp0 (t)), t ∈ [T0 + τ0 + L2 + 2bf , kp0 + T0 ],

(8.238)

268

8 Continuous-Time Nonautonomous Problems on Half-Axis

I f (T0 + τ0 + L2 + bf , T0 + τ0 + L2 + 2bf , x, u) ≤ I f (T0 + τ0 + L2 + bf , T0 + τ0 + L2 + 2bf , xf , uf ) + Δ/8.

(8.239)

It follows from (8.219), (8.220), (8.227), (8.228), and (8.233) that I f (T0 , T0 + kp0 , x, u) ≥ I f (T0 , T0 + kp0 , xkp0 , ukp0 ).

(8.240)

By (8.229), (8.230), (8.232), (8.236)–(8.240), and property (ii), 0 ≤ I f (T0 , T0 + kp0 , x, u) − I f (T0 , T0 + kp0 , xkp0 , ukp0 ) = I f (T0 , T0 + τ0 , y, v) + I f (T0 + τ0 , T0 + τ0 + L2 , x∗ , u∗ ) + I f (T0 + τ0 + L2 , T0 + τ0 + L2 + bf , xf , uf ) + Δ/8 + I f (T0 + τ0 + L2 + bf , T0 + τ0 + L2 + 2bf , xf , uf ) + Δ/8 − I f (T0 , T0 + τ0 + L2 , xkp0 , ukp0 ) − I f (T0 + τ0 + L2 , T0 + τ0 + L2 + bf , xkp0 , ukp0 ) − I f (T0 + τ0 + L2 + bf , T0 + τ0 + L2 + 2bf , xkp0 , ukp0 ) ≤ I f (T0 , T0 + τ0 , y, v) + I f (T0 + τ0 , T0 + τ0 + L2 , x∗ , u∗ ) + I f (T0 + τ0 + L2 , T0 + τ0 + L2 + bf , xf , uf ) + Δ/8 + I f (T0 + τ0 + L2 + bf , T0 + τ0 + L2 + 2bf , xf , uf ) + Δ/8 − I f (T0 , T0 + τ0 + L2 , xkp0 , ukp0 ) − I f (T0 + τ0 + L2 , T0 + τ0 + L2 + 2bf , xkp0 , ukp0 ) + Δ/8 = I f (T0 , T0 + τ0 , y, v) + I f (T0 + τ0 , T0 + τ0 + L2 , x∗ , u∗ ) − I f (T0 , T0 + τ0 + L2 , xkp0 , ukp0 ) + Δ/2 < I f (T0 , T0 + τ0 , x∗ , u∗ ) − 2Δ + I f (T0 + τ0 , T0 + τ0 + L2 , x∗ , u∗ ) − I f (T0 , T0 + τ0 + L2 , xkp0 , ukp0 ) + Δ/2 ≤ −2Δ + Δ/2 + Δ/2, a contradiction. The contradiction we have reached completes the proof of Proposition 8.38.

8.18 Proof of Theorem 8.29

269

8.18 Proof of Theorem 8.29 Clearly, (i) implies (ii). In view of Assertion 2 of Theorem 8.5, (ii) implies (iii). By (P1), (iii) implies (iv). Evidently, (iv) implies (v). We show that (v) implies (iii). Assume that (x∗ , u∗ ) is (f )-minimal and satisfies lim inf ρE (x∗ (t), xf (t)) = 0. t→∞

(8.241)

Since (x, ˜ u) ˜ is (f )-good, there exists S0 > 0 such that for all numbers T > S, ˜ u) ˜ − I f (S, T , xf , uf )| ≤ S0 . |I f (S, T , x,

(8.242)

By (A2), there exists δ > 0 such that the following property holds: (a) For each (T , z) ∈ A satisfying ρE (z, xf (T )) ≤ δ, there exist τ1 ∈ (0, bf ] and (x1 , u1 ) ∈ X(T , T + τ1 ) that satisfy x1 (T ) = z, x1 (T + τ1 ) = xf (T + τ1 ), I f (T , T + τ1 , x1 , u1 ) ≤ I f (T , T + τ1 , xf , uf ) + 1, and if T ≥ bf , then there exist τ2 ∈ (0, bf ] and (x2 , u2 ) ∈ X(T − τ2 , T ) such that x2 (T − τ2 ) = xf (T − τ2 ), x2 (T ) = z, I f (T − τ2 , T , x2 , u2 ) ≤ I f (T − τ2 , T , xf , uf ) + 1. In view of (8.241) and (P1), there exists an increasing sequence {tk }∞ k=1 ⊂ (S, ∞) such that lim tk = ∞,

k→∞

ρE (x∗ (tk + 2bf ), xf (tk + 2bf )) ≤ δ for all integers k ≥ 1, ˜ xf (t)) ≤ δ for all t ≥ t0 . ρE (x(t),

(8.243) (8.244) (8.245)

Let k ≥ 1 be an integer. By property (a) and (8.244) and (8.245), there exists (y, v) ∈ X(S, tk + 2bf ) such that u(t)), ˜ t ∈ [S, tk ], (y(t), v(t)) = (x(t), ˜ y(tk + bf ) = xf (tk + bf ),

270

8 Continuous-Time Nonautonomous Problems on Half-Axis

I f (tk , tk + bf , y, v) ≤ I f (tk , tk + bf , xf , uf ) + 1, y(tk + 2bf ) = x∗ (tk + 2bf ), I f (tk + bf , tk + 2bf , y, v) ≤ I f (tk + bf , tk + 2bf , xf , uf ) + 1. The relations above and (8.242) imply that I f (S, tk + 2bf , x∗ , u∗ ) ≤ I f (S, tk + 2bf , y, v) ≤ I f (S, tk , x, ˜ u) ˜ + I f (tk , tk + 2bf , xf , uf ) + 2 ≤ I f (S, tk + 2bf , xf , uf ) + 2 + 2S0 . Together with Theorem 8.5, this implies that (x∗ , u∗ ) is (f )-good and (iii) holds. We show that (iii) implies (i). Assume that (x∗ , u∗ ) is (f )-minimal and (f )-good. (P1) implies that lim ρE (x∗ (t), xf (t)) = 0.

t→∞

(8.246)

Since (x∗ , u∗ ) is (f )-good, there exists S0 > 0 such that |I f (S, T , x∗ , u∗ ) − I f (S, T , xf , uf )| ≤ S0 for all T > S.

(8.247)

Let (x, u) ∈ X(S, ∞) satisfy x(S) = x∗ (S).

(8.248)

We show that lim sup[I f (S, T , x∗ , u∗ ) − I f (S, T , x, u)] ≤ 0. T →∞

In view of Theorem 8.5 and (8.247), we may assume that (x, u) is (f )-good. (P1) implies that lim ρE (x(t), xf (t)) = 0.

t→∞

(8.249)

Let  > 0. By (A2) and Lemma 8.37, there exist δ ∈ (0, ) and L1 > 0 such that the following properties hold: (b) For each (T , z) ∈ A satisfying ρE (z, xf (T )) ≤ δ, there exist τ1 ∈ (0, bf ] and (x1 , u1 ) ∈ X(T , T + τ1 ) satisfying x1 (T ) = z , x1 (T + τ1 ) = xf (T + τ1 ),

8.18 Proof of Theorem 8.29

271

I f (T , T + τ1 , x1 , u1 ) ≤ I f (T , T + τ1 , xf , uf ) + /8, and if T ≥ bf , then there exist τ2 ∈ (0, bf ] and (x2 , u2 ) ∈ X(T − τ2 , T ) satisfying x2 (T − τ2 ) = xf (T − τ2 ) , x2 (T ) = z, I f (T − τ2 , T , x2 , u2 ) ≤ I f (T − τ2 , T , xf , uf ) + /8. (c) For each T1 ≥ L1 , each T2 > T1 , and each (y, v) ∈ X(T1 , T2 ) that satisfy ρE (y(Ti ), xf (Ti )) ≤ δ, i = 1, 2, we have I f (T1 , T2 , y, v) ≥ I f (T1 , T2 , xf , uf ) − /8. It follows from (8.246) and (8.249) that there exists τ0 > 0 such that ρE (x(t), xf (t)) ≤ δ, ρE (x∗ (t), xf (t)) ≤ δ for all t ≥ τ0 .

(8.250)

Let T ≥ τ0 + L1 .

(8.251)

Property (b) and (8.250) and (8.251) imply that there exists (y, v) ∈ X(S, T +2bf ), which satisfies (y(t), v(t)) = (x(t), u(t)), t ∈ [S, T ], y(T + bf ) = xf (T + bf ),

(8.252)

I f (T , T + bf , y, v) ≤ I f (T , T + bf , xf , uf ) + /8,

(8.253)

y(T + 2bf ) = x∗ (T + 2bf ), I f (T + bf , T + 2bf , y, v) ≤ I f (T + bf , T + 2bf , xf , uf ) + /8.

(8.254) (8.255)

By property (c) and (8.250) and (8.251), I f (T , T + 2bf , x∗ , u∗ ) ≥ I f (T , T + 2bf , xf , uf ) − /8. It follows from (8.248), (8.252)–(8.256) that I f (S, T , x∗ , u∗ ) + I f (T , T + 2bf , xf , uf ) − /8 ≤ I f (S, T + 2bf , x∗ , u∗ ) ≤ I f (S, T + 2bf , y, v)

(8.256)

272

8 Continuous-Time Nonautonomous Problems on Half-Axis

= I f (S, T , x, u) + I f (T , T + 2bf , xf , uf ) + /4, I f (S, T , x∗ , u∗ ) ≤ I f (S, T , x, u) +  for all T ≥ τ0 + L1 . Since  is any positive number, we conclude that lim sup[I f (S, T , x∗ , u∗ ) − I f (S, T , x, u)] ≤ 0, T →∞

(x∗ , u∗ ) is (f )-overtaking optimal, and (i) holds. Assertion 2 of Theorem 8.29 is proved. Assertion 2 of Theorem 8.29 and Proposition 8.38 imply Assertion 1 of Theorem 8.29.

8.19 Proof of Theorem 8.33 In view of Theorem 8.27, in order to show that (P1), (P2) hold, it is sufficient to show that fr has (P3). Let , M > 0. Theorem 8.5 implies that there exists S0 > 0 such that for each T2 > T1 ≥ 0 and each (x, u) ∈ X(T1 , T2 ), I f (T1 , T2 , x, u) + S0 ≥ I f (T1 , T2 , xf , uf ).

(8.257)

Property (i) of Sect. 8.9 implies there exists δ ∈ (0, ) such that if y, z ∈ E satisfies φ(y, z) ≤ δ, then ρE (y, z) ≤ .

(8.258)

Set L = δ −1 r −1 (S0 + M).

(8.259)

Assume that T1 ≥ 0, T2 ≥ T1 + L, (x, u) ∈ X(T1 , T2 ) satisfies I fr (T1 , T2 , x, u) ≤ I fr (T1 , T2 , xf , uf ) + M. By (8.257), (8.258), the equality φ(0, 0) = 0, and the definition of fr , I f (T1 , T2 , xf , uf ) + M = M + I fr (T1 , T2 , xf , uf )  ≥ I (T1 , T2 , x, u) + r f

T2

φ(x(t), xf (t))dt T1

 ≥ I (T1 , T2 , xf , uf ) − S0 + r f

T2

φ(x(t), xf (t))dt, T1

(8.260)

8.19 Proof of Theorem 8.33

273

r −1 (S0 + M) ≥



T2

φ(x(t), xf (t))dt T1

≥ δmes({t ∈ [T1 , T2 ] : φ(x(t), xf (t)) ≥ δ}) ≥ δmes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) and mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) ≤ δ −1 r −1 (S0 + M) ≤ L. Thus fr has (P3) and (P1), (P2) too. Assume that (xf , uf ) ∈ X(0, ∞) is (f )-minimal. It is not difficult to see that (xf , uf ) is (fr )-minimal too. Let (x, u) ∈ X(0, ∞) be (fr )-good, x(0) = xf (0).

(8.261)

lim ρE (x(t), xf (t)) = 0.

(8.262)

Therefore, t→∞

Let  > 0. By (A4), there exists δ > 0 such that the following property holds: (i) For each (T , z) ∈ A satisfying ρE (z, xf (T )) ≤ δ, there exist τ ∈ (0, bf ], (y, v) ∈ X(T , T + τ ) such that y(T ) = z, y(T + τ ) = xf (T + τ ), I fr (T , T + τ, y, v) ≤ I fr (T , T + τ, xf , uf ) + /8. In view of (8.262), there exists T0 > 0 such that ρE (x(t), xf (t)) ≤ δ for all t ≥ T0 .

(8.263)

Let T ≥ T0 . Property (i) and (8.263) imply that there exist τ ∈ (0, bf ], (y, v) ∈ X(T , T + τ ) such that y(T ) = x(T ), y(T + τ ) = xf (T + τ ), I fr (T , T + τ, y, v) ≤ I f (T , T + τ, xf , uf ) + /8.

(8.264)

Set y(t) = x(t), v(t) = u(t), t ∈ [0, T ].

(8.265)

274

8 Continuous-Time Nonautonomous Problems on Half-Axis

Clearly, (y, v) ∈ X(0, T + τ ). Since (xf , uf ) is (f )-minimal, it follows from (8.261), (8.264), and (8.265) that I f (0, T + τ, xf , uf ) ≤ I f (0, T + τ, y, v).

(8.266)

By the definition of fr and (8.264)–(8.266), I fr (0, T , xf , uf ) ≤ I f (0, T , y, v) + I f (T , T + τ, y, v) − I f (T , T + τ, xf , uf ) = I f (0, T , x, u) + I fr (T , T + τ, y, v)  −r

T +τ

φ(y(t), xf (t))dt − I f (T , T + τ, xf , uf )

T



T +τ

≤ I f (0, T , x, u) − r

φ(y(t), xf (t))dt + /8

T

 = I (0, T , x, u) − r fr

T



T +τ

φ(x(t), xf (t))dt − r

0

φ(y(t), xf (t))dt + /8.

T

Since the relation above holds for any T ≥ T0 , we obtain  lim sup(I fr (0, T , xf , uf )−I fr (0, T , x, u)) ≤ −r lim

T →∞ 0

T →∞

T

φ(x(t), xf (t))dt+/8.

Since  is any positive number, we have  lim sup(I (0, T , xf , uf ) − I (0, T , x, u)) ≤ −r lim fr

f

T →∞

T →∞ 0

T

φ(x(t), xf (t))dt.

(8.267) This implies that (xf , uf ) ∈ X(0, ∞) is (fr )-overtaking optimal. Assume that (x, u) is (fr )-overtaking optimal. Then (8.267) implies that x(t) = xf (t) for all t ≥ 0. Theorem 8.33 is proved.

8.20 The Strong Turnpike Property We showed (see Proposition 8.3) that (A0)’ holds for the third class of problems. By Lemma 6.44 of [148], (A0)’ holds for the first class of problems with A(t) = 0, t ≥ 0. Here we consider the first class of problems with A(t) = 0, t ≥ 0, and the third class of problems and study them simultaneously. We prove the following result.

8.21 Auxiliary Results for Theorem 8.39

275

Theorem 8.39 Let f have LSC property and (xf , uf ) be (f )-overtaking optimal. Then f has STP if and only if (P1) and (P2) hold and the following property holds: (a) For each (f )-overtaking optimal pair of sequences (y, v) ∈ X(0, ∞) satisfying y(0) = xf (0), the equality y(t) = xf (t) holds for all t ≥ 0. The next result follows from (A2), (P1), and STP. Theorem 8.40 Assume that f has STP and  > 0. Then there exists δ > 0 such that for every T1 ≥ 0 and every (f )-overtaking optimal pair (x, u) ∈ X(T1 , ∞) satisfying ρE (x(T1 ), xf (T1 )) ≤ δ, the inequality ρE (x(t), xf (t)) ≤  holds for all t ≥ T1 .

8.21 Auxiliary Results for Theorem 8.39 For the first class, we assume that θ0 = 0 and ρE (x, y) = x − y, x, y ∈ E. For the third class, we assume that G(t, x, u) = G(t, x, u), (t, x, u) ∈ M. Assumption (A0)’ implies the following result. Lemma 8.41 Let M1 > 0, 0 < τ0 < τ1 . Then there exists M2 > 0 such that for each T1 ≥ 0, each T2 ∈ (T1 + τ0 , T1 + τ1 ], and each (x, u) ∈ X(T1 , T2 ) satisfying I f (T1 , T2 , x, u) ≤ M1 the following inequality holds: ρE (x(t), θ0 ) ≤ M2 for all t ∈ [T1 , T2 ]. Lemma 8.42 Let M1 > 0,  ∈ (0, 1), 0 < τ0 < τ1 . Then there exists δ > 0 such that for each T1 ≥ 0, each T2 ∈ [T1 + τ0 , T1 + τ1 ], each (x, u) ∈ X(T1 , T2 ) satisfying I f (T1 , T2 , x, u) ≤ M0 and each t1 , t2 ∈ [T1 , T2 ] satisfying |t1 − t2 | ≤ δ the inequality ρE (x(t1 ), x(t2 )) ≤  holds. Proof Lemma 8.3 implies that there exists M2 > 0 such that the following property holds: (i) For each T1 ≥ 0, each T2 ∈ [T1 + τ0 , T1 + τ1 ], and each (x, u) ∈ X(T1 , T2 ) satisfying I f (T1 , T2 , x, u) ≤ M1 , we have

276

8 Continuous-Time Nonautonomous Problems on Half-Axis

ρE (x(t), θ0 ) ≤ M2 for all t ∈ [T1 , T2 ]. In view of (8.3), (8.5), and (8.56), for the first class of problems, there exists h0 > 0 such that for the first class of problems f (t, x, u) ≥ 4 −1 (M1 + a0 τ1 + 8)(G(t, x, u) − a0 x)+

(8.268)

for each (t, x, u) ∈ M satisfying G(t, x, u) − a0 x ≥ h0 and that for the third class of problems f (t, x, u) ≥ 4 −1 (M1 + a0 τ1 + 8)(G(t, x, u) − a0 ρE (x, θ0 )+

(8.269)

for each (t, x, u) ∈ M satisfying G(t, x, u) − a0 ρE (x, θ0 ) ≥ h0 . Choose a number δ ∈ (0, (4a0 M2 + 4h0 + 4)−1 ).

(8.270)

Let T1 ≥ 0, T2 ∈ [T1 + τ0 , T1 + τ1 ], (x, u) ∈ X(T1 , T2 ), I f (T1 , T2 , x, u) ≤ M1 ,

(8.271)

t1 , t2 ∈ [T1 , T2 ], 0 < t2 − t2 ≤ δ.

(8.272)

Property (i) and (8.271) imply that ρE ((x(t), θ0 ) ≤ M2 for all t ∈ [T1 , T2 ].

(8.273)

Define Ω1 = {t ∈ [t1 , t2 ] : G(t, x(t), u(t)) − a0 ρE (x(t), θ0 ) ≥ h0 }, Ω2 = [t1 , t2 ] \ Ω1 .

(8.274) (8.275)

By (8.18) and (8.268)–(8.275),  ρE (x(t2 ), x(t1 )) ≤

t2

G(s, x(s), u(s))ds

t1

 ≤ a0

t2

t1

 ρE (x(t), θ0 )dt +

t2

t1

(G(t, x(t), u(t)) − a0 ρE (x(t), θ0 ))+ dt

8.21 Auxiliary Results for Theorem 8.39

277

 ≤ a0 δM2 + δh0 + Ω1

(G(t, x(t), u(t)) − a0 ρE (x(t), θ0 ))+ dt −1



≤ a0 δM2 + δh0 + (4(M1 + a0 τ1 + 8))

f (t, x(t), u(t))dt Ω1

≤ a0 δM2 + δh0 + 4−1  < . Lemma 8.42 is proved. Lemma 8.43 Let Δ > 0. Then there exists δ (T1 , z1 ), (T2 , z2 ) ∈ A satisfying

> 0 such that for each

T2 ≥ T1 + 2bf , ρE (zi , xf (Ti )) ≤ δ, i = 1, 2,

(8.276)

the following inequality holds: U f (T1 , T2 , z1 , z2 ) ≤ I f (T1 , T2 , xf , uf ) + Δ. Proof Let δ > 0 be as guaranteed by (A2) with  = Δ/4. Let (T1 , z1 ), (T2 , z2 ) ∈ A satisfy (8.276). By (8.276), the choice of δ, and (A2) with  = Δ/4, there exists (y, v) ∈ X(T1 , T2 ) such that y(T1 ) = z1 , y(T2 ) = z2 , y(t) = xf (t), v(t) = uf (t), t ∈ [T1 + bf , T2 − bf ], I f (T1 , T1 + bf , y, v) ≤ I f (T1 , T1 + bf , xf , uf ) + Δ/4, I f (T2 − bf , T2 , y, v) ≤ I f (T2 − bf , T2 , xf , uf ) + Δ/4. By the relations above, U f (T1 , T2 , z1 , z2 ) ≤ I f (T1 , T2 , y, v) ≤ I f (T1 , T2 , xf , uf ) + Δ/2. Lemma 8.43 is proved. Lemma 8.44 Let Δ ∈ (0, 1). Then there exists δ > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + 3bf , each (x, u) ∈ X(T1 , T2 ) satisfying ρE (x(Ti ), xf (Ti )) ≤ δ, i = 1, 2,

278

8 Continuous-Time Nonautonomous Problems on Half-Axis

I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ, and each τ1 ∈ [T1 , T1 + δ] and each τ2 ∈ [T2 − δ, T2 ], the inequality I f (τ1 , τ2 , x, u) ≤ I f (τ1 , τ2 , xf , uf ) + Δ holds. Proof Lemma 8.43 implies that there exists δ0 ∈ (0, min{4−1 Δ, 2−1 bf }) such that the following property holds: (ii) For each (T1 , z1 ), (T2 , z2 ) ∈ A satisfying T2 ≥ T1 + 2bf , ρE (zi , xf (Ti )) ≤ δ0 , i = 1, 2, we have U f (T1 , T2 , z1 , z2 ) ≤ I f (T1 , T2 , xf , uf ) + Δ/4. Theorem 8.5 implies that there exists M0 > 0 such that the following property holds: (iii) For each pair of numbers T2 > T1 ≥ 0 and each (x, u) ∈ X(T1 , T2 ), I f (T1 , T2 , x, u) + M0 ≥ I f (T1 , T2 , xf , uf ). Lemma 8.42 implies that there exists δ1 ∈ (0, δ0 ) such that the following property holds: (iv) For each S ≥ 0, each (y, v) ∈ X(S, S + bf ) satisfying I f (S, S + bf , y, v) ≤ |Δf |(bf + 2) + 3a0 + 2M0 + 1, and each t1 , t2 ∈ [S, S + bf ] satisfying |t1 − t2 | ≤ δ1 , we have ρE (y(t1 ), y(t2 )) ≤ δ0 /8. Set δ = δ1 /8.

(8.277)

T1 ≥ 0, T2 ≥ T1 + 3bf ,

(8.278)

Assume that

8.21 Auxiliary Results for Theorem 8.39

279

(x, u) ∈ X(T1 , T2 ), ρE (x(Ti ), xf (Ti )) ≤ δ, i = 1, 2, I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ, τ1 ∈ [T1 , T1 + δ], τ2 ∈ [T2 − δ, T2 ].

(8.279) (8.280) (8.281)

Property (ii) and (8.278)–(8.280) imply that I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + 1. Property (iii) and the relation above imply that for each s1 , s2 ∈ [T1 , T2 ] satisfying s1 < s2 , I f (s1 , s2 , x, u) = I f (T1 , T2 , x, u) − I f (T1 , s1 , x, u) − I f (s2 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + 1 − I f (T1 , s1 , xf , uf ) + M0 − I f (s2 , T2 , xf , uf ) + M0 = I f (T1 , T2 , xf , uf ) + 2M0 + 1.

(8.282)

In view of (8.5) and (8.74), I f (T1 , T1 + bf , xf , uf ) ≤ I f (T1 , T1  + bf  + 2, xf , uf ) + 3a0 ≤ (bf  + 2)Δf + 3a0 ,

(8.283)

I f (T2 − bf , T2 , xf , uf ) ≤ I f (T2  − bf  − 1, T2  + 1, xf , uf ) + 3a0 ≤ (bf  + 2)Δf + 3a0 .

(8.284)

It follows from (8.282)–(8.284) that I f (T1 , T1 + bf , x, u), I f (T2 − bf , T2 , x, u) ≤ Δf (bf  + 2) + 3a0 + 2M0 + 1. (8.285) Property (iv), (8.277), (8.281), and (8.285) imply that ρE (xf (Ti ), xf (τi )) ≤ δ0 /8, i = 1, 2,

(8.286)

ρE (x(Ti ), x(τi )) ≤ δ0 /8, i = 1, 2.

(8.287)

By (8.277), (8.279), (8.286), and (8.287), for i = 1, 2, ρE (xf (τi ), x(τi )) ≤ ρE (xf (τi ), xf (Ti )) + ρE (xf (Ti ), x(Ti )) + ρE (x(Ti ), x(τi ))

280

8 Continuous-Time Nonautonomous Problems on Half-Axis

≤ δ0 /8 + δ0 /8 + δ ≤ δ0 /2.

(8.288)

Property (ii), (8.278), (8.279), (8.281), and (8.288) imply that U f (τ1 , τ2 , x(τ1 ), x(τ2 )) ≤ I f (τ1 , τ2 , xf , uf ) + Δ/4.

(8.289)

By (8.280), (8.281), and (8.289), I f (τ1 , τ2 , x, u) ≤ U f (τ1 , τ2 , x(τ1 ), x(τ2 )) + δ ≤ I f (τ1 , τ2 , xf , uf ) + Δ/4 + δ ≤ I f (τ1 , τ2 , xf , uf ) + Δ. Lemma 8.44 is proved.

8.22 Proof of Theorem 8.39 Here we again consider the first class of problems with A(t) = 0, t = 0, and the third class of problems and study them simultaneously. For the first class, θ0 = 0, ρE (x, y) = x − y, x, y ∈ E, and for the third class, G(t, x, u) = G(t, x, u), (t, x, u) ∈ M. Assume that STP holds. Theorem 8.25 implies that (P1) and (P2) hold. Let (x, ˜ u) ˜ ∈ X(0, ∞) be (f )-overtaking optimal and x(0) ˜ = xf (0).

(8.290)

(P1), STP, and (8.290) imply that x(t) ˜ = xf (t) for all t ≥ 0 and (P4) holds. Assume that (P1), (P2) and (a) of Theorem 8.32 hold. Lemma 8.45 Let  > 0. Then there exists δ > 0 such that for each pair of numbers T1 ≥ 0, T2 ≥ T1 + 3bf and each (x, u) ∈ X(T1 , T2 ) satisfying ρE (x(Ti ), xf (Ti )) ≤ δ, i = 1, 2, I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ, the inequality ρE (x(t), xf (t)) ≤  for all t ∈ [T1 , T2 ]. Proof Lemma 8.36 and (A2) imply that there exist δ0 ∈ (0, min{1, /4}) and L0 > 0 such that the following properties hold: (i) (A2) holds with  = 1 and δ = δ0 . (ii) For each pair of numbers T1 ≥ L0 , T2 ≥ T1 + 2bf and each (x, u) ∈ X(T1 , T2 ) satisfying

8.22 Proof of Theorem 8.39

281

ρE (x(Ti ), xf (Ti )) ≤ δ0 , i = 1, 2 I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + δ0 , we have ρE (x(t), xf (t)) ≤ , t ∈ [T1 , T2 ]. Theorem 8.5 implies that there exists S0 > 0 such that for each T2 > T1 ≥ 0 and each (x, u) ∈ X(T1 , T2 ), I f (T1 , T2 , x, u) + S0 ≥ I f (T1 , T2 , xf , uf ).

(8.291)

It follows from Theorem 8.26 that there exists L1 > 0 such that the following property holds: (iii) For each T ≥ 0 and each (x, u) ∈ X(T , T + L1 ) satisfying I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + 2S0 + 3, we have min{ρE (x(t), xf (t)) : t ∈ [T , T + L1 ]} ≤ δ0 . Consider a sequence {δi }∞ i=1 ⊂ (0, 1) such that δi < 2−1 δi−1 , i = 1, 2, . . . .

(8.292)

Assume that the lemma does not hold. Then for each integer i ≥ 1, there exist Ti,1 ≥ 0, Ti,2 ≥ Ti,1 + 3bf

(8.293)

and (xi , ui ) ∈ X(Ti,1 , Ti,2 ) such that ρE (xi (Ti,1 ), xf (Ti,1 )) ≤ δi , ρE (xi (Ti,2 ), xf (Ti,2 )) ≤ δi ,

(8.294)

I f (Ti,1 , Ti,2 , xi , ui ) ≤ U f (Ti,1 , Ti,2 , xi (Ti,1 ), xi (Ti,2 )) + δi

(8.295)

and ti ∈ [Ti,1 , Ti,2 ] for which ρE (x(ti ), xf (ti )) > .

(8.296)

Let i be a natural number. Property (ii) and (8.292)–(8.296) imply that Ti,1 < L0 .

(8.297)

282

8 Continuous-Time Nonautonomous Problems on Half-Axis

We show that ti ≤ 2bf + 1 + L1 + L0 .

(8.298)

Property (i), (A2) with  = 1, δ = δ0 , (8.292)–(8.294) imply that there exists (yi , vi ) ∈ X(Ti,1 , Ti,2 ) such that yi (Ti,1 ) = xi (Ti,1 ), yi (Ti,2 ) = xi (Ti,2 ),

(8.299)

yi (t) = xf (t), vi (t) = uf (t), t ∈ [Ti,1 + bf , Ti,2 − bf ],

(8.300)

I f (Ti,1 , Ti,1 + bf , yi , vi ) ≤ I f (Ti,1 , Ti,1 + bf , xf , uf ) + 1,

(8.301)

I f (Ti,2 − bf , Ti,2 , yi , vi ) ≤ I f (Ti,2 − bf , Ti,2 , xf , uf ) + 1.

(8.302)

It follows from (8.295) and (8.299)–(8.302) that I f (Ti,1 , Ti,2 , xi , ui ) ≤ 1 + I f (Ti,1 , Ti,2 , yi , vi ) ≤ I f (Ti,1 , Ti,2 , xf , uf ) + 3. (8.303) In view of (8.291), (8.303), for each pair of integers S1 , S2 ∈ [Ti,1 , Ti,2 ] satisfying S1 < S2 , I f (S1 , S2 , xi , ui ) = I f (Ti,1 , Ti,2 , xi , ui ) −I f (Ti,1 , S1 , xi , ui ) −I f (S2 , Ti,2 , xi , ui ) ≤ I f (Ti,1 , Ti,2 , xf , uf ) + 3 − I f (Ti,1 , S1 , xf , uf ) + S0 − I f (S2 , Ti,2 , xf , uf ) + S0 = I f (S1 , S2 , xf , uf ) + 3 + 2S0 .

(8.304)

Assume that (8.298) does not hold. Then ti > 2bf + 1 + L1 + L0 .

(8.305)

Consider the restriction of (xi , ui ) to the interval [ti − 2bf − 1 − L1 , ti − 1 − 2bf ] ⊂ [L0 , ∞).

(8.306)

By property (iii) and (8.304), there exists t˜ ∈ [ti − 2bf − 1 − L1 , ti − 1 − 2bf ]

(8.307)

ρE (xf (t˜), x(t˜)) ≤ δ0 .

(8.308)

such that

8.22 Proof of Theorem 8.39

283

Property (ii), (8.292), (8.294), (8.295), and (8.306)–(8.308) imply that ρE (x(t), xf (t)) ≤ , t ∈ [t˜, Ti,2 ], and in particular, ρE (x(ti ), xf (ti )) ≤ . This contradicts (8.296). The contradiction we have reached proves (8.298). In view of (8.74) and (8.304), there exists M1 > 0 such that the following property holds: (iv) For each integer i ≥ 1 and each S2 > S1 ≥ 0 satisfying S1 , S2 ∈ [Ti,1 , Ti,2 ], S2 − S1 ≤ 2bf , I f (S1 , S2 , xi , ui ) ≤ M1 . Lemma 8.41 and property (iv) imply that there exists M2 > 0 such that ρE (xi (t), θ0 ) ≤ M2 , t ∈ [Ti,1 , Ti,2 ], i = 1, 2, . . . .

(8.309)

In view of (8.207) and (8.298), extracting a subsequence and re-indexing, we may assume without loss of generality that there exists T˜1 = lim Ti,1 ∈ [0, L0 ],

(8.310)

t˜ = lim ti ∈ [T˜1 , L0 + L1 + 2bf + 1],

(8.311)

T˜2 = lim Ti,2 ∈ [t˜, ∞].

(8.312)

i→∞

i→∞

i→∞

There exist a strictly decreasing sequence {T1(k) }∞ k=1 and a strictly increasing (k) ∞ sequence {T2 }k=1 such that (k) (k) T˜1 = lim T1 , T˜2 = lim T2 ,

(8.313)

T1(k) < T2(k) .

(8.314)

k→∞

k→∞

for all integers k ≥ 1,

If T˜2 = ∞, then we may assume that T2(k) = Tk,2 for all integers k ≥ 1.

(8.315)

284

8 Continuous-Time Nonautonomous Problems on Half-Axis

By LSC property and property (iv), extracting a subsequence and re-indexing, we may assume without loss of generality that there exist  x : (T˜1 , T˜2 ) → E, ˜ ˜ ˜ ˜  u : (T1 , T2 ) → F such that for each t ∈ (T1 , T2 ),  x (t) = lim xi (t),

(8.316)

( x,  u) ∈ X(T1(k) , T2(k) ),

(8.317)

i→∞

for each integer k ≥ 1,

I f (T1(1) , T2(1) ,  x,  u) ≤ lim inf I f (T1(1) , T2(1) , xi , ui ), i→∞

(8.318)

and for each integer k ≥ 1, (k+1)

I f (T1

(k)

(k)

(k+1)

, T1 ,  x,  u) ≤ lim inf I f (T1 i→∞

(k+1)

I f (T2 , T2

(k)

(k)

, T1 , xi , ui ), (k+1)

, x,  u) ≤ lim inf I f (T2 , T2 i→∞

, xi , ui ).

(8.319) (8.320)

By (8.309) and (8.316), ρE ( x (t), θ0 ) ≤ M2 , t ∈ (T˜1 , T˜2 ).

(8.321)

xf (T˜1 ) = lim  x (t).

(8.322)

We show that t→T˜1+

Let γ > 0. Lemma 8.42, property (iv), and (8.74) imply that there exists δ > 0 such that the following property holds: (v) For each integer i ≥ 1 and each S1 , S2 ∈ [Ti,1 , Ti,2 ] satisfying |S2 − S1 | ≤ 2δ, we have ρE (xi (S1 ), xi (S2 )) ≤ γ /2. (vi) For each S1 , S2 ∈ [0, ∞) satisfying |S2 − S1 | ≤ 2δ, we have ρE (xf (S1 ), xf (S2 )) ≤ γ /2. Let t ∈ (T˜1 , T˜1 + 2δ).

(8.323)

8.22 Proof of Theorem 8.39

285

In view of property (v), (8.294), (8.313), and (8.323), for all sufficiently large natural numbers k, t ∈ (Tk,1 , Tk,1 + 2δ), ρE (xk (t), xk (Tk,1 )) ≤ γ /2, ρE (xk (t), xf (Tk,1 )) ≤ ρE (xk (t), xk (Tk,1 )) + ρE (xk (Tk,1 ), xf (Tk,1 )) ≤ γ /2 + δk . (8.324) Property (vi) and (8.313) imply that for all sufficiently large natural numbers k, ρE (xf (Tk,1 ), xf (T˜1 )) ≤ γ /2.

(8.325)

It follows from (8.292), (8.316), (8.324), (8.325) that for all t satisfying (8.323), x (t), xf (T˜1 )) = lim ρE (xk (t), xf (T˜1 )) ρE ( k→∞

≤ lim sup ρE (xk (t), xf (Tk,1 )) + ρE (xf (Tk,1 ), xf (T˜1 )) ≤ γ . k→∞

Since γ is an arbitrary positive number, we conclude that x (t), xf (T˜1 )) = 0 lim ρE (

t→T˜1+

and (8.322) holds. Analogously, we can show that if T˜2 < ∞, then x (t). xf (T˜2 ) = lim 

(8.326)

u(T˜1 ) = uf (T˜1 ).  x (T˜1 ) = xf (T˜1 ), 

(8.327)

u(T˜2 ) = uf (T˜2 ).  x (T˜2 ) = xf (T˜2 ), 

(8.328)

x (t˜), xf (t˜)) ≥ . t˜ > T˜1 , ρE (

(8.329)

t→T˜2−

Set

If T˜2 < ∞, then we set

We show that

Let γ ∈ (0, /4), and let δ > 0 be such that properties (v) and (vi) hold. Let k ≥ 1 be an integer. If tk − Tk,1 ≤ δ, then in view of properties (v), (vi) and the choice of δ,

286

8 Continuous-Time Nonautonomous Problems on Half-Axis

ρE (xf (tk ), xf (Tk,1 )) ≤ γ /2, ρE (xk (tk ), xk (Tk,1 )) ≤ γ /2, and together with (8.294) and (8.296), these relations imply that  < ρE (xf (tk ), xk (tk )) ≤ ρE (xf (tk ), xf (Tk,1 )) + ρE (xf (Tk,1 ), xk (Tk,1 )) + ρE (xk (Tk,1 ), xk (tk )) ≤ γ /2 + δk + γ /2 ≤ γ + /4 < , a contradiction. The contradiction we have reached proves that tk − Tk,1 > δ.

(8.330)

If Tk,2 − tk ≤ δ, then in view of properties (v), (vi) and the choice of δ, ρE (xf (tk ), xf (Tk,2 )) ≤ γ /2, ρE (xk (tk ), xk (Tk,2 )) ≤ γ /2, and together with (8.294) and (8.296), these relations imply that  < ρE (xf (tk ), xk (tk )) ≤ ρE (xf (tk ), xf (Tk,2 )) + ρE (xf (Tk,2 ), xk (Tk,2 )) + ρE (xk (Tk,2 ), xk (tk )) ≤ γ /2 + δk + γ /2 ≤ γ + /4 < , a contradiction. The contradiction we have reached proves that Tk,2 − tk ≥ δ.

(8.331)

By (8.330) and (8.331), for all natural numbers k, Tk,2 − tk ≥ δ, tk − Tk,1 ≥ δ.

(8.332)

It follows from (8.310), (8.312), and (8.332) that T˜2 − t˜ ≥ δ, t˜ − T˜1 ≥ δ.

(8.333)

Properties (v) and (vi), (8.298), (8.311), and (8.316) imply that ρE ( x (t˜), xf (t˜)) = lim ρE (xk (t˜), xf (t˜)) = lim ρE (xk (tk ), xf (tk )) ≥  k→∞

k→∞

and (8.329) holds. In view of (8.317), for every τ ∈ (T˜1 , T˜2 ), the function G(t,  x (t),  u(t)), t ∈ [T˜1 , τ ] is strongly measurable for the first class of problems and just measurable for the third class of problems. Let k0 be a natural number and τ = T2(k0 ) .

(8.334)

8.22 Proof of Theorem 8.39

287

We show that the function G(t,  x (t),  u(t)), t ∈ [T˜1 , τ ], is (Bochner) integrable. By (8.5), (8.20), and (8.56), there exists γ0 > 0 such that f (t, x, u) ≥ 8(G(t, x, u) − a0 ρE (x, θ0 ))+ for all (t, x, u) ∈ M satisfying G(t, x, u) − a0 ρE (x, θ0 ) ≥ γ0 .

(8.335)

Let k ≥ 1 be an integer and T1(k) < τ . By (8.309), 



τ (k)

G(t,  x (t),  u(t))dt ≤

T1

τ (k)

a0 ρE ( x (t), θ0 )dt

T1



τ

+

(k)

T1

(G(t,  x (t),  u(t)) − a0 ρE ( x (t), θ0 ))+ dt



(k)

≤ a0 M2 (τ − T1 ) +

τ (k)

T1

(G(t,  x (t),  u(t)) − a0 ρE ( x (t), θ0 ))+ dt.

(8.336)

Set (k)

E1 = {t ∈ [T1 , τ ] : G(t,  x (t),  u(t)) ≥ a0 ρE ( x (t), θ0 ) + γ0 }, (k)

E2 = [T1 , τ ] \ E1 .

(8.338)

It follows from (8.5), (8.304), (8.318), (8.320), (8.334)–(8.338) that 

τ

(k)

(k)

T1

G(t,  x (t),  u(t))dt ≤ a0 M2 (τ − T1 )

 + E1

(G(t,  x (t),  u(t)) − a0 ρE ( x (t), θ0 ))+ dt

 + E2

(G(t,  x (t),  u(t)) − a0 ρE ( x (t), θ0 ))+ dt (k)

(k)

≤ a0 M2 (τ − T1 ) + γ0 (τ − T1 )  + E1

≤ a0 M2 (τ

(G(t,  x (t),  u(t)) − a0 ρE ( x (t), θ0 ))+ dt

− T1(k) ) + γ0 (τ

− T1(k) ) +

 f (t,  x (t),  u(t))dt E1

(8.337)

288

8 Continuous-Time Nonautonomous Problems on Half-Axis (k)

(k)

(k)

(k)

≤ a0 M2 (τ − T1 ) + γ0 (τ − T1 ) + I f (T1 , τ,  x,  u) + a0 (τ − T1 ) (k0 )

≤ (a0 M2 + γ0 + a0 )(T2 (k0 )

(k0 )

(k)

(k)

≤ (T2 ≤ (T2

(k)

(k0 )

− T1 ) + I f (T1 , T2 (k)

(k0 )

− T1 )(a0 M2 + γ0 + a0 ) + lim inf I f (T1 , T2 i→∞

(k)

(k)

(k0 )

− T1 )(a0 M2 + γ0 + a0 ) + I f (T1 , T2

, x,  u) , xi , ui )

, xf , uf ) + 3 + 2S0 .

(8.339)

By (8.313) and (8.339),  lim

(k0 )

T2

k→∞ T (k) 1 (k0 )

≤ (T2

G(t,  x (t),  u(t))dt

(k ) − T˜1 )(a0 M2 + γ0 + a0 ) + I f (T˜1 , T2 0 , xf , uf ) + 3 + 2S0 .

Fatou’s lemma and (8.313) imply that 

(k0 )

T2

T˜1

G(t,  x (t),  u(t))dt < ∞,

G(t,  x (t),  u(t)), t ∈ [T˜1 , T2(k0 ) ] is Bochner integrable for all integers k0 ≥ 1, and 

(k0 )

T2

(k ) G(t,  x (t),  u(t))dt ≤ I f (T˜1 , T2 0 , xf , uf ) + 3 + 2S0

T˜1

+ (T2(k0 ) − T˜1 )(a0 M2 + γ0 + a0 ).

(8.340)

If T˜2 = ∞, then by (8.340), G(t,  x (·),  u(·)) is (Bochner) integrable on [T˜1 , τ ] for any τ > T˜1 .

(8.341)

If T˜2 < ∞, then by (8.313) and (8.340), G(·,  x (·),  u(·)) is strongly measurable on [T˜1 , T˜2 ] and  lim

(k)

T2

k→∞ T˜1

G(t,  x (t),  u(t))dt ≤ I f (T˜1 , T˜2 , xf , uf ) + 3 + 2S0 + (T˜2 − T˜1 )(a0 M2 + γ0 + a0 ),

and in view of Fatou’s lemma

8.22 Proof of Theorem 8.39

289

G(·,  x (·),  u(·)) is integrable on [T˜1 , T˜2 ].

(8.342)

By (8.55), (8.322), (8.326), and (8.342), ( x,  u) ∈ X(T˜1 , T˜2 ).

(8.343)

Assume that T˜2 = ∞. By (8.55), (8.322), and (8.341), for every τ > T˜1 , ( x,  u) ∈ X(T˜1 , τ ).

(8.344)

Assume that T˜2 < ∞. It follows from (8.327) and (8.328) that x (T˜i ), i = 1, 2, ρE ( x (t˜), xf (t˜)) ≥ . xf (T˜i ) = 

(8.345)

Property (a) and (8.345) imply that x,  u). I f (T˜1 , T˜2 , xf , uf ) < I f (T˜1 , T˜2 , 

(8.346)

Let Δ > 0. Lemma 8.44 implies that there exists δ > 0 such that the following property holds: (vii) For each τ1 ≥ 0, each τ2 ≥ τ1 + 3bf , each (x, u) ∈ X(τ1 , τ2 ) satisfying ρE (x(τi ), xf (τi )) ≤ δ, i = 1, 2, I f (τ1 , τ2 , x, u) ≤ U f (τ1 , τ2 , x(τ1 ), x(τ2 )) + δ, each t1 ∈ [τ1 , τ1 + δ], each t2 ∈ [τ2 − δ, τ2 ], I f (t1 , t2 , x, u) ≤ I f (t1 , t2 , xf , uf ) + Δ/2. In view of (8.292), there exists an integer k0 ≥ 1 such that δk < δ for all integers k ≥ k0 .

(8.347)

Let q ≥ 1 be an integer such that for all integers i ≥ q, (i) (i) |T˜1 − T1 | ≤ δ/2, |T˜2 − T2 | ≤ δ/2.

(8.348)

By (8.310), (8.312), (8.313), and (8.348), there exists an integer k1 ≥ k0 such that for each integer k ≥ k1 , (q)

Tk,1 < T1

(q)

≤ Tk,1 + δ, Tk,2 − δ ≤ T2

< Tk,2 .

(8.349)

290

8 Continuous-Time Nonautonomous Problems on Half-Axis

Property (vii), (8.293)–(8.295), (8.347), and (8.349) imply that for each integer k ≥ k1 , (q)

(q)

(q)

(q)

I f (T1 , T2 , xk , uk ) ≤ I f (T1 , T2 , xf , uf ) + Δ/2.

(8.350)

In view of (8.318)–(8.320) and (8.350), (q)

(q)

(q)

(q)

I f (T1 , T2 ,  x,  u) ≤ lim inf I f (T1 , T2 , xk , uk ) k→∞

(q)

(q)

≤ I f (T1 , T2 , xf , uf ) + Δ/2

(8.351)

for every integer q ≥ 1 such that (8.348) is true for all integers i ≥ q. Fatou’s lemma and (8.5) and (8.313) imply that I f (T˜1 , T˜2 ,  x,  u) ≤ I f (T˜1 , T˜2 , xf , uf ) + Δ/2. Since Δ is any positive number, we conclude that I f (T˜1 , T˜2 ,  x,  u) ≤ I f (T˜1 , T˜2 , xf , uf ). This contradicts (8.346). The contradiction we have reached proves T˜2 = ∞. Recall (see (8.327) and (8.329)) that  x (T˜1 ) = xf (T˜1 ), ρE ( x (t˜), xf (t˜)) ≥ .

(8.352)

Let Δ > 0. Lemma 8.44 implies that there exists δ ∈ (0, bf /4) such that the following property holds: (viii) For each τ1 ≥ 0, each τ2 ≥ τ1 + 3bf , each (x, u) ∈ X(τ1 , τ2 ) satisfying ρE (x(τi ), xf (τi )) ≤ δ, i = 1, 2, I f (τ1 , τ2 , x, u) ≤ U f (τ1 , τ2 , x(τ1 ), x(τ2 )) + δ, each t1 ∈ [τ1 , τ1 + δ], each t2 ∈ [τ2 − δ, τ2 ], I f (t1 , t2 , x, u) ≤ I f (t1 , t2 , xf , uf ) + Δ/2. Recall (see (8.315)) that for all integers k ≥ 1, (k)

T2

= Tk,2 .

In view of (8.292), there exists an integer k0 ≥ 1 such that

(8.353)

8.22 Proof of Theorem 8.39

291

δk < δ for all integers k ≥ k0 .

(8.354)

Property (viii), (8.293)–(8.295), (8.353), and (8.354) imply that the following property holds: (ix) For each integer k ≥ k0 , each τ1 ∈ [Tk,1 , Tk,1 + δ], each τ2 ∈ [Tk,2 − δ, Tk,2 ], I f (τ1 , τ2 , xk , uk ) ≤ I f (τ1 , τ2 , xf , uf ) + Δ/2. By (8.318)–(8.320) and (8.353), for each integer k ≥ 1 and each integer s ≥ k, (s)

I f (T1 , Tk,2 ,  x,  u) (s)

(k)

(s)

(k)

= I f (T1 , T2 ,  x,  u) ≤ lim inf I f (T1 , T2 , xi , ui ) i→∞

(s)

= lim inf I f (T1 , Tk,2 , xi , ui ). i→∞

(8.355)

In view of (8.304) and (8.355), for each integer k ≥ 1 and each integer s ≥ k, x,  u) ≤ I f (T1(s) , Tk,2 , xf , uf ) + 3 + 2S0 . I f (T1(s) , Tk,2 , 

(8.356)

Fatou’s lemma, (8.5), (8.313), and (8.356) imply that for all integers k ≥ 1, x,  u) ≤ I f (T˜1 , Tk,2 , xf , uf ) + 4 + 2S0 . I f (T˜1 , Tk,2 ,  Together with Theorem 8.5 and (8.352), this implies that ( x,  u) is (f )-good. Property (P1) implies that lim ρE ( x (t), xf (t)) = 0.

t→∞

Therefore, there exists τ0 > 1 + T˜1 + 3bf such that x (t), xf (t)) ≤ δ/4 for all t ≥ τ0 . ρE (

(8.357)

By (8.310)–(8.313) and (8.315), there exists an integer q ≥ 1 such that x,  u)| < Δ/8, |I f (T˜1 , T1 , xf , uf )| < Δ/8, Tq,2 > τ0 , |I f (T˜1 , T1 ,  (8.358) for all integers i ≥ q, (q)

(q)

|T˜1 − T1(i) | ≤ δ/2.

(8.359)

292

8 Continuous-Time Nonautonomous Problems on Half-Axis

It follows from (8.310), (8.313), (8.316), and (8.359) that there exists an integer k1 ≥ k0 + q such that for each integer k ≥ k1 , (q)

Tk,1 < T1

≤ Tk,1 + δ, ρE ( x (Tq,2 ), xk (Tq,2 )) ≤ δ/4.

(8.360)

Assume that an integer k ≥ k1 . Then (8.360) holds. By (8.293)–(8.295), (8.315), (8.360) and the choice of k0 , (q)

(q)

(q)

I f (T1 , Tq,2 , xk , uk ) ≤ U f (T1 , Tq,2 , xk (T1 ), xk (Tq,2 )) + δ, ρE (xk (Tk,1 ), xf (Tk,1 )) ≤ δk ≤ δ.

(8.361) (8.362)

In view of (8.357)–(8.360), ρE (xk (Tq,2 ), xf (Tq,2 )) ≤ ρE (xk (Tq,2 ),  x (Tq,2 )) + ρE ( x (Tq,2 ), xf (Tq,2 )) ≤ δ/4 + δ/4.

(8.363) (q)

Property (x) applied with τ1 = Tk,1 , τ2 = Tq,2 , (x, u) = (xk , uk ), t1 = T1 , t2 = Tq,2 , and (8.359)–(8.363) imply that (q)

(q)

I f (T1 , Tq,2 , xk , uk ) ≤ I f (T1 , Tq,2 , xf , uf ) + Δ/2.

(8.364)

In view of (8.364), (q)

(q)

I f (T1 , Tq,2 ,  x,  u) ≤ lim inf I f (T1 , Tq,2 , xk , uk ) k→∞

(q)

≤ I f (T1 , Tq,2 , xf , uf ) + Δ/2.

(8.365)

By (8.365) and (8.358), I f (T˜1 , Tq,2 ,  x,  u) ≤ I f (T˜1 , Tq,2 , xf , uf ) + Δ.

(8.366)

Since the relation above holds for any integer q ≥ 1 satisfying (8.358), we have lim inf(I f (T˜1 , T ,  x,  u) − I f (T˜1 , T , xf , uf )) ≤ Δ. T →∞

Since Δ is any positive number, we conclude that lim inf(I f (T˜1 , T ,  x,  u) − I f (T˜1 , T , xf , uf )) ≤ 0. T →∞

For all t ∈ [0, T˜1 ] \ {T˜1 }, set

(8.367)

8.23 Examples

293

 x (t) = xf (t),  u(t) = uf (t). In view of (8.367), ( x,  u) ∈ X(0, ∞) is (f )-weakly optimal. Theorem 8.29 and (a) imply that ( x,  u) ∈ X(0, ∞) is (f )-overtaking optimal and that  x (t) = xf (t), t ≥ 0. This contradicts (8.345). The contradiction we have reached completes the proof of Lemma 8.45. Completion of Theorem 8.39 By Theorem 8.25, TP holds. Lemma 8.45 and TP imply STP.

8.23 Examples In this section, we present a family of problems that belong to the second class of problems and for which the results of this chapter hold. We use the notation introduced in Sects. 8.1–8.3. Let (E, ·, · )E be a Hilbert space equipped with an inner product ·, · E that induces the norm ·E , and let (F, ·, · F ) be a Hilbert space equipped with an inner product ·, · F that induces the norm  · F . For simplicity, we set ·, · E = ·, · ,  · E =  · , ·, · F = ·, · ,  · F =  · , if E, F are understood. We suppose that A0 is a nonempty subset of E and U0 : A0 → 2F is a point to set mapping with a graph M0 = {(x, u) : x ∈ A0 , u ∈ U0 (x)}. We suppose that M0 is a Borel measurable subset of E × F . Define A = [0, ∞) × A0 , U : A → 2F by U (t, x) = U0 (x), (t, x) ∈ A, M = [0, ∞) × M0 . Let a linear operator A : D(A) → E generate a C0 semigroup S(t) = eAt , t ≥ 0 on E, E1d = D(A∗ ), E−1 = D(A∗ ) , and let B ∈ L(F, E−1 ) be an admissible control operator for eAt , t ≥ 0. For T2 > T1 ≥ 0, we consider the following control system: x (t) = Ax(t) + Bu(t), t ∈ [T1 , T2 ] a.e.

294

8 Continuous-Time Nonautonomous Problems on Half-Axis

Assume that (xf , uf ) ∈ X(0, ∞) and sup{xf (t) : t ∈ [0, ∞)} < ∞. Recall that for every T > 0, ΦT ∈ L(L2 (0, ∞; F ), E) is defined by 

T

ΦT u =

S(T − s)Bu(s)ds, u ∈ L2 (0, T ; F ).

0

The following result is proved in Chapter 6 of [148]. Theorem 8.46 Assume that T0 > 0, Ran(ΦT0 ) = E. Then there exists a constant c > 0 such that for each T ≥ 0 and each z0 , z1 ∈ E, there exist u ∈ L2 (T , T + T0 ; F ) and z ∈ C 0 ([T , T + T0 ]); E), which is a solution of the initial value problem z (t) = Az(t) + Bu(t), t ∈ [T , T + T0 ] a.e. , z(T ) = z0 in E−1 and satisfies z(T + T0 ) = z1 and such that the following inequalities hold: z(t) − xf (t), u(t) − uf (t) ≤ c(z1 − xf (T + T0 ) + z0 − xf (T )), t ∈ [T , T + T0 ]. Assume that there exists Tf > 0 such that RanΦTf = E. In other words, the pair (A, B) is exactly controllable. We suppose that there exists r∗ > 0 such that for each t ≥ 0, {(x, u) ∈ E × F : x − xf (t) ≤ r∗ , u − uf (t) ≤ r∗ } ⊂ M0 . Assume that L : [0, ∞) × E × F → [0, ∞) is a Borelian function, L(t, xf (t), uf (t)) = 0, t ∈ [0, ∞), ψ0 : [0, ∞) → [0, ∞) is an increasing function, K1 , a1 > 0, such that lim ψ0 (t) = ∞,

t→∞

L(t, x, u) ≥ −a1 + max{ψ0 (x)x, K1 u2 } for all (t, x, u) ∈ M, μ ∈ R 1 , p¯ ∈ D(A∗ ). Let for all (t, x, u) ∈ M.

8.23 Examples

295

It is not difficult to see that f is a Borelian function, and there exist a0 , K0 > 0 and an increasing function ψ : [0, ∞) → [0, ∞) such that limt→∞ ψ(t) = ∞, and for all (t, x, u) ∈ M, f (t, x, u) ≥ −a0 + max{ψ(x), K0 u2 }. We suppose that the following property holds: (a) For each  > 0, there exists δ > 0 such that for each (t, x, u) ∈ M satisfying x − xf (t) + u − uf (t) ≤ δ we have f (t, x, u) ≤ f (t, xf (t), uf (t)) + . It follows from property (a) and Theorem 8.46 that (A3) holds. Let 0 ≤ T1 < T2 , (x, u) ∈ X(T1 , T2 ). It is not difficult to see that I f (T1 , T2 , x, u) − I f (T1 , T2 , xf , uf )  =

T2

 L(t, x(t), u(t))dt +

T1

T2

∗

x(t), A p dt ¯ +

T1

 −

T2 T1

xf (t), A∗ p dt ¯ −



T2 T1



T2 T1

Bu(t), p ¯ E−1 ,Ed dt

Buf (t), p ¯ E−1 ,Ed dt

≥ x(T2 ) − x(T1 ), p ¯ − xf (T2 ) − xf (T1 ), p . ¯ This implies that (A1) holds. We can easily obtain particular cases of this example with the pairs (A, B) considered in Sect. 7.6.

Chapter 9

Stability and Genericity Results

In this chapter we continue to study the turnpike phenomenon for the continuoustime optimal control problems on subintervals of half-axis in metric spaces which are discussed in Chap. 8. For these optimal control problems the turnpike is not a singleton. We show that the turnpike phenomenon is stable under small perturbations of an objective function. Using the Baire category approach we show that for some classes of problems a typical (generic) problem has a turnpike property.

9.1 Preliminaries We continue to consider the class of problems introduced in Sect. 8.1. Let .(E, ρE ) be a complete metric space and .(F, ρF ) be a metric space. We suppose that .A is a nonempty subset of .[0, ∞) × E, .U : A → 2F is a point to set mapping with a graph M = {(t, x, u) : (t, x) ∈ A, u ∈ U (t, x)}.

.

We suppose that .M is a Borel measurable subset of .[0, ∞) × E × F . Let .θ0 ∈ E and .θ1 ∈ F . Assume that for each pair of numbers .T1 , T2 satisfying .0 ≤ T1 < T2 < ∞ we are given a set .X(T1 , T2 ) of pairs .(x, u) (called as trajectory–control pairs) such that .x : [T1 , T2 ] → E is a continuous function, .u : [T1 , T2 ] → F is a Lebesgue measurable function satisfying (t, x(t)) ∈ A, t ∈ [T1 , T2 ],

.

u(t) ∈ U(t, x(t)), t ∈ [T1 , T2 ] almost everywhere (a.e.).

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0_9

297

298

9 Stability and Genericity Results

We suppose that the following property holds: if .0 ≤ T1 ≤ S1 < S2 ≤ T2 < ∞, .(x, u) ∈ X(T1 , T2 ), .x˜ is the restriction of x to .[S1 , S2 ] and .u ˜ is the restriction of u to .[S1 , S2 ], then .(x, ˜ u) ˜ ∈ X(S1 , S2 ); for each triplet of nonnegative numbers .T1 < T2 < T3 , each .(x1 , u1 ) ∈ X(T1 , T2 ) and each .(x2 , u2 ) ∈ X(T2 , T3 ] satisfying x1 (T2 ) = x2 (T2 )

.

a pair .(x3 , u3 ) ∈ X(T1 , T3 ), where x3 (t) = x1 (t), t ∈ [T1 , T2 ], x3 (t) = x2 (t), t ∈ (T2 , T3 ],

.

u3 (t) = u1 (t), t ∈ [T1 , T2 ], u3 (t) = u2 (t), t ∈ (T2 , T3 ].

.

Let .a0 > 0 and let .ψ : [0, ∞) → [0, ∞) be an increasing function such that ψ(t) → ∞ as t → ∞

.

(9.1)

and let .ψ1 : M → [0, ∞) satisfy ψ1 (t, x, u) ≥ max{ψ(ρE (x, θ0 )), ψ(ρF (x, θ1 ))}, (t, x, u) ∈ M.

.

(9.2)

Denote by .Mψ the set of all borelian functions .g : M → R 1 such that for each .(t, x, u) ∈ M, g(t, x, u) ≥ ψ1 (t, x, u) − a0 .

.

(9.3)

Recall that for each pair of numbers .T2 > T1 ≥ 0, each .(x, u) ∈ X(T1 , T2 ) and each g ∈ Mψ ,

.

 I g (T1 , T2 , x, u) =

T2

.

g(t, x(t), u(t))dt ∈ (−∞, ∞].

T1

We consider functionals of the form .I g (T1 , T2 , x, u), where .0 ≤ T1 < T2 , .(x, u) ∈ X(T1 , T2 ) and .g ∈ Mψ . Let .g ∈ Mψ . Recall that for each pair of numbers .T2 > T1 ≥ 0 and each pair of points .(T1 , y), (T2 , z) ∈ A we define U g (T1 , T2 , y, z) = inf{I g (T1 , T2 , x, u) :

.

(x, u) ∈ X(T1 , T2 ), x(T1 ) = y, x(T2 ) = z},

.

U g (T1 , T2 , y) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 ), x(T1 ) = y},

.

g (T1 , T2 , z) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 ), x(T2 ) = z}, U

.

9.1 Preliminaries

299

U g (T1 , T2 ) = inf{I g (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 )}.

.

We equip the space .Mψ with the uniformity determine by the base (8.10) and assume that (A0) holds. Note that all the results of Chap. 8 hold. Proposition 9.1 Assume that .g ∈ Mψ , .D,  > 0 and .c, M0 > 0. Then there exists a neighborhood V of g in .Mψ such that for each .T1 ≥ 0 each .T2 ∈ (T1 , T1 + c], each .h ∈ V and each .(x, u) ∈ X(T1 , T2 ) which satisfies ρE (x(T1 ), θ0 ) ≤ M0 ,

.

min{I g (T1 , T2 , x, u), I h (T1 , T2 , x, u)} ≤ D

.

the inequality |I g (T1 , T2 , x, u) − I h (T1 , T2 , x, u)| ≤ 

.

holds. Proof By (A0), there exists .S > 0 such that the following property holds: (i) for each .h ∈ Mψ , each .T1 ≥ 0 each .T2 ∈ (T1 , T1 + c] and each .(x, u) ∈ X(T1 , T2 ) which satisfies ρE (x(T1 ), θ0 ) ≤ M0 , I h (T1 , T2 , x, u) ≤ D + 1

.

we have ρE (x(t), θ0 ) ≤ S, t ∈ [T1 , T2 ].

.

(9.4)

By (9.1), there exist .δ ∈ (0, 1), .N > S and .Γ > 1 such that ψ(N) > 4a0 , δc ≤ 8−1 ,

(9.5)

(Γ − 1)(D + a0 c + c) ≤ /8.

(9.6)

V = {h ∈ Mψ : (g, h) ∈ E(N, δ, Γ )}

(9.7)

.

.

Define .

(see (8.10)). Assume that h ∈ V , T1 ≥ 0, T2 ∈ (T1 , T1 + c], (x, u) ∈ X(T1 , T2 )

.

ρE (x(T1 ), θ0 ) ≤ M0 ,

.

(9.8) (9.9)

300

9 Stability and Genericity Results .

min{I g (T1 , T2 , x, u), I h (T1 , T2 , x, u)} ≤ D.

(9.10)

Property (i), (9.8), and (9.10) imply (9.4). Set E1 = {t ∈ [T1 , T2 ] : ρF (u(t), θ1 ) ≤ N}, E2 = [T1 , T2 ] \ E1 .

.

(9.11)

It follows from (8.10), (9.4), (9.7), (9.8), and (9.11) that for all .t ∈ E1 , |g(t, x(t), u(t)) − h(t, x(t), u(t))| ≤ δ.

.

(9.12)

Define ξ(t) = min{g(t, x(t), u(t)), h(t, x(t), u(t))}, t ∈ [T1 , T2 ].

.

(9.13)

It follows from (8.10), (9.2), (9.4), (9.5), (9.7), and (9.8) that for each .t ∈ E2 , (g(t, x(t), u(t)) + 1)(h(t, x(t), u(t)) + 1)−1 ∈ [Γ −1 , Γ ],

.

|g(t, x(t), u(t)) − h(t, x(t), u(t))| ≤ (Γ − 1)(ξ(t) + 1).

.

(9.14)

By (9.5), (9.6), (9.8) and (9.10)–(9.14), |I g (T1 , T2 , x, u) − I h (T1 , T2 , x, u)|  . ≤ δc + (Γ − 1) ξ(t)dt + (Γ − 1)c .

E2 .

≤ δc + (Γ − 1)(D + a0 c + c) ≤ .

Proposition 9.1 is proved.

9.2 Stability of TP We continue to use the notation, definitions, and assumptions introduced in Chap. 8. Assume that .f ∈ Mψ , .(xf , uf ) ∈ X(0, ∞), .bf > 0, (8.73), (8.74) are true, (A1), (A2), and TP hold. We prove the following result which shows the stability of the turnpike phenomenon. Theorem 9.2 Assume that .L > 0 and . ∈ (0, 1). Then there exist .L1 > L, .δ ∈ (0, ) and a neighborhood V of f in .Mψ such that for each .g ∈ V and each pair of integers .T1 ≥ 0, .T2 ≥ T1 + 4L1 ,

9.3 Auxiliary Results for Theorem 9.2

301

I g (S1 , S2 , xf , uf ) is finite for each S2 > S1 ≥ 0,

.

U g (T1 , T2 ) is finite, .U g (T1 , T2 , z) is finite for each .(z, T1 ) ∈ AL satisfying g .ρE (z, θ0 ) ≤ L, .U (T1 , T2 , z1 , z2 ) is finite for each .z1 , z2 ∈ E satisfying .(T1 , z1 ) ∈  AL , .(T2 , z2 ) ∈ AL and .ρE (z1 , θ0 ) ≤ L and that for each .(x, u) ∈ X(T1 , T2 ) which satisfies at least one of the following conditions: .

(i) I g (T1 , T2 , x, u) ≤ min{U g (T1 , T2 ) + L, U g (T1 , T2 , x(T1 ), x(T2 )) + δ};

.

(ii) I g (T1 , T2 , x, u) ≤ U g (T1 , T2 , x(T1 ))+δ, (T1 , x(T1 )) ∈ AL , ρE (x(T1 ), θ0 ) ≤ L;

.

(iii) I g (T1 , T2 , x, u) ≤ U g (T1 , T2 , x(T1 ), x(T2 )) + δ,

.

L , ρE (x(T1 ), θ0 ) ≤ L (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A

.

there exist .τ1 ∈ [T1 , T1 + L1 ], .τ2 ∈ [T2 − L1 , T2 ] such that ρE (x(t), xf (t)) ≤ , t ∈ [τ1 , τ2 ].

.

Moreover, if .ρE (x(T2 ), xf (T2 )) ≤ δ, then .τ2 = T2 and if .T1 ≥ L1 and ρE (x(T1 ), xf (T1 ) ≤ δ, then .τ1 = T1 .

.

9.3 Auxiliary Results for Theorem 9.2 Lemma 9.3 Assume that M ≥ 1,  ∈ (0, 1) and L > 0. Then there exists an integer L0 ≥ 4 + L such that for each L1 > L0 there exists a neighborhood V of f in Mψ such that for each g ∈ V and each pair of numbers T1 ≥ 0, T2 ∈ [T1 +L0 , T1 +L1 ], I g (T1 , T2 , xf , uf )

.

is finite and for each (x, u)) ∈ X(T1 , T2 ) which satisfies I g (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M

.

the inequality mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) ≤ L0 /2

.

holds.

302

9 Stability and Genericity Results

Proof Theorems 8.25 and 8.26 and WTP imply that there exists an integer L0 ≥ 4 + L such that the following property holds: (a) for each pair of numbers T1 ≥ 0, T2 ≥ T1 + L0 /2, and for each (x, u) ∈ X(T1 , T2 ) which satisfies I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M + 4 + 8a0 + 4Δf

.

the inequality mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) < L0 /4

.

holds. Let L1 > L0 . Choose a positive number M0 such that ψ(M0 ) > M + (L1 + 2)(a0 + 2) + (L1 + 2)(Δf + 1).

.

(9.15)

Proposition 9.1 implies that there exists a neighborhood V of f in Mψ such that the following property holds: (b) for each pair of numbers T1 ≥ 0, T2 ∈ (T1 , T1 + L1 ], each g ∈ V and each (x, u) ∈ X(T1 , T2 ) which satisfies ρE (x(T1 ), θ0 ) ≤ M0 ,

.

.

min{I g (T1 , T2 , x, u), I f (T1 , T2 , x, u)} ≤ (L1 + 2)Δf + M + 4c0 + 4

the inequality |I g (T1 , T2 , x, u) − I f (T1 , T2 , x, u)| ≤ 1

.

holds. Assume that g ∈ V , T1 ≥ 0, T2 ∈ (T1 + L0 , T1 + L1 ],

.

(9.16)

(x, u) ∈ X(T1 , T2 ) and that I g (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M.

.

(9.17)

We show that there exist S1 ∈ [T1 , T1 + 1], S2 ∈ [T2 − 1, T2 ]

.

(9.18)

9.3 Auxiliary Results for Theorem 9.2

303

such that ρE (x(Si ), θ0 ) ≤ M0 , i = 1, 2.

.

(9.19)

Assume the contrary. Then in view of (9.2), (9.3), (9.16), and (9.17), I f (T1 , T2 , xf , uf ) + M ≥ I g (T1 , T2 , x, u)

.

.

≥ −a0 + ψ(M0 ) − a0 (T2 − T1 − 2) ≥ ψ(M0 ) − a0 L1 .

Together with (8.74), (9.2), (9.3), and (9.16) this implies that ψ(M0 ) ≤ a0 L1 + M + (L1 + 2)Δf + 2a0 .

.

This contradicts (9.15). The contradiction we have reached proves that there exist numbers S1 , S2 satisfying (9.18) and (9.19). In view of (9.16) and (9.18), S2 − S1 ≥ T2 − T1 − 2 ≥ L0 − 2 ≥ L0 /2.

.

(9.20)

Equations (9.3), (9.16), and (9.17) imply that I g (S1 , S2 , x, u) = I g (T1 , T2 , x, u) − I g (T1 , S1 , x, u) − I g (S2 , T2 , x, u)

.

.

≤ M + 2a0 + I g (T1 , T2 , xf , uf )

(9.21)

and I g (S1 , S2 , x, u) ≤ M + 2a0 + (L1 + 2)Δf + 2a0 .

.

Property (b), the relation above and Eqs. (9.16), (9.19) and (9.20) imply that |I f (S1 , S2 , x, u) − I g (S1 , S2 , x, u)| ≤ 1.

.

(9.22)

By (8.74), (9.3), (9.18), (9.21), and (9.22), I f (S1 , S2 , x, u) ≤ I g (S1 , S2 , x, u) + 1

.

.

.

≤ M + 2a0 + 1 + I f (T1 , T2 , xf , uf )

≤ M + 2a0 + 1 + I f (S1 , S2 , xf , uf ) + 4(Δf + a0 ).

Property (a), (9.20), and (9.23) imply that mes({t ∈ [S1 , S2 ] : ρE (x(t), xf (t)) > }) < L0 /4.

.

(9.23)

304

9 Stability and Genericity Results

Together with (9.18) this implies that mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) < L0 /2.

.

Lemma 9.3 is proved. Lemma 9.4 Assume that M ≥ 8, L > sup{ρE (θ0 , xf (t)) : t ∈ [0, ∞)}

.

(9.24)

be an integer and  > 0. Then there exist L0 > L + 4, L1 > 8L0 (2 + L(a0 + 1))

.

and a neighborhood V of f in Mψ such that for each g ∈ V , I g (S1 , S2 , xf , uf ) is finite for each S2 > S1 ≥ 0

.

and for each pair of numbers T1 ≥ 0, T2 ≥ T1 + L1 and each (x, u) ∈ X(T1 , T2 ) which satisfies at least one of the following conditions: (i) I g (T1 , T2 , x, u) ≤ U g (T1 , T2 ) + M;

.

(ii) I g (T1 , T2 , x, u) ≤ U g (T1 , T2 , x(T1 ))+M, (T1 , x(T1 )) ∈ AL , ρE (x(T1 ), θ0 ) ≤ L;

.

(iii) I g (T1 , T2 , x, u) ≤ U g (T1 , T2 , x(T1 ), x(T2 )) + M,

.

L , ρE (x(T1 ), θ0 ) ≤ L; (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A

.

(iv) L g (T1 , T2 , x(T2 )) + M, (T2 , x(T2 )) ∈ A I g (T1 , T2 , x, u) ≤ U

.

the value I g (T1 , T2 , x, u) is finite and the inequality mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) ≤ }) ≥ L0 /2

.

holds.

(9.25)

9.3 Auxiliary Results for Theorem 9.2

305

Proof Lemma 9.3 implies that there exist an integer L0 ≥ 4+L and a neighborhood V0 of f in Mψ such that the following property holds: (a) for each g ∈ V0 and each pair of numbers T1 ≥ 0, T2 ∈ [T1 +L0 , T1 +8(L0 +1)] I g (T1 , T2 , xf , uf )

.

is finite and for each (x, u) ∈ X(T1 , T2 ) which satisfies I g (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M

.

the inequality mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) ≤ L0 /2

.

holds. Proposition 9.1 implies that there exists a neighborhood V ⊂ V0 of f in Mψ such that the following property holds: (b) for each pair of numbers T1 ≥ 0, T2 ∈ (T1 , T1 + 8L0 ], each g ∈ V and each (x, u) ∈ X(T1 , T2 ) which satisfies ρE (x(T1 ), θ0 ) ≤ L,

.

.

min{I g (T1 , T2 , x, u), I f (T1 , T2 , x, u)} ≤ 2L0 (Δf + 2) + 2L0 (1 + a0 )

the inequality |I g (T1 , T2 , x, u) − I f (T1 , T2 , x, u)| ≤ 1/8

.

holds. Choose L1 > 8L0 (2 + L(a0 + 1)).

.

(9.26)

Property (b), (8.74), and (9.24) imply that for each g ∈ V , each S1 ≥ 0 and each S2 > S1 I g (S1 , S2 , xf , uf )

.

is finite. Assume that g ∈ V , T1 ≥ 0, T2 ≥ T1 + L1 , (x, u) ∈ X(T1 , T2 )

.

(9.27)

306

9 Stability and Genericity Results

and at least one of the following conditions (i)–(iii) holds. Define ( x,  u) ∈ X(T1 , T2 ). If condition (i) holds, then we set  x (t) = xf (t),  u(t) = uf (t), t ∈ [T1 , T2 ].

.

(9.28)

If condition (ii) holds, then there exist ( xt ,  u) ∈ X(T1 , T2 ) and τ1 ∈ (0, L] such that  x (T1 ) = x(T1 ),  xt = xf (t),  u(t) = uf (t), t ∈ [T1 + τ1 , T2 ],

.

I f (T1 , T1 + τ1 ,  x,  u) ≤ L.

.

(9.29) (9.30)

If condition (iii) holds, then there exist ( xt ,  u) ∈ X(T1 , T2 ) and τ1 , τ2 ∈ (0, L] such that  x (T1 ) = x(T1 ),  x (T2 ) = x(T2 ),

.

(9.31)

 xt = xf (t),  u(t) = uf (t), t ∈ [T1 + τ1 , T2 − τ2 ],

(9.32)

I f (T1 , T1 + τ1 ,  x,  u) ≤ L, I f (T2 − τ2 , T2 ,  x,  u) ≤ L.

(9.33)

.

.

If condition (iv) holds, then there exist ( x,  u) ∈ X(T1 , T2 ) and τ1 ∈ (0, L] such that  x (T2 ) = x(T2 ), I f (T2 − τ1 , T2 ,  x,  u) ≤ L,

(9.34)

 xt = xf (t),  u(t) = uf (t), t ∈ [T1 , T2 − τ1 ].

(9.35)

.

.

There exists a natural number q such that qL0 ≤ T2 − T1 < (q + 1)L0 .

.

(9.36)

By (9.27) and (9.36), −1 q ≥ 8 and q ≥ (T2 − T1 )L−1 0 − 1 ≥ L1 L0 − 1.

.

(9.37)

It follows from property (b), the relation L0 ≥ L + 4, (8.74), (9.24), (9.27)–(9.29), (9.32), and (9.35)–(9.37) that for all i = 1, . . . , q − 2, |I g (T1 + iL0 , T1 + (i + 1)L0 ,  x,  u) − I f (T1 + iL0 , T1 + (i + 1)L0 , xf , uf )| ≤ 8−1 . (9.38) By (9.3) and (9.28)–(9.36),

.

I f (T1 +(q −1)L0 , T2 ,  x,  u) ≤ I f (T1 +(q −1)L0 , T2 ,  x,  u)+L(1+a0 ).

.

Equations (9.24), (9.27), (9.28)–(9.36), and (9.39) imply that

(9.39)

9.3 Auxiliary Results for Theorem 9.2

307

I g (T1 + (q − 1)L0 , T2 ,  x,  u) ≤ I f (T1 + (q − 1)L0 , T2 , xf , uf ) + L(a0 + 1) + 1. (9.40) In view of (9.3) and (9.28)–(9.35) and the inequality L0 > L + 4,

.

I f (T1 , T1 + L0 ,  x,  u) ≤ I f (T1 , T1 + L0 , xf , uf ) + L(1 + a0 ).

.

(9.41)

Property (b) and Eqs. (8.74), (9.27) and (9.41) imply that I g (T1 , T1 + L0 ,  x,  u) ≤ I f (T1 , T1 + L0 ,  x,  u) + 1

.

≤ I f (T1 , T1 + L0 , xf , uf ) + L(a0 + 1) + 1.

.

(9.42)

By (9.38), (9.40), and (9.42), I g (T1 , T2 ,  x,  u) ≤ I f (T1 , T2 , xf , uf ) + (q − 2) + 2L(a0 + 1) + 2.

.

(9.43)

It follows from the construction of ( x,  u) that in all the cases (i)–(iv), I g (T1 , T2 , x, u) ≤ I g (T1 , T2 , xf , uf ) + M

.

(9.44)

and it is finite. In order to complete the proof of our lemma we need only to show that (9.24) holds. Assume the contrary. Then mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) ≤ }) < L0 /2.

.

(9.45)

Property (a), (9.27), and (9.45) imply that for all i = 0, . . . , q − 2, I g (T1 + iL0 , T1 + (i + 1)L0 , x, u) > I f (T1 + iL0 , T1 + (i + 1)L0 , xf , uf ) + M,

.

I g (T1 + (q − 1)L0 , T2 , x, u) ≥ I f (T1 + (q − 1)L0 , T2 , xf , uf ) + M.

.

By the equations above I g (T1 , T2 , x, u) ≥ I f (T1 , T2 , xf , uf ) + Mq.

.

(9.46)

In view of (9.43), (9.44), and (9.46), I f (T1 , T2 , xf , uf ) ≤ I g (T1 , T2 , x, u) − qM

.

.

≤ I g (T1 , T2 ,  x,  u)−qM +M ≤ M(1−q)+I f (T1 , T2 , xf , uf )+q +1+2L(a0 +1), M(q − 1) ≤ q + 1 + 2(a0 + 1)L, (M − 1)(q − 1) ≤ 2(1 + a0 )L + 2

.

and

308

9 Stability and Genericity Results

q ≤ 1 + L(a0 + 1).

.

(9.47)

Equations (9.27), (9.36), and (9.47) imply that L1 ≤ L0 (q + 1) ≤ L0 (2 + L(a0 + 1)).

.

This contradicts (9.26). The contradiction we have reached proves (9.24) completes the proof of Lemma 9.4.

9.4 Proof of Theorem 9.2 By (A2), we may assume without loss of generality that L > 8 + 4bf + sup{ρE (θ0 , xf (t)) : t ∈ [0, ∞)}

.

(9.48)

and that the following property holds: (a) if .(T , z) ∈ A and .ρE (z, xf (t)) ≤ , then .z ∈ AL and if in addition .T ≥ L, then L . .z ∈ A Lemma 8.36 and (A2) imply that there exist .δ0 ∈ (0, /4) and .L¯ 0 > 0 such that the following properties hold: (b) for each .T1 ≥ L¯ 0 , each .T2 ≥ T1 + 2bf and each .(x, u) ∈ X(T1 , T2 ) which satisfies ρE (xTi , xf (Ti )) ≤ 4δ0 , i = 1, 2,

.

I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + 4δ0

.

the inequality .ρE (x(t), xf (t)) ≤  holds for all .t ∈ [T1 , T2 ]; (c) for each .(Ti , zi ) ∈ A, .i = 1, 2 satisfying .ρE (zi , xf (Ti )) ≤ δ0 , .i = 1, 2 and .T2 ≥ bf there exist .τ1 , τ2 ∈ (0, bf ] and .(x1 , u1 ) ∈ X(T1 , T1 + τ1 ), .(x2 , u2 ) ∈ X(T2 − τ2 , T2 ) which satisfies x1 (T1 ) = z1 , x1 (T1 + τ1 ) = xf (T1 + τ1 ),

.

I f (T1 , T1 + τ1 , x1 , u1 ) ≤ I f (T1 , T1 + τ1 , xf , uf ) + 1,

.

x2 (T2 ) = z2 , x2 (T2 − τ2 ) = xf (T2 − τ2 ),

.

I f (T2 − τ2 , T2 , x2 , u2 ) ≤ I f (T2 − τ2 , T2 , xf , uf ) + 1.

.

Lemma 9.4 implies that there exists an integer .L0 > L + 4 + 2L¯ 0 ,

9.4 Proof of Theorem 9.2

309

L1 > 8L0 (2 + L(a0 + 1))

.

and a neighborhood .V0 of f in .Mψ such that the following property holds: (d) for each .g ∈ V0 , I g (S1 , S2 , xf , uf ) is finite for each S2 > S1 ≥ 0

.

and for each pair of numbers .T1 ≥ 0, .T2 ≥ T1 + L1 , .U g (T1 , T2 ) is finite, g .U (T1 , T2 , z) is finite for each .(T1 , z) ∈ AL satisfying .ρE (z, θ0 ) ≤ L, g L , .U g (T1 , T2 , z1 , z2 ) is finite for  .U (T1 , T2 , z) is finite for each .(T2 , z) ∈ A L and .ρE (z1 , θ0 ) ≤ L each .z1 , z2 ∈ E satisfying .(T1 , z1 ) ∈ AL , .(T2 , z2 ) ∈ A and for each .(x, u) ∈ X(T1 , T2 ) which satisfies at least one of the following conditions: I g (T1 , T2 , x, u) ≤ U g (T1 , T2 ) + L;

.

I g (T1 , T2 , x, u)≤U g (T1 , T2 , x(T1 ))+L, (T1 , x(T1 )) ∈ AL , ρE (x(T1 ), θ0 )≤L;

.

I g (T1 , T2 , x, u) ≤ U g (T1 , T2 , x(T1 ), x(T2 )) + L,

.

L , ρE (x(T1 ), θ0 ) ≤ L; (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A

.

(iv) L g (T1 , T2 , x(T2 )) + L, (T2 , x(T2 )) ∈ A I g (T1 , T2 , x, u) ≤ U

.

the value .I g (T1 , T2 , x, u) is finite and the inequality mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≥ L0 /2

.

holds. Proposition 9.1 implies that there exists a neighborhood .V ⊂ V0 of f in .Mψ such that the following property holds: (e) for each .S1 ≥ 0 each .S2 ∈ (S1 , S1 + L1 ], each .g ∈ V and each .(x, u) ∈ X(S1 , S2 ) which satisfies ρE (x(S1 ), θ0 ) ≤ L,

.

.

min{I g (S1 , S2 , x, u), I f (S1 , S2 , x, u)} ≤ (L1 + 2)Δf + 4 + 2a0

the inequality |I g (S1 , S2 , x, u) − I f (S1 , S2 , x, u)| ≤ δ0

.

is valid.

310

9 Stability and Genericity Results

Fix δ ∈ (0, δ0 /2).

(9.49)

g ∈ V , T1 ≥ 0, T2 ≥ T1 + 4L1 .

(9.50)

.

Assume that .

Property (d) and (9.50) imply that I g (S1 , S2 , xf , uf ) is finite for each S2 > S1 ≥ 0,

.

U g (T1 , T2 ) is finite, .U g (T1 , T2 , z) is finite for each .(T1 , z) ∈ AL satisfying L , .U g (T1 , T2 , z1 , z2 ) g (T1 , T2 , z) is finite for each .(T2 , z) ∈ A .ρE (z, θ0 ) ≤ L, .U L and is finite for each .z1 , z2 ∈ E satisfying .(T1 , z1 ) ∈ AL , .(T2 , z2 ) ∈ A .ρE (z1 , θ0 ) ≤ L. Assume that .(x, u) ∈ X(T1 , T2 ) satisfies at least one of the following conditions: .

(i) I g (T1 , T2 , x, u) ≤ min{U g (T1 , T2 ) + L, U g (T1 , T2 , x(T1 ), x(T2 )) + δ};

.

(ii) I g (T1 , T2 , x, u)≤U g (T1 , T2 , x(T1 ))+δ, (T1 , x(T1 )) ∈ AL , ρE (x(T1 ), θ0 )≤L;

.

(iii) I g (T1 , T2 , x, u) ≤ U g (T1 , T2 , x(T1 ), x(T2 )) + δ,

.

L , ρE (x(T1 ), θ0 ) ≤ L. (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A

.

In order to complete the proof of the theorem it is sufficient to show that there exist .τ1 ∈ [T1 , T1 + L1 ], .τ2 ∈ [T2 − L1 , T2 ] such that ρE (x(t), xf (t)) ≤ , t ∈ [τ1 , τ2 ];

.

if .ρE (x(T2 ), xf (T2 )) ≤ δ, then .τ2 ρE (x(T1 ), xf (T1 )) ≤ δ, then .τ1 = T1 .

=

T2 and if .T1

≥

L1 and

.

Property (d) and (9.50) imply that mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≥ L0 /2.

.

Set

(9.51)

9.4 Proof of Theorem 9.2

311

S1 = min{t ∈ [max{T1 , L¯ 0 }, T2 ] : ρE (x(t), xf (t)) ≤ δ0 }.

.

(9.52)

(Note that if .T1 ≥ L¯ 0 and .ρE (x(T1 ), xf (T1 )) ≤ δ, then .S1 = T1 .) In view of (9.48) and (9.51), .S1 is well-defined. If .S1 ≥ T1 + L1 , then it follows from properties (a), (d) applied to .(x, u) ∈ X(T1 , S1 ), (i)–(iii), (9.50) and (9.52) that mes({t ∈ [T1 , S1 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≥ L0 /2.

.

On the other hand, in view of (9.52), mes({t ∈ [T1 , S1 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≤ L¯ 0

.

and .L¯ 0 ≥ L0 /2. This contradicts (9.48). The contradiction we have reached proves that S1 ≤ T1 + L1 .

(9.53)

S2 = max{t ∈ [S1 , T2 ] : ρE (x(t), xf (t)) ≤ δ0 }.

(9.54)

.

Set .

If .S2 ≤ T2 − L1 , then it follows from properties (a), (d) applied to .(x, u) ∈ X(S2 , T2 ), and (9.50) that mes({t ∈ [S2 , T2 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≥ L0 /2

.

and this contradicts (9.54). Thus S2 ≥ T2 − L1 .

.

(9.55)

In order to complete the proof of the theorem it is sufficient to show that ρE (x(t), xf (t)) ≤ , t ∈ [S1 , S2 ].

.

Assume the contrary. Then there exists τ0 ∈ [S1 , S2 ]

(9.56)

ρE (x(τ0 ), xf (τ0 )) > .

(9.57)

.

such that .

In view of (9.50), (9.53), and (9.55), S2 − S1 ≥ T2 − T1 − 2L1 ≥ 2L1 .

.

(9.58)

312

9 Stability and Genericity Results

By (9.58) and (9.59), there exist numbers .S3 , S4 such that S4 = S3 + L0 /4, S1 ≤ S3 ≤ τ0 ≤ S4 ≤ S2 .

(9.59)

S5 = max{t ∈ [S1 , S3 ] : ρE (x(t), xf (t)) ≤ δ0 },

(9.60)

S6 = min{t ∈ [S4 , S2 ] : ρE (x(t), xf (t)) ≤ δ0 }.

(9.61)

.

Set .

.

It follows from (9.52), (9.54) and (9.59)–(9.61) that S6 − S5 ≥ S4 − S3 = L0 /4.

.

(9.62)

Assume that S6 − S5 ≥ L1 .

.

Properties (a) and (d) applied to .(x, u) ∈ X(S5 , S6 ), conditions (i)–(iii) and Eqs. (9.50), (9.60) and (9.61) imply that mes({t ∈ [S5 , S6 ] : ρE (x(t), xf (t)) ≤ δ}) ≥ L0 /2.

.

On the other hand, by (9.59)–(9.61), mes({t ∈ [S5 , S6 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≤ S4 − S3 = L0 /4.

.

The contradiction we have reached proves that S6 − S5 ≤ L1 .

.

Together with (9.62) this implies that L0 /4 ≤ S6 − S5 ≤ L1 .

.

(9.63)

Property (c) and Eqs. (9.48), (9.60), (9.61), and (9.63) imply that there exist .(y, v) ∈ X(S5 , S6 ) and .t1 , t2 ∈ (0, bf ] such that y(S5 ) = x(S5 ), y(S6 ) = x(S6 ),

.

y(t) = xf (t), v(t) = uf (t), t ∈ [S5 + t1 , S6 − t2 ],

.

I f (S5 , S5 + t1 , y, v) ≤ I f (S5 , S5 + t1 , xf , uf ) + 1,

.

9.4 Proof of Theorem 9.2

313

I f (S6 − t2 , S6 , y, v) ≤ I f (S6 − t2 , S6 , xf , uf ) + 1.

.

(9.64)

By (8.74), (9.3), (9.63), and (9.64), I f (S5 , S6 , y, v) ≤ I f (S5 , S6 , xf , uf ) + 2 ≤ 2 + (L1 + 2)Δf + 2a0 .

.

(9.65)

In view of conditions (i)–(iii), I f (S5 , S6 , x, u) ≤ U g (S5 , S6 , x(S5 ), x(S6 )) + δ0 .

.

(9.66)

It follows from (9.64) and (9.65) that U g (S5 , S6 , x(S5 ), x(S6 )) = inf{I f (S5 , S6 , ξ, η) :

.

(ξ, η) ∈ X(S5 , S6 ), I f (S5 , S6 , ξ, η) ≤ 2 + (L1 + 2)Δf + 2a0 }.

.

(9.67)

Property (e) and Eqs. (9.50), (9.60), (9.63)–(9.65), and (9.67) imply that U g (S5 , S6 , x(S5 ), x(S6 )) ≤ inf{I g (S5 , S6 , ξ, η) : (ξ, η) ∈ X(S5 , S6 ),

.

I f (S5 , S6 , ξ, η) ≤ 2 + (L1 + 2)Δf + 2a0 }

.

.

≤ δ0 + inf{I g (S5 , S6 , ξ, η) : (ξ, η) ∈ X(S5 , S6 ), I f (S5 , S6 , ξ, η) ≤ 2 + (L1 + 2)Δf + 2a0 }

.

.

≤ δ0 + U f (S5 , S6 , x(S5 ), x(S6 )).

By (9.66)–(9.68), I g (S5 , S6 , x, u) ≤ (L1 + 2)Δf + 2a0 + 3.

.

Together with property (e) and Eqs. (9.50), (9.60) and (9.63) this implies that |I g (S5 , S6 , x, u) − I f (S5 , S6 , x, u)| ≤ δ0 .

.

Combined with (9.65) and (9.68) this implies that I f (S5 , S6 , x, u) ≤ I g (S5 , S6 , x, u) + δ0

.

.

≤ U g (S5 , S6 , x(S5 ), x(S6 )) + 2δ0 ≤ U f (S5 , S6 , x(S5 ), x(S6 )) + 3δ0 .

Property (b), (9.60), (9.61), (9.63) and the equation above imply that

(9.68)

314

9 Stability and Genericity Results

ρE (x(t), xf (t)) ≤ , t ∈ {S5 , S6 ].

.

Together with (9.59)–(9.61) this implies that ρE (x(τ0 ), xf (τ0 )) ≤ .

.

This contradicts (9.57). The contradiction we have reached competes the proof of Theorem 9.2.

9.5 Problems with Discount We use the notation, definitions, and assumptions introduced in Sects. 9.1 and 9.2. Let .g ∈ Mψ and .α : [0, ∞) → [0, ∞) be a Borelian bounded function. For each pair of numbers .T2 > T1 ≥ 0 and each .(x, u) ∈ X(T1 , T2 ) define  I αg (T1 , T2 , x, u) =

T2

.

α(t)g(t, x(t), u(t))dt ∈ (−∞, ∞]

T1

and for each pair of numbers .T2 > T1 ≥ 0 and each pair of points .(T1 , y), (T2 , z) ∈ A we define U αg (T1 , T2 , y, z) = inf{I αg (T1 , T2 , x, u) :

.

(x, u) ∈ X(T1 , T2 ), x(T1 ) = y, x(T2 ) = z},

.

U αg (T1 , T2 , y) = inf{I αg (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 ), x(T1 ) = y},

.

αg (T1 , T2 , z) = inf{I αg (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 ), x(T2 ) = z}, U

.

U αg (T1 , T2 ) = inf{I αg (T1 , T2 , x, u) : (x, u) ∈ X(T1 , T2 )}.

.

(9.69)

We prove the following stability result. Theorem 9.5 Assume that .L > 0 and . ∈ (0, 1). Then there exist .L1 > L, .δ ∈ (0, ) .λ ∈ (0, 1) and a neighborhood V of f in .Mψ such that for each .g ∈ V , each pair of numbers .T1 ≥ 0, .T2 ≥ T1 + 4L1 , each borelian function .α : [0, ∞) → [0, 1] which satisfies α(t1 )α(t2 )−1 ≥ λ for each t1 , t2 ∈ [T1 , T2 ] satisfying |t2 − t1 | ≤ L1

.

U αg (T1 , T2 ) is finite, .U αg (T1 , T2 , z) is finite for each .(z, T1 ) ∈ AL satisfying αg (T , T , z , z ) is finite for each .z , z ∈ E satisfying .(T , z ) ∈ .ρE (z, θ0 ) ≤ L, .U 1 2 1 2 1 2 1 1 .

9.6 Auxiliary Results for Theorem 9.5

315

L and .ρE (z1 , θ0 ) ≤ L and that for each .(x, u) ∈ X(T1 , T2 ) which AL , .(T2 , z2 ) ∈ A satisfies at least one of the following conditions: (i) I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 ) + δ inf{α(t) : t ∈ [T1 , T2 ]};

.

(ii) I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 , x(T1 ))

.

.

+ δ inf{α(t) : t ∈ [T1 , T2 ]}, (T1 , x(T1 )) ∈ AL , ρE (x(T1 ), θ0 ) ≤ L;

(iii) I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 , x(T1 ), x(T2 )) + δ inf{α(t) : t ∈ [T1 , T2 ]},

.

L , ρE (x(T1 ), θ0 ) ≤ L (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A

.

there exist .τ1 ∈ [T1 , T1 + L1 ], .τ2 ∈ [T2 − L1 , T2 ] such that ρE (x(t), xf (t)) ≤ , t ∈ [τ1 , τ2 ].

.

Moreover, if .ρE (x(T2 ), xf (T2 )) ≤ δ, then .τ2 = T2 and if .T1 ≥ L1 and ρE (x(T1 ), xf (T1 )) ≤ δ, then .τ1 = T1 .

.

9.6 Auxiliary Results for Theorem 9.5 Lemma 9.6 Assume that M ≥ 2,  ∈ (0, 1) and L > 0. Then there exists an integer L0 ≥ 4 + L such that for each L1 > L0 there exist λ ∈ (0, 1) and a neighborhood V of f in Mψ such that for each g ∈ V and each pair of numbers T1 ≥ 0, T2 ∈ [T1 + L0 , T1 + L1 ], each borelian function α : [0, ∞) → [0, 1] which satisfies α(t1 )α(t2 )−1 ≥ λ for each t1 , t2 ∈ [T1 , T2 ] satisfying |t2 − t1 | ≤ L1

.

(9.70)

and for each (x, u) ∈ X(T1 , T2 ) which satisfies I αg (T1 , T2 , x, u) ≤ I αf (T1 , T2 , xf , uf )+4−1 M inf{α(t) : t ∈ [T1 , T2 ]}

.

(9.71)

the inequality mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) ≤ L0 /2

.

holds.

(9.72)

316

9 Stability and Genericity Results

Proof Lemma 9.3 implies that there exists an integer L0 ≥ 4 + L such that for each L1 > L0 the following property holds: (a) there exists a neighborhood V of f in Mψ such that for each g ∈ V and each pair of numbers T1 ≥ 0, T2 ∈ [T1 + L0 , T1 + L1 ], I g (T1 , T2 , xf , uf ) is finite and for each (x, u) ∈ X(T1 , T2 ) which satisfies I g (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M

.

inequality (9.72) holds. Assume that L1 > L0 and a neighborhood V of f in Mψ be as guaranteed by property (a). Choose λ ∈ (0, 1) such that (1 − λ)(Δf + 1)(L1 + 2)(a0 + 1) < 1.

(9.73)

g ∈ V , T1 ≥ 0, T2 ∈ [T1 + L0 , T1 + L1 ],

(9.74)

.

Assume that .

a borelian function α : [0, ∞) → [0, 1] satisfies (9.70) and that (x, u) ∈ X(T1 , T2 ) satisfies (9.71). In order to complete the proof of the lemma it is sufficient to show that (9.72) holds. Assume then contrary. Then by property (a) and (9.74), I g (T1 , T2 , x, u) > M + I f (T1 , T2 , xf , uf ).

(9.75)

.

By (9.3) and (9.75),  I αg (T1 , T2 , x, u) = −a0

T2

.

 α(t)dt + (I αg (T1 , T2 , x, u) + a0

T1

 .

= −a0

T2 T1

 .

T2

≥ −a0

T2

(α(t)g(t, x(t), u(t)) + a0 )dt

T1

 α(t)dt + inf{α(t) : t ∈ [T1 , T2 ]}

T1

 .

≥ −a0

T2

α(t)dt) T1

 α(t)dt +

T2

T2

(g(t, x(t), u(t)) + a0 )dt

T1

 α(t)dt + inf{α(t) : t ∈ [T1 , T2 ]}

T1

T2

(f (t, xf (t), uf (t)) + a0 )dt

T1 .

+ M inf{α(t) : t ∈ [T1 , T2 ]}.

(9.76)

9.6 Auxiliary Results for Theorem 9.5

317

By (8.74), (9.3), (9.70), (9.73), and (9.74), 

T2

sup{α(t) : t ∈ [T1 , T2 ]}

.

(f (t, xf (t), uf (t)) + a0 )dt

T1

 .

T2

− inf{α(t) : t ∈ [T1 , T2 ]}

(f (t, xf (t), uf (t)) + a0 )dt

T1

 .

≤ (1 − λ) sup{α(t) : t ∈ [T1 , T2 ]}

T2

(f (t, xf (t), uf (t)) + a0 )dt

T1 .

≤ (1 − λ) sup{α(t) : t ∈ [T1 , T2 ]}(Δf (L1 + 2) + 2a0 + a0 L1 ) .

≤ sup{α(t) : t ∈ [T1 , T2 ]}.

(9.77)

By (9.3), (9.76), and (9.77),  I αg (T1 , T2 , x, u) ≥ −a0

T2

α(t)dt

.

T1

 .

+ sup{α(t) : t ∈ [T1 , T2 ]}

T2

(f (t, xf (t), uf (t)) + a0 )dt

T1 .

.

− sup{α(t) : t ∈ [T1 , T2 ]} + M inf{α(t) : t ∈ [T1 , T2 ]}

≥ I αf (T1 , T2 , xf , uf ) − sup{α(t) : t ∈ [T1 , T2 ]} + M inf{α(t) : t ∈ [T1 , T2 ]} .

≥ I αf (T1 , T2 , xf , uf ) + 2−1 M inf{α(t) : t ∈ [T1 , T2 ]}.

This contradicts (9.71). The contradiction we have reached proves Lemma 9.6. Lemma 9.7 Assume that  > 0 and L > sup{ρE (θ0 , xf (t)) : t ∈ [0, ∞)}.

.

(9.78)

Then there exists an integer L0 ≥ L + 4, L1 > 8L0 + 8, λ ∈ (0, 1) and a neighborhood V of f in Mψ such that for each g ∈ V and each pair of numbers T1 ≥ 0, T2 ≥ T1 + L1 , each borelian function α : [0, ∞) → [0, 1] which satisfies α(t1 )α(t2 )−1 ≥ λ for each t1 , t2 ∈ [T1 , T2 ] satisfying |t2 − t1 | ≤ L1

.

(9.79)

the following assertion holds. Assume that (x, u) ∈ X(T1 , T2 ) satisfies at least one of the following conditions:

318

9 Stability and Genericity Results

(i) I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 ) + 4 inf{α(t) : t ∈ [T1 , T2 ]};

.

(ii) (T1 , x(T1 )) ∈ AL , ρE (x(T1 ), θ0 ) ≤ L,

.

I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 , x(T1 )) + 4 inf{α(t) : t ∈ [T1 , T2 ]};

.

(iii) L , (T2 , x(T2 )) ∈ A

.

αg (T1 , T2 , x(T2 )) + 4 inf{α(t) : t ∈ [T1 , T2 ]}; I αg (T1 , T2 , x, u) ≤ U

.

(iv) L , ρE (x(T1 ), θ0 ) ≤ L, (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A

.

I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 , x(T1 ), x(T2 )) + 4 inf{α(t) : t ∈ [T1 , T2 ]}.

.

Then I αg (T1 , T2 , x, u) is finite and mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) ≤ }) ≥ L0 /2.

.

(9.80)

Proof Lemma 9.6 implies that there exist an integer L0 ≥ 4 + L, λ0 ∈ (0, 1) and a neighborhood V0 of f in Mψ such that the following property holds: (a) for each g ∈ V0 , each pair of numbers T1 ≥ 0, T2 ∈ [T1 + L0 , T1 + 8(L0 + 1)], each borelian function α : [0, ∞) → [0, 1] which satisfies α(t1 )α(t2 )−1 ≥ λ0 for each t1 , t2 ∈ [T1 , T2 ]

.

and for each (x, u) ∈ X(T1 , T2 ) which satisfies I αg (T1 , T2 , x, u) ≤ I αf (T1 , T2 , xf , uf ) + 8 inf{α(t) : t ∈ [T1 , T2 ]}

.

the inequality mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) ≤ L0 /2

.

holds. We may assume without loss of generality that

(9.81)

9.6 Auxiliary Results for Theorem 9.5

319

λ0 > 1/2, (1 − λ0 )((2L0 Δf + 2a0 + 2L0 a0 ) < 8−1 .

.

(9.82)

Proposition 9.1 implies that there exists a neighborhood V ⊂ V0 of f in Mψ such that the following property holds: (b) for each pair of numbers T1 ≥ 0, T2 ∈ (T1 , T1 + 8(L0 + 1)], each g ∈ V and each (x, u) ∈ X(T1 , T2 ) which satisfies ρE (x(T1 ), θ0 ) ≤ L + 4,

.

.

min{I g (T1 , T2 , x, u), I f (T1 , T2 , x, u)} ≤ 2L0 (Δf + 1) + 2L(1 + a0 )

the inequality |I g (T1 , T2 , x, u) − I f (T1 , T2 , x, u)| ≤ 1/8

.

holds. Choose integers q0 ≥ 4 + 16((L0 + 1)(a0 + 1) + 2),

.

L1 > L0 (8 + L(a0 + 1)) + L0 (2q0 + 4)

.

(9.83)

and λ ∈ (λ0 , 1)

.

such that λq0 > 2−1 .

.

(9.84)

Assume that g ∈ V , T1 ≥ 0, T2 ≥ T1 + L1 ,

.

a borelian function α : [0, ∞) → [0, 1] satisfies (9.79), (x, u) ∈ X(T1 , T2 ) and at least one of cases (i)–(iv) holds. Define ( x,  u) ∈ X(T1 , T2 ). If case (i) holds, then we set  x (t) = xf (t),  u(t) = uf (t), t ∈ [T1 , T2 ],

.

(9.85)

If case (ii) holds, then there exist ( x,  u) ∈ X(T1 , T2 ) and τ1 ∈ (0, L] such that  x (T1 ) = x(T1 ),  x (t) = xf (t),  u(t) = uf (t), t ∈ [T1 + τ1 , T2 ],

.

320

9 Stability and Genericity Results

I f (T1 , T1 + τ1 ,  x,  u) ≤ L.

.

(9.86)

If case (iv) holds, then there exist ( x,  u) ∈ X(T1 , T2 ) and τ1 , τ2 ∈ (0, L] such that  x (T1 ) = x(T1 ),  x (T2 ) = x(T2 ),

.

 x (t) = xf (t),  u(t) = uf (t), t ∈ [T1 + τ1 , T2 − τ2 ],

.

I f (T1 , T1 + τ1 ,  x,  u) ≤ L, I f (T2 − τ2 , T2 ,  x,  u) ≤ L.

.

(9.87)

If case (iii) holds, then there exist ( x,  u) ∈ X(T1 , T2 ) and τ1 ∈ (0, L] such that  x (T2 ) = x(T2 ), I f (T2 − τ1 , T2 ,  x,  u) ≤ L,

.

 x (t) = xf (t),  u(t) = uf (t), t ∈ [T1 , T2 − τ1 ].

.

(9.88)

There exists a natural number q such that qL0 ≤ T2 − T1 < (q + 1)L0 .

.

(9.89)

By (9.83), (9.87), and (9.89), −1 q ≥ (T2 − T1 )L−1 0 − 1 ≥ L1 L0 − 1 ≥ 8.

.

(9.90)

It follows from property (b), (8.74), (9.3), (9.76), (9.78), (9.84)–(9.90) and the relation L0 ≥ L + 4 that for all i = 1, . . . , q − 2, |I g (T1 +iL0 , T1 +(i+1)L0 ,  x,  u)−I f (T1 +iL0 , T1 +(i+1)L0 , xf , uf )| ≤ 8−1 . (9.91) By (9.3), (9.78) and (9.85)–(9.89),

.

I f (T1 + (q − 1)L0 , T2 ,  x,  u) ≤ I f (T1 + (q − 1)L2 , T2 ,  x,  u) + L(1 + a0 ). (9.92) Property (b) and Eqs. (8.74), (9.3), (9.84), and (9.92) imply that .

I g (T1 + (q − 1)L0 , T2 ,  x,  u) ≤ I f (T1 + (q − 1)L0 , T2 , xf , uf ) + L(a0 + 1) + 1. (9.93) In view of (9.3), (9.78), (9.83) and (9.85)–(9.88),

.

I f (T1 , T1 + L0 ,  x,  u) ≤ I f (T1 , T1 + L0 , xf , uf ) + L(1 + a0 ).

.

(9.94)

Property (b) and Eqs. (8.74), (9.3), (9.76), (9.78), (9.84), and (9.94) imply that

9.6 Auxiliary Results for Theorem 9.5

321

I g (T1 , T1 + L0 ,  x,  u) ≤ I f (T1 , T1 + L0 ,  x,  u) + 1

.

≤ I f (T1 , T1 + L0 , xf , uf ) + L(1 + a0 ) + 1.

(9.95)

{S1 , S2 } ∈ {(T1 + iL0 , T1 + (i + 1)L0 : i = 1, . . . , q − 2}.

(9.96)

.

Let .

In view of (9.3), (9.79), (9.82)–(9.84), and (9.91), 

T2

I αg (S1 , S2 ,  x,  u) =

.

 (α(t)g(t,  x (t),  u(t)) + a0 )dt − a0

T1

≤ sup{α(t) : t ∈ [S1 , S2 ]}

S2

 (g(t,  x (t),  u(t)) + a0 )dt − a0

S1

≤ sup{α(t) : t ∈ [S1 , S2 ]}

S2

α(t)dt S1

 .

α(t)dt S1

 .

S2

S2

(f (t, xf (t), uf (t)) + a0 )dt

S1

.

+ 8−1 sup{α(t) : t ∈ [S1 , S2 ]} − a0



S2

α(t)dt S1

 .

≤

S2

(f (t, xf (t), uf (t)) + a0 )α(t)dt

S1

 .

+ (1 − λ) sup{α(t) : t ∈ [S1 , S2 ]}

S2

(f (t, xf (t), uf (t)) + a0 )dt

S1

.

+ 8−1 sup{α(t) : t ∈ [S1 , S2 ]} − a0



S2

α(t)dt S1

.

≤ I αf (S1 , S2 , xf , uf ) + (1 − λ) sup{α(t) : t ∈ [S1 , S2 ]}((L0 + 2)Δf + 2a0 ) .

.

+ 8−1 sup{α(t) : t ∈ [S1 , S2 ]}

≤ I αf (S1 , S2 , xf , uf )+(1−λ)λ−1 inf{α(t) : t ∈ [S1 , S2 ]}((L0 +2)Δf +2a0 ) .

.

+ 8−1 sup{α(t) : t ∈ [S1 , S2 ]}

≤ I αf (S1 , S2 , xf , uf ) + 2−1 inf{α(t) : t ∈ [S1 , S2 ]}.

Thus for all i = 1, . . . , q − 2,

322

9 Stability and Genericity Results

I αg (T1 + iL0 , T1 + (i + 1)L0 ,  x,  u)

.

.

≤ I αf (T1 + iL0 , T1 + (i + 1)L0 , xf , uf )

+ 2−1 inf{α(t) : t ∈ [T1 + iL0 , T1 + (i + 1)L0 ]}.

.

(9.97)

By (8.74), (9.3), (9.79), (9.82), (9.84), (9.93), and (9.95),  I αg (S1 , S2 ,  x,  u) =

S2

.

 (α(t)g(t,  x (t),  u(t)) + a0 )dt − a0

α(t)dt

S1

S1

 .

≤ sup{α(t) : t ∈ [S1 , S2 ]}

S2

S2

 (g(t,  x (t),  u(t)) + a0 )dt − a0

S1

α(t)dt S1

 ≤ sup{α(t) : t ∈ [S1 , S2 ]}(

.

S2

S2

(f (t, xf (t), uf (t)) + a0 )dt

S1

 .

+ L(a0 + 1) + 1) − a0

S2

α(t)dt S1

 .

≤ sup{α(t) : t ∈ [S1 , S2 ]}(1 + a(L0 + 1)) − a0

S2

α(t)dt S1

 +I

.

αf

(S1 , S2 , xf , uf ) + a0

S2

α(t)dt S1

 .

+(

S2

(f (t, xf (t), uf (t)) + a0 )dt) sup{α(t) : t ∈ [S1 , S2 ]}(1 − λ)

S1 .

≤ I αf (S1 , S2 , xf , uf ) + λ−1 sup{α(t) : t ∈ [S1 , S2 ]}(1 + a(L0 + 1)) .

+ (Δf (L0 + 2) + 2a0 )λ−1 (1 − λ) inf{α(t) : t ∈ [S1 , S2 ]} .

≤ I αf (S1 , S2 , xf , uf ) + 2−1 inf{α(t) : t ∈ [S1 , S2 ]}

.

+ 2 inf{α(t) : t ∈ [S1 , S2 ]}(1 + a(L0 + 2)).

It follows from (9.89) and (9.98) that I αg (T1 , T2 ,  x,  u) ≤ I αf (T1 , T2 , xf , uf )

.

(9.98)

9.6 Auxiliary Results for Theorem 9.5

.

q−2 

+ 2−1

323

inf{α(t) : t ∈ [T1 + iL0 , T1 + (i + 1)L0 ]}

i=0

+ 2−1 inf{α(t) : t ∈ [T1 + (q − 1)L0 , T2 ]}

.

+ (1 + a0 (L0 + 2))(2 inf{α(t) : t ∈ [T1 , T1 + L0 ]}

.

.

+ 2 inf{α(t) : t ∈ [T1 + (q − 1)L0 , T2 ]}).

(9.99)

In view of (9.99), I αg (T1 , T2 ,  x,  u) is finite. Equations (9.86) and (9.88) imply that I αg (T1 , T2 , x, u) is finite too. In order to complete the proof of our lemma we need only to show that (9.80) holds. Assume the contrary. Then mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) ≤ }) < L0 /2.

.

(9.100)

Property (a), (9.79), (9.83), (9.84), (9.89), and (9.100) imply that for all i = 0, . . . , q − 2, I αg (T1 + iL0 , T1 + (i + 1)L0 , x, u) > I αf (T1 + iL0 , T1 + (i + 1)L0 , xf , uf )

.

.

+ 8 inf{α(t) : t ∈ [T1 + iL0 , T1 + (i + 1)L0 ]},

(9.101)

I αg (T1 + (q − 1)L0 , T2 , x, u) > I αf (T1 + (q − 1)L0 , T2 , xf , uf )

.

.

+ 8 inf{α(t) : t ∈ [T1 + (q − 1)L0 , T2 ]}.

(9.102)

By (9.98), (9.101), and (9.102), I αg (T1 , T2 , x, u) > I αf (T1 , T2 , xf , uf )

.

.

+ 8(

q−2 

inf{α(t) : t ∈ [T1 + iL0 , T1 + (i + 1)L0 ]}

i=0 .

.

+ inf{α(t) : t ∈ [T1 + (q − 1)L0 , T2 ]})

≥ I αg (T1 , T2 ,  x,  u) + 7(

q−2 

inf{α(t) : t ∈ [T1 + iL0 , T1 + (i + 1)L0 ]}

i=0 .

+ inf{α(t) : t ∈ [T1 + (q − 1)L0 , T2 ]}) .

− (2 inf{α(t) : t ∈ [T1 , T1 + L0 ]}

324

9 Stability and Genericity Results

+ 2 inf{α(t) : t ∈ [T1 + (q − 1)L0 , T2 ]})(2 + (a0 + 1)(L0 + 2)).

.

(9.103)

Properties (a)–(d) and Eqs. (9.85), (9.88) imply that I αg (T1 , T2 , x, u) ≤ I αg (T1 , T2 ,  x,  u) + 4 inf{α(t) : t ∈ [T1 , T2 ]}.

.

(9.104)

It follows from (9.103) and (9.104) that 4(2 + (a0 + 1)(L0 + 2))(inf{α(t) : t ∈ [T1 , T1 + L0 ]}

.

.

≥

.

+ inf{α(t) : t ∈ [T1 + (q − 1)L0 , T2 ]})

q−2 

inf{α(t) : t ∈ [T1 + iL0 , T1 + (i + 1)L0 ]}

i=0 .

+ inf{α(t) : t ∈ [T1 + (q − 1)L0 , T2 ]}.

(9.105)

There exists j ∈ {0, . . . , q − 2}

.

(9.106)

such that .

.

inf{α(t) : t ∈ [T1 + iL0 , T1 + (i + 1)L0 ]}

≤ inf{α(t) : t ∈ [T1 + j L0 , T1 + (j + 1)L0 ]}, i = 0, . . . , q − 2.

(9.107)

In view of (9.83) and (9.90), q − 2 ≥ L1 L−1 0 − 3 ≥ 2q0 + 2.

.

(9.108)

By (9.106) and (9.108), there exists an integer j0 ≥ 0 such that j0 + q0 ≤ q − 2, j0 ≤ j ≤ j0 + q0 .

.

It follows from (9.79), (9.83), (9.107), and (9.109) that q−2  .

inf{α(t) : t ∈ [T1 + iL0 , T1 + (i + 1)L0 ]}

i=0

.

≥

j0 +q 0 −1  i=j0

inf{α(t) : t ∈ [T1 + iL0 , T1 + (i + 1)L0 ]}

(9.109)

9.7 Proof of Theorem 9.5 .

325

≥ q0 λq0 inf{α(t) : t ∈ [T1 + j L0 , T1 + (j + 1)L0 ]}.

(9.110)

Equations (9.83), (9.84), (9.105), (9.107), and (9.110) imply that 8(2 + ((a0 + 1)(L0 + 1)) min{αt : t ∈ [T1 + j L0 , T1 + (j + 1)L0 ]}λ−1

.

.

≥ λq0 q0 min{αt : t ∈ [T1 + j L0 , Ti + (j + 1)L0 ]}.

Together with (9.84) this implies that 16(2 + (a0 + 1)(L0 + 1)) ≥ q0 .

.

This contradicts (9.82). The contradiction we have reached proves Lemma 9.7.

9.7 Proof of Theorem 9.5 We may assume without loss of generality that L > 8bf + sup{ρE (θ0 , xf (t)) : t ∈ [0, ∞)}.

.

(9.111)

Lemma 8.36 and (A2) imply that there exist .δ0 ∈ (0, ) and .L¯ 0 > 0 such that the following properties hold: (a) for each .T1 ≥ L¯ 0 , each .T2 ≥ T1 + 2bf and each .(x, u) ∈ X(T1 , T2 ) which satisfies ρE (x(Ti ), xf (Ti )) ≤ 4δ0 , i = 1, 2,

.

I f (T1 , T2 , x, u) ≤ U f (T1 , T2 , x(T1 ), x(T2 )) + 8δ0

.

the inequality .ρE (x(t), xf (t)) ≤  holds for all .t ∈ [T1 , T2 ]; (b) for each .(Ti , zi ) ∈ A, .i = 1, 2 satisfying .ρE (zi , xf (Ti )) ≤ δ0 , .i = 1, 2 and .T2 ≥ bf there exist .τ1 , τ2 ∈ (0, bf ] and .(x1 , u1 ) ∈ X(T1 , T1 + τ1 ), .(x2 , u2 ) ∈ X(T2 − τ2 , T2 ) which satisfies x1 (T1 ) = z1 , x1 (T1 + τ1 ) = xf (T1 + τ1 ),

.

I f (T1 , T1 + τ1 , x1 , u1 ) ≤ I f (T1 , T1 + τ1 , xf , uf ) + 1,

.

x2 (T2 ) = z2 , x2 (T2 − τ2 ) = xf (T2 − τ2 ),

.

I f (T2 − τ2 , T2 , x2 , u2 ) ≤ I f (T2 − τ2 , T2 , xf , uf ) + 1.

.

326

9 Stability and Genericity Results

Let δ ∈ (0, δ0 /2).

.

(9.112)

Lemma 9.7 implies that there exist an integer L0 ≥ L + 4 + 2L¯ 0 , L1 > 8L0 + 8,

.

(9.113)

λ ∈ (1/2, 1) and a neighborhood .V0 of f in .Mψ such that

.

4λ−1 (1 − λ)a0 L1 < δ0 , 4λ−1 (1 − λ)(2 + (L1 + 2)Δf + 2a0 ) < δ0

.

(9.114)

and that the following property holds: (c) for each .g ∈ V0 and each pair of numbers .T1 ≥ 0, .T2 ≥ T1 + L1 , each borelian function .α : [0, ∞) → (0, 1] which satisfies α(t1 )α(t2 )−1 ≥ λ for each t1 , t2 ∈ [T1 , T2 ] satisfying |t2 − t1 | ≤ L1

.

U αg (T1 , T2 ) is finite, .U αg (T1 , T2 , ξ ) is finite for each .(T1 , ξ ) ∈ AL satL , αg (T1 , T2 , ξ ) is finite for each .(T2 , ξ ) ∈ A isfying .ρE (ξ, θ0 ) ≤ L, .U αg  .U (T1 , T2 , ξ1 , ξ2 ) is finite for each .(T1 , ξ1 ) ∈ AL , .(T2 , ξ2 ) ∈ AL satisfying .ρE (z1 , θ0 ) ≤ L and if .(x, u) ∈ X(T1 , T2 ) satisfies at least one of the following conditions: .

I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 ) + 4 inf{α(t) : t ∈ [T1 , T2 ]};

.

(T1 , x(T1 )) ∈ AL , ρE (x(T1 ), θ0 ) ≤ L,

.

I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 , x(T1 )) + 4 inf{α(t) : t ∈ [T1 , T2 ]};

.

L , (T2 , x(T2 )) ∈ A

.

αg (T1 , T2 , x(T2 )) + 4 inf{α(t) : t ∈ [T1 , T2 ]}; I αg (T1 , T2 , x, u) ≤ U

.

L , ρE (x(T1 ), θ0 ) ≤ L, (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A

.

I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 , x(T1 ), x(T2 )) + 4 inf{α(t) : t ∈ [T1 , T2 ]},

.

then .I αg (T1 , T2 , x, u) is finite and

9.7 Proof of Theorem 9.5

327

mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≥ L0 /2.

.

Proposition 9.1 implies that there exists a neighborhood .V ⊂ V0 of f in .Mψ such that the following property holds: (d) for each pair of numbers .S1 ≥ 0, .S2 ∈ (S1 , S1 + L1 ], each .g ∈ V and each .(x, u) ∈ X(S1 , S2 ) which satisfies ρE (x(T1 ), θ0 ) ≤ L,

.

.

min{I g (S1 , S2 , x, u), I f (S1 , S2 , x, u)} ≤ (L1 + 2)(Δf + 4 + 2a0 )

the inequality |I g (S1 , S2 , x, u) − I f (S1 , S2 , x, u)| ≤ δ0

.

is valid. Assume that g ∈ V , T1 ≥ 0, T2 ≥ T1 + 4L1 ,

.

(9.115)

a borelian function .α : [0, ∞) → [0, 1] satisfies α(t1 )α(t2 )−1 ≥ λ for each t1 , t2 ∈ [T1 , T2 ] satisfying |t2 − t1 | ≤ L1 .

.

(9.116)

Property (c), (9.115), and (9.116) imply that .U αg (T1 , T2 ) is finite, for each L , .U αg (T1 , T2 , ξ1 ), .(T1 , ξ1 ) ∈ AL satisfying .ρE (ξ1 , θ0 ) ≤ L and each .(T2 , ξ2 ) ∈ A αg αg  (T1 , T2 , ξ2 ) and .U (T1 , T2 , ξ1 , ξ2 ) are finite. .U Assume that .(x, u) ∈ X(T1 , T2 ) satisfies at least one of the following conditions: (i) I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 ) + δ inf{α(t) : t ∈ [T1 , T2 ]};

.

(ii) (T1 , x(T1 )) ∈ AL , ρE (x(T1 ), θ0 ) ≤ L,

.

I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 , x(T1 )) + δ inf{α(t) : t ∈ [T1 , T2 ]};

.

(iii) L , ρE (x(T1 ), θ0 ) ≤ L, (T1 , x(T1 )) ∈ AL , (T2 , x(T2 )) ∈ A

.

I αg (T1 , T2 , x, u) ≤ U αg (T1 , T2 , x(T1 ), x(T2 )) + δ inf{α(t) : t ∈ [T1 , T2 ]}.

.

328

9 Stability and Genericity Results

Property (c), conditions (i)–(iii), (9.115), and (9.116) imply that mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≥ L0 /2.

(9.117)

S1 = min{t ∈ [max{T1 , L¯ 0 }, T2 ] : ρE (x(t), xf (t)) ≤ δ0 }.

(9.118)

.

Set .

(Note that if .T1 ≥ L¯ 0 and .ρE (x(T1 ), xf (T1 )) ≤ δ, then .S1 = T1 .) Clearly, in view of (9.112) and (9.117), .S1 is well-defined. Assume that .S1 ≥ T1 + L1 . It follows from property (c) applied to .(x, u) ∈ X(T1 , S1 ), conditions (i)–(iii) and Eqs. (9.115), (9.116) that mes({t ∈ [T1 , S1 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≥ L0 /2.

.

On the other hand in view of (9.112) and (9.118), mes({t ∈ [T1 , S1 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≤ L¯ 0 < L0 /2.

.

The contradiction we have reached proves that S1 ≤ T1 + L1 .

.

(9.119)

Set S2 = max{t ∈ [S1 , T2 ] : ρE (x(t), xf (t)) ≤ δ0 }.

.

(9.120)

If .S2 ≤ T2 − L1 , then it follows from property (c) applied to .(x, u) ∈ X(S2 , T2 ), conditions (i)–(iii) and Eqs. (9.115)–(9.117) that mes({t ∈ [S2 , T2 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≥ L0 /2.

.

This contradicts (9.120). Thus S2 ≥ T2 − L1 .

.

(9.121)

In order to complete the proof of Theorem 9.5 it is sufficient to show that ρE (x(t), xf (t)) ≤ , t ∈ [S1 , S2 ].

.

Assume the contrary. Then there exists an integer τ0 ∈ [S1 , S2 ]

.

such that

(9.122)

9.7 Proof of Theorem 9.5

329

ρE (x(τ0 ), xf (τ0 )) > .

.

(9.123)

In view of (9.115) and (9.121), S2 − S1 ≥ T2 − T1 − 2L1 ≥ 2L1 .

.

(9.124)

By (9.112), (9.122), and (9.124), there exist numbers .S3 , S4 such that S1 ≤ S3 ≤ τ0 ≤ S4 ≤ S2 ,

(9.125)

S4 = S3 + L0 /4.

(9.126)

.

.

Set S5 = max{t ∈ [S1 , S3 ] : ρE (x(t), xf (t)) ≤ δ0 },

(9.127)

S6 = min{t ∈ [S4 , S2 ] : ρE (x(t), xf (t)) ≤ δ0 }.

(9.128)

.

.

It follows from (9.126)–(9.128) that S6 − S5 ≥ S4 − S3 = L0 /4.

(9.129)

S6 − S5 ≥ L1 .

(9.130)

.

Assume that .

Property (c) applied to .(x, u) ∈ X(S5 , S6 ), conditions (i)–(iii) and Eqs. (9.101), (9.115) and (9.116) imply that mes({t ∈ [S5 , S6 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≥ L0 /2.

.

On the other hand it follows from (9.126)–(9.128) that mes({t ∈ [S5 , S6 ] : ρE (x(t), xf (t)) ≤ δ0 }) ≤ S4 − S3 = L0 /4.

.

The contradiction we have reached proves that S6 − S5 ≤ L1 .

.

Together with (9.126)–(9.128) this implies that L0 /4 ≤ S6 − S5 ≤ L1 .

.

(9.131)

Property (b) and Eqs. (9.111), (9.112), (9.127), (9.128), and (9.131) imply that there exist .(y, v) ∈ X(S5 , S6 ) and .t1 , t2 ∈ (0, bf ] such that

330

9 Stability and Genericity Results

y(S5 ) = x(S5 ), y(S6 ) = x(S6 ),

.

y(t) = xf (t), v(t) = uf (t), t ∈ [S5 + t1 , S6 − t2 ],

.

I f (S5 , S5 + t1 , y, v) ≤ I f (S5 , S + t1 , xf , uf ) + 1,

.

I f (S6 − t2 , S6 , y, v) ≤ I f (S6 − t2 , S6 , xf , uf ) + 1.

(9.132)

.

Equations (8.74), (9.3), (9.131), and (9.132) imply that I f (S5 , S6 , y, v) ≤ I f (S5 , S6 , xf , uf ) + 2 ≤ 2 + (L1 + 2)Δf + 2a0 .

(9.133)

.

Property (d) and (9.111), (9.115), (9.127), (9.131), and (9.133) imply that I f (S5 , S6 , y, v) ≤ 3 + (L1 + 2)Δf + 2a0 .

(9.134)

.

By (9.3), (9.116), (9.131), and (9.134),  I αg (S5 , S6 , x, u) =

S6

.

 (α(t)(g(t, x(t), u(t)) + a0 )dt − a0

S5

≥ inf{α(t) : t ∈ [S5 , S6 ]}

S6

 α(t)(g(t, x(t), u(t)) + a0 )dt − a0

S5

S6

α(t)dt S5

 .

α(t)dt S5

 .

S6

≥ I g (S5 , S6 , x, u) inf{α(t) : t ∈ [S5 , S6 ]} − a0

S6

α(t)dt.

(9.135)

S5

Property (d), conditions (i)–(iii) and Eqs. (9.111), (9.115), (9.127), (9.133), and (9.135) imply that I g (S5 , S6 , x, u) inf{α(t) : t ∈ [S5 , S6 ]}

.

 .

≤I

αg

(S5 , S6 , x, u) + (1 − λ)a0

S6

α(t)dt S5

.

≤ U αg (S5 , S6 , x(S5 ), x(S6 )) + δ0 inf{α(t) : t ∈ [S5 , S6 ]}  .

+ (1 − λ)a0

S6

α(t)dt.

(9.136)

S5

Conditions (i)–(iii) imply that I αg (S5 , S6 , x, u) ≤ U αg (S5 , S6 , x(S5 ), x(S6 )) + δ0 inf{α(t) : t ∈ [T1 , T2 ]}. (9.137)

.

9.7 Proof of Theorem 9.5

331

By (9.3), (9.116), (9.131), and (9.132), U αg (S5 , S6 , x(S5 ), x(S6 )) ≤ inf{I αg (S5 , S6 , ξ, η) : (ξ, η) ∈ X(S5 , S6 ),

.

I f (S5 , S6 , ξ, η) ≤ I f (S5 , S6 , y, v), ξ(S5 ) = x(S5 ), ξ(S6 ) = x(S6 )}

.

 .

= inf{



S6

S6

α(t)(g(t, x(t), u(t)) + a0 )dt − a0

S5

α(t)dt :

S5

(ξ, η) ∈ X(S5 , S6 ), I f (S5 , S6 , ξ, η) ≤ I f (S5 , S6 , y, v),

.

ξ(S5 ) = x(S5 ), ξ(S6 ) = x(S6 )}

.

.

≤ inf{λ−1 inf{α(t) : t ∈ [S5 , S6 ]}



S6

(g(t, x(t), u(t)) + a0 )dt

S5

 .

− a0

S6

α(t)dt :

S5

(ξ, η) ∈ X(S5 , S6 ), I f (S5 , S6 , ξ, η) ≤ I f (S5 , S6 , y, v),

.

ξ(S5 ) = x(S5 ), ξ(S6 ) = x(S6 )}

.

.

≤ λ−1 inf{α(t) : t ∈ [S5 , S6 ]} inf{I g (S5 , S6 , ξ, η) : (ξ, η) ∈ X(S5 , S6 ), I f (S5 , S6 , ξ, η) ≤ I f (S5 , S6 , y, v), ξ(S5 ) = x(S5 ), ξ(S6 ) = x(S6 )}

.

.

.

+ a0 inf{α(t) : t ∈ [S5 , S6 ]}(λ−1 − 1)(S6 − S5 )

≤ λ−1 inf{α(t) : t ∈ [S5 , S6 ]} inf{I f (S5 , S6 , ξ, η) + δ0 : (ξ, η) ∈ X(S5 , S6 ), ξ(S5 ) = x(S5 ), ξ(S6 ) = x(S6 ),

.

I f (S5 , S6 , ξ, η) ≤ I f (S5 , S6 , y, v)}

.

.

+ a0 inf{α(t) : t ∈ [S5 , S6 ]}(λ−1 − 1)(S6 − S5 ) .

≤ λ−1 inf{α(t) : t ∈ [S5 , S6 ]}I f (S5 , S6 , y, v)

+ a0 inf{α(t) : t ∈ [S5 , S6 ]}(λ−1 − 1)L1 + δ0 λ−1 inf{α(t) : t ∈ [S5 , S6 ]}. (9.138) By (9.132) and (9.138), .

U αg (S5 , S6 , x(S5 ), x(S6 ))

.

332 .

9 Stability and Genericity Results

≤ λ−1 inf{α(t) : t ∈ [S5 , S6 ]} inf{I f (S5 , S6 , ξ, η) + δ0 : (ξ, η) ∈ X(S5 , S6 ), I f (S5 , S6 , ξ, η) ≤ I f (S5 , S6 , y, v), ξ(S5 ) = x(S5 ), ξ(S6 ) = x(S6 )}

.

.

+ a0 inf{α(t) : t ∈ [S5 , S6 ]}(λ−1 − 1)L1

≤ λ−1 inf{α(t) : t ∈ [S5 , S6 ]}U f (S5 , S6 , x(S5 ), x(S6 ))

.

.

.

+ δ0 λ−1 inf{α(t) : t ∈ [S5 , S6 ]}

+ a0 inf{α(t) : t ∈ [S5 , S6 ]}L1 (λ−1 − 1).

(9.139)

In view of (9.132) and (9.133), U αg (S5 , S6 , x(S5 ), x(S6 )) ≤ 2 + 2a0 + (L1 + 2)Δf .

.

(9.140)

It follows from (9.137) and (9.139) that I αg (S5 , S6 , x, u) ≤ U αg (S5 , S6 , x(S5 ), x(S6 )) + δ inf{α(t) : t ∈ [T1 , T2 ]}

.

.

.

≤ δ inf{α(t) : t ∈ [S5 , S6 ]}

+ λ−1 inf{α(t) : t ∈ [S5 , S6 ]}U f (S5 , S6 , x(S5 ), x(S6 ) .

.

+ δ0 λ−1 inf{α(t) : t ∈ [S5 , S6 ]}

+ a0 L1 λ−1 (1 − λ) inf{α(t) : t ∈ [S5 , S6 ]}.

By (9.13), (9.135), and (9.141), I g (S5 , S6 , x, u) inf{α(t) : t ∈ [S5 , S6 ]}

.

 .

≤ a0 (1 − λ)

S6

α(t)dt + I αg (S5 , S6 , x, u)

S5 .

.

≤ a0 L1 λ−1 (1 − λ) inf{α(t) : t ∈ [S5 , S6 ]}

+ inf{α(t) : t ∈ [S5 , S6 ]}(δ + δ0 λ−1 + a0 L1 λ−1 (1 − λ) .

+ U f (S5 , S6 , x(S5 ), x(S6 )λ−1 ).

Together with (9.113), (9.114), and (9.140) this implies that I g (S5 , S6 , x, u) ≤ 2a0 L1 λ−1 (1 − λ) + δ + δ0 λ−1

.

(9.141)

9.8 Genericity .

333

+ U f (S5 , S6 , x(S5 ), x(S6 )) + λ−1 (1 − λ)(2 + 2a0 + Δf (L1 + 2)) .

≤ U f (S5 , S6 , x(S5 ), x(S6 )) + δ + 2δ0 + 2δ0 (2 + δ0 ) ≤ U f (S5 , S6 , x(S5 ), x(S6 )) + 4δ0 .

.

(9.142)

Property (d), (9.111), (9.115), (9.127), (9.131), (9.140), and (9.142) imply that |I f (S5 , S6 , x, u) − I g (S5 , S6 , x, u)| ≤ δ0 .

.

Together with (9.142) this implies that I f (S5 , S6 , x, u) ≤ U f (S5 , S6 , x(S5 ), x(S6 )) + 5δ0 .

.

The equation above and (9.111), (9.118), (9.127), (9.128) and (9.131) this implies that ρE (x(t), xf (t)) ≤ , t ∈ [S5 , S6 ].

.

Combined with (9.125)–(9.128) this implies that ρE (x(τ0 ), xf (τ0 )) ≤ .

.

This contradicts (9.123). The contradiction we have reached completes the proof of Theorem 9.5.

9.8 Genericity Assume that .M0 is a borelian measurable subset of .[0, ∞) × E × F such that M ⊂ M0 . Let .M0 be a set of borelian functions .g : M0 → R 1 such that for every .(t, x, u) ∈ M0 , .

g(t, x, u) ≥ ψ1 (t, x, u) − a0 .

.

(9.143)

For each .g ∈ M0 denote by .R(g) the restriction of g to .M which belongs to .Mψ . For each pair of numbers .T2 > T1 ≥ 0, each .(x, u) ∈ X(T1 , T2 ), each .g ∈ M0 and each .(T1 , y), (T2 , z) ∈ A define I g (T1 , T2 , x, u) = I R(g) (T1 , T2 , x, u), U g (T1 , T2 , y, z) = U R(g) (T1 , T2 , y, z),

.

U g (T1 , T2 ) = U R(g) (T1 , T2 ), U g (T1 , T2 , y) = U R(g) (T1 , T2 , y),

.

R(g) (T1 , T2 , z). g (T1 , T2 , z) = U U

.

334

9 Stability and Genericity Results

Recall that the set .Mψ is equipped with the uniformity which is defined by the base (8.10) and that assumption (A0) holds. This uniformity induces the topology in .Mψ denoted by .τw . A set .V ⊂ M0 is open in a weak topology if .V = R−1 (W ), where W is an open set in .Mψ equipped with the topology .τw . Assume that .M0 is equipped with the metric .dM which induces a topology which is stronger than the weak topology and is called a strong topology. Denote by .Mreg the set of all .f ∈ M0 such that there exist .(xf , uf ) ∈ X(0, ∞) and .bf > 0 satisfying (8.73) and (8.74) and such that (A1), (A4), (TP) hold for .R(f ). ¯ reg the closure of .Mreg in .M0 equipped with the strong topology. Denote by .M We prove the following result. Theorem 9.8 Let .M > 0 and . ∈ (0, 1). Then there exists an open (in the weak topology) and everywhere dense (in the strong topology) set .F satisfying ¯ reg Mreg ⊂ F ⊂ M

.

and such that for each .g ∈ F there exists an open set in the weak topology .V ⊂ M0 containing g, .δ ∈ (0, 1), .L > 0 and .(x∗ , u∗ ) ∈ X(0, ∞) such that for each .h ∈ V , each pair of numbers .T1 ≥ 0, .T2 ≥ T1 + 2L, .U h (T1 , T2 ) is finite and for each .(x, u) ∈ X(T1 , T2 ) which satisfies I h (T1 , T2 , x, u) ≤ min{U h (T1 , T2 ) + M, U h (T1 , T2 , x(T1 ), x(T2 )) + δ}

.

there exist .τ1 ∈ [T1 , T1 + L], .τ2 ∈ [T2 − L, T2 ] such that ρE (x(t), xf (t)) ≤ , t ∈ [τ1 , τ2 ].

.

Moreover, if .ρE (x(T2 ), xf (T2 )) ≤ δ, then .τ2 = T2 and if .ρE (x(T1 ), xf (T1 )) ≤ δ, T1 ≥ L, then .τ1 = T1 .

.

Proof Let .f ∈ Mreg . Theorem 9.2 implies that there exists an open neighborhood V (f ) of f in .M0 in the weak topology and .L(f ) > 0, δ(f ) ∈ (0, 1) such that the following property holds: for each .h ∈ V (f ), each pair of numbers .T1 ≥ 0, .T2 ≥ T1 + 2L(f ), .U h (T1 , T2 ) is finite and for each .(x, u) ∈ X(T1 , T2 ) which satisfies

.

I h (T1 , T2 , x, u) ≤ min{U h (T1 , T2 ) + M, U h (T1 , T2 , x(T1 ), x(T2 )) + δ(f )}

.

there exist .τ1 ∈ [T1 , T1 + L(f )], .τ2 ∈ [T2 − L(f ), T2 ] such that ρE (x(t), xf (t)) ≤ , t ∈ [τ1 , τ2 ];

.

9.9 Proof of Theorem 9.9

335

moreover, if .ρE (x(T2 ), xf (T2 )) ≤ δ(f ), then .τ2 = T2 and if .ρE (x(T1 ), xf (T1 )) ≤ δ(f ), .T1 ≥ L, then .τ1 = T1 . Define ¯ reg . F = ∪{V (f ) : f ∈ Mreg } ∩ M

.

It is not difficult to see that the set .F is open in the weak topology, everywhere ¯ reg and that our assertion holds. This completes dense in the strong topology of .M the proof of Theorem 9.8. The next generic result is proved in Sect. 9.9. Theorem 9.9 There exists an everywhere dense (in the strong topology) set .F ⊂ ¯ reg which is a countable intersection of open (in the weak topology) subsets of M ¯ .Mreg such that Mreg ⊂ F

.

and that for each .f ∈ F there exist .(x∗ , u∗ ) ∈ X(0, ∞) such that the following assertion holds. For each .M,  > 0 there exists a neighborhood V of f in .M0 in the weak topology and .δ, L > 0 such that for each .g ∈ V , each pair of integers .T1 ≥ 0, g .T2 ≥ T1 + 2L, .U (T1 , T2 ) is finite and for each .(x, u) ∈ X(T1 , T2 ) which satisfies I g (T1 , T2 , x, u) ≤ min{U g (T1 , T2 ) + M, U g (T1 , T2 , x(T1 ), x(T2 )) + δ}

.

there exist .τ1 ∈ [T1 , T1 + L], .τ2 ∈ [T2 − L, T2 ] such that ρE (x(t), xf (t)) ≤ , t ∈ [τ1 , τ2 ].

.

9.9 Proof of Theorem 9.9 Let .f ∈ Mreg and .n ≥ 1 be an integer. By (A4), there exists Δ(f, n) ∈ (0, 2−n )

.

(9.144)

such that the following property holds: (a) for each .(Ti , zi ) ∈ A, .i = 1, 2 satisfying .ρE (zi , xf (Ti )) ≤ Δ(f, n), .i = 1, 2 and .T2 ≥ bf there exist .(x1 , u1 ) ∈ X(T1 , T1 + bf ), .(x2 , u2 ) ∈ X(T2 − bf , T2 ) such that x1 (T1 ) = z1 , x1 (T1 + bf ) = xf (T1 + bf ),

.

336

9 Stability and Genericity Results

x2 (T2 ) = z2 , x2 (T2 − bf ) = xf (T2 − bf ),

.

I f (T1 , T1 + bf , x1 , u1 ) ≤ I f (T1 , T1 + bf , xf , uf ) + 2−n ,

.

I f (T2 − bf , T2 , x2 , u2 ) ≤ I f (T2 − bf , T2 , xf , uf ) + 2−n ,

.

ρE (x1 (t), xf (t)) ≤ 2−n , t ∈ [T1 , T1 + bf ],

.

ρ(x2 (t), xf (t)) ≤ 2−n , t ∈ [T2 − bf , T2 ].

.

Theorem 9.2, Proposition 9.1 and Eqs. (8.73), (8.74) imply that there exists an open neighborhood .U (f, n) of f in .M0 in the weak topology and L(f, n) ≥ n, δ(f, n) ∈ (0, 2−n )

.

(9.145)

such that the following properties hold: (b) for each .h ∈ U (f, n), each pair of numbers .T1 ≥ 0, .T2 ≥ T1 + 2L(f, n), I h (T1 , T2 , xf , uf )

.

is finite and for each .(x, u) ∈ X(T1 , T2 ) which satisfies I h (T1 , T2 , x, u) ≤ min{U h (T1 , T2 ) + n, U h (T1 , T2 , x(T1 ), x(T2 )) + δ(f, n)}

.

there exist .τ1 ∈ [T1 , T1 + L(f, n)], .τ2 ∈ [T2 − L(f, n), T2 ] such that ρE (x(t), xf (t)) ≤ 2−n Δ(f, n), t ∈ [τ1 , τ2 ];

.

moreover, if ρE (x(T2 ), xf (T2 )) ≤ δ(f, n),

.

then .τ2 = T2 and if .ρE (x(T1 ), xf (T1 )) ≤ δ(f, n), .T1 ≥ L(f, n), then .τ1 = T1 ; (c) for each .h ∈ U (f, n), each .T ≥ 0 and each .(x, u) ∈ X(T , T + bf ) which satisfies x(T ) = xf (T ),

.

I f (T , T + bf , x, u) ≤ I f (T , T + bf , xf , uf ) + 4

.

we have |I h (T , T + bf , x, u) − I f (T , T + bf , x, u)| ≤ 2−n Δ(f, n)

.

is true.

9.9 Proof of Theorem 9.9

337

Define ¯ F = (∩∞ n=1 ∪ {U (f, n) : f ∈ Mreg }) ∩ Mreg .

.

(9.146)

It is not difficult to see that the set .F is a countable intersection of open in ¯ reg . the weak topology, everywhere dense in the strong topology subsets of .M Assume that f ∈ F.

.

(9.147)

By (9.146) and (9.147), for each integer .n ≥ 1 there exists f (n) ∈ Mreg

.

such that f ∈ U (f (n) , n).

.

(9.148)

Property (b) and (9.148) imply that the following property holds: (d) for each integer .n ≥ 1, each pair of numbers .T1 ≥ 0, .T2 ≥ T1 + 2L(f (n) , n), f .U (T1 , T2 ) is finite and for each .(x, u) ∈ X(T1 , T2 ) which satisfies I f (T1 , T2 , x, u) ≤ min{U f (T1 , T2 ) + n, U f (T1 , T2 , x(T1 ), x(T2 )) + δ(f, n)}

.

we have ρE (x(t), xf (n) (t)) ≤ 2−n Δ(f, n), t ∈ [T1 + L(f (n) , n), T2 − L(f (n) , n)].

.

Property (d) implies that the following property holds: (e) for each pair of integers .n, m ≥ 1, each pair of numbers .T1 ≥ 0, T2 ≥ T1 + 2 max{L(f (n) , n), L(f (m) , m)}

.

and each .(x, u) ∈ X(T1 , T2 ) which satisfies I f (T1 , T2 , x, u)

.

.

≤ min{U f (T1 , T2 ) + 1, U f (T1 , T2 , x(T1 ), x(T2 )) + min{δ(f (i) , i) : i = n, m}

we have for .i = n, m, ρE (x(t), xf (i) (t)) ≤ 2−i Δ(f (i) , i), t ∈ [T1 + L(f (i) , i), T2 − L(f (i) , i)]

.

338

9 Stability and Genericity Results

and for every .T ≥ 2 max{L(f (n) , n), L(f (m) , m)]} and every t ∈ [max{L(f (n) , n), L(f (m) , m)}, T − max{L(f (n) , n), L(f (m) , m)}]

.

we have ρE (xf (m) (t), xf (n) (t)) ≤ 2−n Δ(f (n) , n) + 2−m Δ(f (m) , m).

.

Property (e) implies that the following property holds: (f) for each pair of integers .n, m ≥ 1 and each integer t ≥ max{L(f (n) , n), L(f (m) , m)}

.

we have ρE (xf (m) (t), xf (n) (t)) ≤ 2−n Δ(f (n) , n) + 2−m Δ(f (m) , m).

.

Consider a strictly increasing sequence of natural numbers .{nk }∞ k=1 such that n1 = 1

.

(9.149)

and that for each integer .k ≥ 1, we have nk+1 > L(f (nk ) , nk ) + nk (bf (nk ) + 1),

.

(9.150)

2−nk+1 Δ(f (nk+1 ) , nk+1 ) < 2−nk Δ(f (nk ) , nk ).

(9.151)

x∗ (t) = xf (n1 ) (t), u∗ (t) = uf (n1 ) (t), t ∈ [0, L(f (n2 ) , n2 )].

(9.152)

.

Set .

In view of property (b) and (9.152), I f (0, L(f (n2 ) , n2 ), x∗ , u∗ )

.

is finite. Assume that .k ≥ 1 is an integer and that we defined (x∗ , u∗ ) ∈ X(0, L(f (nk+1 ) ), nk+1 ))

.

such that x∗ (L(f (nk+1 ) , nk+1 )) = xf (nk ) (L(f (nk+1 ) , nk+1 )).

.

(9.153)

9.9 Proof of Theorem 9.9

339

(Clearly, our assumption holds for .k = 1.) Property (f), Eqs. (9.145), (9.150), and (9.151) imply that ρE (xf (nk ) (L(f (nk+1 ) , nk+1 ) + bf (nk ) ), xf (nk+1 ) (L(f (nk+1 ) , nk+1 ) + bf (nk ) ))

.

.

.

≤ 2−nk Δ(f (nk ) , nk ) + 2−nk+1 Δ(f (nk+1 ) , nk+1 )

≤ 2−nk +1 Δ(f (nk ) , nk ) ≤ Δ(f (nk ) , nk ).

(9.154)

Property (a) and Eqs. (9.153) and (9.154) imply that there exists (x∗ , u∗ ) ∈ X(L(f (nk+1 ) , nk+1 ), L(f (nk+1 ) , nk+1 ) + bf (nk ) )

.

such that x∗ (L(f (nk+1 ) , nk+1 )) = xf (nk ) (L(f (nk+1 ) , nk+1 )),

(9.155)

x∗ (L(f (nk+1 ) , nk+1 )+bf (nk ) ) = xf (nk+1 ) (L(f (nk+1 ) , nk+1 )+bf (nk ) ),

(9.156)

.

.

If

.

(nk )

(L(f (nk+1 ) , nk+1 ), L(f (nk+1 ) , nk+1 ) + bf (nk ) ), x∗ , u∗ )

(L(f (nk+1 ) , nk+1 ), L(f (nk+1 ) , nk+1 ) + bf (nk ) ), xf (nk ) , uf (nk ) ) + 2−nk (9.157) and that for every .t ∈ [L(f (nk+1 ) , nk+1 ), L(f (nk+1 ) , nk+1 ) + bf (nk ) )], .

≤ If

(nk )

ρE (x∗ (t), xf (nk ) (t)) ≤ 2−nk .

.

(9.158)

Set x∗ (t) = xf (nk+1 ) (t), u∗ (t) = uf (nk+1 ) (t),

.

t ∈ [L(f (nk+1 ) , nk+1 ) + bf (nk ) , L(f (nk+2 ) , nk+2 )].

.

(9.159)

It is not difficult to see that (x∗ , u∗ ) ∈ X(0, L(f (nk+2 ) , nk+2 )).

.

Properties (b), (c) and Eqs. (9.157), (9.159) imply that I f (0, L(f (nk+2 ) , nk+2 ), x∗ , u∗ )

.

is finite and the assumption made for k also holds for .k + 1. Therefore by induction we defined .(x∗ , u∗ ) ∈ X(0, ∞) such that .I f (0, T , x∗ , u∗ ) is finite

340

9 Stability and Genericity Results

for every .T > 0, (9.152) holds and for each integer .k ≥ 1 Eqs. (9.153) and (9.156)–(9.159) are true. Assume that .k ≥ 2 is an integer and that t ≥ L(f (nk ) , nk ).

.

(9.160)

By (9.60), (9.145) and (9.150), there exists an integer q≥k

(9.161)

t ∈ [L(f (nq ) , nq ), L(f (nq+1 ) , nq+1 )].

(9.162)

.

such that .

It follows from property (f) and Eqs. (9.145), (9.150), (9.161), and (9.162) that ρE (xf (nk ) (t), xf (nq ) (t)) ≤ 2−nk +1 Δ(f (nk ) , nk ).

.

(9.163)

If t ∈ [L(f (nq ) , nq ), L(f (nq ) , nq ) + bf (nq−1 ) ],

.

then it follows from property (f), (9.144) and (9.157) that ρE (x∗ (t), xf (nq ) (t)) ≤ ρE (x∗ (t), xf (nq−1 ) (t)) + ρE (xf (nq−1 ) (t), xf (nq ) (t))

.

.

≤ 2−nq−1 + 2−nq−1 +1 Δ(f (nq−1 ) , nq−1 ) ≤ 2−nq−1 +1 .

By (9.159), for each t ∈ [L(f (nq ) , nq ) + bf (nq−1 ) , L(f (nq+1 ) , nq+1 )]

.

we have x∗ (t) = xf (nq ) (t).

.

Thus in all the cases ρE (x∗ (t), xf (nq ) (t)) ≤ 2−nq−1 +1 .

.

In view of (9.161), (9.163) and the relation above, ρE (x∗ (t), xf (nk ) (t)) ≤ ρE (xf (nk ) (t), xf (nq ) (t)) + ρE (xf (nq ) (t), x∗ (t))

.

.

≤ 2−nk +1 Δ(f (nk ) , nk )+2−nq−1 ≤ 2−nk +1 Δ(f (nk ) , nk )+2−nk−1 +1 ≤ 3·2−nk−1 .

9.9 Proof of Theorem 9.9

341

Thus we have shown that the following property holds: (g) for each integer .k ≥ 2 and each .t ≥ L(f (nk ) , nk ) we have ρE (x∗ (t), xf (nk ) (t)) ≤ 3 · 2−nk−1 .

.

Assume that .M > 1 and . ∈ (0, 1). Choose an integer .k > 4 such that nk > M, 4 · 2−nk−1 < 

(9.164)

.

δ = δ(f (nk ) , nk ).

(9.165)

g ∈ U (f (nk ) , nk ),

(9.166)

T1 ≥ 0, T2 ≥ T1 + 2L(f (nk ) , nk )

(9.167)

.

and let

Assume that .

.

and that .(x, u) ∈ X(T1 , T2 ) satisfies I g (T1 , T2 , x, u) ≤ min{U g (T1 , T2 )+M, U g (T1 , T2 , x(T1 ), x(T2 ))+δ}.

.

(9.168)

By (9.164), (9.165), (9.167), and (9.168), I g (T1 , T2 , x, u) ≤ min{U g (T1 , T2 ) + nk , U g (T1 , T2 , x(T1 ), x(T2 )) + δ(f (nk ) , nk )}. (9.169) Property (b) and Eqs. (9.166) and (9.169) imply that there exist

.

τ1 ∈ [T1 , T1 + L(f (nk ) , nk )], τ2 ∈ [T2 − L(f (nk ) , nk ), T2 ]

.

such that ρE (x(t), xf (nk ) (t)) ≤ 2−nk Δ(f (nk ) , nk ), t ∈ [τ1 , τ2 ].

.

(9.170)

Property (g) and Eqs. (9.164) and (9.170) imply that for each .t ∈ [max{τ1 , L(f (nk ) , nk )}, τ2 ], ρE (x(t), x∗ (t)) ≤ ρE (x(t, )xf (nk ) (t)) + ρE (xf (nk ) (t), x∗ (t))

.

.

≤ 2−nk Δ(f (nk ) , nk ) + 3 · 2−nk−1 ≤ 4 · 2−nk−1 < .

This completes the proof of Theorem 9.9.

342

9 Stability and Genericity Results

9.10 A Turnpike Result We continue to use the notation, definitions, and assumptions introduced in the previous sections of this chapter. We also assume that the following assumption holds: (A5) For each .f ∈ Mψ , each .M,  > 0 there exists .δ ∈ (0, ) such that for each −1 , T + 1] and each .(x , u ) ∈ X(T , T ), .i = 1, 2 .T1 ≥ 0, each .T2 ∈ [T1 + 2 1 i i 1 2 satisfying I f (T1 , T2 , xi , ui ) ≤ M, i = 1, 2

.

and each .τ ∈ [T1 , T2 ] satisfying 0 < T2 − τ ≤ δ, ρE (x1 (τ ), x2 (τ )) ≤ δ

.

the inequality .ρE (x1 (T2 ), x2 (T2 )) ≤  holds. In the sequel we show that it holds for our three classes of optimal control problems. Denote by .M0,reg the set of all .f ∈ M0 such that there exist .(xf , uf ) ∈ X(0, ∞) and .bf > 0 satisfying (8.73) and (8.74) and such that (A1), (A4) hold. Clearly, .Mreg ⊂ M0,reg . We prove the following turnpike result. Theorem 9.10 Assume that .f ∈ Mψ , .(xf , uf ) ∈ X(0, ∞) and .bf > 0 and that (8.73) and (8.74), (A1), (A4) hold. Then the integrand f has (TP) if and only if the following property holds: (P3) for each . > 0 and each .M > 0 there exists .L > 0 such that for each .T1 ≥ 0, each .T2 ≥ T1 + L and each .(x, u) ∈ X(T1 , T2 ) which satisfies I f (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M

.

the inequality mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) ≤ L

.

is valid. Proof In view of Theorems 8.25 and 8.26, TP implies WTP. Evidently, WTP implies (P3). Thus (TP) implies (P3).Clearly, (P3) implies (P2). In view of Theorem 8.25, in order to complete our proof it is sufficient to show that (P3) implies (P1). Assume that (P3) holds and that .(x, u) ∈ X(0, ∞) is .(f )-good. We show that .

lim ρ(x(t), xf (t)) = 0.

t→∞

9.10 A Turnpike Result

343

Assume the contrary. Then there exists . ∈ (0, 1/4) and a sequence of numbers {tk }∞ k=1 such that

.

.

t1 ≥ 8, tk+1 ≥ tk + 8 for all integers k ≥ 1,

(9.171)

ρE (x(tk ), xf (tk )) >  for all integers k ≥ 1.

(9.172)

.

Theorem 8.5 implies that there exists .M1 > 0 such that ρE (xf (t), θ0 ) ≤ M1 , ρE (x(t), θ0 ) ≤ M1 , t ∈ [0, ∞),

(9.173)

|I f (0, t, x, u) − I f (0, t, xf , uf )| < M1 , t ∈ (0, ∞).

(9.174)

.

.

Property (P3) and (9.174) imply that the following property holds: (a) for each .γ > 0 there exists .Lγ > 0 such that for each .T > Lγ , we have mes({t ∈ [0, T ] : ρ(x(t), xf (t)) > γ }) < Lγ .

.

(9.175)

In view of (9.76), there exists .M2 > 0 such that I f (S1 , S2 , xf , uf ) ≤ M2 for every S1 ≥ 0 and every S2 ∈ [S1 , S1 + 1]. (9.176) By (9.174), for all .S > T ≥ 0, .

|I f (T , S, x, u) − I f (T , S, xf , uf )| ≤ 2M1 .

.

(9.177)

Equations (9.176) and (9.177) imply that I f (S1 , S2 , x, u) ≤ M2 + 2M1 for all S1 ≥ 0 and all S2 ∈ (S1 , S1 + 1]. (9.178) Assumption (A5) implies that there exists .δ ∈ (0, ) such that the following property holds: (b) for each .S1 ≥ 0, each .S2 ∈ [S1 + 2−1 , S1 + 1], each .(xi , ui ) ∈ X(S1 , S2 ), .i = 1, 2 satisfying .

I f (S1 , S2 , xi , ui ) ≤ 2M1 + M2 , i = 1, 2

.

and each .τ ∈ [S1 , S2 ] satisfying 0 < S2 − τ ≤ δ, ρE (x1 (τ ), x2 (τ )) ≤ δ

.

the inequality .ρE (x1 (S2 ), x2 (S2 )) ≤  holds.

344

9 Stability and Genericity Results

Let .k ≥ 1 be an integer. It follows from (9.171), (9.172), (9.176), (9.178) and property (b) applied with .S1 = tk − 1, .S2 = tk , .x1 = x, .x2 = xf we obtain that for each .τ ∈ [tk − δ, tk ) we have ρE (x(t), xf (t)) > δ.

.

Therefore .

∪∞ k=1 [tk − δ, tk ) ⊂ {t ∈ [0, ∞) : ρE (x(t), xf (t)) > δ}.

This contradicts property (a). The contradiction we have reached proves that .

lim ρ(x(t), xf (t)) = 0

t→∞

and property (P1) holds. This completes the proof of Theorem 9.10. Assume that .φ : E × E → [0, 1] is a continuous function satisfying .φ(x, x) = 0 for all .x ∈ E and such that the following properties hold: (i) for each . > 0 there exists .δ > 0 such that if .x, y ∈ E and .φ(x, y) ≤ δ, then .ρE (x, y) ≤ . (ii) for each . > 0 there exists .δ > 0 such that if .x, y ∈ E and .ρE (x, y) ≤ δ, then .φ(x, y) ≤ . Let .f ∈ M0,reg , .(xf , uf ) ∈ X(0, ∞) and .bf > 0 and that (8.73) and (8.74), (A1), (A4) hold and let .r ∈ (0, 1). Define f

fr (t, x, u) = f (t, x, u) + rφ(x, xt ), (t, x, u) ∈ M0 .

.

We suppose that for each .f ∈ Mreg and each .r ∈ (0, 1), .fr ∈ M0 and .fr → f as r → 0+ in the strong topology.

.

Theorem 9.11 Let .f ∈ Mreg and .r ∈ (0, 1). Then the function .fr has TP and (xfr , ufr ) = (xf , uf ), .bfr = bf .

.

Proof It is not difficult to see that (8.73), (8.74), (A2), (A4) hold. In view of Theorem 9.10, it is sufficient to show that .fr has (P3). Let ., M > 0. Theorem 8.5.implies that there exists .M0 > 0 such that for each .S2 > S1 ≥ 0 and each .(x, u) ∈ X(S1 , S2 ), I f (S1 , S2 , x, u) + M0 ≥ I f (S1 , S2 , xf , uf ).

.

Property (i) implies that there exists .δ ∈ (0, ) such that for (x, y) ∈ E × E satisfying ρ(x, y) >  we have φ(x, y) > δ.

.

(9.179)

9.10 A Turnpike Result

345

Set L = (rδ)−1 (M0 + M).

.

(9.180)

Assume that .T1 ≥ 0, .T2 ≥ T1 + L, .(x, u) ∈ X(T1 , T2 ) satisfies I fr (T1 , T2 , x, u) ≤ I fr (T1 , T2 , xf , uf ) + M.

.

(9.181)

By (9.179), (9.181) and the definition of .fr , 

T2

I f (T1 , T2 , x, u) + r

φ(x(t), xf (t))dt

.

T1 .

≤ I fr (T1 , T2 , x, u) ≤ I f (T1 , T2 , xf , uf ) + M .

≤ I f (T1 , T2 , x, u) + M0 + M.

Together with (9.180) and the choice of .δ this implies that  M0 + M ≥ r

T2

φ(x(t), xf (t))dt

.

T1 .

≥ rδmes({t ∈ [T1 , T2 ] : φE (x(t), xf (t)) > })

and mes({t ∈ [T1 , T2 ] : ρE (x(t), xf (t)) > }) ≤ δ −1 r −1 (M0 + M) = L.

.

Thus .fr has (P3). Theorem 9.11 is proved. Theorem 9.11 implies that {fr : f ∈ M0,reg , r ∈ (0, 1)} ⊂ Mreg ⊂ M0,reg

.

and .fr → f as .r → 0+ for every .f ∈ M0,reg . Clearly, ¯ 0,reg ¯ reg = M M

.

in the strong topology. Note that Theorems 9.8 and 9.9 hold in our case.

346

9 Stability and Genericity Results

9.11 Spaces od Smooths Integrands We continue to use the notation, definitions, and assume the assumptions introduced in the previous section. In particular, we assume that (A5) holds. Define φ(x, y) = min{ρ(x, y), 1}, x, y ∈ E

.

Clearly, .φ : E × E → [0, 1] is a continuous function, (i), (ii) hold, .φ is a Lipschitz function with a Lipschitz constant 1. We equip the space .Mψ with the uniformity determined by the base Es () = {(h, g) ∈ Mψ × Mψ : |h(t, x, u) − g(t, x.u)| ≤ 

.

.

.

for each (t, x, u) ∈ M}

∩ {(h, g) ∈ Mψ × Mψ :

|(h − g)(t, x1 , u) − (h − g)(t, x2 , u)| ≤ ρE (x1 , x2 )

.

.

for each (t, x1 , u), (t, x2 , u) ∈ M} .

∩ {(h, g) :∈ Mψ × Mψ :

(h − g)(t, x, u1 ) = (h − g)(t, x, u2 ) for each (t, x, u1 ), (t, x, u2 ) ∈ M},

.

where . > 0. Clearly, this uniform space .Mψ is Hausdorff and has a countable base. Therefore it is metrizable (by a metric .ds ). It is not difficult to see that the uniform space .(Mψ , ds ) is complete and that this uniformity is stronger than the uniformity determined by the base (8.10). It is equipped with a topology induced by the uniformity which is called sometimes as a strong topology. Denote by .Ml the set of all .f ∈ Mψ such that .f (t, ·, ·) is a lower semicontinuous function for every .t ≥ 0 on the set .{(x, u) ∈ E × F : (t, x, u) ∈ M}, by .Mc the set of all .f ∈ Mψ such that .f (t, ·, ·) is a continuous function for every .t ≥ 0 on the set .{(x, u) ∈ E × F : (t, x, u) ∈ M}, by .ML the set of all .f ∈ Mψ such that .f (t, ·, ·) is a Lipschitz function for every .t ≥ 0 on the set .{(x, u) ∈ E × F : (t, x, u) ∈ M} and by .MlL the set of all .f ∈ Mψ such that .f (t, ·, ·) is a Lipschitz function on all bounded subsets of the set .{(x, u) ∈ E × F : (t, x, u) ∈ M} for every .t ≥ 0. Clearly, all these sets are closed in the metric space .(Mψ , ds ). Clearly, .M0 can be any of the spaces .Mψ , Ml , Mc , ML , MlL and in any of them we have .fr → f as + .r → 0 for each .f ∈ M0 . Now we consider spaces of smooth integrands using the notation, definitions, ˜ ψ the set of all and assumptions of Sect. 6.15. Let .k ≥ 1 be an integer. Denote by .M 1 borelian functions .g : [0, ∞) × E × F → R , which satisfy

9.11 Spaces od Smooths Integrands

347

g(t, x, u) ≥ ψ1 (t, x, u) − a0 for each (t, x, u) ∈ [0, ∞) × E × F

.

(see (8.3)–(8.5)). Assume that .(E, ·), .(F, ·) are Banach spaces, .ρE (x, y) = x−y, .x, y ∈ E, .ρF (x, y) = x − y, x, y ∈ F , .E × F is a Banach space, (x, y) = x + y, x ∈ E, y ∈ F.

.

We assume that there exists a function .λ∗ ∈ C k (E, R 1 ) such that (see (6.144) with .X = E) 0 ≤ λ∗ (x) ≤ 1, x ∈ E,

.

{x ∈ E : λ∗ (x) > 0} ⊂ {x ∈ X : x ≤ 1},

.

λ∗ (z) > 0 for some z ∈ E,

.

(j )

sup{λ∗ (z) : z ∈ E, j = 0, . . . , k} < ∞.

.

Then (see Sect. 6.15) there exists .φ0 ∈ C k (X; R 1 ) such that φ0 (0) = 0, φ0 (z) ∈ [0, 1], z ∈ E,

.

.

(i)

sup{φ0 (z) : z ∈ E, i = 0, . . . , k} < ∞

and that the following property holds: for each . > 0 there exists .δ > 0 such that if .x ∈ E and .φ(x) ≤ δ, then .x ≤ . ˜ ψ such that for each .t ≥ 0, .f (t, ·, ·) ∈ ˜ the set of all .f ∈ M Denote by .M k ˜ ˜ C (E × F ). Denote by .Ml the set of all lower semicontinuous functions .f ∈ M ˜ ˜ ˜ and by .Mc the set of all continuous functions .f ∈ M. The space .M is equipped with the uniformity determined by the base ˜ ×M ˜ : for each t ≥ 0, E() = {(h, g) :∈ M

.

(k) dE×F (g1 (t, ·, ·), g2 (t, ·, ·)) ≤ }

.

where . > 0. This uniformity is metrizable (by a metric .ds ) and complete and ˜ l , .M ˜ c are closed subsets of .M. ˜ Clearly, .M0 can be induces a topology. Clearly, .M ˜ ˜ ˜ any of the spaces .M, Ml , Mc with the metric .ds . Let .M0 can be any of the spaces ˜ c , .M0 = M, .φ(x, y) = φ0 (x − y), .x, y ∈ E. Consider .M0,reg . It is ˜ M ˜ l, M .M, not difficult to see that .fr → f as .r → 0+ for each .f ∈ Mreg . Thus Theorems 9.8 ¯ 0,reg . ¯ reg = M and 9.9 hold in our case with .M

348

9 Stability and Genericity Results

9.12 Assumption (A5) for the First Class of Problems Consider the first class of problems. (A5) follows from the next result. Proposition 9.12 Let .M > 0,  ∈ (0, 1), .≤ τ < τ0 < τ1 . Then there exists .δ > 0 such that for each .T1 ≥ 0, each .T2 ∈ [T1 + τ0 , T1 + τ1 ], each .g ∈ Mψ , each .(xi , ui ) ∈ X(T1 , T2 ), .i = 1, 2 satisfying I f (T1 , T2 , xi , ui ) ≤ M, i = 1, 2

.

and each .t1 , t2 ∈ [T1 + τ, T2 ] satisfying 0 < t2 − t1 ≤ δ, (x1 (t1 ) − x2 (t1 ) ≤ δ

.

the inequality .x1 (t2 ) − x2 (t2 ) ≤  holds. Proof Proposition 8.1 implies that there exists .M1 > M such that the following property holds: (i) for each .g ∈ Mψ , each .T1 ≥ 0, each .T2 ∈ [T1 + τ0 , T1 + τ1 ] and each .(x, u) ∈ X(T1 , T2 ) satisfying I g (T1 , T2 , x, u) ≤ M

.

we have x(t) ≤ M1 , t ∈ [T1 + τ, T2 ].

.

Set 1 = 4−1 (2Δ1 + 1)−1

.

(9.182)

(see Sect. 8.2). By (8.5) and (8.20), there exists .h0 > 0 such that for each .(t, x, u) ∈ M satisfying .G(t, x, u) − a0 x ≥ h0 and each .g ∈ Mψ we have g(t, x, u) ≥ 41−1 (M1 + M2 + a0 + 8 + a0 τ0 )(G(t, x, u) − a0 x+ )

.

(9.183)

Choose .δ ∈ (0, /4) such that δΔ1 (2a0 M0 + 2h0 + 2) < /4.

(9.184)

g ∈ Mψ , 0 ≤ T1 , T2 ∈ [T1 + τ0 , T1 + τ1 ],

(9.185)

.

Assume that .

9.12 Assumption (A5) for the First Class of Problems

349

(xi , ui ) ∈ X(T1 , T2 ), i = 1, 2

(9.186)

I f (T1 , T2 , xi , ui ) ≤ M, i = 1, 2,

(9.187)

.

T1 ≤ t1 < t2 ≤ T2 , t1 , t2 ∈ [T1 + τ, T2 ],

(9.188)

0 < t2 − t1 ≤ δ, (x1 (t1 ) − x2 (t1 ) ≤ δ.

(9.189)

.

satisfy .

.

In order to complete the proof it is sufficient to show that the inequality .(x1 (t2 ) − x2 (t2 ) ≤ . Property (i), (9.185)–(9.187) imply that for .i = 1, 2, xi (t) ≤ M1 , t ∈ [T1 + τ, T2 ].

.

(9.190)

Property (v) from Sect. 8.2 and Eq. (8.18) imply that for .i = 1, 2,  xi (t2 ) = U (t2 , t1 )xi (t1 ) +

t2

U (t2 , s)G(s, xi (s), ui (s))ds.

.

(9.191)

t1

It follows from (9.189) and (9.191) that x1 (t2 ) − x2 (t2 ) ≤ U (t2 , t1 )x1 (t1 ) − x2 (t1 )

.

 .

+

t2

U (t2 , s)G(s, x1 (s), u1 (s))ds

t1

 .

+

t2

U (t2 , s)G(s, x2 (s), u2 (s))ds

t1

 .

≤ Δδ δ + Δδ

t2

G(s, x1 (s), u1 (s))ds

t1

 .

+ Δδ

t2

G(s, x2 (s), u2 (s))ds.

(9.192)

t1

Let .i ∈ {1, 2}. Set Ω1 = {t ∈ [t1 , t2 ] : G(t, xi (t), ui (t)) − a0 xi (t) ≥ h0 }, Ω2 = [t1 , t2 ] \ Ω1 . (9.193) By (8.5), (9.187)–(9.190) and (9.193), .

350

9 Stability and Genericity Results





t2

G(s, xi (s), ui (s))ds ≤ a0

.

t1

xi (s)ds

t1

 .

t2

+

t2 t1

(G(s, xi (s), ui (s)) − a0 xi (s))+ ds 

.

≤ a0 δM1 + h0 δ + Ω1

.

−1

≤ a0 δM1 + h0 δ + 4

(G(s, xi (s), ui (s)) − a0 xi (s))+ ds −1



1 (M1 + M + a0 (τ1 + 1) + 8)

g(s, xi (s), ui (s))ds Ω1

.

≤ δ(a0 M1 + h0 ) + (4−1 1 (M1 + M + a0 (τ1 + 1) + 8)−1 )(I g (T1 , T2 , xi , ui ) + a0 τ1 ) .

≤ δ(a0 M1 + h0 ) + (4−1 1 (M1 + M0 + a0 (τ1 + 1) + 8)−1 )(M + a0 τ1 ) .

≤ δ(a0 M1 + h0 ) + 4−1 1 .

(9.194)

In view of (9.182), (9.184), (9.192), and (9.194), x1 (t2 ) − x2 (t2 ) ≤ Δ1 δ + 2Δ1 δ(a0 M1 + h0 ) + 2−1 1 Δ1 < .

.

Proposition 9.12 is proved.

9.13 Assumption (A5) for the Second Class of Problems Consider the second class of problems. We assume that for this class of problem for each .(t, x, u) ∈ M, ψ1 (t, x, u) = max{ψ(x), ψ(u2 )}.

.

(9.195)

(A5) follows from the next result. Proposition 9.13 Let .M0 > 0,  ∈ (0, 1), .0 < τ0 < τ1 . Then there exists .δ > 0 such that for each .T1 ≥ 0, each .T2 ∈ [T1 + τ0 , T1 + τ1 ], each .g ∈ Mψ , each .(xi , ui ) ∈ X(T1 , T2 ), .i = 1, 2 satisfying I g (T1 , T2 , xi , ui ) ≤ M, i = 1, 2

.

and each .t1 , t2 ∈ [T1 , T2 ] satisfying 0 < t2 − t1 ≤ δ, x1 (t1 ) − x2 (t1 ) ≤ δ

.

(9.196)

9.13 Assumption (A5) for the Second Class of Problems

351

the inequality x1 (t2 ) − x2 (t2 ) ≤ 

.

(9.197)

holds. Proof Recall that .B ∈ L(F, E−1 ) is an admissible operator for the semigroup S(t) = eAt , .t ≥ 0. Proposition 7.1 implies that there exist exist .M∗ ≥ 1, .ω∗ ∈ R 1 such that

.

eAt  ≤ M∗ eω∗ t , t ∈ [0, ∞)

.

(9.198)

and for each .τ ≥ 0, .Φτ ∈ L(L2 (0, τ ; F ), E) is defined by  Φτ u =

τ

eA(τ −s) Bu(s)ds, u ∈ L2 (0, τ ; F )

.

(9.199)

0

(see Sect. 7.5). Choose .Λ > 1 such that Λ−1/2 (Φ1  + 1)(M + a0 τ1 + 1) < /8.

.

(9.200)

By (9.195), there exists .c0 > 0 such that the following property holds: (i) For each .(t, x, u) ∈ M satisfying .u ≥ c0 each .g ∈ Mψ and each .t ≥ 0, g(t, x, u) ≥ Λu2 .

(9.201)

M∗ e|ω∗ | δ < /8, Φ1 c0 δ 1/2 < /8.

(9.202)

.

Fix .δ ∈ (0, 1) such that .

Assume that .g ∈ Mψ , .T1 ≥ 0, .T2 ∈ [T1 + τ0 , T1 + τ1 ], .(xi , ui ) ∈ X(T1 , T2 ), i = 1, 2 satisfy

.

I g (T1 , T2 , xi , ui ) ≤ M, i = 1, 2,

.

(9.203)

T1 ≤ t1 < t2 ≤ T2 and that (9.196) holds. In order to complete the proof it is sufficient to show that (9.197) is true. It is clear that .(xi , ui ) ∈ X(t1 , t2 ), .i = 1, 2.. In view of (8.24), for .i = 1, 2,

.

xi (t2 ) = eA(t2 −t1 ) xi (t1 ) +



t2

.

t1

eA(t2 −s) Bui (s)ds.

(9.204)

352

9 Stability and Genericity Results

Equations (9.129), (9.196), (9.198), (9.204) imply that x2 (t2 ) − x1 (t2 )

.

= eA(t2 −t1 ) (x1 (t1 ) − x2 (t1 )) + 

.



t2

eA(t2 −s) Bu1 (s)ds

t1

 .

t2

−

eA(t2 −s) Bu2 (s)ds

t1

.

≤ eA(t2 −t1 ) (x1 (t1 ) − x2 (t1 )) + 



t2

eA(t2 −s) Bu1 (s)ds

t1

 .

t2

+

eA(t2 −s) Bu2 (s)ds

t1

.

≤ δM∗ e|ω∗ |δ +   .



t2 −t1

+

t2 −t1

eA(t2 −t1 −s) Bu1 (t1 + s)ds

0

eA(t2 −t1 −s) Bu2 (t1 + s)ds

0 .

≤ δM∗ e|ω∗ | + Φt2 −t1 u1 (t1 + ·)L2 (0,t2 −t1 ;F ) + Φt2 −t1 u2 (t1 + ·)L2 (0,t2 −t1 ;F )

.

.

≤ δM∗ e|ω∗ + Φ1 (



t2

 u1 (s)2 ds)1/2 + Φ1 (

t1

t2

u2 (s)2 ds)1/2 .

(9.205)

t1

Let .i ∈ {1, 2} and Ωi = {t ∈ [t1 , t2 ] : ui (t) ≥ c0 }.

(9.206)

.

Property (i) and Eqs. (9.195), (9.196), (9.203), and (9.206) imply that  Φ1 (



t2



ui (s)2 ds)1/2 = Φ1 (

.

t1

ui (s)2 ds + Ω

 .

ui (s)2 ds)1/2



≤= Φ1 (

ui (s)2 ds)1/2 + Φ1 ( Ω

 .

[t1 ,t2 ]\Ω

[t1 ,t2 ]\Ω

ui (s)2 ds)1/2

Λ−1 G(s, xi (s), ui (s))ds)1/2 + Φ1 c0 δ 1/2

≤ Φ1 ( Ω

9.14 Assumption (A5) for the Third Class of Problems

353

≤ |Φ1 c0 δ 1/2 + Φ1 (Λ−1 I g (T1 , T2 , xi , ui ) + a0 τ1 )1/2

.

.

≤ Φ1 c0 δ 1/2 + Φ1 Λ−1/2 (M1 + a0 τ1 )1/2 .

(9.207)

By (9.200), (9.202), (9.205), and (9.207), x2 (t2 ) − x1 (t2 ) ≤ δM∗ e|ω∗ | + 2Φ1 c0 δ 1/2

.

.

+ 2Φ1 Λ−1/2 (M1 + a0 τ1 )1/2 < .

Proposition 9.13 is proved.

9.14 Assumption (A5) for the Third Class of Problems Consider the third class of problems. (A5) follows from the next result. Proposition 9.14 Let .M > 0,  ∈ (0, 1), .0 < τ0 ≤ τ1 . Then there exists .δ > 0 such that for each .T1 ≥ 0, each .T2 ∈ [T1 + τ0 , T1 + τ1 ], each .g ∈ Mψ , each .(xi , ui ) ∈ X(T1 , T2 ), .i = 1, 2 satisfying I f (T1 , T2 , xi , ui ) ≤ M, i = 1, 2

.

(9.208)

and each .t1 , t2 ∈ [T1 + τ, T2 ] satisfying 0 < |t2 − t1 | ≤ δ

.

the inequality .ρE ((x1 (t2 ), x2 (t2 )) ≤  holds. Proof Proposition 8.3 and (8.56) imply that there exists .M1 > M such that the following property holds: (i) for each .g ∈ Mψ , each .T1 ≥ 0, each .T2 ∈ [T1 + τ0 , T1 + τ1 ] and each .(x, u) ∈ X(T1 , T2 ) satisfying I g (T1 , T2 , x, u) ≤ M

.

we have ρE (x(t), θ0 ) ≤ M1 , t ∈ [T1 , T2 ].

.

By (8.54), there exists .h0 > 0 such that for each .(t, x, u) ∈ M satisfying G(t, x, u) − a0 ρE (x, θ0 ) ≥ h0 and each .g ∈ Mψ we have

.

g(t, x, u) ≥ 41−1 (M1 + M + a0 + 8 + a0 (τ0 + 1) + 8(G(t, x, u) − a0 ρE (x, θ0 )+ ). (9.209)

.

354

9 Stability and Genericity Results

Choose .δ ∈ (0, 1) such that δ < /8, δ(a0 M1 + h0 ) < /4.

(9.210)

.

Assume that g ∈ Mψ , 0 ≤ T1 , T2 ∈ [T1 + τ0 , T1 + τ1 ],

.

(x, u) ∈ X(T1 , T2 ), (9.208) is true, and .t1 , t2 ∈ [T1 + τ, T2 ] satisfy

.

0 < t2 − t1 ≤ δ.

(9.211)

.

Property (i), (8.55), and (9.208) imply that ρE (x(t), θ0 ) ≤ M1 , t ∈ [T1 , T2 ].

(9.212)

.

Clearly, 

t2

ρE (x(t2 ), x(t1 )) ≤

G(s, x(s), u(s))ds.

.

(9.213)

t1

Set Ω = {t ∈ [T1 , T2 ] : G(t, x(s), u(s)) − a0 ρE (x(s), θ0 ) ≥ h0 }

.

(9.214)

By (9.208)–(9.210) and (9.212)–(9.214),  ρE (x(t2 ), x(t1 )) ≤

t2

G(s, x(s), u(s))ds

.

t1

 .

≤ a0

t2

 ρE (x(s), θ0 )ds +

t1

t2 t1

(G(s, x(s), u(s)) − a0 ρE (x(s), θ0 )+ ds

 .

≤ a0 δM1 + h0 δ + Ω

.

−1

≤ δ(a0 M1 + h0 ) + 4

(G(s, x(s), u(s)) − a0 ρE (x(s), θ0 ))+ ds −1



(M1 + M + a0 (τ0 + 1)) + 8

G(s, x(s), u(s))ds Ω

.

≤ δ(a0 M1 + h0 ) + 4−1 (M1 + M + a0 (τ0 + 1) + 8)−1 (I g (T1 , T2 , x, u) + a0 τ1 ) .

≤ δ(a0 M1 + h0 ) + 4−1  < .

Proposition 9.14 is proved.

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Index

A Absolutely continuous function, 2 Adjoint operator, 17 Admissible control operator, 17 Approximate solution, 1

B Banach space, 14 Base, 3 Bilinear form, 22 Bochner integrable function, 14 Bochner integral, 21 Borelian function, 12 Borel set, 6 Bounded operator, 197

C Cardinality, 6 Compact set, 8 Complete metric space, 1, 3 Continuous function, 14 Control system, 12 Convex process, 38 Cost function, 2 Countable base, 12 .C0 semigroup, 17

D Domain space, 14 Duality pairing, 6 Dual space, 14

E Eigenvalue, 198 Euclidean norm, 2 Evolution equation, 200 F Frechet differentiable norm, 195

G Generator, 17 Good function, 4 Graph, 11 Graph norm, 16 H Half-axis, 129 Hausdorff space, 12 Hilbert space, 6 Homogeneous Cauchy problem, 14 I Identity mapping, 14 Increasing function, 3 Infinite dimensional optimal control, 197 Inner product, 6 Integral functional, 2 Integrand, 1 K Korteweg-de Vries equation, 211

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. J. Zaslavski, Turnpike Phenomenon in Metric Spaces, Springer Optimization and Its Applications 201, https://doi.org/10.1007/978-3-031-27208-0

361

362 L Lebesgue integral, 21 Lebesgue measurable function, 6 Lebesgue measurable set, 6 Lebesgue measure, 6 Linear operator, 14 Lipschitz function, 27 Locally absolutely continuous functions, 201 Locally convex Hausdorff space, 24 Lower semicontinuity property, 2 Lower semicontinuous function, 124 Lyapunov function, 7

M Measurable function, 22 Metric, 5 Metric space, 3 Metrizable uniformity, 3 Mild solution, 15

Index Relative topology, 23 Resolvent, 198

S Set-valued mapping, 21 Simple function, 21 Sobolev space, 201 Spectrum, 198 Strong derivative, 23 Strongly continuous group of operators, 199 Strongly continuous semigroup of operators, 198 Strongly measurable function, 21

T Topology, 3 Trajectory, 5 Trajectory-control pair, 11 Turnpike, 4 Turnpike property, 2

N Normed space, 6

O Optimal control problem, 11

U Uniformity, 3 Uniform space, 190 Upper semicontinuous function, 24

P Point spectrum, 198 Product topology, 202 Proper mapping, 24

V Variational principle, 195 Vector space, 123 Von Neumann-gale model, 38

R Range, 14 Reflexive Banach space, 23 Relatively compact subset, 23

W Weak solution, 200 Weak topology, 24 Well-posed problem, 14