Turnpike Theory of Continuous-Time Linear Optimal Control Problems 3319191403, 978-3-319-19140-9, 978-3-319-19141-6, 3319191411

Table of contents :
Front Matter....Pages i-ix
Introduction....Pages 1-19
Linear Control Systems with Periodic Convex Integrands....Pages 21-84
Linear Control Systems with Nonconvex Integrands....Pages 85-125
Stability Properties....Pages 127-162
Linear Control Systems with Discounting....Pages 163-190
Dynamic Zero-Sum Games with Linear Constraints....Pages 191-207
Genericity Results....Pages 209-231
Variational Problems with Extended-Valued Integrands....Pages 233-268
Dynamic Games with Extended-Valued Integrands....Pages 269-289
Back Matter....Pages 291-296

Citation preview

Springer Optimization and Its Applications  104

Alexander J. Zaslavski

Turnpike Theory of ContinuousTime Linear Optimal Control Problems

Springer Optimization and Its Applications VOLUME 104 Managing Editor Panos M. Pardalos (University of Florida) Editor–Combinatorial Optimization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (University of Chicago) C.A. Floudas (Princeton University) F. Giannessi (University of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (McMaster University) Y. Ye (Stanford University)

Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences. The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximation techniques and heuristic approaches.

More information about this series at http://www.springer.com/series/7393

Alexander J. Zaslavski

Turnpike Theory of Continuous-Time Linear Optimal Control Problems

123

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

ISSN 1931-6828 ISSN 1931-6836 (electronic) Springer Optimization and Its Applications ISBN 978-3-319-19140-9 ISBN 978-3-319-19141-6 (eBook) DOI 10.1007/978-3-319-19141-6 Library of Congress Control Number: 2015941448 Mathematics Subject Classification (2010): 49J15, 49K15, 49K40, 49N05, 49N70, 91A05, 91A23, 93C15 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)

Preface

In this monograph, we study the structure of approximate solutions of linear optimal control problems with nonsmooth integrands. These problems are governed by differential equations whose right-hand side is linear with respect to a state and control variables. We establish a number of new results on properties of approximate solutions which are independent of the length of the interval, for all large intervals. As a matter of fact, these results provide a full description of the structure of approximate solutions of linear optimal control problems. In our 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. Now it is well known that the turnpike property is a general phenomenon, which holds for large classes of variational and optimal control problems. In our research, using the Baire category (generic) approach, it was shown that the turnpike property holds for a generic (typical) variational problem [44] and for a generic optimal control problem [52]. 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 Gı 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. Individual (non-generic) turnpike results and sufficient and necessary conditions for the turnpike phenomenon are of great interest because of their numerous applications in engineering and the economic theory. In particular, we are interested in the cases when a turnpike has a simple structure (a singleton or a periodic trajectory). v

vi

Preface

In this case, it is possible to find the turnpike (or at least its approximations) numerically. In our research which was summarized in [51], we obtained a number of individual (non-generic) turnpike results for variational problems. In our more recent book [53], we studied the turnpike phenomenon for discrete-time optimal control problems, which describe a general model of economic dynamics and for autonomous variational problems with extended-valued integrands. For these problems, the turnpike property was established with the turnpike being a singleton. In [53], for problems which satisfy concavity (convexity) assumption common in the literature, we also studied the structure of approximate solutions in the regions containing end points and obtained a full description of the structure of approximate solutions. In this monograph, we are also interested in individual turnpike results but for linear optimal control problems which have important applications in engineering. We study two large classes of problems. The first class studied in Chap. 2 consists of linear control problems with periodic nonsmooth convex integrands. We show that for these problems the turnpike property holds and the turnpike is a periodic trajectory-control pair. The second class studied in Chaps. 3–5 consists of linear control problems with autonomous nonconvex nonsmooth integrands. It is shown that for this class of problems the turnpike phenomenon takes place with the turnpike being a singleton. For these two classes of problems, we study the structure of approximate solutions in the regions containing end points and obtain a full description of the structure of approximate solutions. We show that the structure of approximate solutions is stable under small perturbations of integrands. This stability is an important property from the view of practice if we are interested to find a turnpike or its approximations numerically. In the other chapters of the book, we establish a turnpike property for dynamic zero-sum games with linear constraints (see Chap. 6), obtain the description of the structure of variational problems with extended-valued integrands (see Chap. 8), and study the turnpike phenomenon for dynamic games with extended-valued integrands (see Chap. 9). Haifa, Israel February 28, 2015

Alexander J. Zaslavski

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Turnpike Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Problems with Periodic Convex Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Nonconvex Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Problems of the Calculus of Variations with Extended-Valued Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 10 16

2

Linear Control Systems with Periodic Convex Integrands . . . . . . . . . . . . . . 2.1 Preliminaries and Turnpike Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stability of the Turnpike Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Structure of Solutions in the Regions Close to the End Points . . . . . . 2.4 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Proofs of Theorems 2.9 and 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Auxiliary Results for Theorem 2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Proof of Theorem 2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Basic Lemma for Theorem 2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Proof of Theorem 2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Proof of Theorem 2.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Proofs of Propositions 2.18 and 2.21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Auxiliary Results for Theorem 2.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 The Basic Lemma for Theorem 2.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Proof of Theorem 2.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Proof of Theorem 2.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Proof of Theorem 2.27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 27 29 35 46 47 50 53 63 65 69 71 73 75 80 80

3

Linear Control Systems with Nonconvex Integrands . . . . . . . . . . . . . . . . . . . . 85 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.2 Turnpike Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.3 Structure of Solutions in the Regions Close to the End Points . . . . . . 92 3.4 Auxiliary Results and the Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . 97 3.5 Auxiliary Results for Theorems 3.7, 3.9 and 3.10 . . . . . . . . . . . . . . . . . . . 100 3.6 Proof of Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 vii

viii

Contents

3.7 Proof of Theorem 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Proof of Theorem 3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Proofs of Propositions 3.14, 3.16, 3.17, and 3.22 . . . . . . . . . . . . . . . . . . . . 3.10 The Basic Lemma for Theorem 3.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Proofs of Theorems 3.23 and 3.24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Proof of Proposition 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108 109 111 116 118 123

4

Stability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Two Auxiliary Results and Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . 4.3 Basic Lemma for Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Auxiliary Results for Theorem 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Proofs of Theorems 4.4 and 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Proof of Theorem 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 132 135 144 147 151 153 158

5

Linear Control Systems with Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Auxiliary Results for Theorems 5.1 and 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 166 175 180

6

Dynamic Zero-Sum Games with Linear Constraints . . . . . . . . . . . . . . . . . . . . 6.1 Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Proof of Theorem 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 198 201 205

7

Genericity Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Proof of Proposition 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Auxiliary Results for Theorem 7.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Proof of Theorem 7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209 209 213 216 227

8

Variational Problems with Extended-Valued Integrands. . . . . . . . . . . . . . . . 8.1 Existence of Solutions and Their Turnpike Properties . . . . . . . . . . . . . . . 8.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Proof of Theorem 8.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Proofs of Propositions 8.14, 8.15, 8.17, 8.19, 8.20 and 8.24 . . . . . . . . 8.7 The Basic Lemma for Theorem 8.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Proof of Theorem 8.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Proof of Theorem 8.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Structure of Solutions of Problem (P1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Proof of Theorem 8.37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

233 233 239 243 244 246 249 257 259 263 264 265

Contents

9

Dynamic Games with Extended-Valued Integrands . . . . . . . . . . . . . . . . . . . . . 9.1 Preliminaries and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 An Auxiliary Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Proof of Theorem 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Auxiliary Results for Theorem 9.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Proof of Theorem 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

269 269 275 276 278 284 287

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Chapter 1

Introduction

The study of the existence and the structure of solutions of optimal control problems defined on infinite intervals and on sufficiently large intervals has recently been a rapidly growing area of research. See, for example, [2–4, 6–11, 13, 14, 16, 19, 20, 22, 25, 27, 32–34, 36, 37, 46–49, 51, 52] and the references mentioned therein. These problems arise in engineering [1, 23, 44, 56, 57], in models of economic growth [12, 13, 17, 22, 26, 31, 35, 39–41, 44, 45, 53, 55], in the game theory [18, 21, 43, 44, 50, 53, 54], in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals [5, 42], and in the theory of thermodynamical equilibrium for materials [15, 24, 28–30]. In this chapter we explain the turnpike phenomenon for a simple class of variational problems, discuss certain turnpike results obtained in our previous research, and describe the structure of the book.

1.1 The Turnpike Phenomenon Denote by j  j the Euclidean norm in the n-dimensional Euclidean space Rn which is induced by the inner product h; i in Rn . Assume that a function f W Rn  Rn ! R1 is strictly convex and differentiable and satisfies f .y; z/=.jyj C jzj/ ! 1 as jyj C jzj ! 1: In this section we study a family of variational problems Z 0

T

f .v.t/; v 0 .t//dt ! min;

© Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6_1

(P0 )

1

2

1 Introduction

v W Œ0; T ! Rn is an absolutely continuous function such that v.0/ D y; v.T/ D z; where T is a positive number and y; z 2 Rn . More precisely, we analyze the structure of minimizers of the problem (P0 ) when the values at the endpoints y; z and T vary and T is sufficiently large. Consider the following optimization problem f .y; 0/ ! min; y 2 Rn :

(P1 )

The strict convexity of f and the growth condition imply that the problem (P1 ) possesses a unique minimizer yN which satisfies @f [email protected]; 0/ D 0: Define a function L W Rn  Rn ! R1 by L.y; z/ D f .y; z/  f .Ny; 0/ hrf .Ny; 0/; .y; z/  .Ny; 0/i D f .y; z/  f .Ny; 0/  h.@f =@z/.Ny; 0/; zi for all y; z 2 Rn . It is not difficult to see that the function L W Rn  Rn ! R1 is differentiable and strictly convex and satisfies L.y; z/=.jyj C jzj/ ! 1 as jyj C jzj ! 1: In view of the strict convexity of the functions f and L we have [38] L.y; z/  0 for all .y; z/ 2 Rn  Rn and L.y; z/ D 0 if and only if y D yN ; z D 0: Consider an auxiliary variational problem Z

T 0

L.v.t/; v 0 .t//dt ! min;

v W Œ0; T ! Rn is an absolutely continuous function such that v.0/ D y; v.T/ D z;

(P2 )

1.1 The Turnpike Phenomenon

3

where T > 0 and y; z 2 Rn . It is not difficult to see that for every T > 0 and every absolutely continuous function x W Œ0; T ! Rn the following equalities hold: Z

T 0

L.x.t/; x0 .t//dt Z

T

D Z

0 T

D 0

Œf .x.t/; x0 .t//  f .Ny; 0/  h.@f =@z/.Ny; 0/; x0 .t/idt f .x.t/; x0 .t//dt C Tf .Ny; 0/  h.@f =@z/.Ny; 0/; x.T/  x.0/i:

The equations above imply that a function x W Œ0; T ! Rn is a solution of the problem (P0 ) if and only if it is a solution of the problem (P2 ). This means that the problems (P0 ) and (P2 ) are equivalent. Note that the point .Ny; 0/ is the unique solution of the minimization problem L.y; z/ ! min; y; z 2 Rn : This optimization problem is also well-posed. More precisely, we claim that the following property holds: n n (C) for every sequence f.yi ; zi /g1 iD1  R  R satisfying limi!1 L.yi ; zi / D 0 we have limi!1 .yi ; zi / D .Ny; 0/: n n Indeed, let a sequence f.yi ; zi /g1 iD1  R  R be such that

lim L.yi ; zi / D 0:

i!1

It follows from the growth condition that the sequence f.yi ; zi /g1 iD1 is bounded. Let .y; z/ be its limit point. By the continuity of L, L.y; z/ D lim L.yi ; zi / D 0 i!1

and since .Ny; 0/ is the unique point of minimum of the function L we have .y; z/ D .Ny; 0/: Since .y; z/ is any limit point of the sequence f.yi ; zi /g1 iD1 we conclude that .Ny; 0/ D limi!1 .yi ; zi /, as claimed. Let y; z 2 Rn , T > 2 be a real number and let an absolutely continuous function xN W Œ0; T ! Rn be an optimal solution of the problem (P0 ). Evidently, the function xN is also an optimal solution of the problem (P2 ). We claim that Z

T 0

L.Nx.t/; xN 0 .t//dt  2c0 .jyj; jzj/

where c0 .jyj; jzj/ is a positive constant depending only on jyj and jzj.

4

1 Introduction

Set x.t/ D y C t.Ny  y/; t 2 Œ0; 1; x.t/ D yN ; t 2 Œ1; T  1; x.t/ D yN C .t  .T  1//.z  yN /; t 2 ŒT  1; T: Clearly, x W Œ0; T ! Rn is an absolutely continuous function satisfying x.0/ D y and x.T/ D z. Since xN is a solution of the problem (P2) we have Z 0

T

L.Nx.t/; xN 0 .t//dt 

Z Z

T 0 1

D 0

L.x.t/; x0 .t//dt Z 1

Z

T

C Z

1

L.Ny; 0/dt

L.x.t/; z  yN /dt

T1

D

T1

L.x.t/; yN  y/dt C

Z

L.x.t/; yN  y/dt C

0

T

L.x.t/; z  yN /dt:

T1

It is easy to see that the integrals Z

1

Z L.x.t/; yN  y/dt and

0

T

L.x.t/; z  yN /dt

T1

do not exceed a constant c0 .jyj; jzj/ > 0 which depends only on the norms jyj, jzj. Hence Z

T 0

L.Nx.t/; xN 0 .t//dt  2c0 .jyj; jzj/:

It should be mentioned that here the constant c0 .jyj; jzj/ does not depend on T. Denote by mes.E/ the Lebesgue measure of a Lebesgue measurable set E  R1 . Now let  > 0 be a real number. Property (C) implies the existence of a real number ı > 0 such that if a point .y; z/ 2 Rn  Rn satisfies L.y; z/  ı, then jy  yN j C jzj  . It follows from the choice of ı and the relation Z

T 0

L.Nx.t/; xN 0 .t//dt  2c0 .jyj; jzj/

that mesft 2 Œ0; T W j.Nx.t/; xN 0 .t//  .Ny; 0/j > g  mesft 2 Œ0; T W L.Nx.t/; xN 0 .t// > ıg

1.2 Problems with Periodic Convex Integrands

 ı 1

Z

T 0

5

L.Nx.t/; xN 0 .t//dt  ı 1 2c0 .jyj; jzj/

and mesft 2 Œ0; T W jNx.t/  yN j > g  ı 1 2c0 .jyj; jzj/: In view of the inequality above the minimizer xN spends most of the time in the -neighborhood of the point yN . More precisely, the Lebesgue measure of the set of all points t such that xN .t/ does not belong to this -neighborhood, does not exceed the constant 2ı 1 c0 .jyj; jzj/ depending only on the norms jyj; jzj and . Note that it does not depend on the length of the interval T. Following the tradition, the point yN is called the turnpike. Moreover, it can be shown that the set ft 2 Œ0; T W jNx.t/  yN j > g is a subset of the union of two intervals Œ0; 1  [ ŒT  2 ; T, where 0 < 1 ; 2  2ı 1 c0 .jyj; jzj/. The main goal of this book is to study the turnpike phenomenon and the structure of approximate solutions in regions close to endpoints for linear optimal control systems.

1.2 Problems with Periodic Convex Integrands In Chap. 2 of this book we study the structure of approximate optimal trajectories of linear control systems with periodic convex integrands and show that these systems possess a turnpike property. This means that approximate optimal trajectories are determined mainly by the integrand, and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. We also study the stability of the turnpike phenomenon under small perturbations of integrands and study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals. More precisely, we study the structure of approximate optimal trajectories of linear control systems governed by the equation x0 .t/ D Ax.t/ C Bu.t/;

(1.1)

with periodic convex integrands f W Œ0; 1/Rn Rm ! R1 , where A and B are given matrices of dimensions n  n and n  m, x.t/ 2 Rn , u.t/ 2 Rm and the admissible controls are Lebesgue measurable functions. We assume that the integrand f is a Borel measurable function and that the linear system (1.1) is controllable which means that the rank of the matrix .B; AB; : : : An1 B/ is n.

6

1 Introduction

The performance of the above control system is measured on any finite interval ŒT1 ; T2   Œ0; 1/ by the integral functional Z I.T1 ; T2 ; x; u/ D

T2

f .t; x.t/; u.t//dt:

T1

We denote by j  j the Euclidean norm and by h; i the inner product in the ndimensional Euclidean space Rn . Artstein and Leizarowitz [1] analyzed the existence and structure of solutions of the linear system (1.1) with an integrand f .t; x; u/ D .x   .t//0 Q.x   .t// C u0 Pu .t 2 Œ0; 1/; x 2 Rn ; u 2 Rm /; where P is a given positive definite symmetric matrix, Q is a positive semidefinite symmetric matrix, the pair .A; Q/ is observable and  W Œ0; 1/ ! Rn is a measurable function satisfying  .t C T/ D  .t/ .t 2 Œ0; 1// for some constant T > 0. Artstein and Leizarowitz [1] showed the existence of a unique solution for the infinite horizon tracking of the periodic trajectory  and established a turnpike property for finite time optimizers. Their methods are based on explicit expressions for optimal solutions to tracking on finite intervals. In Chap. 6 of [44] and in [57] we extended the results of [1] to an integrand f W Œ0; 1/  Rn  Rn ! R1 which satisfies the following assumptions: (i) f .t C ; x; u/ D f .t; x; u/ for all t 2 Œ0; 1/; all x 2 Rn and all u 2 Rm for some constant  > 0 depending only on f ; (ii) for any t 2 Œ0; 1/ the function f .t; ; / W Rn  Rm ! R1 is strictly convex; (iii) the function f is bounded on any bounded subset of Œ0; 1/  Rn  Rm ; (iv) f .t; x; u/ ! 1 as jxj ! 1 uniformly in .t; u/ 2 Œ0; 1/  Rm ; (v) f .t; x; u/juj1 ! 1 as juj ! 1 uniformly in .t; x/ 2 Œ0; 1/  Rn . In this section we also suppose that the assumptions above hold for the integrand f . Remark 1.1. It is not difficult to see that if  is a positive number and if assumptions (i)–(v) hold with f D fi , i D 1; 2 and with the same  > 0, where f1 ; f2 W Œ0; 1/  Rn Rm ! R1 are measurable functions, then assumptions (i)–(v) hold with f D f1 and f D f1 C f2 . Example 1.2. Let  > 0. It is not difficult to see that assumptions (i)–(v) hold with a function f W Œ0; 1/  Rn  Rm ! R1 defined by f .t; x; u/ D g.t/.x   .t//0 Q.x   .t// Ch.t/u0 Pu C H.t/ .t 2 Œ0; 1/; x 2 Rn ; u 2 Rm /;

1.2 Problems with Periodic Convex Integrands

7

where  W Œ0; 1/ ! Rn is a measurable and bounded on Œ0; 1/ function, P; Q are positive definite symmetric matrices, H; h; g W Œ0; 1/ ! R1 are measurable bounded functions such that for all t  0,  .t C  / D  .t/; g.t C / D g.t/; h.t C / D h.t/; H.t C  / D H.t/ and that inffg.t/ W t 2 Œ0; 1/g > 0; inffh.t/ W t 2 Œ0; 1/g > 0: Example 1.3. Let  > 0. Assume that h W Œ0; 1/ ! R1 is a bounded measurable function such that h.t C / D h.t/ for all t  0; inffh.t/ W t 2 Œ0; 1/g > 0 and that a strictly convex function g D g.x; u/ 2 C1 .RnCm / has the following properties: g.x; u/  maxf .jxj/; maxfj@[email protected]; u/j; j@[email protected]; u/jg 

.juj/jujg; 0 .jxj/.1

C

juj/juj;

x 2 Rn , u 2 Rm , where W Œ0; 1/ ! .0; 1/, 0 W .0; 1/ ! Œ0; 1/ are monotone increasing functions, .t/ ! 1 as t ! 1; for any  > 0 there exists ı./ > 0 such that if x1 ; x2 2 Rn and if u1 ; u2 2 Rm satisfy jx1  x2 j C ju1  u2 j  , then g.21 .x1 C x2 /; 21 .u1 C u2 //  21 Œg.x1 ; u1 / C g.x2 ; u2 /  ı./: It is not difficult to see that the integrand f .t; x; u/ D h.t/g.x; u/, t 2 Œ0; 1/, x 2 Rn , u 2 Rm satisfies assumptions (i)–(v) and f is not taken from Remark 1.1 and Example 1.2. Using Remark 1.1 and Examples 1.2 and 1.3, we can easily construct numerous examples of integrands satisfying assumptions (i)–(v). Example 1.4. Let  > 0 and k be a natural number. Assume that i W Œ0; 1/ ! Rn , i D 1; : : : ; k are measurable and bounded on Œ0; 1/ functions, Pi ; Qi , i D 1; : : : ; k are positive definite symmetric matrices, hi ; i D 1; : : : ; k, gi ; i D 1; : : : ; k, H W Œ0; 1/ ! R1 are measurable bounded functions such that for all t  0 and all i D 1; : : : ; k, i .t C  / D i .t/; gi .t C / D gi .t/; hi .t C  / D hi .t/; H.t C  / D H.t/

8

1 Introduction

and that inffgi .t/ W t 2 Œ0; 1/g > 0; inffhi .t/ W t 2 Œ0; 1/g > 0: In view of Remark 1.1 and Example 1.2, assumptions (i)–(v) hold with a function f W Œ0; 1/  Rn  Rm ! R1 defined by f .t; x; u/ D

k X

gi .t/.x  i .t//0 Qi .x  i .t//

iD1

C

k X

hi .t/u0 Pi u C H.t/ .t 2 Œ0; 1/; x 2 Rn ; u 2 Rm /:

iD1

We consider the following optimal control problems: I.0; T; x; u/ ! min;

(P1 )

x W Œ0; T ! Rn ; u W Œ0; T ! Rm is a trajectory-control pair such that x.0/ D y; x.T/ D z; I.0; T; x; u/ ! min;

(P2 )

x W Œ0; T ! Rn ; u W Œ0; T ! Rm is a trajectory-control pair such that x.0/ D y; I.0; T; x; u/ ! min;

(P3 )

x W Œ0; T ! Rn ; u W Œ0; T ! Rm is a trajectory-control; where y; z 2 Rn and T > 0. The study of these problems is based on the properties of solutions of the corresponding infinite horizon optimal control problem associated with the control system (1.1) and the integrand f . In [57] (see also Chap. 6 of [44]) we were interested in a turnpike property of the approximate solutions of problems (P2 ). It was shown that there exists a trajectorycontrol pair xf W Œ0;  ! Rn ; uf W Œ0;  ! Rm which is the unique solution of the minimization problem I.0; ; x; u/ ! min; x W Œ0;  ! Rn ; u W Œ0;   ! Rm is a trajectory-control pair such that x.0/ D x./: Put .f / D  1 I.0; ; xf ; uf /:

1.2 Problems with Periodic Convex Integrands

9

It was shown in [57] (see also Chap. 6 of [44]) that for any trajectory-control pair x W Œ0; 1/ ! Rn ; u W Œ0; 1/ ! Rm either I.0; T; x; u/  T.f / ! 1 as T ! 1 or supfjI.0; T; x; u/  T.f /j W T > 0g < 1:

(1.2)

Moreover, if (1.2) holds, then supfjx.i C t/  xf .t/j W t 2 Œ0; g ! 0 as i ! 1 over the integers. We say that a trajectory-control pair x W Œ0; 1/ ! Rn ; u W Œ0; 1/ ! Rm is good [44, 53] if supfjI.0; T; x; u/  T.f /j W T > 0g < 1: We say that a trajectory-control pair xQ W Œ0; 1/ ! Rn ; uQ W Œ0; 1/ ! Rm is overtaking optimal [44, 53] if lim supŒI.0; T; xQ ; uQ /  I.0; T; x; u/  0 T!1

for each trajectory-control pair x W Œ0; 1/ ! Rn ; u W Œ0; 1/ ! Rm satisfying x.0/ D xQ .0/. The following existence result was obtained in [57] (see also Chap. 6 of [44]). Theorem 1.5. Let x0 2 Rn . Then there exists a unique overtaking optimal trajectory-control pair xQ W Œ0; 1/ ! Rn ; uQ W Œ0; 1/ ! Rm satisfying xQ .0/ D x0 . The next theorem, which was also obtained in [57], establishes the turnpike property for approximate solutions of problems (P2 ) with the turnpike xf ./. Theorem 1.6. Let M;  > 0. Then there exist an integer N  1 and ı > 0 such that for each T > 2N and each trajectory-control pair x W Œ0; T ! Rn ; u W Œ0; T ! Rm which satisfies jx.0/j  M; I.0; T; x; u/  inffI.0; T; y; v/ W y W Œ0; T ! Rn ; v W Œ0; T is a trajectory-control pair; y.0/ D x.0/g C ı the inequality supfjx.i C t/  xf .t/j W t 2 Œ0;  g  

(1.3)

10

1 Introduction

holds for all integers i 2 ŒN;  1 T  N. Moreover if jx.0/  xf .0/j  ı, then inequality (1.3) holds for all integers i 2 Œ0;  1 T  N. In Chap. 2 we establish the turnpike property of the approximate solutions of problems (P1 ) and (P3 ). We show the stability of the turnpike phenomenon under small perturbations of the integrand f and study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals. For the problems (P2 ) and (P3 ) we show that in regions close to the right endpoint T of the time interval these approximate solutions are determined only by the integrand, and are essentially independent of the choice of interval and the endpoint value y. For the problems (P3 ), approximate solutions are determined only by the integrand also in regions close to the left endpoint 0 of the time interval. The study of these problems is based on the properties of solutions of the corresponding infinite horizon optimal control problem associated with the control system (1.1) and the integrand f .

1.3 Nonconvex Optimal Control Problems In Chap. 3 we study the existence and structure of optimal trajectories of linear control systems with autonomous integrands. For these control systems we establish the existence of optimal trajectories over an infinite horizon and show that the turnpike phenomenon holds. We also study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals. It is shown that in these regions optimal trajectories converge to solutions of the corresponding infinite horizon optimal control problem which depend only on the integrand. For this class of optimal control problems the turnpike property holds with the turnpike being a singleton. It should be mentioned that there are many linear optimal control problems for which the turnpike is a singleton. Let us consider a few examples. We use the notation, definitions, and assumptions introduced in Sect. 1.2. Assume that an integrand f .t; x; u/, .t; x; u/ 2 Œ0; 1/  Rn  Rm satisfies assumptions (ii)– (v) and does not depend on the variable t. As a matter of fact, we can consider this function as f W Rn  Rm ! R1 . Clearly, f satisfies the assumption (i) with any  > 0. In view of the results discussed in Sect. 1.2, the turnpike phenomenon holds and the turnpike xf ./ is a constant. Remark 1.7. If integrands f1 ; f2 do not depend on t, assumptions (ii)–(v) hold with f D fi , i D 1; 2, where f1 ; f2 W Rn  Rm ! R1 are measurable functions, and if  is a positive number, then in view of Remark 1.1, assumptions (ii)–(v) hold with f D f1 and f D f1 C f2 , the turnpike property holds for these integrands and their turnpikes are singletons.

1.3 Nonconvex Optimal Control Problems

11

Example 1.8. Assume that a strictly convex function g D g.x; u/ 2 C1 .RnCm / has the following properties: g.x; u/  maxf .jxj/; maxfj@[email protected]; u/j; j@[email protected]; u/jg 

.juj/jujg; 0 .jxj/.1

C

juj/juj;

x 2 Rn , u 2 Rm , where W Œ0; 1/ ! .0; 1/, 0 W .0; 1/ ! Œ0; 1/ are monotone increasing functions, .t/ ! 1 as t ! 1; for any  > 0 there exists ı./ > 0 such that if x1 ; x2 2 Rn and if u1 ; u2 2 Rm satisfy jx1  x2 j C ju1  u2 j  , then g.21 .x1 C x2 /; 21 .u1 C u2 //  21 Œg.x1 ; u1 / C g.x2 ; u2 /  ı./: In view of Example 1.3, the function g.x; u/, x 2 Rn , u 2 Rm satisfies assumptions (ii)–(v), possesses the turnpike property and the turnpike is a singleton. Example 1.9. Let k be a natural number. Assume that i 2 Rn , i D 1; : : : ; k, Pi ; Qi , i D 1; : : : ; k are positive definite symmetric matrices and let f .x; u/ D

k k X X .x  i /0 Qi .x  i / C u0 Pi u; x 2 Rn ; u 2 Rm : iD1

iD1

In view of Example 1.4, assumptions (ii)–(v) hold for the function f which possesses the turnpike property and the turnpike is a singleton. Using Remark 1.7 and Examples 1.8 and 1.9, we can easily construct numerous examples of integrands which do not depend on the variable t and satisfy assumptions (ii)–(v). For these integrands the turnpike property holds and the turnpike is a singleton. It is clear that in the examples above the integrands are convex functions. In our book we study the existence and structure of optimal trajectories of linear control systems with autonomous integrands which are not necessarily convex. More precisely, we study the structure of approximate optimal trajectories of linear control systems governed by the equation x0 .t/ D Ax.t/ C Bu.t/;

(1.4)

with integrands f W Rn  Rm ! R1 satisfying the assumptions below, where n; m are natural numbers, A and B are given matrices of dimensions nn and nm, x.t/ 2 Rn , u.t/ 2 Rm and the admissible controls are Lebesgue measurable functions. We assume that the linear system (1.4) is controllable and that the integrand f is a continuous function. For every s 2 R1 set sC D maxfs; 0g. For every nonempty set X and every function h W X ! R1 [ f1g set inf.h/ D inffh.x/ W x 2 Xg:

12

1 Introduction

Let a0 > 0 and

W Œ0; 1/ ! Œ0; 1/ be an increasing function such that .t/ D 1:

lim

t!1

Suppose that f W Rn  Rm ! R1 is a continuous function such that the following assumption holds: (A1) (i) for each .x; u/ 2 Rn  Rm , f .x; u/  maxf .jxj/;

.juj/;

.ŒjAx C Buj  a0 jxjC /ŒjAx C Buj  a0 jxjC g  a0 I (ii) for each x 2 Rn the function f .x; / W Rm ! R1 is convex; (iii) for each M;  > 0 there exist ; ı > 0 such that jf .x1 ; u1 /  f .x2 ; u2 /j   maxff .x1 ; u1 /; f .x2 ; u2 /g for each u1 ; u2 2 Rm and each x1 ; x2 2 Rn which satisfy jxi j  M; jui j  ; i D 1; 2;

maxfjx1  x2 j; ju1  u2 jg  ıI

(iv) for each K > 0 there exists a constant aK > 0 and an increasing function K

W Œ0; 1/ ! Œ0; 1/

such that K .t/

! 1 as t ! 1

and f .x; u/ 

K .juj/juj

 aK

for each u 2 Rm and each x 2 Rn satisfying jxj  K. Remark 1.10. A function h satisfies (A1) if h 2 C1 .Rn  Rm /, (A1)(i), (A1)(ii), (A1)(iv) hold, and for each K > 0 there exists an increasing function Q W Œ0; 1/ ! Œ0; 1/ such that for each x 2 Rn satisfying jxj  K and each u 2 Rm , maxfj@[email protected]; u/j; j@[email protected]; u/jg  Q .jxj/.1 C

K .juj/juj/:

1.3 Nonconvex Optimal Control Problems

13

The performance of the above control system is measured on any finite interval ŒT1 ; T2   Œ0; 1/ and for any trajectory-control pair x W ŒT1 ; T2  ! Rn ; u W ŒT1 ; T2  ! Rm by the integral functional Z I.T1 ; T2 ; x; u/ D

T2

f .x.t/; u.t//dt:

T1

In Chap. 3 we consider optimal control problems (P1 ), (P2 ), and (P3 ) introduced in Sect. 1.2. A number .f / WD infflim inf T 1 I.0; T; x; u/ W T!1

x W Œ0; 1/ ! Rn ; u W Œ0; 1/ ! Rm is a trajectory-control pairg is called the minimal long-run average cost growth rate of f . By (A1)(i), 1 < .f /. We say that a trajectory-control pair xQ W Œ0; 1/ ! Rn ; uQ W Œ0; 1/ ! Rm is overtaking optimal [44, 53] if lim supŒI.0; T; xQ ; uQ /  I.0; T; x; u/  0 T!1

for each trajectory-control pair x W Œ0; 1/ ! Rn ; u W Œ0; 1/ ! Rm satisfying x.0/ D xQ .0/. Let .xf ; uf / 2 Rn  Rm satisfy Axf C Buf D 0: Clearly, .f /  f .xf ; uf /: We suppose that the following assumption holds. (A2)

.f / D f .xf ; uf / and if .x; u/ 2 Rn  Rm satisfies Ax C Bu D 0; .f / D f .x; u/;

then x D xf . In Chap. 3 we will show that for each trajectory-control pair x W Œ0; 1/ ! Rn ; u W Œ0; 1/ ! Rm either I.0; T; x; u/  T.f / ! 1 as T ! 1 or supfjI.0; T; x; u/  T.f /j W T > 0g < 1:

14

1 Introduction

A trajectory-control pair x W Œ0; 1/ ! Rn ; u W Œ0; 1/ ! Rm is called good [44, 53] if supfjI.0; T; x; u/  T.f /j W T > 0g < 1: We suppose that the following assumption holds. (A3)

For each good trajectory-control pair x W Œ0; 1/ ! Rn ; u W Œ0; 1/ ! Rm , lim x.t/ D xf :

t!1

Let us consider examples of integrands satisfying assumptions (A1)–(A3). Example 1.11. Assume that a continuous strictly convex function h W Rn Rm ! R1 satisfies assumption (A1) (with f D h) and h.x; u/=juj ! 1 as juj ! 1 uniformly in x 2 Rn : Then the function h satisfies assumptions (A2) and (A3) (with f D h). This follows from Corollary 2.11 of Chap. 2. Let us consider another example of an integrand which satisfies (A1)–(A3). Example 1.12. Let c 2 R1 , a1 > 0, l 2 Rn , .x ; u / 2 Rn Rm satisfy Ax CBu D 0 and let 0 W Œ0; 1/ ! Œ0; 1/ be an increasing function such that limt!1 0 .t/ D 1. Assume that a continuous function L W Rn  Rm ! Œ0; 1/ satisfies for each .x; u/ 2 Rn  Rm , L.x; u/  maxf

0 .jxj/;

0 .juj/jujg

 a1 C jljjAx C Buj;

L.x; u/ D 0 if and only if x D x ; u D u ; for each x 2 Rn , the function L.x; / W Rm ! R1 is convex and for each M;  > 0 there exist ; ı > 0 such that jL.x1 ; u1 /  L.x2 ; u2 /j   maxfL.x1 ; u1 /; L.x2 ; u2 /g for each x1 ; x2 2 Rn and each u1 ; u2 2 Rm which satisfy jxi j  M; jui j  ; i D 1; 2; jx1  x2 j; ju1  u2 j  ı: For every .x; u/ 2 Rn  Rm set h.x; u/ D L.x; u/ C c C hl; Ax C Bui:

1.3 Nonconvex Optimal Control Problems

15

It is not difficult to see that for each .x; u/ 2 Rn  Rm , h.x; u/  maxf

0 .jxj/;

0 .juj/jujg

 a1  jcj

and that h satisfies (A1) holds under the appropriate choice of a0 > 0, In Sect. 3.12 we prove that

.

.h/ D h.x ; u / D c; (A2) holds for f D h and for any good trajectory-control pair x W Œ0; 1/ ! Rn ; u W Œ0; 1/ ! Rm ; lim x.t/ D x

t!1

(see Proposition 3.6). In Chap. 3 we study the existence and structure of optimal trajectories of linear control system (1.4) with the integrand f . For these control systems we establish the existence of optimal trajectories over an infinite horizon and show that the turnpike phenomenon holds for approximate solutions of problems (P1 ), (P2 ) and (P3 ). For problems (P2 ) and (P3 ) we show that, in regions close to the right endpoint T of the time interval, their approximate solutions are determined only by the integrand, and are essentially independent of the choice of interval and the endpoint value y. For problems (P3 ), approximate solutions are determined only by the integrand also in regions close to the left endpoint 0 of the time interval. It is shown that in the regions closed to the endpoints optimal closed trajectories converge to solutions of the corresponding infinite horizon optimal control problem which depend only on the integrand. In Chaps. 4, 5, and 7 we continue to study the class of linear optimal control problems considered in this section. In Chap. 4 we show that the turnpike phenomenon and the convergence, in the regions close to the endpoints of time intervals, are stable under small perturbations of the integrand f . Linear control systems with discounting are studied in Chap. 5. In Chap. 7 we show that for a typical (in the sense of Baire category) integrand the values of approximate solutions at the end points converge to the limit which is a unique solution of the corresponding minimization problem associated with the integrand. In Chap. 6 we study the existence and turnpike properties of approximate solutions for a class of dynamic continuous-time two-player zero-sum games without using convexityconcavity assumptions and with linear control constraints. We describe the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals and show that approximate solutions are determined mainly by the objective function, and are essentially independent of the choice of interval and endpoint conditions.

16

1 Introduction

1.4 Problems of the Calculus of Variations with Extended-Valued Integrands In Chap. 8 we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous extended-valued integrand. In our recent research we showed that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. In Chap. 8 our goal is to study the structure of approximate solutions in regions close to the endpoints of the time intervals. More precisely, in Chap. 8 we consider the following variational problems Z

T

0

f .v.t/; v 0 .t//dt ! min;

(P1 )

v W Œ0; T ! Rn is an absolutely continuous (a.c.) function such that v.0/ D x; v.T/ D y; Z

T

0

f .v.t/; v 0 .t//dt ! min;

(P2 )

v W Œ0; T ! Rn is an a. c. function such that v.0/ D x and Z 0

T

f .v.t/; v 0 .t//dt ! min;

(P3 )

v W Œ0; T ! Rn is an a. c. function, where x; y 2 Rn . Here Rn is the n-dimensional Euclidean space with the Euclidean norm j  j and f W Rn  Rn ! R1 [ f1g is an extended-valued integrand. The problems (P1 ) and (P2 ) were studied in [46, 49, 53], where it was shown, under certain assumptions, that the turnpike property holds and that the turnpike xN is a unique solution of the minimization problem f .x; 0/ ! min, x 2 Rn . In this book we study the structure of approximate solutions of the problems (P2 ) and (P3 ) in regions close to the endpoints of the time intervals. It is shown that in regions close to the right endpoint T of the time interval these approximate solutions are determined only by the integrand, and are essentially independent of the choice of interval and endpoint value x. For the problems (P3 ), approximate solutions are determined only by the integrand also in regions close to the left endpoint 0 of the time interval. More precisely, we define fN .x; y/ D f .x; y/ for all x; y 2 Rn and consider the set P.fN / of all solutions of a corresponding infinite horizon variational problem

1.4 Problems of the Calculus of Variations with Extended-Valued Integrands

17

associated with the integrand fN . For a given pair of real positive numbers , , we show that if T is large enough and v W Œ0; T ! Rn is an approximate solution of the problem (P2 ), then jv.T  t/  y.t/j   for all t 2 Œ0;  , where y./ 2 P.fN /. In this section we describe the class of integrands which is considered in Chap. 8. For each function f W X ! R1 [ f1g, where X is a nonempty, set dom.f / D fx 2 X W f .x/ < 1g: Let a be a real positive number, such that

W Œ0; 1/ ! Œ0; 1/ be an increasing function .t/ D 1

lim

t!1

and let f W Rn  Rn ! R1 [ f1g be a lower semicontinuous function such that the set dom.f / D f.x; y/ 2 Rn  Rn W f .x; y/ < 1g is nonempty, convex, and closed and that f .x; y/  maxf .jxj/;

.jyj/jyjg  a for each x; y 2 Rn :

We suppose that there exists a point xN 2 Rn such that f .Nx; 0/  f .x; 0/ for each x 2 Rn

(1.5)

and that the following assumptions hold: (A1) .Nx; 0/ is an interior point of the set dom.f / and the function f is continuous at the point .Nx; 0/; (A2) for each M > 0 there exists cM > 0 such that Z 0

T

f .v.t/; v 0 .t//dt  Tf .Nx; 0/  cM

for each real number T > 0 and each a. c. function v W Œ0; T ! Rn satisfying jv.0/j  M; (A3) for each x 2 Rn the function f .x; / W Rn ! R1 [ f1g is convex. It should be mentioned that inequality (1.5) and assumptions (A1)–(A3) are common in the literature and hold for many infinite horizon optimal control problems. In particular, we need inequality (1.5) and assumption (A2) in the cases when the problems (P1 ) and (P2 ) possess the turnpike property and the point xN is its turnpike. Assumption (A2) means that the constant function v.t/ N D xN , t 2 Œ0; 1/

18

1 Introduction

is an approximate solution of the infinite horizon variational problem with the integrand f related to the problems (P1 ) and (P2 ). We say that an a. c. function v W Œ0; 1/ ! Rn is .f /-good [44, 51] if ˇ Z ˇ sup ˇˇ

T 0

ˇ  ˇ f .v.t/; v 0 .t//dt  Tf .Nx; 0/ˇˇ W T 2 .0; 1/ < 1:

The following result was obtained in [46]. Proposition 1.13. Let v W Œ0; 1/ ! Rn be an a.c. function. Then either the function v is .f /-good or Z

T 0

f .v.t/; v 0 .t//dt  Tf .Nx; 0/ ! 1 as T ! 1:

Moreover, if the function v is .f /-good, then supfjv.t/j W t 2 Œ0; 1/g < 1. We suppose that the following assumption holds: (A4) (the asymptotic turnpike property) for each .f /-good function v W Œ0; 1/ ! Rn , limt!1 jv.t/  xN j D 0. We consider two examples of integrands f which satisfy assumptions (A1)–(A4). Example 1.14. Let a0 be a positive number, function satisfying lim

t!1

0 .t/

0

W Œ0; 1/ ! Œ0; 1/ be an increasing

D1

and let L W Rn  Rn ! Œ0; 1 be a lower semicontinuous function such that dom.L/ WD f.x; y/ 2 Rn  Rn W L.x; y/ < 1g is nonempty, convex, closed set and L.x; y/  maxf

0 .jxj/;

0 .jyj/jyjg

 a0 for each x; y 2 Rn :

Assume that for each point x 2 Rn the function L.x; / W Rn ! R1 [ f1g is convex and that there exists a point xN 2 Rn such that L.x; y/ D 0 if and only if .x; y/ D .Nx; 0/; .Nx; 0/ is an interior point of dom.L/ and that L is continuous at the point .Nx; 0/. Let  2 R1 and l 2 Rn . Define f .x; y/ D L.x; y/ C  C hl; yi; x; y 2 Rn :

1.4 Problems of the Calculus of Variations with Extended-Valued Integrands

19

It was shown in [46] and in Sect. 3.9 of [53] that all the assumptions introduced in this section hold for f . Example 1.15. Let a be a positive number, W Œ0; 1/ ! Œ0; 1/ be an increasing function such that limt!1 .t/ D 1 and let f W Rn  Rn ! R1 [ f1g be a convex lower semicontinuous function such that the set dom.f / is nonempty, convex and closed and that f .x; y/  maxf .jxj/;

.jyj/jyjg  a for each x; y 2 Rn :

We suppose that there exists a point xN 2 Rn such that f .Nx; 0/  f .x; 0/ for each x 2 Rn and that .Nx; 0/ is an interior point of the set dom.f /. It is known that the function f is continuous at the point .Nx; 0/. It is a well-known fact of convex analysis [38] that there exists a point l 2 Rn such that f .x; y/  f .Nx; 0/ C hl; yi for each x; y 2 Rn : We assume that for each pair of points .x1 ; y1 /, .x2 ; y2 / 2 dom.f / satisfying .x1 ; y1 / 6D .x2 ; y2 / and each number ˛ 2 .0; 1/, we have f .˛.x1 ; y1 / C .1  ˛/.x2 ; y2 // < ˛f .x1 ; y1 / C .1  ˛/f .x2 ; y2 /: Put L.x; y/ D f .x; y/  f .Nx; 0/  hl; yi for each x; y 2 Rn : It is not difficult to see that there exist a positive number a0 and an increasing function 0 W Œ0; 1/ ! Œ0; 1/ such that L.x; y/  maxf

0 .jxj/;

0 .jyj/jyjg

 a0 for all x; y 2 Rn :

It is easy to see that L is a convex, lower semicontinuous function and that the equality L.x; y/ D 0 holds if and only if .x; y/ D .Nx; 0/. Now it is easy to see that our example is a particular case of Example 1.14 and all the assumptions introduced in this section hold for f . In the last chapter of the book we study a class of dynamic continuous-time two-player zero-sum unconstrained games with extended-valued integrands. We do not assume convexity-concavity assumptions and establish the existence and the turnpike property of approximate solutions. The results of Chap. 8 were obtained in [55]. The results which are proved in all other chapters are new.

Chapter 2

Linear Control Systems with Periodic Convex Integrands

We study the structure of approximate optimal trajectories of linear control systems with periodic convex integrands and show that these systems possess a turnpike property. To have this property means, roughly speaking, that the approximate optimal trajectories are determined mainly by the integrand, and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. We also show the stability of the turnpike phenomenon under small perturbations of integrands and study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals.

2.1 Preliminaries and Turnpike Results In this chapter we study the structure of approximate optimal trajectories of linear control systems x0 .t/ D Ax.t/ C Bu.t/;

(2.1)

x.0/ D x0 with periodic convex integrands f W Œ0; 1/Rn Rm ! R1 , where A and B are given matrices of dimensions n  n and n  m, x.t/ 2 Rn , u.t/ 2 Rm and the admissible controls are Lebesgue measurable functions. We assume that the linear system (2.1) is controllable and that the integrand f is a Borel measurable function. We denote by j  j the Euclidean norm and by h; i the inner product in the n-dimensional Euclidean space Rn . Denote by Z the set of all integers. For every

© Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6_2

21

22

2 Linear Control Systems with Periodic Convex Integrands

z 2 R1 denote by bzc the largest integer which does not exceed z: bzc D maxfi 2 Z W i  zg. The performance of the above control system is measured on any finite interval ŒT1 ; T2   Œ0; 1/ by the integral functional Z I .T1 ; T2 ; x; u/ D f

T2

f .t; x.t/; u.t//dt:

(2.2)

T1

We suppose that the integrand f W Œ0; 1/  Rn  Rm ! R1 satisfies the following Assumption (A) (i) f .t C ; x; u/ D f .t; x; u/ for all t 2 Œ0; 1/; all x 2 Rn and all u 2 Rm for some constant  > 0 depending only on f ; (ii) for any t 2 Œ0; 1/ the function f .t; ; / W Rn  Rm ! R1 is strictly convex; (iii) the function f is bounded on any bounded subset of Œ0; 1/  Rn  Rm ; (iv) f .t; x; u/ ! 1 as jxj ! 1 uniformly in .t; u/ 2 Œ0; 1/  Rm ; (v) f .t; x; u/juj1 ! 1 as juj ! 1 uniformly in .t; x/ 2 Œ0; 1/  Rn . Assumption (A) implies that f is bounded below on Œ0; 1/  Rn  Rm . Let T2 > T1  0. A pair of an absolutely continuous (a.c.) function x W ŒT1 ; T2  ! Rn and a Lebesgue measurable function u W ŒT1 ; T2  ! Rm is called an .A; B/trajectory-control pair if for almost every (a. e.) t 2 ŒT1 ; T2  (2.1) holds. Denote by X.A; B; T1 ; T2 / the set of all .A; B/-trajectory-control pairs x W ŒT1 ; T2  ! Rn , u W ŒT1 ; T2  ! Rm . Let J D Œa; 1/ be an infinite closed subinterval of Œ0; 1/. A pair of functions x W J ! Rn and u W J ! Rm is called an .A; B/-trajectory-control pair if it is an .A; B/-trajectory-control pair on any bounded closed subinterval of J. Denote by X.A; B; a; 1/ the set of all .A; B/-trajectory-control pairs x W J ! Rn , u W J ! Rm . In this chapter we study the structure of approximate optimal trajectories of the linear control system (2.1) with the integrand f and show that the turnpike property holds. To have this property means, roughly speaking, that the approximate optimal trajectories on sufficiently large intervals are determined mainly by the integrand, and are essentially independent of the choice of time intervals and data, except in regions close to the endpoints of the time intervals. We also show the stability of the turnpike phenomenon under small perturbations of the integrand and study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals. More precisely, we consider the following optimal control problems I f .0; T; x; u/ ! min;

(P1 )

.x; u/ 2 X.A; B; 0; T/ such that x.0/ D y; x.T/ D z; I f .0; T; x; u/ ! min;

(P2 )

2.1 Preliminaries and Turnpike Results

23

.x; u/ 2 X.A; B; 0; T/ such that x.0/ D y; I f .0; T; x; u/ ! min;

(P3 )

.x; u/ 2 X.A; B; 0; T/; where y; z 2 Rn and T > 0. The study of these problems is based on the properties of solutions of the corresponding infinite horizon optimal control problem associated with the control system (2.1) and the integrand f . In [57] we were interested in a turnpike property of the approximate solutions of problems (P2 ). In this chapter we establish the turnpike property of the approximate solutions of problems (P1 ) and (P3 ). We show the stability of the turnpike phenomenon under small perturbations of the integrand f and study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals. For the problems (P2 ) and (P3 ) we show that in regions close to the right endpoint T of the time interval these approximate solutions are determined only by the integrand, and are essentially independent of the choice of interval and the endpoint value y. For the problems (P3 ), approximate solutions are determined only by the integrand also in regions close to the left endpoint 0 of the time interval. The following result was obtained in [57] (see also Chap. 6 of [44]). Proposition 2.1. There exists .xf ; uf / 2 X.A; B; 0;  / which is the unique solution of the following minimization problem I f .0; ; x; u/ ! min; .x; u/ 2 X.A; B; 0; / such that x.0/ D x./: Let a trajectory-control pair .xf ; uf / 2 X.A; B; 0;  / be as guaranteed by Proposition 2.1. Put .f / D  1 I f .0; ; xf ; uf /: The following results were obtained in [57] (see also Chap. 6 of [44]). Theorem 2.2. For any .x; u/ 2 X.A; B; 0; 1/ either (i) I f .0; T; x; u/  T.f / ! 1 as T ! 1 or (ii) supfjI f .0; T; x; u/  T.f /j W T > 0g < 1: Moreover, if relation (ii) holds, then supfjx.i C t/  xf .t/j W t 2 Œ0; g ! 0 as i ! 1; where i 2 Z:

(2.3)

24

2 Linear Control Systems with Periodic Convex Integrands

We say that .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-good [44, 53] if supfjI f .0; T; x; u/  T.f /j W T > 0g < 1: The second statement of Theorem 2.2 describes the asymptotic behavior of .f ; A; B/good trajectory-control pairs, shows that the corresponding infinite horizon optimal control problem has the turnpike property, and that the function xf is its turnpike. We say that .Qx; uQ / 2 X.A; B; 0; 1/ is .f ; A; B/-overtaking optimal [44, 53] if for each .x; u/ 2 X.A; B; 0; 1/ satisfying x.0/ D xQ .0/, lim supŒI f .0; T; xQ ; uQ /  I f .0; T; x; u/  0: T!1

Theorem 2.3. Let x0 2 Rn . Then there exists an .f ; A; B/-overtaking optimal trajectory-control pair .Qx; uQ / 2 X.A; B; 0; 1/ satisfying xQ .0/ D x0 . Moreover, if .x; u/ 2 X.A; B; 0; 1/ n f.Qx; uQ /g satisfies x.0/ D x0 , then there are T0 > 0 and  > 0 such that I f .0; T; x; u/  I f .0; T; xQ ; uQ / C  for all T  T0 : The next result describes the limit behavior of overtaking optimal trajectories. Theorem 2.4. Let M;  > 0. Then there exists a natural number N such that for any .f ; A; B/-overtaking optimal trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ which satisfies jx.0/j  M the relation supfjx.i C t/  xf .t/j W t 2 Œ0;  g  

(2.4)

holds for all integers i  N. Moreover, there exists ı > 0 such that for any .f ; A; B/overtaking optimal trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ satisfying jx.0/  xf .0/j  ı, the relation (2.4) holds for all integers i  0. Let T > 0 and y; z 2 Rn . Set .f ; y; z; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.0/ D y; x.T/ D zg;

(2.5)

.f ; y; T/ D inffI .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.0/ D yg;

(2.6)

.f O ; z; T/ D inffI .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.T/ D zg;

(2.7)

f

f

.f ; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/g: It follows from assumption (A) and Proposition 2.28 that 1 < .f ; y; z; T/; .f ; y; T/; O .f ; z; T/; .f ; T/ < 1:

(2.8)

2.1 Preliminaries and Turnpike Results

25

The next theorem establishes the turnpike property for approximate solutions of problems (P2 ) with the turnpike xf ./. Theorem 2.5. Let M;  > 0. Then there exist an integer N  1 and ı > 0 such that for each T > 2N and each .x; u/ 2 X.A; B; 0; T/ which satisfies jx.0/j  M; I f .0; T; x; u/  .f ; x.0/; T/ C ı the inequality supfjx.i C t/  xf .t/j W t 2 Œ0;  g  

(2.9)

holds for all integers i 2 ŒN;  1 T  N. Moreover if jx.0/  xf .0/j  ı, then inequality (2.9) holds for all integers i 2 Œ0;  1 T  N. Theorems 2.2–2.5 were obtained in [57] (see also Chap. 6 of [44]). Note that under assumptions of Theorem 2.5, if jx.b 1 Tc/xf .0/j  ı, then inequality (2.9) holds for all integers i 2 ŒN;  1 T  1. The next two results establish the turnpike property for approximate solutions of problems (P1 ) and (P3 ) respectively with the turnpike xf ./. Theorem 2.6. Let M;  > 0. Then there exist an integer N  1 and ı > 0 such that for each T > 2N and each .x; u/ 2 X.A; B; 0; T/ which satisfies jx.0/j; jx.T/j  M; I f .0; T; x; u/  .f ; x.0/; x.T/; T/ C ı inequality (2.9) holds for all integers i 2 ŒN;  1 T N. Moreover if jx.0/xf .0/j  ı, then inequality (2.9) holds for all integers i 2 Œ0;  1 T  N and if jx.b 1 Tc /  xf .0/j  ı, then inequality (2.9) holds for all integers i 2 ŒN;  1 T  1. Theorem 2.7. Let  > 0. Then there exist an integer N  1 and ı > 0 such that for each T > 2N and each .x; u/ 2 X.A; B; 0; T/ which satisfies I f .0; T; x; u/  .f ; T/ C ı inequality (2.9) holds for all integers i 2 ŒN;  1 T N. Moreover if jx.0/xf .0/j  ı, then inequality (2.9) holds for all integers i 2 Œ0;  1 T  N and if jx.b 1 Tc /  xf .0/j  ı, then inequality (2.9) holds for all integers i 2 ŒN;  1 T  1. Theorems 2.5–2.7 are partial cases of Theorem 2.13 stated in Sect. 2.2 which is one of the main results of the chapter. The next theorem establishes a weak version of the turnpike property for approximate solutions of problems (P1 ), (P2 ), and (P3 ) with the turnpike xf ./. Theorem 2.8. Let ; M0 ; M1 > 0. Then there exist natural numbers Q; l such that for each T > Ql and each .x; u/ 2 X.A; B; 0; T/ which satisfies at least one of the following conditions:

26

2 Linear Control Systems with Periodic Convex Integrands

jx.0/j; jx.T/j  M0 ; I f .0; T; x; u/  .f ; x.0/; x.T/; T/ C M1 I jx.0/j  M0 ; I f .0; T; x; u/  .f ; x.0/; T/ C M1 I I f .0; T; x; u/  .f ; T/ C M1 there exist strictly increasing sequences of nonnegative integers fai giD1 ; fbi giD1  Œ0;  1 T q

q

such that q  Q, 0  bi  ai  l for all i D 1; : : : ; q; bi  aiC1 for all integers i satisfying 1  i < q and that for each integer i 2 q Œ0;  1 T  1 n [jD1 Œaj ; bj , jx.i C t/  xf .t/j  ; t 2 Œ0;  : Theorem 2.8 is a partial case of Theorem 2.14, our stability result (see Sect. 2.2). We say that .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-minimal [5, 53] if for each T > 0, I f .0; T; x; u/ D .f ; x.0/; x.T/; T/:

(2.10)

The next result which is proved in Sect. 2.5 shows the equivalence of the optimality criterions introduced above. Theorem 2.9. Assume that .x; u/ 2 X.A; B; 0; 1/. Then the following conditions are equivalent: (i) .x; u/ is .f ; A; B/-overtaking optimal; (ii) .x; u/ is .f ; A; B/-minimal and .f ; A; B/-good; (iii) .x; u/ is .f ; A; B/-minimal and maxfjx.i C t/  xf .t/j W t 2 Œ0; g ! 0 as integers i ! 1I (iv) .x; u/ is .f ; A; B/-minimal and lim inft!1 jx.t/j < 1. The following result is also proved in Sect. 2.5. It shows that if the integrand f does not depend on the variable t, then xf ./ is a constant function. Theorem 2.10. Assume that for each x 2 Rn , each u 2 Rm , and each t1 ; t2  0, f .t1 ; x; u/ D f .t2 ; x; u/. Then xf .t/ D xf .0/ for all t 2 Œ0;   and xf .0/ does not depend of  . Corollary 2.11. Assume that for each x 2 Rn , each u 2 Rm , and each t1 ; t2  0, f .t1 ; x; u/ D f .t2 ; x; u/. Then for all t 2 Œ0; , xf .t/ D x and uf .t/ D u where .x ; u / 2 Rn  Rm is a unique solution of the minimization problem f .x; u/ ! min; .x; u/ 2 Rn  Rm ; Ax C Bu D 0:

2.2 Stability of the Turnpike Phenomenon

27

2.2 Stability of the Turnpike Phenomenon In this section we state Theorems 2.12–2.14 which show that the turnpike phenomenon is stable under small perturbations of the integrand f . We use the notation, definitions, and assumptions introduced in Sect. 2.1. Recall that f W Œ0; 1/  Rn  Rm ! R1 is a Borel measurable function satisfying assumption (A). Let a > 0 and W Œ0; 1/ ! Œ0; 1/ be an increasing function such that lim

t!1

.t/ D 1:

(2.11)

We suppose that for all .t; x; u/ 2 Œ0; 1/  Rn  Rm , f .t; x; u/  maxf .jxj/;

.juj/jujg  a:

(2.12)

Denote by M the set of all Borel measurable functions g W Œ0; 1/  Rn  Rm ! R1 which are bounded on all bounded subsets of Œ0; 1/  Rn  Rm and such that for all .t; x; u/ 2 Œ0; 1/  Rn  Rm , g.t; x; u/  maxf .jxj/;

.juj/jujg  a:

(2.13)

For the set M we consider the uniformity which is determined by the following base: E.N; ; / D f.g1 ; g2 / 2 M  M W jg1 .t; x; u/  g2 .t; x; u/j   for each t  0; each x 2 Rn satisfying jxj  N and each u 2 Rm satisfying juj  Ng \f.g1 ; g2 / 2 M  M W .jg1 .t; x; u/j C 1/.jg2 .t; x; u/j C 1/1 2 Œ1 ;  for each t  0; each x 2 Rn satisfying jxj  N and each u 2 Rm g;

(2.14)

where N > 0,  > 0,  > 1. It is not difficult to see that the space M with this uniformity is metrizable and complete. Let T2 > T1  0, y; z 2 Rn , and g 2 M. For each pair of Lebesgue measurable functions x W ŒT1 ; T2  ! Rn , u W ŒT1 ; T2  ! Rm set Z I .T1 ; T2 ; x; u/ D g

T2

g.t; x.t/; u.t//dt

(2.15)

T1

and set .g; y; z; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T1 / D y; x.T2 / D zg;

(2.16)

28

2 Linear Control Systems with Periodic Convex Integrands

.g; y; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T1 / D yg;

(2.17)

O .g; z; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T2 / D zg; .g; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 /g:

(2.18) (2.19)

Since any g 2 M is bounded on all the bounded subsets of Œ0; 1/  Rn  Rm it follows from Proposition 2.28 and (2.13) that all the values defined above are finite. In this chapter we prove the following three stability results. Theorem 2.12. Let ; M > 0. Then there exist an integer L0  1 and ı0 > 0 such that for each integer L1  L0 there exists a neighborhood U of f in M such that the following assertion holds. Assume that T > 2L1  , g 2 U , .x; u/ 2 X.A; B; 0; T/ and that a finite sequence q of integers fSi giD0 satisfy S0 D 0; SiC1  Si 2 ŒL0 ; L1 ; i D 0; : : : ; q  1; Sq  2 .T  L1 ; T;

(2.20)

I g .Si ; SiC1 ; x; u/  .SiC1  Si /.f / C M for each integer i 2 Œ0; q  1, I g .Si ; SiC2 ; x; u/  .g; x.Si /; x.SiC2  /; Si ; SiC2  / C ı0 for each nonnegative integer i  q  2 and I g .Sq2 ; T; x; u/  .g; x.Sq2 /; x.T/; Sq2 ; T/ C ı0 : Then there exist integers p1 ; p2 2 Œ0;  1 T such that p1  p2 , p1  2L0 , p2 >  1 T  2L1 and that for all integers i D p1 ; : : : ; p2  1, maxfjx.i C t/  xf .t/j W t 2 Œ0;  g  : Moreover if jx.0/  xf .0/j  ı0 , then p1 D 0 and if jx.b 1 Tc /  xf .0/j  ı0 , then p2 D Œ 1 T. Theorem 2.13. Let  2 .0; 1/; M0 ; M1 > 0. Then there exist an integer L  1, ı 2 .0; / and a neighborhood U of f in M such that for each T > 2L , each g 2 U , and each .x; u/ 2 X.A; B; 0; T/ which satisfies for each S 2 Œ0; T  L , I g .S; S C L; x; u/  .g; x.S/; x.S C L /; S; S C L / C ı

2.3 Structure of Solutions in the Regions Close to the End Points

29

and satisfies at least one of the following conditions: (a) jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C M1 I (b) jx.0/j  M0 ; I g .0; T; x; u/  .g; x.0/; 0; T/ C M1 I (c) I g .0; T; x; u/  .g; 0; T/ C M1 there exist integers p1 2 Œ0; L, p2 2 Œb 1 Tc  L;  1 T such that for all integers i D p1 ; : : : ; p2  1, jx.i C t/  xf .t/j   for all t 2 Œ0;  : Moreover if jx.0/  xf .0/j  ı, then p1 D 0 and if jx.b 1 Tc /  xf .0/j  ı, then p2 D b 1 Tc. Denote by Card.A/ the cardinality of the set A. Theorem 2.14. Let  2 .0; 1/; M0 ; M1 > 0. Then there exist an integer L  1 and a neighborhood U of f in M such that for each T > L, each g 2 U , and each .x; u/ 2 X.A; B; 0; T/ which satisfies at least one of the following conditions: (a) jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C M1 I (b) jx.0/j  M0 ; I g .0; T; x; u/  .g; x.0/; 0; T/ C M1 I (c) I g .0; T; x; u/  .g; 0; T/ C M1 the following inequality holds: Card.fi 2 f0; : : : ; b 1 Tc  1g W maxfjx.i C t/  xf .t/j W t 2 Œ0;  g > g/  L:

2.3 Structure of Solutions in the Regions Close to the End Points In this section we state results which describe the structure of solutions of problems (P1 ), (P2 ) and (P3 ) in the regions close to the end points. Combined with the turnpike results of Sect. 2.2 they provide the full description of the structure of their solutions. We use the notation, definitions, and assumptions introduced in Sects. 2.1 and 2.2. By Theorem 2.14 for each z 2 Rn there exists a unique .f ; A; B/-overtaking optimal pair . .z/ ; .z/ / 2 X.A; B; 0; 1/ such that  .z/ .0/ D z. Let z 2 Rn . Set  f .z/ D lim inf ŒI f .0; T;  .z/ ; .z/ /  T .f /: T!1; T2Z

(2.21)

30

2 Linear Control Systems with Periodic Convex Integrands

In view of Theorems 2.2, 2.3, and 2.9,  f .z/ is a finite number. Definition (2.21) and the definition of .f ; A; B/-overtaking optimal pairs imply the following result. Proposition 2.15. 1. Let .x; u/ 2 X.A; B; 0; 1/ be .f ; A; B/-good. Then  f .x.0//  lim inf ŒI f .0; T; x; u/  T .f / T!1; T2Z

and for each pair of integers S > T  0,  f .x.T //  I f .T; S; x; u/  .S  T/ .f / C  f .x.S//:

(2.22)

2. Let S > T  0 be integers and .x; u/ 2 X.A; B; T; S /. Then (2.22) holds. The next result follows from definition (2.21). Proposition 2.16. Let .x; u/ 2 X.A; B; 0; 1/ be .f ; A; B/-overtaking optimal. Then for each pair of integers S > T  0,  f .x.T// D I f .T; S; x; u/  .S  T/.f / C  f .x.S//: Theorems 2.3–2.5 and (2.21), (2.3) imply the following result. Proposition 2.17.  f .xf .0// D 0. The following result is proved in Sect. 2.11. Proposition 2.18. The function  f is continuous at xf .0/. Proposition 2.19. Let .x; u/ 2 X.A; B; 0; 1/ be .f ; A; B/-overtaking optimal. Then  f .x.0// D

lim

ŒI f .0; T; x; u/  T .f /:

T!1; T2Z

Proof. It follows from Propositions 2.16–2.18 and Theorems 2.2 and 2.9 that  f .x.0// D D

lim

. f .x.0//   f .x.T///

lim

ŒI f .0; T; x; u/  T .f /:

T!1; T2Z

T!1; T2Z

Proposition 2.19 is proved. Proposition 2.20. The function  f is strictly convex and continuous. Proof. It is sufficient to show that the function  f is strictly convex. Let y; z 2 Rn , y 6D z and ˛ 2 .0; 1/. Consider .˛ .y/ C .1  ˛/ .z/ ; ˛ .y/ C .1  ˛/ .z/ / 2 X.A; B; 0; 1/ which satisfies .˛ .y/ C.1˛/ .z/ /.0/ D ˛yC.1˛/z and in view of (A) and Theorem 2.2, is .f ; A; B/-good. By Propositions 2.15 and 2.19, assumption (A) and the relation y 6D z,

2.3 Structure of Solutions in the Regions Close to the End Points

31

 f .˛y C .1  ˛/z/ D lim inf ŒI f .0; T; ˛ .y/ C .1  ˛/ .z/ ; ˛ .y/ T!1; T2Z

C .1  ˛/ .z/ /  T.f / < lim inf Œ˛.I f .0; T;  .y/ ; .y/ /  T .f // T!1; T2Z

C .1  ˛/.I f .0; T;  .z/ ; .z/ /  T .f // D˛

lim

ŒI f .0; T;  .y/ ; .y/ /  T .f /

T!1; T2Z

C .1  ˛/

lim

ŒI f .0; T;  .z/ ; .z/ /  T .f /

T!1; T2Z

D ˛ f .y/ C .1  ˛/ f .z/: Proposition 2.20 is proved. The next result is proved in Sect. 2.11. Proposition 2.21. For each M > 0 the set fx 2 Rn W  f .x/  Mg is bounded. Set inf. f / D inff f .z/ W z 2 Rn g:

(2.23)

By Propositions 2.20 and 2.21, inf. f / is finite and there exists a unique f 2 Rn such that  f . f / D inf. f /. Proposition 2.22. Let .x; u/ 2 X.A; B; 0; 1/ be .f ; A; B/-good such that for all integers T > 0, I f .0; T; x; u/  T.f / D  f .x.0//   f .x.T//:

(2.24)

Then .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-overtaking optimal. Proof. Theorem 2.3 implies that there exists an .f ; A; B/-overtaking optimal pair .x1 ; u1 / 2 X.A; B; 0; 1/ such that x1 .0/ D x.0/: By Proposition 2.16, for each integer T  1, I f .0; T; x1 ; u1 /  T.f / D  f .x1 .0//   f .x1 .T//: It follows from the equality above, (2.24), Theorems 2.2 and 2.9, and Propositions 2.17 and 2.18 that for all integers T > 0, I f .0; T; x; u/  I f .0; T; x1 ; u1 / D  f .x1 .T//   f .x.T // ! 0 as T ! 1:

32

2 Linear Control Systems with Periodic Convex Integrands

Thus lim

ŒI f .0; T; x; u/  I f .0; T; x1 ; u1 / D 0

T!1; T2Z

and in view of Theorem 2.3 this implies the pair .x; u/ is .f ; A; B/-overtaking optimal. Proposition 2.22 is proved. Consider a linear control system x0 .t/ D Ax.t/  Bu.t/; x.0/ D x0 which is also controllable. There exists a Borel measurable function fN W Œ0; 1/  Rn  Rm ! R1 such that for all .x; u/ 2 Rn  Rm , fN .t C ; x; u/ D fN .t; x; u/ for all t  0; fN .t; x; u/ D f .  t; x; u/ for all t 2 Œ0;  :

(2.25)

Evidently, fN satisfies assumption (A). For fN we use all the notation and definitions introduced for f . It is clear that all the results obtained for the triplet .f ; A; B/ also hold for the triplet .fN ; A; B/. Assume that integers S2 > S1  0 and that .x; u/ 2 X.A; B; S1 ; S2  /. For all t 2 ŒS1 ; S2   set xN .t/ D x.S2   t C S1 /; uN .t/ D u.S2   t C S1  /:

(2.26)

In view of (2.26) for a. e. t 2 ŒS1 ; S2 , xN 0 .t/ D x0 .S2   t C S1 / D Ax.S2   t C S1  /  Bu.S2   t C S1  / D ANx.t/  BNu.t/ and .Nx; uN / 2 X.A; B; S1 ; S2 /. By (2.25) and (2.26), Z

S2  S1 

fN .t; xN .t/; uN .t//dt D

Z Z

D

S2  S1  S2  S1 

fN .t; x.S2   t C S1  /; u.S2   t C S1  //dt f .S2   t C S1 ; x.S2   t C S1  /;

u.S2   t C S1 //dt Z S2  D f .t; x.t/; u.t//dt: S1 

(2.27)

2.3 Structure of Solutions in the Regions Close to the End Points

33

For each pair T2 > T1  0 and each .x; u/ 2 X.A; B; T1 ; T2 / set Z

fN

I .T1 ; T2 ; x; u/ D

T2

fN .t; x.t/; u.t//dt:

T1

For each y; z 2 Rn and each T > 0 set N

 .fN ; y; z; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.0/ D y; x.T/ D zg; N  .fN ; y; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.0/ D yg; N

O  .fN ; z; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.T/ D zg; N  .fN ; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/g:

(2.28)

Relations (2.26) and (2.27) imply the following result. Proposition 2.23. Let S2 > S1  0 be integers, M  0 and that .xi ; ui / 2 X.A; B; S1 ; S2  /, i D 1; 2. Then I f .S1 ; S2 ; x1 ; u1 /  I f .S1 ; S2 ; x2 ; u2 /  M if and only if I fN .S1 ; S2 ; xN 1 ; uN 1 /  I fN .S1 ; S2 ; xN 2 ; uN 2 /  M: Proposition 2.23 implies the following result. Proposition 2.24. Let S2 > S1  0 be integers and .x; u/ 2 X.A; B; S1 ; S2  /: Then the following assertion holds: I f .S1 ; S2 ; x; u/  .f ; .S2  S1 // C M N if and only if I f .S1 ; S2 ; xN ; uN /   .fN ; .S2  S1 // C MI

I f .S1 ; S2 ; x; u/  .f ; x.S1 /; x.S2 /; .S2  S1 // C M N if and only if I f .S1 ; S2 ; xN ; uN /   .fN ; xN .S1 /; xN .S2  /; .S2  S1 // C MI

I f .S1 ; S2 ; x; u/  .f ; x.S1 /; .S2  S1 // C M N if and only if I f .S1 ; S2 ; xN ; uN /  O  .fN ; xN .S2  /; .S2  S1 // C MI

I f .S1 ; S2 ; x; u/  O .f ; x.S2 /; .S2  S1 // C M N

if and only if I f .S1 ; S2 ; xN ; uN /   .fN ; xN .S1  /; .S2  S1 // C M:

34

2 Linear Control Systems with Periodic Convex Integrands

By Proposition 2.1, .xf ; uf / 2 X.A; B; 0; / is the unique solution of the minimization problem I f .0; ; x; u/ ! min; .x; u/ 2 X.A; B; 0; / such that x.0/ D x./: Analogously there exists .xfN ; ufN / 2 X.A; B; 0; / which is the unique solution of the minimization problem N

I f .0; ; x; u/ ! min; .x; u/ 2 X.A; B; 0;  / such that x.0/ D x./: In view of Proposition 2.23 and (2.27), for all t 2 Œ0;  , xfN .t/ D xf .  t/; ufN .t/ D uf .  t/; .fN / D .f /:

(2.29)

For each z 2 Rn , set N

N

 f .z/ D lim inf ŒI f .0; T; x; u/  T .f /; T!1; T2Z

(2.30)

where .x; u/ 2 X.A; B; 0; 1/ is the unique .fN ; A; B/-overtaking optimal pair such that x.0/ D z. Let .x ; u / 2 X.A; B; 0; 1/ be the unique .f ; A; B/-overtaking optimal pair such that  f .x .0// D inf. f / and .Nx ; uN  / 2 X.A; B; 0; 1/ be the unique .fN ; A; B/-overtaking optimal pair such that  fN .Nx .0// D inf. fN /. The following three theorems describe the structure of solutions of problems (P1 ), (P2 ), and (P3 ) in the regions closed to the end points. Theorem 2.25. Let L0 > 0 be an integer,  2 .0; 1/; M > 0. Then there exist ı > 0, a neighborhood U of f in M and an integer L1 > L0 such that for each integer T  L1 , each g 2 U and each .x; u/ 2 X.A; B; 0; T / which satisfies jx.0/j  M; I g .0; T; x; u/  .g; x.0/; 0; T / C ı the following inequality holds: jx.T  t/  xN  .t/j   for all t 2 Œ0; L0  : Theorem 2.26. Let L0 > 0 be an integer,  > 0. Then there exist ı > 0, a neighborhood U of f in M and an integer L1 > L0 such that for each integer T  L1 , each g 2 U and each .x; u/ 2 X.A; B; 0; T / which satisfies I g .0; T; x; u/  .g; 0; T / C ı

2.4 Auxiliary Results

35

the following inequalities hold for all t 2 Œ0; L0 : jx.T  t/  xN  .t/j  ; jx.t/  x .t/j  : Theorem 2.27. Let L0 > 0 be an integer,  > 0; M0 > 0. Then there exist ı > 0, a neighborhood U of f in M and an integer L1 > L0 such that for each integer T  L1 , each g 2 U and each .x; u/ 2 X.A; B; 0; T / which satisfies jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T /; 0; T / C ı the inequalities N jx.T  t/  .t/j  ; jx.t/  .t/j   hold for all t 2 Œ0; L0 , where .; / 2 X.A; B; 0; 1/ is the unique .f ; A; B/overtaking optimal pair such that .0/ D x.0/ and N / .; N 2 X.A; B; 0; 1/ N is the unique .fN ; A; B/-overtaking optimal pair such that .0/ D x.T /.

2.4 Auxiliary Results In the sequel we use the following auxiliary results. Proposition 2.28 (Proposition 6.2.1 of [44]). For every yQ ; zQ 2 Rn and every T > 0 there exists a solution x./, y./ of the system x0 D Ax C BBt y; y0 D x  At y with the boundary conditions x.0/ D yQ , x.T/ D zQ (where Bt denotes the transpose of B). Proposition 2.29 (Proposition 6.2.2 of [44]). Let M1 > 0 and 0 < 0 < 1 . Then there exists a positive number M2 such that for each T 2 Œ0 ; 1  and each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  M1 the inequality jx.t/j  M2 holds for all t 2 Œ0; T. Proposition 2.30 (Proposition 6.2.4 of [44]). Let M1 and T be positive numbers and let F be the set of all .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  M1 . Then 1 for every sequence f.xi ; ui /g1 iD1  F there exist a subsequence f.xik ; uik /gkD1 and 0 0 .x; u/ 2 F such that xik .t/ ! x.t/ as k ! 1 uniformly in Œ0; T, xik ! x as k ! 1 weakly in L1 .Rn I .0; T//, and uik ! u as k ! 1 weakly in L1 .Rm I .0; T//.

36

2 Linear Control Systems with Periodic Convex Integrands

For each y; z 2 Rn define v.y; z/ D inffI f .0; ; x; u/ W .x; u/ 2 X.A; B; 0;  / such that x.0/ D y; x. / D zg:

(2.31)

It was shown in Sect. 6.2 of [44] that the function v is convex, satisfies 1 < v.y; z/ < 1 for each y; z 2 Rn ; v.y; z/ ! 1 as jyj C jzj ! 1

(2.32)

and that there exists zf 2 Rn such that v.zf ; zf / < v.z; z/ for all z 2 Rn n fzf g;

(2.33)

xf .0/ D zf ; .f / D  1 v.zf ; zf /:

(2.34)

Proposition 2.31 (Proposition 6.2.5 of [44]). There exists pf 2 Rn such that the function f W Rn  Rn ! R1 defined by

f .y; z/ D v.y; z/  v.zf ; zf /  hpf ; y  zi; y; z 2 Rn is strictly convex and

f .zf ; zf / D 0; f .y; z/ > 0 for all .y; z/ 2 Rn  Rn n f.zf ; zf /g:

(2.35)

Define a function f0 W Rn  Rm ! R1 by f0 .x; u/ D supff .t; x; u/ W t 2 Œ0; 1/g; .x; u/ 2 Rn  Rm :

(2.36)

In view of assumption (A), the function f0 is well defined, convex, and bounded on bounded subsets of Rn  Rm . For all y; z 2 Rn set Z v0 .y; z/ D inf

0



f0 .x.t/; u.t//dt W .x; u/ 2 X.A; B; 0;  / 

such that x.0/ D y; x. / D z :

(2.37)

By (2.37), convexity of f0 , Proposition 2.28 and assumption (A), the function v0 W Rn  Rn ! R1 is well defined, convex, and continuous. Proposition 2.32 (Corollary 6.2.1 of [44]). Let x1 ; x2 2 Rn . Then there is a unique .x; u/ 2 X.A; B; 0;  / such that x.0/ D x1 , x. / D x2 and I f .0; ; x; u/ D v.x1 ; x2 /:

2.4 Auxiliary Results

37

Proposition 2.33. Let  > 0. Then there exist ı > 0 such that for each integer k  1 and each y; z 2 Rn satisfying jy  xf .0/j; jz  xf .0/j  ı, .f ; y; z; k/  k.f / C : Proof. Since the function v is continuous there exists ı 2 .0; / such that for each y; z 2 Rn satisfying jy  xf .0/j; jz  xf .0/j  ı;

(2.38)

we have jv.y; z/  .f /j D jv.y; z/  v.xf .0/; xf .0//j  =4: Let k  1 be an integer and y; z 2 Rn satisfy (2.38). Assume that k D 1. By (2.31) and the choice of ı [see (2.38)], .f ; y; z; / D v.y; z/  .f / C : Assume that k > 1. By Proposition 2.32, there exists .x; u/ 2 X.A; B; 0; k / such that x.0/ D y; x. / D xf .0/; I f .0; ; x; u/ D v.y; xf .0//; x..k  1// D xf .0/; x.k/ D z; I f ..k  1/; k; x; u/ D v.xf .0/; z/; and that for each integer i satisfying 1  i < k  1, x.i C t/ D xf .t/; u.i C t/ D uf .t/; t 2 Œ0;  : By the definition above, the choice of ı [see (2.38)], .f ; y; z; k /  I f .0; k; x; u/ D v.xf .0/; z/ C v.y; xf .0// C  .f /.k  2/  2..f / C =4/ C .f /.k  2/  k .f / C =2: Proposition 2.33 is proved. Proposition 2.34. There exists M > 0 such that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/, I f .0; T; x; u/  T.f /  M : Proof. By (A) there is c0 > 0 such that f .t; x; u/  c0 for all .t; x; u/ 2 Œ0; 1/  Rn  Rm :

(2.39)

38

2 Linear Control Systems with Periodic Convex Integrands

In view of Proposition 2.29 there exists M0 > 0 such that for each T 2 Œ=2; 4  and each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  4 .j.f /j C 1/ we have jx.t/j  M0 for all t 2 Œ0; T:

(2.40)

Let pf 2 Rn be as guaranteed by Proposition 2.31. Choose M > 4c0  C 4j.f /j C 1 C 4jpf jM0 :

(2.41)

Let T > 0 and .x; u/ 2 X.A; B; 0; T/. If T  4, then by (2.39) and (2.41), I f .0; T; x; u/  4c0   T.f /  4j.f /j  4c0   T.f /  M and in this case Proposition 2.34 holds. Assume that T > 4:

(2.42)

jx.i/j > M0 for all integers i 2 Œ1;  1 T  1I

(2.43)

There are two cases:

minfjx.i/j W an integer i 2 Œ1;  1 T  1g  M0 :

(2.44)

Assume that (2.43) holds. Set S0 D 0; Si D i for all integers i 2 Œ0; b 1 Tc  1; Sb 1 Tc D T:

(2.45)

By (2.45), Sb 1 Tc  Sb 1 Tc1 D T  .b 1 Tc  1/ 2 Œ; 2 : By the choice of M0 (see (2.40)), (2.42), (2.43), (2.45) and (2.46), I f .Si ; SiC1 ; x; u/ > 4 .j.f /j C 1/; i D 0; : : : ; bT=c  1: This implies that I f .0; T; x; u/  4 .j.f /j C 1/bT=c > 2.j.f /j C 1/T: Thus in this case Proposition 2.34 holds.

(2.46)

2.4 Auxiliary Results

39

Assume that (2.44) holds. Then there exist integers j1 ; j2 such that 1  j1  j2   1 T  1;

(2.47)

jx.j1 /j; jx.j2 /j  M0 ;

(2.48)

jx.i/j > M0

(2.49)

for each integer i satisfying 1  i < j1 and for each integer i satisfying j2 < i  T=  1. We will estimate I f .0; j1 ; x; u/, I f .j2 ; T; x; u/ and I f .j1 ; j2 ; x; u/. By (2.39), I f .0; ; x; u/  c0 :

(2.50)

In view of (2.49) and the choice of j1 and M0 [see (2.40)], for each integer i satisfying 1  i < j1 , I f .i; .i C 1/; x; u/ > 4 .j.f /j C 1/: Together with (2.50) this implies that I f .0; j1 ; x; u/  j1 .f /  c0   j.f /j:

(2.51)

I f .b 1 Tc  ; T; x; u/  2c0 :

(2.52)

By (2.39),

It follows from (2.49) and the choice of j2 and M0 [see (2.40)] that for each integer i satisfying j2  i < T=  1, I f .i; .i C 1/; x; u/  4 .j.f /j C 1/: Together with (2.52) this implies that I f .j2 ; T; x; u/  .T  j2 /.f /  2c0   2j.f /j:

(2.53)

If j1 D j2 , then (2.51) and (2.53) imply that I f .0; T; x; u/  T.f /  3c0   3j.f /j > M : Therefore we may assume without loss of generality that j1 < j2 . We estimate I f .j1 ; j2 ; x; u/. By (2.31), the choice of pf , Proposition 2.31, (2.34), (2.42), and (2.48),

40

2 Linear Control Systems with Periodic Convex Integrands j2 1

I f .j1 ; j2 ; x; u/  .j2  j1 / .f / 

X

v.x.i/; x..i C 1/ //  .j2  j1 /v.zf ; zf /

iDj1

 hpf ; x.j1 /  x.j2  /i  2jpf jM0 :

(2.54)

It follows from (2.41), (2.54), (2.51), and (2.53) that I f .0; T; x; u/  T.f / D I f .0; j1 ; x; u/  j1 .f / C I f .j1 ; j2 ; x; u/  .j2  j1 / .f / C I f .j2 ; T; x; u/  .T  j2  /.f /  c0    j.f /j  2jpf jM0  2c0   2 j.f /j  3c0   3j.f /j  2jpf jM0 > M : Proposition 2.34 is proved. Proposition 2.35. Let M0 > 0. Then there exists M > 0 such that for each T  3 and each y; z 2 Rn satisfying jyj; jzj  M0 , .f ; y; z; T/  T.f / C M: Proof. We may assume without loss of generality that M0  jxf .t/j for all t 2 Œ0;  :

(2.55)

Since the function v0 is continuous there exists M1 > 0 such that jv0 .y; z/j  M1 for all y; z 2 Rn satisfying jyj; jzj  M0 :

(2.56)

In view of assumption (A), there exists c0 > 0 such that f .t; x; u/  c0 for all .t; x; u/ 2 Œ0; 1/  Rn  Rm :

(2.57)

Choose M > 2M1 C 2c0  C 2 C 2 j.f /j:

(2.58)

T  3; y; z 2 Rn ; jyj; jzj  M0 :

(2.59)

 2 Œ0; /

(2.60)

Assume that

There exists

2.4 Auxiliary Results

41

such that .T  / 1 2 Z:

(2.61)

By (2.37), there exists .x1 ; u1 / 2 X.A; B; 0; / such that x1 .0/ D y; x1 . / D xf .0/;

Z

 0

f0 .x1 .t/; u1 .t//dt  v0 .y; xf .0// C 1:

(2.62) (2.63)

By (2.37), there exists .x2 ; u2 / 2 X.A; B; T  ; T/ such that Z

T

x2 .T  / D xf ./; x2 .T/ D z;

(2.64)

f0 .x2 .t/; u2 .t//dt  v0 .xf ./; z/ C 1:

(2.65)

T

It follows from (2.36), (2.55), (2.56), (2.59), and (2.62)–(2.65) that Z Z

 0

T

Z f .t; x1 .t/; u1 .t//dt  Z f .t; x2 .t/; u2 .t//dt 

T

 0

f0 .x1 .t/; u1 .t//dt  M1 C 1;

T

f0 .x2 .t/; u2 .t//dt  M1 C 1:

(2.66)

T

Define x.t/ D x1 .t/; u.t/ D u1 .t/; t 2 Œ0;  ; x.t/ D xf .t   b 1 tc/; u.t/ D uf .t   b 1 tc/; t 2 .; T   ; x.t/ D x2 .t/; u.t/ D u2 .t/; t 2 .T  ; T: By (2.61), (2.62), (2.64), (2.67), .x; u/ 2 X.A; B; 0; T/; x.0/ D y; x.T/ D z: In view of (2.3), (2.47), (2.58), (2.66), and (2.67), I f .0; T; x; u/ D I f .0; ; x1 ; u1 / C I f .T  ; T; x2 ; u2 / C I f .; T  ; x; u/  2M1 C 2 C .b 1 Tc  1/.f / C c0  < T.f / C M: Proposition 2.35 is proved.

(2.67)

42

2 Linear Control Systems with Periodic Convex Integrands

Proposition 2.36. Let M;  > 0. Then there exists a natural number L such that for each .x; u/ 2 X.A; B; 0; L/ satisfying I f .0; L; x; u/  L.f / C M there exists an integer i 2 Œ0; L  1 such that supfjx. i C t/  xf .t/j W t 2 Œ0;  g  : Proof. Assume that the proposition does not hold. Then there exist a strictly increasing sequence of natural numbers fLk g1 kD1 such that Lk  k for all integers k  1 and a sequence .xk ; uk / 2 X.A; B; 0; Lk /, k D 1; 2; : : : such that for each integer k  1 and each integer i 2 Œ0; Lk  1, I f .0; Lk ; xk ; uk /  Lk .f / C M;

(2.68)

supfjxk .i C t/  xf .t/j W t 2 Œ0;  g > :

(2.69)

By Proposition 2.34, there exists M > 0 such that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/, I f .0; T; x; u/  T.f /  M :

(2.70)

Let p  1 be an integer. It follows from (2.68) and (2.70) that for each integer k > p, I f .0; p; xk ; uk / D I f .0; Lk ; xk ; uk /  I f .p; Lk ; xk ; uk /  Lk .f / C M  .Lk   p /.f / C M  p.f / C M C M :

(2.71)

By (2.71) and Proposition 2.30, extracting a subsequence and re-indexing if necessary, we may assume without loss of generality that there exists .x; u/ 2 X.A; B; 0; 1/ such that for each integer p  1, xk .t/ ! x.t/ as k ! 1 uniformly on Œ0; p ;

(2.72)

xk0 ! x0 as k ! 1 weakly in L1 .Rn I .0; p//; uk ! u as k ! 1 weakly in L1 .Rm I .0; p//; I f .0; p; x; u/  p.f / C M C M :

(2.73)

In view of (2.73) and Theorem 2.2, .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-good and there exists an integer i0  1 such that for each integer i  i0 , supfjx. i C t/  xf .t/j W t 2 Œ0;  g  =4:

(2.74)

2.4 Auxiliary Results

43

Relation (2.72) implies that there exists an integer k0 > i0 C 4 such that for each integer k  k0 , jxk .t/  x.t/j  =4 for all t 2 Œ0; .i0 C 4/ :

(2.75)

By (2.74) and (2.75), for each integer k  k0 and each t 2 Œ0;  , jxf .t/  xk .i0  C t/j  jxf .t/  x.i0  C t/j C jx.i0  C t/  xk .i0  C t/j  =2: This contradicts (2.69). The contradiction we have reached proves Proposition 2.36. Proposition 2.37. Any .f ; A; B/-overtaking optimal .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-good. Proof. Let .x; u/ 2 X.A; B; 0; 1/ be .f ; A; B/-overtaking optimal. By Theorem 2.4, there is a number M0 such that M0 > jx.t/j for all t  0. Together with Proposition 2.35 this implies the existence of M1 > 0 such that for each T  3, I f .0; T; x; u/  T.f /CM1 : In view of Theorem 2.2, the pair .x; u/ is .f ; A; B/-good. Proposition 2.38. Let M;  > 0. Then there exists a natural number L such that for each T  L, each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  T.f / C M

(2.76)

and each integer S satisfying ŒS; .S C L/   Œ0; T

(2.77)

there exists an integer i 2 ŒS; S C L  1 such that jx.i C t/  xf .t/j   for all t 2 Œ0;  :

(2.78)

Proof. By Proposition 2.34, there exists M > 0 such that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/, I f .0; T; x; u/  T.f /  M :

(2.79)

By Proposition 2.36, there exists a natural number L such that the following property holds: (i) for each .x; u/ 2 X.A; B; 0; L/ satisfying I f .0; L; x; u/  L.f / C M C 2M

44

2 Linear Control Systems with Periodic Convex Integrands

there exists an integer i 2 Œ0; L  1 such that jx. i C t/  xf .t/j   for all t 2 Œ0;  : Assume that T  L, an .x; u/ 2 X.A; B; 0; T/ satisfies (2.76) and an integer S satisfies (2.77). By the choice of M [see (2.79)], I f .0; S; x; u/  S .f /  M ; I f . .S C L/; T; x; u/  .T   .S C L//.f /  M : Together with (2.76) this implies that I f .S;  .S C L/; x; u/ D I f .0; T; x; u/  I f .0; S; x; u/  I f ..S C L/; T; x; u/  T.f / C M  S.f / C M  .T  .S C L//.f / C M  L.f / C 2M : By the inequality above and property (i), there exists an integer i such that Œi; .i C 1/  ŒS; .S C L/  and (2.78) holds. Proposition 2.38 is proved. Proposition 2.39. Let  2 .0; 1/. Then there exists ı > 0 such that for each integer p  1, each .x; u/ 2 X.A; B; 0; p/ satisfying jx.0/  xf .0/j; jx.p/  xf .0/j  ı; I f .0; p; x; u/  .f ; x.0/; x.p /; p / C ı and each integer i 2 Œ0; p  1, the inequality jx.i C t/  xf .t/j   holds for all t 2 Œ0;  . Proof. By the continuity of v, (2.34) and Proposition 2.33, for each integer k  1, there is ık 2 .0; 4k /

(2.80)

such that the following properties hold: (ii) for each y; z 2 Rn satisfying jy  xf .0/j; jz  xf .0/j  ık we have jv.y; z/  .f /j  4k ; (iii) for each integer p  1 and each y; z 2 Rn satisfying jy  xf .0/j; jz  xf .0/j  ık we have .f ; y; z; p/  p.f / C 4k : We may assume without loss of generality that the sequence fık g1 kD1 is decreasing. Assume that the proposition does not hold. Then for each natural number k there exist an integer pk  1 and .xk ; uk / 2 X.A; B; 0; pk  / satisfying jxk .0/  xf .0/j  ık ; jxk .pk /  xf .0/j  ık ;

(2.81)

I f .0; pk ; xk ; uk /  .f ; xk .0/; xk .pk  /; pk  / C ık ;

(2.82)

supfsupfjxk .i C t/  xf .t/j W t 2 Œ0; g W i D 0; : : : ; pk  1g > :

(2.83)

2.4 Auxiliary Results

45

By property (iii) and (2.80)–(2.82), for each integer k  1, I f .0; pk ; xk ; uk /  pk .f / C 2  4k :

(2.84)

In view of Proposition 2.32 there exists .x; u/ 2 X.A; B; 0; 1/ such that x.t/ D x1 .t/; u.t/ D u1 .t/; t 2 Œ0; p1 ; x..p1 C 1/ / D x2 .0/; I f .p1 ; .p1 C 1/; x; u/ D v.x1 .p1  /; x2 .0//

(2.85) (2.86)

and for each integer k  1, X  k x .pi C 1/ C t D xkC1 .t/;

(2.87)

iD1

X  k u .pi C 1/ C t D ukC1 .t/;

(2.88)

iD1

x

X  kC1 .pi C 1/ D xkC2 .0/;

(2.89)

iD1

 X  X  kC1 kC1 I .pi C 1/  1 ; .pi C 1/; x; u D v.xkC1 .pkC1  /; xkC2 .0//: f

iD1

iD1

(2.90) By (2.81), (2.84)–(2.90) and property (ii), for each integer k  2,  X   X k k I 0; .pi C 1/ ; x; u D .I f .0; pi ; xi ; ui / C v.xi .pi  /; xiC1 .0/// f

iD1

iD1



k X Œpi .f / C 2  4i C .f / C 4i  iD1

 .f /

k X

.pi C 1/ C 6:

iD1

Since the relation above holds for any integer k  2 it follows from Theorem 2.2 that the pair .x; u/ is .f ; A; B/-good and supfjx.i C t/  xf .t/j W t 2 Œ0; g ! 0 as i ! 1:

46

2 Linear Control Systems with Periodic Convex Integrands

Thus there exists an integer i0  1 such that for each integer i  i0 ,   jx.i C t/  xf .t/j; t 2 Œ0;  : In view of (2.85)–(2.90) this contradicts (2.83). The contradiction we have reached proves Proposition 2.39.

2.5 Proofs of Theorems 2.9 and 2.10 Proof of Theorem 2.9. In view of Proposition 2.37, (i) implies (ii). By Theorem 2.2, (ii) implies (iii). Clearly, (iv) follows from (iii). Let us show that (iv) implies (i). Assume that .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-minimal and that lim inf jx.t/j < 1: t!1

(2.91)

We show that .x; u/ is .f ; A; B/-overtaking optimal. Since .x; u/ is .f ; A; B/-minimal it follows from (2.91), Proposition 2.35, and Theorem 2.2 that .x; u/ is .f ; A; B/-good and that lim

i!1; i2Z

maxfjx.i C t/  xf .t/j W t 2 Œ0;  g D 0:

(2.92)

Assume that .x; u/ is not .f ; A; B/-overtaking optimal. By Theorem 2.3, there exist an .f ; A; B/-overtaking optimal .Qx; uQ / 2 X.A; B; 0; 1/,  > 0 and T0 > 0 such that xQ .0/ D x.0/;

(2.93)

I f .0; T; x; u/  I f .0; T; xQ ; uQ / C  for all T  T0 :

(2.94)

In view of Theorem 2.2 and Proposition 2.37, lim

i!1; i2Z

maxfjQx.i C t/  xf .t/j W t 2 Œ0;  g D 0:

(2.95)

Since the function v is continuous there exists ı > 0 such that jv.z1 ; z2 /  v.xf .0/; xf .0//j  =4 for all z1 ; z2 2 Rn satisfying jzi  xf .0/j  ı; i D 1; 2:

(2.96)

By (2.92) and (2.95), there exists a natural number p > T0 = such that for all integers i  p; jQx.i/  xf .0/j; jx.i/  xf .0/j  ı:

(2.97)

2.6 Auxiliary Results for Theorem 2.12

47

Proposition 2.32 implies that there exists .x1 ; u1 / 2 X.A; B; 0; .p C 1/ / such that x1 .t/ D xQ .t/; u1 .t/ D uQ .t/; t 2 Œ0; p; x1 ..p C 1/ / D x..p C 1/ /; I f . p; .p C 1/; x1 ; u1 / D v.Qx.p/; x..p C 1/ //:

(2.98)

By (2.94), (2.96), (2.97), (2.98), and .f ; A; B/-minimality of .x; u/, I f .0; .p C 1/; x1 ; u1 /  I f .0; .p C 1/; x; u/ D I f .0;  p; xQ ; uQ /  I f .0;  p; x; u/ C I f .p;  .p C 1/; x1 ; u1 /  I f .p;  .p C 1/x; u/   C v.Qx.p /; x..p C 1/ //  v.x.p /; x..p C 1/ //   C =2: This contradicts .f ; A; B/-minimality of .x; u/. The contradiction we have reached completes the proof of Theorem 2.9. t u Proof of Theorem 2.10. Since f satisfies assumption (A) with any  > 0 all our results hold for all  > 0. By Theorems 2.2, 2.3, and 2.9, .f / D lim T 1 I f .0; T; x; u/; T!1

for any .f ; A; B/-overtaking optimal .x; u/ 2 X.A; B; 0; 1/. In view of Proposition 2.1, for any  > 0, there exists .x ; u / 2 X.A; B; 0;  / which is a unique solution of the minimization problem I f .0; ; x; u/ ! min; .x; u/ 2 X.A; B; 0; /; x.0/ D x./; .f / D I f .0; ; x ; u /:

(2.99) (2.100)

By (2.99) and (2.100), for any  > 0, x .t/ D x=2 .t/; t 2 Œ0; =2 and x .=2Ct/ D x .t/; t 2 Œ0; =2. Since the relation above holds for any  > 0 we conclude that for any  > 0, x ./ is a constant function. In view of Theorem 2.2, x .0/ does not depend on . Theorem 2.10 is proved. t u

2.6 Auxiliary Results for Theorem 2.12 Proposition 2.40. Let M1 > 0, 0 < 0 < 1 . Then there exists M2 > 0 such that for each g 2 M, each T2 > T1  0 satisfying T2  T1 2 Œ0 ; 1 

(2.101)

48

2 Linear Control Systems with Periodic Convex Integrands

and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying I g .T1 ; T2 ; x; u/  M1

(2.102)

jx.t/j  M2 for all t 2 ŒT1 ; T2 :

(2.103)

ı 2 .0; minf81 0 ; .2kAk C 2/1 g/:

(2.104)

the following inequality holds:

Proof. Fix

By (2.11) and (2.13), there exists c0 > 1 such that g.t; x; u/  8juj.kBk C 1/

(2.105)

for each g 2 M and each .t; x; u/ 2 Œ0; 1/  Rn  Rm satisfying juj  c0 and h0 > 0 such that g.t; x; u/  4M1 .minf1; 0 g/1 ı 1 C 2a1 ı 1

(2.106)

for each g 2 M and each .t; x; u/ 2 Œ0; 1/  Rn  Rm satisfying jxj  h0 . Fix M2 > 2 C 2M1 C 2a1 C 2c0 .1 C 1 /kBk C 2h0 :

(2.107)

Let g 2 M, T2 > T1  0 satisfy (2.101) and let .x; u/ 2 X.A; B; T1 ; T2 / satisfy (2.102). We show that (2.103) holds. Assume the contrary. Then there exists t0 2 ŒT1 ; T2  such that jx.t0 /j > M2 :

(2.108)

By the choice of h0 [see (2.106)], (2.13), (2.102) and (2.104), there exists t1 2 ŒT1 ; T2  satisfying jx.t1 /j  h0 ; jt1  t0 j  ı:

(2.109)

There exists a number t2 such that minft0 ; t1 g  t2  maxft0 ; t1 g; jx.t2 /j  jx.t/j; t 2 Œminft0 ; t1 g; maxft0 ; t1 g:

(2.110)

2.6 Auxiliary Results for Theorem 2.12

49

It follows from (2.1), (2.109), and (2.110) that ˇ Z t2 ˇ Z t2 Z t2 ˇ ˇ jx.t1 /  x.t2 /j D ˇˇ x0 .t/dtj  kAkˇˇ jx.t/jdtj C kBkj ju.t/jdtj t1 t1 t1 Z t2  kAkjx.t2 /jı C kBkj ju.t/jdtj: (2.111) t1

By the choice of c0 (see (2.105)), (2.13), (2.101), (2.102) and (2.109), ˇZ ˇ ˇ ˇ

t2

t1

ˇ ˇZ ˇ ˇ ju.t/jdtˇˇ  ˇˇ

t2

1

1

Œ8 g.t; x.t/; u.t//.kBk C 1/

t1

ˇ ˇ C c0 dtˇˇ

 c0 jt1  t2 j C 81 a1 .kBk C 1/1 C 81 .kBk C 1/1 I g .T1 ; T2 ; x; u/  c0 ı C a1 81 .kBk C 1/1 C 81 .kBk C 1/1 M1 : By this relation, (2.104) and (2.111), jx.t1 /  x.t2 /j  21 jx.t2 /j C kBkc0 ı C a1 C M1 : Combined with (2.108) and (2.109) this implies that 21 M2  h0  kBkc0 ı C a1 C M1 . This contradicts (2.107). The contradiction we have reached proves Proposition 2.40. Proposition 2.41. Let 0 < c1 < c2 ; D;  > 0. Then there exists a neighborhood V of f in M such that for each g 2 V, each T2 > T1  0 satisfying T2  T1 2 Œc1 ; c2  and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  D

(2.112)

the inequality jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j   holds. Proof. By Proposition 2.40, there exists S > 0 such that jx.t/j  S for all t 2 ŒT1 ; T2 

(2.113)

for each g 2 M, each T2 > T1  0 satisfying T2  T1 2 Œc1 ; c2  and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying I g .T1 ; T2 ; x; u/  D C 1. Choose ı 2 .0; 1/, N > S,  > 1 such that ı.c2 C 1/  41 ;

.N/N > 4a; .  1/.c2 C D C a.c2 C 1//  =4

(2.114)

V D fg 2 M W .f ; g/ 2 E.N; ı;  /g:

(2.115)

and set

50

2 Linear Control Systems with Periodic Convex Integrands

Assume that g 2 V; T2 > T1  0; T2  T1 2 Œc1 ; c2 

(2.116)

and that .x; u/ 2 X.A; B; T1 ; T2 / satisfies (2.112). By the choice of S, (2.112) and (2.116), (2.113) is true. Set E1 D ft 2 ŒT1 ; T2  W ju.t/j  Ng; E2 D ŒT1 ; T2  n E1 :

(2.117)

By (2.113), (2.115), (2.116), (2.117), and the inequality N > S, jf .t; x.t/; u.t//  g.t; x.t/; u.t//j  ı; t 2 E1 :

(2.118)

h.t/ D minff .t; x.t/; u.t//; g.t; x.t/; u.t//g; t 2 ŒT1 ; T2 :

(2.119)

Set

In view of (2.13), (2.113), (2.114), (2.115), (2.116), (2.117), (2.119), and the inequality N > S, for all t 2 E2 , .f .t; x.t/; u.t// C 1/.g.t; x.t/; u.t// C 1/1 2 Π1 ;   and jf .t; x.t/; u.t//  g.t; x.t/; u.t//j  .  1/.h.t/ C 1/:

(2.120)

Relations (2.13), (2.112), (2.114), and (2.116)–(2.120) imply that Z jI .T1 ; T2 ; x; u/  I .T1 ; T2 ; x; u/j  f

jf .t; x.t/; u.t//  g.t; x.t/; u.t//jdt

g

E1

Z

C

jf .t; x.t/; u.t//  g.t; x.t/; u.t//jdt E2

Z

 ıc2 C .  1/

.h.t/ C 1/dt  ıc2 E2

C .  1/c2 C .  1/.D C ac2 /  : Proposition 2.41 is proved.

2.7 Proof of Theorem 2.12 By Proposition 2.39, there exists ı0 2 .0; 1=8/ such that the following property holds:

2.7 Proof of Theorem 2.12

(P1)

51

for each integer p  1, each .x; u/ 2 X.A; B; 0; p / satisfying jx.0/  xf .0/j; jx.p/  xf .0/j  4ı0 ; I f .0; p; x; u/  .f ; x.0/; x.p /; p / C 4ı0

and each integer i 2 Œ0; p  1, the inequality jx.i C t/  xf .t/j   holds for all t 2 Œ0;  . By Proposition 2.38, there exists an integer L0  5 such that the following property holds: (P2) for each T  .L0  4/ , each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  T.f / C M C 4 and each integer S satisfying ŒS; .S C L0  4/   Œ0; T there exists an integer i 2 ŒS; S C L0  5 such that jx. i C t/  xf .t/j  ı0 for all t 2 Œ0;  : Let an integer L1  L0 . By Proposition 2.35, there exists a number M0 > 0 such that for each S  3 and each y; z 2 Rn satisfying jyj; jzj  maxfjxf .t/j W t 2 Œ0;  g C 4, .f ; y; z; S/  S.f / C M0 :

(2.121)

By Proposition 2.41, there exists a neighborhood U of f in M such that the following property holds: (P3) for each g 2 U , each T1  0, each T2 2 ŒT1 C 1; T1 C 4L1  and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  .4L1 j.f /j C M C M0 C 1/. C 1/; jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  ı0 : Assume that T > 2L1 , g 2 U , .x; u/ 2 X.A; B; 0; T/ and that a finite sequence q of integers fSi giD0 satisfy S0 D 0; SiC1  Si 2 ŒL0 ; L1 ; i D 0; : : : ; q  1; Sq  2 .T  L1 ; T; I g .Si ; SiC1 ; x; u/  .SiC1  Si /.f / C M

(2.122) (2.123)

for each integer i 2 Œ0; q  1, I g .Si ; SiC2 ; x; u/  .g; x.Si /; x.SiC2 /; Si ; SiC2  / C ı0

(2.124)

for each nonnegative integer i  q  2 and I g .Sq2 ; T; x; u/  .g; x.Sq2 /; x.T/; Sq2 ; T/ C ı0 :

(2.125)

52

2 Linear Control Systems with Periodic Convex Integrands

Let i 2 Œ0; q  1 be an integer. By (2.122), (2.123) and the choice of U (see property (P3)), I f .Si ; SiC1 ; x; u/  I g .Si ; SiC1 ; x; u/ C ı0  .SiC1  Si /.f / C M C 1: The inequality above, (2.122) and property (P2) imply that there exists an integer pi such that pi 2 ŒSi C 3; Si C L0 ; jx.pi /  xf .0/j  ı0 :

(2.126)

Let an integer i 2 Œ0; q  2. In view of (2.122) and (2.126), pi ; piC1 2 ŒSi C 3; SiC2 ; 3  piC1  pi  2L1 :

(2.127)

It follows from (2.124) and (2.127) that I g .pi ; piC1 ; x; u/  .g; x.pi /; x.piC1 /; pi ; piC1  / C ı0

(2.128)

Thus we have shown that there exists a strictly increasing sequence of nonnegative integers fpi gkiD0 where k is a natural number such that p0  L0 ; pk  > T  2 L1 ; jx.pi /  xf .0/j  ı0 ; i D 0; : : : ; k; 3  piC1  pi  2L1 ; i D 0; : : : ; k  1

(2.129)

and (2.128) holds for all i D 0; : : : ; k  1. It is not difficult to see that if jx.0/  xf .0/j  ı0 , then we may assume that p0 D 0 and if jx.bT=c /  xf .0/j  ı0 , then we may assume that pk D bT=c. Let i 2 f0; : : : ; k  1g. By (2.129) and the choice of M0 (see (2.121)), .f ; x.pi  /; x.piC1 /; .piC1  pi / /  .f /.piC1  pi / C M0 :

(2.130)

Combined with (2.129) and the choice of U (see property (P3)) this implies that j .f ; x.pi  /; x.piC1 /; .piC1  pi / /  .g; x.pi  /; x.piC1  /; pi ; piC1  /j  ı0 : The inequality above and (2.128) imply that I g .pi ; piC1 ; x; u/  .f ; x.pi /; x.piC1  /; .piC1  pi // C 2ı0  .f /.piC1  pi / C M0 C 1: Together with (2.129) and the choice of U (see property (P3)) this implies that I f .pi ; piC1 ; x; u/  I g .pi ; piC1 ; x; u/ C ı0  .f ; x.pi /; x.piC1 /; .piC1  pi // C 3ı0 :

2.8 Basic Lemma for Theorem 2.13

53

By the relation above, (2.129) and property (P1), for all j 2 fpi ; : : : ; piC1  1g, jx.j C t/  xf .t/j  ; t 2 Œ0;  : Thus the relation above holds for all j 2 fp0 ; : : : ; pk  1g and Theorem 2.12 is proved. u t

2.8 Basic Lemma for Theorem 2.13 Lemma 2.42. Let  2 .0; 1/; M0 ; M1 > 0. Then there exist an integer L  1 and a neighborhood U of f in M such that the following assertion holds. Assume that T > L, g 2 U , integers S1 ; S2 satisfy 0  S1  S2  L; ŒS1 ; S2   Œ0; T

(2.131)

and .x; u/ 2 X.A; B; 0; T/ satisfies at least one of the following conditions: (a) jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C M1 I (b) jx.0/j  M0 ; I g .0; T; x; u/  .g; x.0/; 0; T/ C M1 I (c) I g .0; T; x; u/  .g; 0; T/ C M1 : Then minfjx.i/  xf .0/j W i D S1 ; : : : ; S2 g  :

(2.132)

Proof. By Proposition 2.38 there exists a natural number L0 such that the following property holds: (P4)

for each T  L0 , each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  T.f / C 16.1 C a/. C 1/

and each integer S satisfying ŒS; .S C L0 /   Œ0; T, minfjx. i/  xf .0/j W i D S; : : : ; S C L0  1g  : We may assume without loss of generality that M0 > supfjxf .t/j W t 2 Œ0;  g C 4:

(2.133)

54

2 Linear Control Systems with Periodic Convex Integrands

We use the functions f0 and v0 introduced in Sect. 2.4 [see (2.36) and (2.37)]. We may assume without loss of generality that M1 > supfjv0 .z1 ; z2 /j W z1 ; z2 2 Rn ; jz1 j; jz2 j  M0 g:

(2.134)

By Proposition 2.35, there exists a number M2 > M1 C M0 such that for each S  3 and each y; z 2 Rn satisfying jyj; jzj  M0 , .f ; y; z; S/  S.f / C M2 :

(2.135)

Choose a natural number l such that minf1; gl > 4 C M1 C 4. C 1/ minf1; g1 C . C 1/j.f /j.2L0 C M2 C 4/ C .2L0 C 18/.1 C a/.1 C / C 2M2 C 18 C a.L0  C  C 1/ C  C a C 1 C a

(2.136)

and set L D 2.L0 C 1/l:

(2.137)

By Proposition 2.41, there exists a neighborhood U of f in M such that the following property holds: (P5) for each g 2 U , each T1  0, each T2 2 ŒT1 C minf1; g; T1 C 4L maxf1; g and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  ..4L C 2/j.f /j C 4M2 C 4 C 16.1 C a//. C 1/; jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  .4L/1 . C 1/1 : Assume that T > L; g 2 U ;

(2.138)

integers S1 ; S2 satisfy (2.131) and .x; u/ 2 X.A; B; 0; T/ satisfies at least one of the conditions (a), (b), (c). We show that (2.132) holds. Assume the contrary. Then jx.i/  xf .0/j > ; i D S1 ; : : : ; S2 :

(2.139)

We may assume without loss of generality that at least one of the following conditions hold: S1 D 0; S2 D bT 1 cI S1  1; jx..S1  1//  xf .0/j  ; S2 D bT 1 cI

(2.140) (2.141)

2.8 Basic Lemma for Theorem 2.13

55

S1 D 0; S2 < bT 1 c; jx..S2 C 1//  xf .0/j  

(2.142)

S1  1; S2 < bT 1 c; jx.j/  xf .0/j  ; j D S1  1; S2 C 1:

(2.143)

In view of (2.131) and (2.137), b.S2  S1 /L01 c  bLL01 c  2l:

(2.144)

It follows from (2.13) that I g .S1 ; S2 ; x; u/ D I g .S1 ; S1  C b.S2  S1 /L01 cL0 ; x; u/ C I g .S1  C b.S2  S1 /L01 cL0 ; S2 ; x; u/ b.S2 S1 /L01 c1



X

I g ..S1 C iL0 /; .S1 C .i C 1/L0 /; x; u/  aL0 :

iD0

(2.145) Let j 2 f0; : : : ; b.S2  S1 /L01 c  1g:

(2.146)

By (2.139), (2.146) and property (P4), I f ..S1 C jL0 /; .S1 C .j C 1/L0 /; x; u/ > L0 .f / C 16.1 C a/. C 1/:

(2.147)

We show that I g ..S1 CjL0 /; .S1 C.jC1/L0 /; x; u/  L0 .f /C16.1Ca/. C1/1:

(2.148)

Assume the contrary. Then I g ..S1 C jL0 /; .S1 C .j C 1/L0 /; x; u/ < L0 .f / C 16.1 C a/. C 1/  1: Combined with the choice of U [see property (P5)], (2.137) and (2.138) this implies that I f ..S1 C jL0 /; .S1 C .j C 1/L0 /; x; u/  1 C I g ..S1 C jL0 /; .S1 C .j C 1/L0 /; x; u/  L0  .f / C 16.1 C a/. C 1/: This contradicts (2.147). The contradiction we have reached proves (2.148). Thus (2.148) holds for all j 2 f0; : : : ; b.S2  S1 /L01 c  1g: Set z0 D x.0/ if jx.0/j  M0 ; z0 D 0 if jx.0/j > M0 ; z1 D x.T/ if jx.T/j  M0 ; z1 D 0 if jx.T/j > M0 :

(2.149)

56

2 Linear Control Systems with Periodic Convex Integrands

It is not difficult to see that there exists .x1 ; u1 / 2 X.A; B; 0; T/ such that: if (2.140) holds, then x1 .0/ D z0 ; x1 . / D xf .0/; I f .0; ; x1 ; u1 /  v0 .z0 ; xf .0// C 1; x1 .T  / D xf .T  b 1 Tc/; x1 .T/ D z1 ; I f .T  ; T; x1 ; u1 /  v0 .xf .T  b 1 Tc /; z1 / C 1; x1 .t/ D xf .t  b 1 tc/; u1 .t/ D uf .t  b 1 tc /; t 2 Œ; T   I if (2.141) holds, then x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 Œ0; .S1  1/; x1 .T   / D xf .T  b 1 Tc /; x1 .T/ D z1 ; I f .T  ; T; x1 ; u1 /  v0 .xf .T  b 1 Tc /; z1 / C 1; x1 .t/ D xf .t  b 1 tc/; u1 .t/ D uf .t  b 1 tc /; t 2 ŒS1 ; T   I I f . .S1  1/;  S1 ; x1 ; u1 /  v0 .x1 . .S1  1//; x1 .S1 // C 1I if (2.142) holds, then x1 .0/ D z0 ; x1 . / D xf .0/; I f .0; ; x1 ; u1 /  v0 .z0 ; xf .0// C 1; x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 Œ .S2 C 1/; T; x1 .t/ D xf .t  Œ 1 t/; u1 .t/ D uf .t  b 1 tc /; t 2 Œ; S2  ; I f .S2 ; .S2 C 1/; x1 ; u1 /  v0 .x1 . S2 /; x1 ..S2 C 1/// C 1I if (2.143) holds, then x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 Œ0; .S1  1/ [ Œ .S2 C 1/; T; x1 .t/ D xf .t  b 1 tc/; u1 .t/ D uf .t  b 1 tc /; t 2 Œ .S1 C 1/; S2 ; I f ..S1  1/; S1 ; x1 ; u1 /  v0 .x1 .S1  1/; x1 .S1  // C 1; I f .S2 ; .S2 C 1/; x1 ; u1 /  v0 .x1 . S2 /; x1 ..S2 C 1/// C 1: In view of (2.149), conditions (a), (b), (c) and the choice of .x1 ; u1 /, I g .0; T; x; u/  I g .0; T; x1 ; u1 / C M:

(2.150)

We consider the cases (2.140)–(2.143) separately and obtain a lower bound for I g .0; T; x; u/  I g .0; T; x1 ; u1 /. Assume that (2.140) holds. By (2.13), (2.140) and (2.148),

2.8 Basic Lemma for Theorem 2.13

57

I g .0; T; x; u/  I g .0; bT 1 c; x; u/   a  I g .0; bbT 1 cL01 cL0 ; x; u/   a.L0 C 1/ bbT 1 cL01 c1

D

X

I g ..jL0 /; .j C 1/L0 ; x; u/  .L0 C 1/a

jD0

 bL0 .f / C 16.1 C a/.1 C /  1cbbT 1 cL01 c  .L0 C 1/ a  bL0 .f / C 16.1 C a/.1 C /  1cbT 1 cL01 L0 .f /  16.1 C a/.1 C /  .L0 C 1/ a  bL0 .f / C 16.1 C a/.1 C /  1cT 1 L01 2.L0 .f / C 16.1 C a/.1 C / C .L0 C 1/ a/  T.f / C TL01 8.1 C a/  2.L0 .f / C .L0 C 17/.1 C a/.1 C //: (2.151) Clearly, I g .0; T; x1 ; u1 / D I g .0; ; x1 ; u1 / C I g .; T  ; x1 ; u1 / C I g .T  ; T; x1 ; u1 /: (2.152) By (2.133), (2.134), (2.140), (2.149), and the choice of .x1 ; u1 / I f .0; ; x1 ; u1 /  v0 .z0 ; xf .0// C 1  M1 C 1; I f .T  ; T; x1 ; u1 /  v0 .xf .T  b 1 Tc /; z1 / C 1  M1 C 1: In view of these inequalities, (2.138) and property (P5), I g .0; ; x1 ; u1 /; I g .T  ; T; x1 ; u1 /  M1 C 5=4:

(2.153)

It follows from (2.3), (2.138), (2.140), and the choice of .x1 ; u1 / that I g .; T  ; x1 ; u1 / D I g .; .bT 1 c  1/; x1 ; u1 / C I g ..bT 1 c  1/; T  ; x1 ; u1 / D

1 c2 bTX

I g .i; .i C 1/; x1 ; u1 /

iD1

C I g ..bT 1 c  1/; T  ; x1 ; u1 /; for all integers i D 1; : : : ; b 1 Tc  2, I f .i; .i C 1/; x1 ; u1 / D I f .0; ; xf ; uf / D  .f /

(2.154)

58

2 Linear Control Systems with Periodic Convex Integrands

and in view of property (P5), I g .i; .i C 1/; x1 ; u1 /  .f / C . C 1/1 .4L/1 : This implies that 1 c2 bTX

I g .i; .i C 1/; x1 ; u1 /  T.f / C . C 1/1 .4L/1 .T=/ C 3 j.f /j:

iD1

(2.155) By (2.13), (2.138), (2.140), property (P5), and the choice of .x1 ; u1 /, I g ..bT 1 c  1/; T  ; x1 ; u1 / Z T D g.t; xf .t  .b 1 Tc  1//; uf .t  .b 1 Tc  1/ //dt Z 

.bT 1 c1/ bT= c

.bT 1 c1/

g.t; xf .t  .b 1 Tc  1//; uf .t  .b 1 Tc  1/ //dt C a

 I f .0; ; xf ; uf / C 1 C a: Combined with (2.152)–(2.155) this implies that I g .0; T; x1 ; u1 /  2.M2 C 1/ C I g .; T  ; x1 ; u1 /  2.M2 C 2/ C T.f / C . C 1/1 .4L/1 .T=/ C 3 j.f /j C .f / C a C 1: The relation above, (2.136)–(2.138), (2.150), and (2.151) imply that M1  I g .0; T; x; u/  I g .0; T; x1 ; u1 /  T.f / C 8TL01 .1 C a/  2ŒL0 .f / C .L0 C 17/.1 C a/. C 1/  T.f /  2M2  4  . C 1/1 .4L/1 T 1  4j.f /j  1  a D TŒL01 8.1 C a/  .4L/1 . C 1/1  1   2L0  j.f /j  2.L0 C 17/.1 C a/. C 1/  2M2  4  4j.f /j  1  a  4TL01 .1 C a/  j.f /j.2L0 C 4/  2.L0 C 17/.1 C a/. C 1/  2M2  5   a  4l  j.f /j.2L0 C 4/  2.L0 C 17/.1 C a/. C 1/  2M2  5  a: This contradicts (2.137). Thus if (2.140) holds we have reached a contradiction.

2.8 Basic Lemma for Theorem 2.13

59

Assume that (2.141) holds. By (2.141) and the choice of .x1 ; u1 /, I g .0; T; x; u/  I g .0; T; x1 ; u1 / D I g . .S1  1/; T; x; u/  I g ..S1  1/; T; x1 ; u1 /: (2.156) It follows from (2.13), (2.136)–(2.138), (2.141), and (2.148) that I g . .S1  1/; T; x; u/  a C I g .S1 ; bT 1 c; x; u/  a  I g .S1 ; S1 C bbT 1  S1 cL01 cL0 ; x; u/  L0 a  a. C 1/ D a.L0  C  C 1/ bbT 1 S1 cL01 c1

C

X

I g .S1  C jL0 ; S1  C .j C 1/L0 ; x; u/

jD0

 a.L0  C  C 1/ C bbT 1  S1 cL01 c.L0  .f / C 16.1 C a/.1 C /  1/  a.L0  C  C 1/  .L0 j.f /j C 16.1 C a/.1 C  // C .bT 1 c  S1 /.f / C .bT 1 c  S1 /L01 .16.1 C a/.1 C  /  1/:

(2.157)

It is clear that I g ..S1 1/; T; x1 ; u1 / D I g . .S1 1/; T ; x1 ; u1 /CI g .T ; T; x1 ; u1 /:

(2.158)

By the choice of .x1 ; u1 /, (2.133), (2.134), (2.141), and (2.149), I f .T  ; T; x1 ; u1 /  v0 .xf .T  b 1 Tc /; z1 / C 1  M1 C 1; I f . .S1  1/;  S1 ; x1 ; u1 /  v0 .x1 . .S1  1//; x1 .S1 // C 1  M1 C 1: Together with (2.138) and property (P5) this implies that I g .T  ; T; x1 ; u1 /; I g . .S1  1/;  S1 ; x1 ; u1 /  M1 C 2:

(2.159)

Clearly, I g ..S1  1/; T  ; x1 ; u1 / D I g .S1 ; bT 1 c  ; x1 ; u1 / C I g ..S1  1/; S1 ; x1 ; u1 / C I g .bT 1 c  ; T  ; x1 ; u1 /:

(2.160)

60

2 Linear Control Systems with Periodic Convex Integrands

It follows from (2.141) and the choice of .x1 ; u1 / that for each j 2 fS1 ; : : : ; b 1 Tc  2g; I f .j; .j C 1/; x1 ; u1 / D I f .0; ; xf ; uf / D  .f / and in view of property (P5), I g .j; .j C 1/; x1 ; u1 /  .f / C . C 1/1 .4L/1 :

(2.161)

By (2.13), (2.138), (2.141), property (P5), and the choice of .x1 ; u1 /, I g ..bT 1 c  1/; T  ; x1 ; u1 / Z T D g.t; xf .t  .b 1 Tc  1//; uf .t  .b 1 Tc  1/ //dt Z 

.bT 1 c1/ bT= c

.bT 1 c1/

g.t; xf .t  .b 1 Tc  1//; uf .t  .b 1 Tc  1/ //dt C a

 I f .0; ; xf ; uf / C 1 C a: In view of the relation above and (2.159)–(2.161), I g ..S1  1/; T  ; x1 ; u1 /  M2 C 2 C .f / C 1 C a C I g .S1 ; bT 1 c  ; x1 ; u1 /  M2 C 4 C .f / C 1 C a C .bT 1 c  S1 /..f / C . C 1/1 .4L/1 / C 2j.f /j: By the relation above, (2.131), (2.137), (2.141), (2.150), and (2.156)–(2.159), M1  I g .0; T; x; u/  I g .0; T; x1 ; u1 / D I g . .S1  1/; T; x; u/  I g . .S1  1/; T; x1 ; u1 /  a.L0  C  C 1/  L0 j.f /j  16.a C 1/. C 1/ C .bT 1 c  S1 /.f / C .bT 1 c  S1 /L01 .16.1 C a/.1 C /  1/  2M2  8  3j.f /j  1  a  .bT 1 c  S1 / .f /  .bT 1 c  S1 /. C 1/1 .4L/1  a.L0  C  C 1/  L0 j.f /j  16.a C 1/. C 1/  2M2  8  3j.f /j  1   a C bbT 1 c  S1 c.L01 .16.1 C a/.1 C  /  1/  .4L/1 .1 C /1 /  a.L0  C  C 1/  L0 j.f /j  16.a C 1/. C 1/

2.8 Basic Lemma for Theorem 2.13

61

 2M2  8  3j.f /j  1  a C bbT 1 c  S1 c4L01 .1 C a/  l  a.L0  C  C 1/  L0 j.f /j  16.a C 1/. C 1/  2M2  8  3j.f /j  1  a: This contradicts (2.136). Thus if (2.141) holds we have reached a contradiction. Assume that (2.142) holds. By (2.142) and the choice of .x1 ; u1 /, I g .0; T; x; u/  I g .0; T; x1 ; u1 / D I g .0; .S2 C 1/; x; u/  I g .0;  .S2 C 1/; x1 ; u1 /: (2.162) By (2.13), I g .0; .S2 C 1/; x; u/  I g .0; S2 ; x; u/   a bS2 L01 c1

X

D

I g .jL0 ; .j C 1/L0 ; x; u/  a.L0  C  /

jD0

 bS2 L01 c.L0 .f / C 16.1 C a/. C 1/  1/  a .L0 C 1/:

(2.163)

By the choice of .x1 ; u1 /, (8.133), (2.134), (2.138), (2.142), (2.149), and property (P5), I f .0; ; x1 ; u1 /  M1 C 1; I g .0; ; x1 ; u1 /  M1 C 2; I f .S2 ;  .S2 C 1/; x1 ; u1 /  M1 C 1; I g .S2 ;  .S2 C 1/; x1 ; u1 /  M1 C 2: (2.164) By (2.164), I g .0; .S2 C 1/; x1 ; u1 / D I g .; .S2 C 1/; x1 ; u1 / C I g .0; ; x1 ; u1 /  M2 C 2 C I g .1 ; .S2 C 1/; x1 ; u1 /:

(2.165)

It follows from (2.142) and the choice of .x1 ; u1 /, (2.138) and property (P5) that for each j 2 f1; : : : ; S2  1g, I f .j; .j C 1/; x1 ; u1 / D I f .0; ; xf ; uf / D  .f /; I g .j; .j C 1/; x1 ; u1 /  .f / C . C 1/1 .4L/1 : These relations imply that I g .; S2 ; x1 ; u1 /  S2 .f / C S2 .4L/1 /. C 1/1 :

(2.166)

62

2 Linear Control Systems with Periodic Convex Integrands

By (2.131), (2.137), (2.142), (2.150), and (2.162)–(2.166), M1  I g .0; T; x; u/  I g .0; T; x1 ; u1 /  bS2 L01 c.L0 .f / C 16.1 C a/. C 1/  1/  a .L0 C 1/  2M2  4  S2 .f /  S2 .4L/1 . C 1/1  S2 .2L0 /1 .16.1 C a/. C 1/  1/  a .L0 C 1/  2M2  4  S2 .4L/1 . C 1/1  L0 j.f /j  4S2 L01  a.L0 C 1/  2M2  4  L0 j.f /j  4l  a.L0 C 1/  L0 j.f /j  2M2  4: This contradicts (2.137). Thus if (2.142) holds we have reached a contradiction. Assume that (2.143) holds. By (2.143) and the choice of .x1 ; u1 /, I g .0; T; x; u/  I g .0; T; x1 ; u1 / D I g . .S1  1/;  .S2 C 1/; x; u/  I g . .S1  1/;  .S2 C 1/; x1 ; u1 /:

(2.167)

By (2.13) and (2.148), I g . .S1  1/; .S2 C 1/; x; u/  I g . S1 ; S2 ; x; u/  2 a  2a C I g .S1 ; S1  C b.S2  S1 /L01 cL0 ; x; u/  L0 a  a.L0 C 2/ C b.S2  S1 /L01 c.L0  .f / C 16.1 C a/. C 1/  1//:

(2.168)

By the choice of .x1 ; u1 /, (2.133), (2.134), (2.143), and property (P5), I f . .S1  1/;  S1 ; x1 ; u1 /  M1 C 1; I f .S2 ;  .S2 C 1/; x1 ; u1 /  M1 C 1; I g . .S1  1/;  S1 ; x1 ; u1 /  M1 C 2; I g .S2 ;  .S2 C 1/; x1 ; u1 /  M1 C 2: (2.169) It follows from (2.141) and the choice of .x1 ; u1 /, (2.138) and property (P5) that for each j 2 fS1 ; : : : ; S2  1g, I f .j; .j C 1/; x1 ; u1 / D I f .0; ; xf ; uf / D  .f /; I g .j; .j C 1/; x1 ; u1 /  .f / C . C 1/1 .4L/1 :

2.9 Proof of Theorem 2.13

63

These relations and (2.169) imply that I g . .S1  1/; .S2 C 1/; x1 ; u1 /  .S2  S1 / .f / C .S2  S1 /.4L/1 /. C 1/1 C 2M1 C 4: (2.170) By (2.131), (2.137), (2.150), (2.167), (2.168), and (2.170), M1  I g . .S1  1/; .S2 C 1/; x; u/  I g . .S1  1/;  .S2 C 1/; x1 ; u1 /   a.2 C L0 / C b.S2  S1 /L01 cL0 .f / C b.S2  S1 /L01 c.16.1 C a/. C 1/  1/  .S2  S1 / .f /  .S2  S1 /.4L/1 . C 1/1  2M2  4  .L0 C 2/j.f /j  a.2 C L0 /  2M2  4  16.1 C a/. C 1/ C .S2  S1 /.L01 .16.1 C a/.1 C /  1/  .4L/1 /  .L0 C 2/ j.f /j  a.2 C L0 /  2M2  16.1 C a/.1 C  /  6 C 4l: This contradicts (2.137). Thus in all the cases we have reached a contradiction which proves (2.132) and Lemma 2.42 itself.

2.9 Proof of Theorem 2.13 By Proposition 2.39, there exists ı0 2 .0; / such that the following property holds: (P6)

for each integer p  1, each .x; u/ 2 X.A; B; 0; p / satisfying

jx.0/  xf .0/j; jx.p/  xf .0/j  ı0 ; I f .0; p; x; u/  .f ; x.0/; x.p /; p / C ı0 and each integer i 2 Œ0; p  1, the inequality jx.i C t/  xf .t/j   holds for all t 2 Œ0;  . By Lemma 2.42, there exist an integer L0  1 and a neighborhood U0 of f in M such that the following property holds: (P7) for each T > L0 , each g 2 U0 , each pair of integers S1 ; S2 satisfying 0  S1  S2  L0 ; S2   T and each .x; u/ 2 X.A; B; 0; T/ for which at least one of the conditions (a), (b), (c) holds, minfjx.i/  xf .0/j W i D S1 ; : : : ; S2 g  ı0 : Fix an integer L  4.L0 C 1/ and ı 2 .0; 41 ı0 /. By assumption (A), Proposition 2.35 and the boundedness of v0 on bounded sets there is M2 > 0 such that the following property holds:

64

2 Linear Control Systems with Periodic Convex Integrands

(P8) for each i 2 f1; : : : ; Lg and each y; z 2 Rn satisfying jyj; jzj < 1Csupfjxf .t/j W t 2 Œ0;  g we have j .f ; y; z; i/j  M2 . By Proposition 2.41, there exists a neighborhood U of f in M such that U  U0 and that the following property holds: (P9) for each g 2 U , each T1  0, each T2 2 ŒT1 C minf1; g; T1 C 4L maxf1; g and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  M2 C 4 the inequality jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  ı holds. Assume that T > 2L; g 2 U ;

(2.171)

.x; u/ 2 X.A; B; 0; T/ satisfies for each S 2 Œ0; T  L , I g .S; S C L; x; u/  .g; x.S/; x.S C L/; S; S C L / C ı

(2.172)

and satisfies at least one of the following conditions: (a) jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C M1 I (b) jx.0/j  M0 ; I g .0; T; x; u/  .g; x.0/; 0; T/ C M1 I (c) I g .0; T; x; u/  .g; 0; T/ C M1 : By conditions (a)–(c) and property (P7), there exists a finite strictly increasing sequence of integers Si , i D 1; : : : ; q such that 0  S1 < L0 ; T  Sq   T  .1 C L0 /; SiC1  Si  L0 C 1; i D 1; : : : ; q  1; (2.173) jx.Si /  xf .0/j  ı0 ; i D 1; : : : ; q:

(2.174)

We may assume without loss of generality that if jx.0/  xf .0/j  ı; then S1 D 0 and if jx.bT 1 c/  xf .0/j  ı; then Sq D bT 1 c:

(2.175)

Assume that an integer i 2 ŒS1 ; Sq /. Then there exists a natural number k 2 f1; : : : ; q  1g such that Sk  i < SkC1 :

(2.176)

2.10 Proof of Theorem 2.14

65

In view of (2.171), (2.173), and the inequality L  4.L0 C1/, there is S 2 Œ0; T L  such that ŒSk ; SkC1   ŒS; S C L :

(2.177)

It follows from (2.172) and (2.177) that I g .Sk ; SkC1 ; x; u/  .g; x.Sk /; x.SkC1 /; Sk ; SkC1  / C ı:

(2.178)

By (2.173), (2.174), and property (P8), .f ; x.Sk /; x.SkC1 /; .SkC1  Sk // < M2 :

(2.179)

By (2.171), (2.173), (2.178), (2.179), and property (P9), .g; x.Sk /; x.SkC1 /; Sk ; SkC1 /  .f ; x.Sk /; x.SkC1 /; .SkC1  Sk // C ı

(2.180)

and I g .Sk ; SkC1 ; x; u/  M2 C 2: In view of the relation above, (2.171), (2.173), (2.178), (2.180), and property (P9), I f .Sk ; SkC1 ; x; u/  I g .Sk ; SkC1 ; x; u/ C ı  .f ; x.Sk /; x.SkC1  /; .SkC1  Sk // C 3ı: Together with (2.174), (2.176), and property (P6), jx.i C t/  xf .t/j   for all t 2 Œ0;  . Theorem 2.13 is proved. u t

2.10 Proof of Theorem 2.14 By Proposition 2.39, there exists ı0 2 .0; / such that the following property holds: (P10)

for each integer p  1, each .x; u/ 2 X.A; B; 0; p / satisfying jx.0/  xf .0/j; jx.p/  xf .0/j  4ı0 ; I .0; p; x; u/  .f ; x.0/; x.p /; p / C 4ı0 f

and each integer i 2 Œ0; p  1, the inequality jx.i C t/  xf .t/j   holds for all t 2 Œ0;  .

66

2 Linear Control Systems with Periodic Convex Integrands

By Lemma 2.42, there exist an integer L0  1 and a neighborhood U0 of f in M such that the following property holds: (P11) for each T > L0 , each g 2 U0 , each pair of integers S1 ; S2 satisfying 0  S1  S2  L0 , S2   T and each .x; u/ 2 X.A; B; 0; T/ which satisfies at least one of the conditions (a), (b) and (c), minfjx.i/  xf .0/j W i D S1 ; : : : ; S2 g  ı0 : Fix an integer L  4.L0 C 1/.9 C 2ı01 M1 /;

(2.181)

ı 2 .0; 41 ı0 /:

(2.182)

By Proposition 2.35 and the boundedness of v0 on bounded sets there is M2 > 0 such that for each i 2 f1; : : : ; Lg and each y; z 2 Rn satisfying jyj; jzj  1 C supfjxf .t/j W t 2 Œ0; g we have j .f ; y; z; i/j  M2 :

(2.183)

By Proposition 2.41, there exists a neighborhood U of f in M such that U  U0 and that the following property holds: (P12) for each g 2 U , each T1  0, each T2 2 ŒT1 Cminf1; g; T1 C4L maxf1; g and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  M2 C 4 the inequality jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  ı holds. Assume that T > L; g 2 U

(2.184)

and that .x; u/ 2 X.A; B; 0; T/ satisfies at least one of the following conditions: (a) jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C M1 I (b) jx.0/j  M0 ; I g .0; T; x; u/  .g; x.0/; 0; T/ C M1 I (c) I g .0; T; x; u/  .g; 0; T/ C M1 : By conditions (a)–(c), (2.184), and property (P11), there exists a finite strictly increasing sequence of integers Si , i D 1; : : : ; q such that 0  S1  L0 ; T  Sq   T  .1 C L0 /; SiC1  Si  L0 C 1; i D 1; : : : ; q  1; (2.185)

2.10 Proof of Theorem 2.14

67

jx.Si /  xf .0/j  ı0 ; i D 1; : : : ; q:

(2.186)

Define by induction a finite strictly increasing sequence of natural numbers i1 ; : : : ; ik 2 f1; : : : ; qg. Set i1 D 1:

(2.187)

Assume that p  1 is an integer and that we defined integers i1 < : : : < ip belonging to f1; : : : ; qg such that for each natural number m < p the following properties hold: (i) I g .Sim ; SimC1 ; x; u/ > .g; x.Sim /; x.SimC1  /; Sim ; SimC1  / C ı0 I

(2.188)

(ii) if imC1 > im C 1, then I g .Sim ; SimC1 1 ; x; u/  .g; x.Sim /; x.SimC1 1  /; Sim ; SimC1 1  / C ı0 : (2.189) (Note that by (2.187) our assumption holds for p D 1.) Let us define ipC1 . If ip D q, then our construction is completed, k D p, ik D q and for each natural number m < p D k, properties (i) and (ii) hold. Assume that ip < q. There are two cases: I g .Sip ; Sq ; x; u/  .g; x.Sip /; x.Sq /; Sip ; Sq  / C ı0 I

(2.190)

I g .Sip ; Sq ; x; u/ > .g; x.Sip /; x.Sq /; Sip ; Sq  / C ı0 :

(2.191)

Assume that (2.190) holds. Then we set k D p C 1, ik D q the construction is completed, for each natural number m < k  1, (2.188) is true and for each natural number m < k, property (ii) holds. Assume that (2.191) holds. Then we set ipC1 D minfj > Sip W j is an integer and I g .Sip ; Sj ; x; u/ > .g; x.Sip /; x.Sj  /; Sip ; Sj  / C ı0 g: (2.192) It is easy to see that the assumption made for p also holds for p C 1. As a result we obtain a finite strictly increasing sequence of integers i1 ; : : : ; ik 2 f1; : : : ; qg such that ik D q, for all integers m satisfying 1  m < k  1, (2.188) holds and for each integer m satisfying 1  m < k, (ii) holds. By conditions (a)–(c) and (2.188), M1  I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ X  fI g .Sij ; SijC1 ; x; u/  .g; x.Sij  /; x.SijC1  /; Sij ; SijC1  / W

68

2 Linear Control Systems with Periodic Convex Integrands

j is an integer satisfying 1  j < k  1g  ı0 .k  2/; k  ı01 M1 C 2:

(2.193)

Set A D fj 2 f1; : : : ; kg W j < k and SijC1  Sij  4.L0 C 1/g:

(2.194)

Let j 2 A:

(2.195)

By (2.185), (2.189), (2.194), (2.195), and property (ii), I g .Sij ; SijC1 1 ; x; u/  .g; x.Sij /; x.SijC1 1  /; Sij ; SijC1 1  / C ı0 ; ijC1 > ij C 3:

(2.196)

Let p 2 fij ; : : : ; ijC1  2g: This implies that fSp ; SpC1 g  fSij ; : : : ; SijC1 1 g and in view of (2.196), I g .Sp ; SpC1 ; x; u/  .g; x.Sp /; x.SpC1 /; Sp ; SpC1  / C ı0 :

(2.197)

By the choice of M2 [see (2.183)], (2.185), and (2.186), .f ; x.Sp /; x.SpC1 /; SpC1   Sp  /  M2 :

(2.198)

Together with (2.181), (2.184), (2.185), and property (P12) this implies that .g; x.Sp /; x.SpC1 /; Sp ; SpC1  /  .f ; x.Sp /; x.SpC1 /; .SpC1  Sp // C ı0 : By (2.181), (2.184), (2.185), (2.197), (2.198), the inequality above and property (P12), I f .Sp ; SpC1 ; x; u/  .f ; x.Sp /; x.SpC1 /; .SpC1  Sp // C 3ı0 :

(2.199)

It follows from (2.186), (2.199), and property (P10) that for all integers m 2 fSp ; : : : ; SpC1  1g, jx.m C t/  xf .t/j  ; t 2 Œ0;  :

(2.200)

2.11 Proofs of Propositions 2.18 and 2.21

69

Thus (2.200) holds for all integers m 2 fSij ; : : : ; SijC1 1  1g: Since j is any integer belonging to A we conclude that (2.200) holds for all m 2 [ffSij ; : : : ; SijC1 1  1g W j 2 Ag and that fm 2 f0; : : : ; bT 1 c  1g W (2.200) does not holdg  f0; : : : ; S1 g [ fSq ; : : : ; bT 1 cg [ ffSij ; : : : ; SijC1 g W j 2 f1; : : : ; kg n Ag [ffSijC1 1 ; : : : ; SijC1 g W j 2 Ag and in view of (2.181), (2.185), (2.193), and (2.194), the cardinality of the righthand side of the inclusion above does not exceed 4.L0 C 1/.2 C 2k/  4.L0 C 1/.8 C 2ı01 M1 / < L: t u

Theorem 2.14 is proved.

2.11 Proofs of Propositions 2.18 and 2.21 Proof of Proposition 2.18. Let  > 0. Since the function v is continuous there exists ı > 0 such that jv.z1 ; z2 /  .f / j  =4

(2.201)

for all z1 ; z2 2 Rn satisfying jzi  xf .0/j  ı; i D 1; 2: Let z1 ; z2 2 Rn ; jzi  xf .0/j  ı; i D 1; 2:

(2.202)

By Proposition 2.32, there exists .x; u/ 2 X.A; B; 0; 1/ x.0/ D z2 ; x. / D z1 ; I f .0; ; x; u/ D v.z2 ; z1 /;

(2.203)

x.t/ D  .z1 / .t  /; u.t/ D .z1 / .t  / for all t  :

(2.204)

By (2.21), (2.202)–(2.204), and Proposition 2.15,  f .z2 / D  f .x.0//  lim inf ŒI f .0; ; x; u/  T.f / T2Z;T!1

70

2 Linear Control Systems with Periodic Convex Integrands

D I f .0; ; x; u/  .f / C lim inf ŒI f .0; T;  .z1 / ; .z1 / /  T.f / T2Z;T!1

 v.z2 ; z1 / C  .z1 /  .f /   f .z1 / C =2: f

t u

Proposition 2.18 is proved.

Proof of Proposition 2.21. Let M > 0. by Proposition 2.18, there exists ı > 0 such that for each x 2 Rn satisfying jx  xf .0/j  ı, j f .x/j  1:

(2.205)

By Proposition 2.38, there exists a natural number L0 such that the following property holds: (i) for each T  L0 , each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  T.f / C M C 1 and each integer S satisfying ŒS; .S C L0 /   Œ0; T there exists an integer i 2 ŒS; S C L0  1 such that jx.i C t/  xf .t/j  ı for all t 2 Œ0;  : By Proposition 2.40, there exists M1 > 0 such that the following property holds: (ii) for each T 2 Œ; .L0 C 1/ and each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/   j.f /j.L0 C 1/ C M C 1 we have jx.t/j  M1 for all t 2 Œ0; T: Assume that x 2 Rn satisfies  f .x/  M:

(2.206)

By (2.21) and (2.206),  f .x/ D lim inf ŒI f .0; T;  .x/ ; .x/ /  T.f /  M: T2Z;T!1

(2.207)

In view of (2.207) and property (i), there exists an integer t0 2 Œ1; L0 C 1 such that j .x/ .t0 /  xf .0/j  ı:

(2.208)

2.12 Auxiliary Results for Theorem 2.25

71

It follows from (2.208) and the choice of ı [see (2.205)] that j f . .x/ .t0 //j  1:

(2.209)

Proposition 2.16, (2.206) and (2.209) imply that I f .0; t0 ;  .x/ ; .x/ /  t0 .f / D  f .x/   f . .x/ .t0  //  M C 1:

(2.210)

By (2.210) and property (ii), j .x/ .t/j  M1 for all t 2 Œ0; t0  . Thus jxj  M1 . Proposition 2.21 is proved. t u

2.12 Auxiliary Results for Theorem 2.25 We continue to use the notation, definitions, and assumptions introduced in Sects. 2.1–2.3. Assume that S2 > S1  0 are integers and g 2 M. There exists a unique function LS1 ;S2 .g/.t; x; u/, .t; x; u/ 2 Œ0; 1/  Rn  Rm such that for each x 2 Rn and each u 2 Rm , LS1 ;S2 .g/.t; x; u/ D g.S2   t C S1 ; x; u/ for each t 2 ŒS1 ; S2  ;

(2.211)

LS1 ;S2 .g/.t C .S2  S1 /; x; u/ D LS1 ;S2 .g/.t; x; u/ for each t  0:

(2.212)

Clearly, LS1 ;S2 .g/ 2 M and LS1 ;S2 is a self-mapping of M. It is easy to see that the following proposition holds. Proposition 2.43. 1. Let V be a neighborhood of fN in M. Then there exists a neighborhood U of f in M such that LS1 ;S2 .g/ 2 V for all g 2 U and all integers S2 > S1  0. 2. Let V be a neighborhood of f in M. Then there exists a neighborhood U of fN in M such that LS1 ;S2 .g/ 2 V for all g 2 U and all integers S2 > S1  0. Let S2 > S1  0 be integers, g 2 M and .x; u/ 2 X.A; B; S1 ; S2  / (X.A; B; S1 ; S2 / respectively). Then in view of (2.26) and (2.211), Z

S2 

S1 

LS1 ;S2 .g/.t; xN .t/; uN .t//dt Z

D

S2 

S1 

Z D

g.S2   t C S1 ; x.S2   t C S1  /; u.S2   t C S1  //dt

S2 

g.t; x.t/; u.t//dt: S1 

(2.213)

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2 Linear Control Systems with Periodic Convex Integrands

Let T2 > T1  0, y; z 2 Rn and g 2 M. For each .x; u/ 2 X.A; B; T1 ; T2 /, put Z I .T1 ; T2 ; x; u/ D g

T2

g.t; x.t/; u.t//dt

(2.214)

T1

and set  .g; y; z; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T1 / D y; x.T2 / D zg;

(2.215)

 .g; y; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T1 / D yg;

(2.216)

O  .g; z; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T2 / D zg;  .g; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 /g:

(2.217) (2.218)

Relation (2.213) implies the following result. Proposition 2.44. Let S2 > S1  0 be integers, g 2 M and .xi ; ui / 2 X.A; B; S1 ; S2  /, i D 1; 2. Then I g .S1 ; S2 ; x1 ; u1 /  I g .S1 ; S2 ; x2 ; u2 /  M

(2.219)

I gN .S1 ; S2 ; xN 1 ; uN 1 /  I gN .S1 ; S2 ; xN 2 ; uN 2 /  M;

(2.220)

if and only if

where gN D LS1 ;S2 .g/. Proposition 2.44 [see (2.219) and (2.220)] implies the following result. Proposition 2.45. Let S2 > S1  0 be integers, M  0, g 2 M, gN D LS1 ;S2 .g/ and .x; u/ 2 X.A; B; S1 ; S2 /. Then the following assertions are equivalent: I g .S1 ; S2 ; x; u/  .g; S1 ; S2  / C M if and only if I gN .S1 ; S2 ; xN ; uN /   .Ng; S1 ; S2 / C MI I g .S1 ; S2 ; x; u/  .g; x.S1 /; x.S2  /; S1 ; S2  / C M

2.13 The Basic Lemma for Theorem 2.25

73

if and only if I gN .S1 ; S2 ; xN ; uN /   .Ng; xN .S1 /; xN .S2  /; S1 ; S2  / C MI I g .S1 ; S2 ; x; u/  .g; O x.S2 /; S1 ; S2  / C M if and only if I gN .S1 ; S2 ; xN ; uN /   .Ng; xN .S1 /; S1 ; S2  / C MI I g .S1 ; S2 ; x; u/  .g; x.S1 /; S1 ; S2  / C M if and only if I gN .S1 ; S2 ; xN ; uN /  O  .Ng; xN .S2 /; S1 ; S2  / C M:

2.13 The Basic Lemma for Theorem 2.25 Let f 2 Rn satisfy  f . f / D inf. f /:

(2.221)

Lemma 2.46. Let S0  1 be an integer,  2 .0; 1/ and .x ; u / 2 X.A; B; 0; 1/ be an .f ; A; B/-overtaking optimal pair satisfying x .0/ D f :

(2.222)

Then there exists ı 2 .0; / such that for each .x; u/ 2 X.A; B; 0; S0  / which satisfies  f .x.0//  inf. f / C ı; I f .0; S0 ; x; u/  S0 .f /   f .x.0// C  f .x.S0  //  ı the inequality jx.t/  x .t/j   holds for all t 2 Œ0; S0  . Proof. Assume that the lemma does not hold. Then there exist a sequence fık g1 kD1  .0; 1 and a sequence f.xk ; uk /g1  X.A; B; 0; S  / such that 0 kD1 lim ık D 0

k!1

74

2 Linear Control Systems with Periodic Convex Integrands

and that for all integers k  1,  f .xk .0//  inf. f / C ık ; I f .0; S0 ; xk ; uk /  S0 .f /   f .xk .0// C  f .xk .S0  //  ık ; supfjxk .t/  x .t/j W t 2 Œ0; S0  g > :

(2.223) (2.224) (2.225)

In view of (2.223) and Proposition 2.21, the sequence fxk .0/g1 kD1 is bounded. By (2.224), the continuity and the boundedness from below of the function  f (see Proposition 2.22) and boundedness of the sequence fxk .0/g1 kD1 , the sequence fI f .0; S0 ; xk ; uk /g1 is bounded. By Proposition 2.30, we may assume without loss kD1 of generality that there exists .x; u/ 2 X.A; B; 0; S0  / such that xk .t/ ! x.t/ as k ! 1 uniformly on Œ0; S0  ; I f .0; S0 ; x; u/  lim inf I f .0; S0 ; xk ; uk /; k!1

(2.226) (2.227)

uk ! u as k ! 1 weakly in L1 .Rm I .0; S0 //: It follows from (2.221), (2.223), (2.226) and the continuity and strict convexity of  f (see Proposition 2.20) that  f .x.0// D lim  f .xk .0// D inf. f /; x.0/ D f : k!1

By (2.224), (2.226), (2.227), and the continuity of  f (see Proposition 2.20),  f .x.S0 // D lim  f .xk .S0 //; k!1

I .0; S0 ; x; u/  S0 .f /   f .x.0// C  f .x.S0  // f

 lim infŒI f .0; S0 ; xk ; uk /  S0 .f /   f .xk .0// C  f .xk .S0 //  0: k!1

In view of the inequality above and Proposition 2.15, I f .0; S0 ; x; u/  S0 .f /   f .x.0// C  f .x.S0  // D 0:

(2.228)

Theorem 2.3 implies that there exists an .f ; A; B/-overtaking optimal pair .Qx; uQ / 2 X.A; B; 0; 1/ such that xQ .0/ D x.S0 /:

(2.229)

x.t/ D xQ .t  S0 /; u.t/ D uQ .t  S0  /:

(2.230)

For all t > S0  set

2.14 Proof of Theorem 2.25

75

It is not difficult to see that the pair .x; u/ 2 X.A; B; 0; 1/ is an .f ; A; B/-good pair. By its definition, (2.228)–(2.230) and Propositions 2.15 and 2.16, I f .0; S; x; u/  S.f /   f .x.0// C  f .x.S// D 0 for all integers S  1: Combined with Proposition 2.23 this implies that .x; u/ is an .f ; A; B/-overtaking optimal pair satisfying x.0/ D f . By Theorem 2.3 and (2.222), x.t/ D x .t/ and u.t/ D u .t/ for all t  0. Together with (2.226) this implies that for all sufficiently large natural numbers k, jxk .t/  x .t/j  =2 for all t 2 Œ0; S0  : This contradicts (2.225). The contradiction we have reached proves Lemma 2.46. Note that Lemma 2.46 can also be applied for the triplet .fN ; A; B/.

2.14 Proof of Theorem 2.25 By Lemma 2.46 applied to the triplet .fN ; A  B/ there exist ı1 2 .0; =4/ such that the following property holds: (P13)

for each .x; u/ 2 X.A; B; 0; L0 / which satisfies N

N

 f .x.0//  inf. f / C ı1 ; N

N

N

I f .0; L0 ; x; u/  L0 .f /   f .x.0// C  f .x.L0  //  ı1 we have jx.t/  xN  .t/j   holds for all t 2 Œ0; L0  : In view of the continuity of  f , Proposition 2.17, and (2.34), there exists ı2 2 .0; ı1 / such that for each z 2 Rn satisfying jz  xf .0/j  2ı2 , N

N

N

j f .z/j D j f .z/   f .xf .0//j  ı1 =8I

(2.231)

for each y; z 2 Rn satisfying jy  xf .0/j  2ı2 , jz  xf .0/j  2ı2 , jv.y; z/  .f /j  ı1 =8:

(2.232)

76

2 Linear Control Systems with Periodic Convex Integrands

By Theorem 2.13, there exist an integer l0  1, ı3 2 .0; ı2 =8/ and a neighborhood U1 of f in M such that the following property holds: (P14) for each integer T > 2l0 , each g 2 U1 and each .x; u/ 2 X.A; B; 0; T / such that jx.0/j  M; I g .0; T; x; u/  .g; x.0/; 0; T / C ı3 we have jx.i/  xf .0/j  ı2 for all i D l0 ; : : : ; T  l0 :

(2.233)

Since the pair .Nx ; uN  / 2 X.A; B; 0; 1/ is .fN ; A; B/-good it follows from Theorem 2.2 and (2.29) that there exists an integer l1  1 such that jNx .i/  xf .0/j  ı2 for all integers i  l1 :

(2.234)

By Proposition 2.41, there exists a neighborhood U  U1 of f in M such that the following property holds: (P15) for each g 2 U , each integer j 2 f1; : : : ; 2L0 C 2l0 C 2l1 C 4g and each .x; u/ 2 X.A; B; 0; j/ satisfying minfI f .0; j; x; u/; I g .0; j; x; u/g N

 .j.f /j C 2/.2L0 C 2l0 C 2l1 C 4/ C j f .Nx .0//j C 2 we have jI f .0; j; x; u/  I g .0; j; x; u/j  ı3 =8: Choose ı > 0 and an integer L1 such that ı  ı3 =4;

(2.235)

L1 > 2L0 C 2l0 C 2l1 C 4:

(2.236)

T  L1 ; g 2 U

(2.237)

Assume that an integer

and .x; u/ 2 X.A; B; 0; T/ satisfies jx.0/j  M; I g .0; T; x; u/  .g; x.0/; 0; T / C ı:

(2.238)

2.14 Proof of Theorem 2.25

77

By property (P14) and (2.235)–(2.238), (2.233) holds. In view of (2.236) and (2.237), ŒT  l0  l1  L0  4; T  l0  l1  L0   Œl0 ; T  l0  l1  L0 :

(2.239)

In view of (2.233) and (2.239), jx.i /  xf .0/j  ı2 for all i 2 fT  l0  l1  L0  4; : : : ; T  l0  l1  L0 g:

(2.240)

Proposition 2.32 implies that there exists .x1 ; u1 / 2 X.A; B; 0; T / such that x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 Œ0; .T  l0  l1  L0  4/;

(2.241)

x1 .t/ D xN  .T  t/; u1 .t/ D uN  .T  t/; t 2 Π.T  l0  l1  L0  3/;  T; (2.242) I f . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x1 ; u1 / D v.x. .T  l0  l1  L0  4//; xN  . .l0 C l1 C L0 C 3///:

(2.243)

By (2.238) and (2.241), ı  I g .0; T; x1 ; u1 /  I g .0; T; x; u/ D I g . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x1 ; u1 / C I g . .T  l0  l1  L0  3/; T; x1 ; u1 /  I g . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x; u/  I g . .T  l0  l1  L0  3/;  T; x; u/:

(2.244)

We show that I g . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x1 ; u1 /  I g . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x; u/  ı1 =8 C ı3 =8 C ı1 =2:

(2.245)

In view of (2.234), (2.240), (2.243), and the choice of ı2 [see (2.232)], I f . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x1 ; u1 /   .f / C ı1 =8: Combined with (2.237) and property (P15) this implies that I g . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x1 ; u1 /   .f / C ı1 =8 C ı3 =8: (2.246)

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2 Linear Control Systems with Periodic Convex Integrands

It follows from (2.240) and the choice of ı2 [see (2.232)] that I f . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x; u/   .f /  ı1 =8:

(2.247)

If I g . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x; u/ <  .f /  ı1 =2; then by property (P15) and (2.237), I f . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x; u/ < .f /  ı1 =2 C ı3 =8 <  .f /  3ı1 =8 and this contradicts (2.247). Thus I g . .T  l0  l1  L0  4/; .T  l0  l1  L0  3/; x; u/   .f /  ı1 =2:

(2.248)

It follows from (2.246) and (2.248) that (2.245) holds. By (2.244) and (2.245), I g . .T  l0  l1  L0  3/;  T; x1 ; u1 /  I g . .T  l0  l1  L0  3/;  T; x; u/  ı  ı1 =8  ı3 =8  ı1 =2:

(2.249)

Since .Nx ; uN  / is an .fN ; A; B/-overtaking optimal pair it follows from (2.27), (2.242), and Proposition 2.16 that N

I f . .T  l0  l1  L0  3/;  T; x1 ; u1 / D I f .0; .l0 C l1 C L0 C 3/; xN  ; uN  / N

D .f / .l0 C l1 C L0 C 3/ C  f .Nx .0// N

  f .Nx ..l0 C l1 C L0 C 3///: (2.250) By (2.234) and the choice of ı2 (see (2.231)), N

j f .Nx . .l0 C l1 C L0 C 3///j  ı1 =8: Together with (2.250) this implies that N

I f . .T  l0  l1  L0  3/;  T; x1 ; u1 /   f .Nx .0// C .f /.l0 C l1 C L0 C 3/ C ı1 =8: (2.251) Property (P15), (2.237), and (2.251) imply that I g . .T  l0  l1  L0  3/;  T; x1 ; u1 / N

  f .Nx .0// C .f / .l0 C l1 C L0 C 3/ C ı1 =8 C ı3 =8:

(2.252)

2.14 Proof of Theorem 2.25

79

In view of (2.249) and (2.252), I g . .T  l0  l1  L0  3/;  T; x; u/ N

  f .Nx .0// C .f / .l0 C l1 C L0 C 3/ C ı C 3ı1 =4 C ı3 =4:

(2.253)

Property (P15) and (2.253) imply that I f . .T  l0  l1  L0  3/; T; x; u/ N

  f .Nx .0// C .f / .l0 C l1 C L0 C 3/ C ı C 3ı1 =4 C 3ı3 =8:

(2.254)

Set xQ .t/ D x.T  t/; uQ .t/ D u.T  t/; t 2 Œ0; T:

(2.255)

Clearly, .Qx; uQ / 2 X.A; B; 0; T/ and in view of (2.27), (2.254), and (2.255), N

I f .0; .l0 C l1 C L0 C 3/; xQ ; uQ / D I f . .T  l0  l1  L0  3/;  T; x; u/ N

  f .Nx .0// C .f / .l0 C l1 C L0 C 3/ C ı C 3ı1 =4 C 3ı3 =8:

(2.256)

It follows from (2.240) and (2.255) that jQx. .l0 C l1 C L0 C 3//  xf .0/j  ı2 :

(2.257)

By (2.257) and the choice of ı2 (see (2.231)), N

j f .Qx. .l0 C l1 C L0 C 3///j  ı1 =8:

(2.258)

By (2.238), (2.256), and Proposition 2.15, N

N

N

N

N

 f .Qx.0//   f .Nx .0// C I f .0; L0 ; xQ ; uQ /  L0 .f /   f .Qx.0// C  f .Qx.L0  // N

N

N

  f .Qx.0//   f .Nx .0// C I f .0; .l0 C l1 C L0 C 3/; xQ ; uQ / N

N

 .f / .l0 C l1 C L0 C 3/   f .Qx.0// C  f .Qx..l0 C l1 C L0 C 3/// N

N

  f .Qx.0//   f .Nx .0// C .f / .l0 C l1 C L0 C 3/ C ı C 3ı3 =8 C 3ı1 =4 N

N

C  f .Nx .0//  .f / .l0 C l1 C L0 C 3/   f .Qx.0// C ı1 =8  ı C 3ı3 =8 C 3ı1 =4 C ı1 =8  ı1 :

80

2 Linear Control Systems with Periodic Convex Integrands

By the relation above, Proposition 2.15 and the relation  fN .Nx .0// D inf. fN /, N

N

 f .Qx.0//   f .Nx .0// C ı1 ; N

N

N

I f .0; L0 ; xQ ; uQ /  L0 .f / C  f .Qx.0// C  f .Qx.L0  //  ı1 :

(2.259) (2.260)

It follows from (2.259), (2.260), and property (P13) that jQx.t/  xN  .t/j   holds for all t 2 Œ0; L0  : Together with (2.245) this implies that jx.T  t/  xN  .t/j   holds for all t 2 Œ0; L0  : t u

Theorem 2.25 is proved.

2.15 Proof of Theorem 2.26 Theorems 2.13 and 2.25 imply the following result. Theorem 2.47. Let L0 > 0 be an integer,  > 0. Then there exist ı > 0, a neighborhood U of f in M and an integer L1 > L0 such that for each integer T  L1 , each g 2 U and each .x; u/ 2 X.A; B; 0; T / which satisfies I g .0; T; x; u/  .g; 0; T / C ı the following inequality holds for all t 2 Œ0; L0 : jx.T  t/  xN  .t/j  : Theorem 2.47 and Propositions 2.43 and 2.45 imply Theorem 2.26.

2.16 Proof of Theorem 2.27 Theorem 2.27 follows from Propositions 2.43 and 2.45 and the next result. Theorem 2.48. Let L0 > 0 be an integer,  > 0; M0 > 0. Then there exist ı > 0, a neighborhood U of f in M and an integer L1 > L0 such that for each integer T  L1 , each g 2 U and each .x; u/ 2 X.A; B; 0; T / which satisfies jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T /; 0; T / C ı

2.16 Proof of Theorem 2.27

81

for all t 2 Œ0; L0  , jx.t/  .t/j  ; where .; / 2 X.A; B; 0; 1/ is the unique .f ; A; B/-overtaking optimal pair such that .0/ D x.0/. Proof. Denote by d the metric of the space M. Assume that Theorem 2.48 does not hold. Then there exist a sequence fık g1 kD1  .0; 1/ such that ık < 4k ; k D 1; 2; : : : ;

(2.261)

Tk  L0 C 2k; k D 1; 2; : : :

(2.262)

a sequence of integers

a sequence fgk g1 kD1  M such that d.gk ; f /  k1 ; k D 1; 2; : : :

(2.263)

and a sequence .xk ; uk / 2 X.A; B; 0; Tk /, k D 1; 2; : : : such that for each natural number k, jxk .0/j; jxk .Tk /j  M0 ; I gk .0; Tk ; xk ; uk /  .gk ; xk .0/; xk .Tk /; 0; Tk  / C ık ; maxfjxk .t/  k .t/j W t 2 Œ0; L0  g > ;

(2.264) (2.265) (2.266)

where the pair .k ; k / 2 X.A; B; 0; 1/ is .f ; A; B/-overtaking optimal and xk .0/ D k .0/:

(2.267)

In view of (2.264), (2.267) and Theorem 2.4, the following property holds: (P16) for each  > 0 there exists a natural number m. / such that for each integer k  1 and each integer p  m. /, jk .p/  xf .0/j  : Proposition 2.16 implies that for each pair of natural number p; k, I f .0; p; k ; k /  p.f / C  f .k .0// C  f .k .p // D 0:

(2.268)

It follows from (2.264), (2.267), (2.268), and the continuity and boundedness from below of the function  f that for each integer p  1 the sequence fI f .0; p; k ; k /g1 kD1 is bounded. By Proposition 2.30, extracting subsequences and

82

2 Linear Control Systems with Periodic Convex Integrands

re-indexing we may assume without loss of generality that there exists .; / 2 X.A; B; 0; 1/ such that for each integer p  1, k .t/ ! .t/ as k ! 1 uniformly on Œ0; p ; I f .0; p; ; /  lim inf I f .0; p; k ; k /: k!1

(2.269) (2.270)

In view of (2.268)–(2.270) and the continuity of  f , for each integer p  1, I f .0; p; ; /  p.f / C  f ..0// C  f ..p //  0: Together with Proposition 2.15 this implies that for each integer p  1, I f .0; p; ; /  p.f /   f ..0// C  f ..p // D 0:

(2.271)

By (2.271), the boundedness from below of  f and Theorem 2.2, the pair .; / 2 X.A; B; 0; 1/ is .f ; A; B/-good. Together with (2.271) and Proposition 2.22 this implies that the pair .; / is .f ; A; B/-overtaking optimal. In view of (2.267) and (2.269), .0/ D lim k .0/ D lim xk .0/: k!1

k!1

(2.272)

Let  > 0. Proposition 2.17, (2.34), and the continuity of  f imply that there exists ı > 0 such that for each y; z 2 Rn satisfying jy  xf .0/j; jz  xf .0/j  ı, j f .y/j  =8; jv.y; z/  .f /j  =8:

(2.273)

By (2.262)–(2.265), (2.267), and Theorems 2.13 and 2.4, there exists an integer k./  1 such that for each integer k  4k./ C 8, jxk .i /  xf .0/j  ı; i D k./ C 1; : : : ; Tk  k./;

(2.274)

jk .i /  xf .0/j  ı for all integers i  k./ C 1:

(2.275)

Let S  1 and k  4k./ C S C 8 be integers. By Proposition 2.32 and (2.262), there exists .Qxk ; uQ k / 2 X.A; B; 0; .S C k./ C 2// such that xQ k .t/ D k .t/; uQ k .t/ D k .t/; t 2 Œ0; .S C k./ C 1/ ;

(2.276)

xQ k ..S C k./ C 2// D xk ..S C k./ C 2/ /;

(2.277)

I f ..S C k./ C 1/; .S C k./ C 2/; xQ k ; uQ k / D v.k ..S C k./ C 1//; xk ..S C k./ C 2/ //:

(2.278)

2.16 Proof of Theorem 2.27

83

In view of (2.267), (2.268), and (2.273)–(2.278), for each integer k  4k./CSC8, I f .0; .S C k./ C 2/; xQ k ; uQ k / D I f .0; .S C k./ C 1/; k ; k / C v.k ..S C k./ C 1/ /; xk ..S C k./ C 2/ //  .f / .S C k./ C 1/ C  f .xk .0//   f .k ..S C k./ C 1/ // C  .f / C =8  .f / .S C k./ C 2/ C  f .xk .0// C =4: (2.279) Proposition 2.41 and (2.263) imply that there exists an integer k1  4k./ C S C 8 such that for all integers k  k1 , I gk .0; .S C k./ C 2/; xQ k ; uQ k /  .f / .S C k./ C 2/ C  f .xk .0// C 3=8: (2.280) By (2.261), (2.276), and (2.267), there exists an integer k2  k1 such that for all integers k  k2 , I gk .0; .SCk./C2/; xk ; uk /  .f / .SCk./C2/C f .xk .0//C=2:

(2.281)

Proposition 2.41, (2.263), and (2.281) imply that there exists an integer k3  k2 such that for all integers k  k3 , I f .0; .SCk./C2/; xk ; uk /  .f / .SCk./C2/C f .xk .0//C5=8:

(2.282)

Since S is any natural number we conclude that for any integer p  1, the sequence fI f .0; p; xk ; uk /g1 kDp is bounded. By Proposition 2.30, extracting subsequences and re-indexing we may assume without loss of generality that there exists .x; u/ 2 X.A; B; 0; 1/ such that for each integer p  1, xk .t/ ! x.t/ as k ! 1 uniformly on Œ0; p ; I .0; p; x; u/  lim inf I .0; p; xk ; uk /: f

f

k!1

(2.283) (2.284)

In view of (2.274), (2.282)–(2.284), and the continuity of  f , I f .0; .S C k./ C 2/; x; u/  .f /.S C k./ C 2/ C  f .x.0// C 5=8; (2.285) jx..S C k./ C 2//  xf .0/j  ı:

(2.286)

84

2 Linear Control Systems with Periodic Convex Integrands

By (2.285), (2.286), and the choice of ı [see (2.273)], I f .0; .S C k./ C 2/; x; u/  .f /.S C k./ C 2/   f .x.0// C  f .x..S C k./ C 2/ //  :

(2.287)

Since S is any natural number it follows from (2.287) and Proposition 2.15 that for any integer p  1, I f .0; p; x; u/  .f /p   f .x.0// C  f .x.p //  :

(2.288)

In view of (2.288), the boundedness from below of the function  f and Theorem 2.2, .x; u/ 2 X.A; B; 0; 1/ is an .f ; A; B/-good pair. Since  is any positive number Proposition 2.15 and (2.288) imply that for any integer p  1, I f .0; p; x; u/  .f /p   f .x.0// C  f .x.p // D 0:

(2.289)

By Proposition 2.22 and (2.289), .x; u/ is an .f ; A; B/-overtaking optimal pair. Since .x; u/ and .; / are .f ; A; B/-overtaking optimal pairs it follows from (2.272), (2.283), and Theorem 2.3 that x.t/ D .t/; u.t/ D .t/; t 2 Œ0; 1/:

(2.290)

By (2.269), (2.283), and (2.290), for all sufficiently large natural numbers k, jxk .t/  k .t/j  =4; t 2 Œ0; L0  : This contradicts (2.266). The contradiction we have reached proves Theorem 2.48.

Chapter 3

Linear Control Systems with Nonconvex Integrands

In this chapter we study the existence and structure of optimal trajectories of linear control systems with autonomous nonconvex integrands. For these control systems we establish the existence of optimal trajectories over an infinite horizon and show that the turnpike phenomenon holds. We also study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals. It is shown that in these regions optimal trajectories converge to solutions of the corresponding infinite horizon optimal control problem which depend only on the integrand.

3.1 Preliminaries In this chapter we study the structure of approximate optimal trajectories of linear control systems described by a differential equation x0 .t/ D Ax.t/ C Bu.t/ for almost every (a. e.) t 2 I;

(3.1)

where I is either R1 or ŒT1 ; 1/ or ŒT1 ; T2  (here 1 < T1 < T2 < 1), n; m are natural numbers, x W I ! Rn is an absolutely continuous (a. c.) function and the control function u W I ! Rm is Lebesgue measurable, and A and B are given matrices of dimensions n  n and n  m with integrands f W Rn  Rm ! R1 . Note that if I is an unbounded interval, then x W I ! Rn is an absolutely continuous function if and only if it is an absolutely continuous function on any bounded subinterval of I. We assume that the linear system (3.1) is controllable and that the integrand f is a continuous function. We denote by j  j the Euclidean norm and by h; i the inner product in the k-dimensional Euclidean space Rk . For every z 2 R1 denote by bzc the largest

© Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6_3

85

86

3 Linear Control Systems with Nonconvex Integrands

integer which does not exceed z: bzc D maxfi 2 Z W i  zg. For every s 2 R1 set sC D maxfs; 0g. For every nonempty set X and every function h W X ! R1 [ f1g set inf.h/ D inffh.x/ W x 2 Xg: Let a0 > 0 and

W Œ0; 1/ ! Œ0; 1/ be an increasing function such that .t/ D 1:

lim

t!1

(3.2)

Suppose that f W Rn  Rm ! R1 is a continuous function such that the following assumption holds: (A1) (i) for each .x; u/ 2 Rn  Rm , f .x; u/  maxf .jxj/;

.juj/;

.ŒjAx C Buj  a0 jxjC /ŒjAx C Buj  a0 jxjC g  a0 I

(3.3)

(ii) for each x 2 Rn the function f .x; / W Rm ! R1 is convex; (iii) for each M;  > 0 there exist ; ı > 0 such that jf .x1 ; u1 /  f .x2 ; u2 /j   maxff .x1 ; u1 /; f .x2 ; u2 /g for each u1 ; u2 2 Rm and each x1 ; x2 2 Rn which satisfy jxi j  M; jui j  ; i D 1; 2;

maxfjx1  x2 j; ju1  u2 jg  ıI

(iv) for each K > 0 there exists a constant aK > 0 and an increasing function K

W Œ0; 1/ ! Œ0; 1/

such that K .t/

! 1 as t ! 1

and f .x; u/ 

K .juj/juj

 aK

for each u 2 Rm and each x 2 Rn satisfying jxj  K.

3.1 Preliminaries

87

Let T1 2 R1 and T2 > T1 . A pair of an absolutely continuous function x W ŒT1 ; T2  ! Rn and a Lebesgue measurable function u W ŒT1 ; T2  ! Rm is called an .A; B/-trajectory-control pair if (3.1) holds with I D ŒT1 ; T2 . Denote by X.A; B; T1 ; T2 / the set of all .A; B/-trajectory-control pairs x W ŒT1 ; T2  ! Rn , u W ŒT1 ; T2  ! Rm . Let T 2 R1 and I D ŒT; 1/ be an infinite closed subinterval of R1 . Denote by X.A; B; T; 1/ the set of all pairs of a.c. functions x W ŒT; 1/ ! Rn and Lebesgue measurable functions u W ŒT; 1/ ! Rm satisfying (3.1). Note that a function h satisfies (A1) if h 2 C1 .Rn Rm /, (A1)(i), (A1)(ii), (A1)(iv) hold, and for each K > 0 there exists an increasing function Q W Œ0; 1/ ! Œ0; 1/ such that for each x 2 Rn satisfying jxj  K and each u 2 Rm , maxfj@[email protected]; u/j; j@[email protected]; u/jg  Q .jxj/.1 C

K .juj/juj/:

The performance of the above control system is measured on any finite interval ŒT1 ; T2   Œ0; 1/ and for any .x; u/ 2 X.A; B; T1 ; T2 / by the integral functional Z T2 f f .x.t/; u.t//dt: (3.4) I .T1 ; T2 ; x; u/ D T1

In this chapter we study the existence and structure of optimal trajectories of linear control system (3.1) with the integrand f . For these control systems we establish the existence of optimal trajectories over an infinite horizon and show that the turnpike phenomenon holds. We also study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals. It is shown that in these regions optimal trajectories converge to solutions of the corresponding infinite horizon optimal control problem which depend only on the integrand. More precisely, we consider the following optimal control problems I f .0; T; x; u/ ! min;

(P1 )

.x; u/ 2 X.A; B; 0; T/ such that x.0/ D y; x.T/ D z; I f .0; T; x; u/ ! min;

(P2 )

.x; u/ 2 X.A; B; 0; T/ such that x.0/ D y; I f .0; T; x; u/ ! min;

(P3 )

.x; u/ 2 X.A; B; 0; T/; where y; z 2 Rn and T > 0. The study of these problems is based on the properties of solutions of the corresponding infinite horizon optimal control problem associated with the control system (3.1) and the integrand f . In this chapter we establish the turnpike property for the approximate solutions of problems (P1 ), (P2 ) and (P3 ). For problems (P2 ) and (P3 ) we show that in regions

88

3 Linear Control Systems with Nonconvex Integrands

close to the right endpoint T of the time interval their approximate solutions are determined only by the integrand, and are essentially independent of the choice of interval and the endpoint value y. For problems (P3 ), approximate solutions are determined only by the integrand also in regions close to the left endpoint 0 of the time interval. A number .f / WD infflim inf T 1 I f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; 1/g T!1

(3.5)

is called the minimal long-run average cost growth rate of f . By (A1)(i), 1 < .f /. Let T > 0 and y; z 2 Rn . Set .f ; y; z; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.0/ D y; x.T/ D zg;

(3.6)

.f ; y; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.0/ D yg;

(3.7)

.f O ; z; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.T/ D zg;

(3.8)

.f ; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/g:

(3.9)

We say that .Qx; uQ / 2 X.A; B; 0; 1/ is .f ; A; B/-overtaking optimal [44, 53] if for each .x; u/ 2 X.A; B; 0; 1/ satisfying x.0/ D xQ .0/, lim supŒI f .0; T; xQ ; uQ /  I f .0; T; x; u/  0: T!1

We say that .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-minimal [44, 53] if for each T > 0, I f .0; T; x; u/ D .f ; x.0/; x.T/; T/: Let .xf ; uf / 2 Rn  Rm satisfy Axf C Buf D 0:

(3.10)

Clearly, .f /  f .xf ; uf /: It is easy to see that the following result holds. Proposition 3.1. Assume that .f / D f .xf ; uf / and let x.t/ D xf , u.t/ D uf for all t 2 Œ0; 1/. Then .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-minimal. The next proposition is proved in Sect. 3.4. Proposition 3.2. Assume that .f / D f .xf ; uf /. Then there exists  > 0 such that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/, I f .0; T; x; u/  Tf .xf ; uf /   :

(3.11)

3.1 Preliminaries

89

It is easy to see that the following proposition holds. Proposition 3.3. Let  > 0 and let for each T > 0 and each .x; u/ 2 X.A; B; 0; T/, relation (3.11) holds. Then .f / D f .xf ; uf /. We suppose that the following assumption holds. (A2)

.f / D f .xf ; uf / and if .x; u/ 2 Rn  Rm satisfies Ax C Bu D 0; .f / D f .x; u/;

then x D xf . Proposition 3.2 imply the following result. Proposition 3.4. For each .x; u/ 2 X.A; B; 0; 1/ either I f .0; T; x; u/  T.f / ! 1 as T ! 1 or supfjI f .0; T; x; u/  T.f /j W T > 0g < 1. A trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ is called .f ; A; B/-good [44, 53] if supfjI f .0; T; x; u/  T.f /j W T > 0g < 1: Proposition 3.4 and Theorem 2.1 of [52] imply the following result. Proposition 3.5. For every .f ; A; B/-good trajectory-control pair .x; u/ X.A; B; 0; 1/,

2

supfjx.t/j W t 2 Œ0; 1/g < 1: We suppose that the following assumption holds. (A3)

For each .f ; A; B/-good trajectory-control pair .x; u/ 2 X.A; B; 0; 1/

the equality limt!1 x.t/ D xf is true. Let us consider examples of integrands satisfying assumptions (A1)–(A3). First note that if a continuous strictly convex function h W Rn  Rm ! R1 satisfies assumption (A1) (with f D h) and h.x; u/=juj ! 1 as juj ! 1 uniformly in x 2 Rn ;

90

3 Linear Control Systems with Nonconvex Integrands

then in view of Corollary 2.11 of Chap. 2, the function h satisfies assumptions (A2) and (A3) (with f D h). Let us consider another example of an integrand which satisfies (A1)–(A3). Let c 2 R1 , a1 > 0, l 2 Rn , .x ; u / 2 Rn  Rm satisfy Ax C Bu D 0 and let 0 W Œ0; 1/ ! Œ0; 1/ be an increasing function such that limt!1 0 .t/ D 1. Assume that a continuous function L W Rn  Rm ! Œ0; 1/ satisfies for each .x; u/ 2 Rn  Rm , L.x; u/  maxf

0 .jxj/;

0 .juj/jujg

 a1 C jljjAx C Buj;

L.x; u/ D 0 if and only if x D x ; u D u ; for each x 2 Rn , the function L.x; / W Rm ! R1 is convex and for each M;  > 0 there exist ; ı > 0 such that jL.x1 ; u1 /  L.x2 ; u2 /j   maxfL.x1 ; u1 /; L.x2 ; u2 /g for each x1 ; x2 2 Rn and each u1 ; u2 2 Rm which satisfy jxi j  M; jui j  ; i D 1; 2; jx1  x2 j; ju1  u2 j  ı: For every .x; u/ 2 Rn  Rm set h.x; u/ D L.x; u/ C c C hl; Ax C Bui: It is not difficult to see that for each .x; u/ 2 Rn  Rm , h.x; u/  maxf

0 .jxj/;

0 .juj/jujg

 a1  jcj

and that h satisfies (A1) under the appropriate choice of a0 > 0, prove the following result.

. In Sect. 3.12 we

Proposition 3.6. .h/ D h.x ; u / D c, (A2) holds for f D h and for any .h; A; B/good trajectory-control pair .x; u/ 2 X.A; B; 0; 1/, lim x.t/ D x :

t!1

3.2 Turnpike Results We use the notation, definitions, and assumptions introduced in Sect. 3.1. The following turnpike result is proved in Sect. 3.6.

3.2 Turnpike Results

91

Theorem 3.7. Let ; M0 ; M1 > 0. Then there exist L > 0, ı 2 .0; / such that for each T > 2L and each .x; u/ 2 X.A; B; 0; T/ which satisfies for each S 2 Œ0; T  L, I f .S; S C L; x; u/  .f ; x.S/; x.S C L/; L/ C ı and satisfies at least one of the following conditions: (a) jx.0/j; jx.T/j  M0 ; I f .0; T; x; u/  .f ; x.0/; x.T/; T/ C M1 I (b) jx.0/j  M0 ; I f .0; T; x; u/  .f ; x.0/; T/ C M1 I (c) I f .0; T; x; u/  .f ; T/ C M1 there exist p1 2 Œ0; L, p2 2 ŒT  L; T such that jx.t/  xf j   for all t 2 Œp1 ; p2 : Moreover if jx.0/  xf j  ı, then p1 D 0 and if jx.T/  xf j  ı, then p2 D T. Theorem 3.7 and Theorem 4.1.1 of [52] imply the following result. Theorem 3.8. Let x0 2 Rn . Then there exists an .f ; A; B/-overtaking optimal trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ satisfying x.0/ D x0 . In the sequel we show (see Proposition 3.36) that any .f ; A; B/-overtaking optimal trajectory-control pair is .f ; A; B/-good. The next result which describes the limit behavior of overtaking optimal trajectories is proved in Sect. 3.7. Theorem 3.9. Let M;  > 0. Then there exists L > 0 such that for any .f ; A; B/overtaking optimal trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ which satisfies jx.0/j  M the inequality jx.t/  xf j   holds for all numbers t  L. Moreover, there exists ı > 0 such that for any .f ; A; B/overtaking optimal trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ satisfying jx.0/  xf j  ı, the inequality jx.t/  xf j   holds for all numbers t  0. We say that .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-minimal [5, 53] if for each T > 0, I f .0; T; x; u/ D .f ; x.0/; x.T/; T/:

(3.11)

92

3 Linear Control Systems with Nonconvex Integrands

The next result which is proved in Sect. 3.8 shows the equivalence of the optimality criterions introduced above. Theorem 3.10. Assume that .x; u/ 2 X.A; B; 0; 1/. Then the following conditions are equivalent: .x; u/ is .f ; A; B/-overtaking optimal; (ii) .x; u/ is .f ; A; B/-minimal and .f ; A; B/-good; (iii) .x; u/ is .f ; A; B/-minimal and

(i)

lim x.t/ D xf I

t!1

(iv)

.x; u/ is .f ; A; B/-minimal and lim inft!1 jx.t/j < 1.

3.3 Structure of Solutions in the Regions Close to the End Points In this section we state results which describe the structure of solutions of problems (P2 ) and (P3 ) in the regions close to the end points. Combined with the turnpike results of Sect. 3.2 they provide the full description of the structure of their solutions. We use the notation, definitions, and assumptions introduced in Sects. 3.1 and 3.2. For each z 2 Rn denote by .z/ the set of all .f ; A; B/-overtaking optimal pairs .x; u/ 2 X.A; B; 0; 1/ such that x.0/ D z which is nonempty in view of Theorem 3.8. Let z 2 Rn . Set  f .z/ D lim infŒI f .0; T; x; u/  T.f /; T!1

(3.12)

where .x; u/ 2 .z/. In view of Proposition 3.4,  f .z/ is finite, well defined, and does not depend on the choice of .x; u/ 2 .z/. Definition (3.12) and the definition of .f ; A; B/-overtaking optimal pairs imply the following result. Proposition 3.11. 1. Let .x; u/ 2 X.A; B; 0; 1/ be .f ; A; B/-good. Then  f .x.0//  lim infŒI f .0; T; x; u/  T.f / T!1

and for each pair of numbers S > T  0,  f .x.T//  I f .T; S; x; u/  .S  T/.f / C  f .x.S//: 2. Let S > T  0 and .x; u/ 2 X.A; B; T; S/. Then (3.13) holds.

(3.13)

3.3 Structure of Solutions in the Regions Close to the End Points

93

The next result follows from definition (3.12). Proposition 3.12. Let .x; u/ 2 X.A; B; 0; 1/ be an .f ; A; B/-overtaking optimal pair. Then for each pair of numbers S > T  0,  f .x.T// D I f .T; S; x; u/  .S  T/.f / C  f .x.S//: Theorems 3.8 and 3.9 and (A2) imply the following result. Proposition 3.13.  f .xf / D 0. The following result is proved in Sect. 3.4. Proposition 3.14. The function  f is continuous at xf . Proposition 3.15. Let .x; u/ 2 X.A; B; 0; 1/ be .f ; A; B/-overtaking optimal. Then  f .x.0// D lim ŒI f .0; T; x; u/  T.f /: T!1

Proof. It follows from Propositions 3.12, 3.14, and (A3) that  f .x.0// D lim . f .x.0//   f .x.T/// T!1

D lim ŒI f .0; T; x; u/  T.f /: T!1

Proposition 3.15 is proved. The next result is proved in Sect. 3.9. Proposition 3.16. For each M > 0 the set fx 2 Rn W  f .x/  Mg is bounded. In Sect. 3.9 we prove the following proposition. Proposition 3.17. The function  f is lower semicontinuous. By Propositions 3.13, 3.16, and 3.17, inf. f / is finite and there exists 2 Rn such that  f . / D inf. f /. Proposition 3.18. Let .x; u/ 2 X.A; B; 0; 1/ be .f ; A; B/-good pair such that for all T > 0, I f .0; T; x; u/  T.f / D  f .x.0//   f .x.T//:

(3.14)

Then .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-overtaking optimal. Proof. Theorem 3.8 implies that there exists an .f ; A; B/-overtaking optimal pair .x1 ; u1 / 2 X.A; B; 0; 1/ such that x1 .0/ D x.0/: By Proposition 3.12, for each integer T  1, I f .0; T; x1 ; u1 /  T.f / D  f .x1 .0//   f .x1 .T//:

94

3 Linear Control Systems with Nonconvex Integrands

It follows from the equality above, (3.14), (A3) and Propositions 3.13 and 3.14 that for all T > 0, I f .0; T; x; u/  I f .0; T; x1 ; u1 / D  f .x1 .T//   f .x.T// ! 0 as T ! 1: Thus lim ŒI f .0; T; x; u/  I f .0; T; x1 ; u1 / D 0:

T!1

Since .x1 ; u1 / is an .f ; A; B/-overtaking optimal pair the equality above implies that .x; u/ is an .f ; A; B/-overtaking optimal pair too. Proposition 3.18 is proved. Consider a linear control system x0 .t/ D Ax.t/  Bu.t/;

(3.15)

x.0/ D x0 which is also controllable. For the triplet .f ; A; B/ we use all the notation and definitions introduced for the triplet .f ; A; B/. It is clear that assumption (A1) holds for the triplet .f ; A; B/. Let T1 2 R1 , T2 > T1 . A pair of an absolutely continuous function x W ŒT1 ; T2  ! Rn and a Lebesgue measurable function u W ŒT1 ; T2  ! Rm is called an .A; B/-trajectory-control pair if (3.15) holds for a. e. t 2 ŒT1 ; T2 . Denote by X.A; B; T1 ; T2 / the set of all .A; B/-trajectory-control pairs x W ŒT1 ; T2  ! Rn , u W ŒT1 ; T2  ! Rm . Let T 2 R1 . Denote by X.A; B; T; 1/ the set of all pairs of a. c. functions x W ŒT; 1/ ! Rn and Lebesgue measurable functions u W ŒT; 1/ ! Rm satisfying (3.15) for a. e. t  T, which are called .A; B/-trajectory-control pairs. Assume that S1 2 R1 , S2 > S1 and that .x; u/ 2 X.A; B; S1 ; S2 /. For all t 2 ŒS1 ; S2  set xN .t/ D x.S2  t C S1 /; uN .t/ D u.S2  t C S1 /:

(3.16)

In view of (3.1) and (3.16) for a. e. t 2 ŒS1 ; S2 , xN 0 .t/ D x0 .S2  t C S1 / D Ax.S2  t C S1 /  Bu.S2  t C S1 / D ANx.t/  BNu.t/; .Nx; uN / 2 X.A; B; S1 ; S2 /:

(3.17)

By (3.16), Z

S2

Z f .Nx.t/; uN .t//dt D

S1

Z D

S2

f .x.S2  t C S1 /; u.S2  t C S1 //dt

S1 S2 S1

f .x.t/; u.t//dt:

(3.18)

3.3 Structure of Solutions in the Regions Close to the End Points

95

For each pair of numbers T2 > T1 and each .x; u/ 2 X.A; B; T1 ; T2 / set Z I .T1 ; T2 ; x; u/ D f

T2

f .x.t/; u.t//dt:

(3.19)

T1

For each y; z 2 Rn and each T > 0 set  .f ; y; z; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.0/ D y; x.T/ D zg;

(3.20)

 .f ; y; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.0/ D yg; (3.21) O  .f ; z; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/ and x.T/ D zg; (3.22)  .f ; T/ D inffI f .0; T; x; u/ W .x; u/ 2 X.A; B; 0; T/g:

(3.23)

Relations (3.17) and (3.18) imply the following result. Proposition 3.19. Let S2 > S1 be real numbers, M  0 and let .xi ; ui / 2 X.A; B; S1 ; S2 /, i D 1; 2. Then I f .S1 ; S2 ; x1 ; u1 /  I f .S1 ; S2 ; x2 ; u2 /  M if and only if I f .S1 ; S2 ; xN 1 ; uN 1 /  I f .S1 ; S2 ; xN 2 ; uN 2 /  M: Proposition 3.19 implies the following result. Proposition 3.20. Let S2 > S1 be real numbers and .x; u/ 2 X.A; B; S1 ; S2 /: Then the following assertions hold: I f .S1 ; S2 ; x; u/  .f ; S2  S1 / C M if and only if I f .S1 ; S2 ; xN ; uN /   .f ; S2  S1 / C MI I f .S1 ; S2 ; x; u/  .f ; x.S1 /; x.S2 /; S2  S1 / C M if and only if I f .S1 ; S2 ; xN ; uN /   .f ; xN .S1 /; xN .S2 /; S2  S1 / C MI I f .S1 ; S2 ; x; u/  .f ; x.S1 /; S2  S1 / C M if and only if I f .S1 ; S2 ; xN ; uN /  O  .f ; xN .S2 /; S2  S1 / C MI I f .S1 ; S2 ; x; u/  O .f ; x.S2 /; S2  S1 / C M if and only if I f .S1 ; S2 ; xN ; uN /   .f ; xN .S1 /; S2  S1 / C M:

96

3 Linear Control Systems with Nonconvex Integrands

By (3.10), Axf  Buf D 0: Set  .f / D infflim inf T 1 I f .0; T; x; u/ 2 X.A; B; 0; 1/g: T!1

(3.24)

It is easy to see that  .f /  f .xf ; uf / D .f /:

(3.25)

In view of (A1),  .f / > 1. Thus  .f / is finite. Proposition 3.21.  .f / D .f / D f .xf ; uf /. Proof. By Proposition 3.2, there exists  > 0 such that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/, I f .0; T; x; u/  T.f /   : Together with (3.17) and (3.18) this implies that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/, I f .0; T; x; u/  T.f /   : This implies that  .f /  .f /. Combined with (3.25) this completes the proof of Proposition 3.21. It follows from Proposition 3.21 that (A2) holds for the triplet .f ; A; B/. The next result is proved in Sect. 3.9. Proposition 3.22. For any .f ; A; B/-good trajectory-control pair .x; u/ 2 X.A; B; 0; 1/, lim x.t/ D xf :

t!1

Therefore .f ; A; B/ satisfies all the assumptions posed for the triplet .f ; A; B/ and all the results stated above for the triplet .f ; A; B/ are also true for .f ; A; B/. For each z 2 Rn , set f .z/ D lim infŒI f .0; T; x; u/  T.f /; T!1

where .x; u/ 2 X.A; B; 0; 1/ an .f ; A; B/-overtaking optimal pair such that x.0/ D z. In Sect. 3.11 we prove the following two theorems which describe the structure of solutions of problems (P2 ) and (P3 ) in the regions closed to the end points.

3.4 Auxiliary Results and the Proof of Proposition 3.2

97

Theorem 3.23. Let L0 > 0,  2 .0; 1/; M > 0. Then there exist ı > 0 and L1 > L0 such that for each T  L1 and each .x; u/ 2 X.A; B; 0; T/ which satisfies jx.0/j  M; I f .0; T; x; u/  .f ; x.0/; T/ C ı there exists an .f ; A; B/-overtaking optimal pair .Nx ; uN  / 2 X.A; B; 0; 1/ such that f .Nx .0// D inf.f /; jx.T  t/  xN  .t/j   for all t 2 Œ0; L0 : Theorem 3.24. Let L0 > 0 and  > 0. Then there exist ı > 0 and L1 > L0 such that for each T  L1 and each .x; u/ 2 X.A; B; 0; T/ which satisfies I f .0; T; x; u/  .f ; 0; T/ C ı there exist an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ and an .f ; A; B/-overtaking optimal pair .Nx ; uN  / 2 X.A; B; 0; 1/ such that  f .x .0// D inf. f /; f .Nx .0// D inf.f / and for all t 2 Œ0; L0 , jx.t/  x .t/j  ; jx.T  t/  xN  .t/j  :

3.4 Auxiliary Results and the Proof of Proposition 3.2 In the sequel we use the following auxiliary results. Proposition 3.25 (Proposition 4.2 of [52]). Let T2 > T1 be real numbers, 1 f f.xj ; uj /g1 jD1  X.A; B; T1 ; T2 / and let the sequence fI .T1 ; T2 ; xj ; uj /gjD1 be 1 bounded. Then there exist a subsequence f.xjk ; ujk /gkD1 and .x; u/ 2 X.A; B; T1 ; T2 / such that xjk .t/ ! x.t/ as k ! 1 uniformly in ŒT1 ; T2 ; ujk ! u as k ! 1 weakly in L1 .Rm I .T1 ; T2 //; I f .T1 ; T2 ; x; u/  lim inf I f .T1 ; T2 ; xjk ; ujk /: k!1

98

3 Linear Control Systems with Nonconvex Integrands

Proposition 3.26 (Proposition 4.3 of [52]). For every yQ ; zQ 2 Rn and every T > 0 there exists a solution x./, y./ of the system x0 D Ax C BBt y; y0 D x  At y with the boundary conditions x.0/ D yQ , x.T/ D zQ (where Bt denotes the transpose of B). Propositions 3.25 and 3.26 and (A1) imply the following result. Proposition 3.27. Let T > 0 and y; z 2 Rn . Then there exists .x; u/ 2 X.A; B; 0; T/ such that x.0/ D y; x.T/ D z; I f .0; T; x; u/ D .f ; y; z; T/: Proposition 3.28 (Proposition 4.5 of [52]). Let M;  > 0. Then supfj .f ; y; z; /j W y; z 2 Rn ; jyj; jzj  Mg < 1: Proposition 3.29 (Proposition 4.6 of [52]). Let M; ;  > 0. Then there exists a number ı > 0 such that for each y1 ; y2 ; z1 ; z2 2 Rn satisfying jyi j; jzi j  M; i D 1; 2; jy1  y2 j; jz1  z2 j  ı the following relation holds: j .f ; y1 ; z1 ; /  .f ; y2 ; z2 ;  /j  : Proposition 3.30 (Proposition 2.7 of [52]). Let M1 > 0 and 0 < 0 < 1 . Then there exists a positive number M2 such that for each T1 2 R1 , each T2 2 ŒT1 C 0 ; T1 C 1  and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying I f .T1 ; T2 ; x; u/  M1 the inequality jx.t/j  M2 holds for all t 2 ŒT1 ; T2 . Proposition 3.31. Let M0 > 0. Then there exists  > 0 such that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/ satisfying jx.0/j; jx.T/j  M0 the following inequality holds: I f .0; T; x; u/  T.f /  : Proof. We may assume without loss of generality that M0 > jxf j:

(3.26)

3.4 Auxiliary Results and the Proof of Proposition 3.2

99

In view of Proposition 3.28, there exists 0 > 0 such that supfj .f ; y; z; 1/j W y; z 2 Rn ; jyj; jzj  M0 g  0 :

(3.27)

Fix   20 C 2j.f /j: Let T > 0; .x; u/ 2 X.A; B; 0; T/; jx.0/j; jx.T/j  M0 :

(3.28)

We show that I f .0; T; x; u/  T.f /  : By Propositions 3.27 and 3.28 there exists .x1 ; u1 / 2 X.A; B; 0; T C 2/ such that x1 .0/ D xf ; I f .0; 1; x1 ; u1 / D .f ; xf ; x.0/; 1/; x1 .t/ D x.t  1/; u1 .t/ D u.t  1/; t 2 Œ1; T C 1; x1 .T C 2/ D xf ; I f .T C 1; T C 2; x1 ; u1 / D .f ; x.T/; xf ; 1/:

(3.29)

It follows from Proposition 3.1 and (3.26)–(3.29) that .T C 2/.f /  I f .0; T C 2; x1 ; u1 / D I f .0; T; x; u/ C .f ; xf ; x.0/; 1/ C .f ; x.T/; xf ; 1/  I f .0; T; x; u/ C 20 : By the relation above and the choice of , I f .0; T; x; u/  T.f / C 2.f /  20  T.f /  : Proposition 3.31 is proved. Proof of Proposition 3.2. By (A1) there exists M0 > 0 such that .M0 / > a0 C 2 C j.f /j:

(3.30)

By Proposition 3.31, there exists  > 0 such that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/ satisfying jx.0/j; jx.T/j  M0

100

3 Linear Control Systems with Nonconvex Integrands

we have I f .0; T; x; u/  T.f /   :

(3.31)

Assume that T > 0 and .x; u/ 2 X.A; B; 0; T/. If jx.t/j  M0 for all t 2 Œ0; T, then it follows from (A1) and (3.30) that I f .0; T; x; u/  T.f /  T. .M0 /  a0  .f //  2T >  : Therefore we may assume without loss of generality that there exist S1 ; S2 2 Œ0; T such that S1 < S2 ; jx.Si /j  M0 ; i D 1; 2; jx.t/j  M0 for all t 2 Œ0; S1  [ ŒS2 ; T: It follows from the relation above, (A1), (3.30) and the choice of  (see (3.31)) that I f .0; T; x; u/  . .M0 /  a0 /S1 C . .M0 /  a0 /.T  S2 / C I f .S1 ; S2 ; x; u/  . .M0 /  a0 /.S1 C T  S2 / C .S2  S1 /.f /   D T.f /  .S1 C T  S2 /.f / C . .M0 /  a0 /.S1 C T  S2 /    T.f /   : Proposition 3.2 is proved.

3.5 Auxiliary Results for Theorems 3.7, 3.9 and 3.10 Proposition 3.32. Let  > 0. Then there exists ı > 0 such that for each T  1 and each y; z 2 Rn satisfying jy  xf j; jz  xf j  ı, .f ; y; z; T/  T.f / C : Proof. By Proposition 3.29, there exists ı y1 ; y2 ; z1 ; z2 2 Rn satisfying

2 .0; 41 / such that for each

jyi j; jzi j  jxf j C 1; i D 1; 2; jy1  y2 j; jz1  z2 j  ı we have j .f ; y1 ; z1 ; 41 /  .f ; y2 ; z2 ; 41 /j  =6:

(3.32)

3.5 Auxiliary Results for Theorems 3.7, 3.9 and 3.10

101

Assume that T  1; y; z 2 Rn ; jy  xf j; jz  xf j  ı:

(3.33)

In view of Proposition 3.27, there exists .x; u/ 2 X.A; B; 0; T/ such that x.0/ D y; x.t/ D xf ; u.t/ D uf ; t 2 Œ41 ; T  41 ; x.T/ D z; I f .0; 41 ; x; u/ D .f ; y; xf ; 41 /; I f .T  41 ; T; x; u/ D .f ; xf ; z; 41 /:

(3.34)

It follows from (3.34) that .f ; y; z; T/  I f .0; T; x; u/ D .f ; y; xf ; 41 / C .f ; xf ; z; 41 / C .T  1=2/.f /:

(3.35)

By (3.33) and the choice of ı (see (3.32)), .f ; y; xf ; 41 /; .f ; xf ; z; 41 /  =6 C .f ; xf ; xf ; 41 / D =6 C 41 .f /: Together with (3.35) this implies that .f ; y; z; T/  T.f / C =3: Proposition 3.32 is proved. Proposition 3.33. Let M0 > 0. Then there exists M > 0 such that for each T  1 and each y; z 2 Rn satisfying jyj; jzj  M0 , .f ; y; z; T/  T.f / C M: Proof. We may assume without loss of generality that M0  jxf j C 1: By Proposition 3.28, there exists M1 > supfj .f ; y; z; 41 /j W y; z 2 Rn ; jyj; jzj  M0 g: Choose a number M > 2M1 C 2j.f /j:

102

3 Linear Control Systems with Nonconvex Integrands

Assume that T  1; y; z 2 Rn ; jyj; jzj  M0 :

(3.36)

In view of Proposition 3.27, there exists .x; u/ 2 X.A; B; 0; T/ such that x.0/ D y; x.t/ D xf ; u.t/ D uf ; t 2 Œ41 ; T  41 ; x.T/ D z; I f .0; 41 ; x; u/ D .f ; y; xf ; 41 /; I f .T  41 ; T; x; u/ D .f ; xf ; z; 41 /:

(3.37)

It follows from (3.36), the choice of M1 , and the relations M0 > jxf j C 1 that j .f ; y; xf ; 41 /j; j .f ; xf ; z; 41 /j  M1 : By the inequalities above, (3.37) and the choice of M, .f ; y; z; T/  I f .0; T; x; u/ D .T  21 /.f / C 2M1 D T.f / C j.f /j C 2M1 < T.f / C M: Proposition 3.33 is proved. Proposition 3.34. Let M;  > 0. Then there exists a natural number L such that for each .x; u/ 2 X.A; B; 0; L/ satisfying I f .0; L; x; u/  L.f / C M the inequality minfjx.t/  xf j W t 2 Œ0; Lg   holds. Proof. Assume that the proposition does not hold. Then there exist a strictly increasing sequence of natural numbers fLk g1 kD1 such that Lk  k for all natural numbers k and a sequence .xk ; uk / 2 X.A; B; 0; Lk /, k D 1; 2; : : : such that for each natural number k we have I f .0; Lk ; xk ; uk /  Lk .f / C M;

(3.38)

minfjxk .t/  xf j W t 2 Œ0; Lk g > :

(3.39)

Proposition 3.2 implies that there exists a positive constant  such that for every positive number T and every trajectory-control pair .x; u/ 2 X.A; B; 0; T/, I f .0; T; x; u/  T.f /   :

(3.40)

3.5 Auxiliary Results for Theorems 3.7, 3.9 and 3.10

103

Let p be a natural number. By (3.38) and (3.40), for each natural number k > p we have I f .0; p; xk ; uk / D I f .0; Lk ; xk ; uk /  I f .p; Lk ; xk ; uk /  Lk .f / C M  .Lk  p/.f / C   p.f / C M C  :

(3.41)

In view of (3.41) and Proposition 3.25, extracting a subsequence and re-indexing if necessary, we may assume without loss of generality that there exists a trajectorycontrol pair .x; u/ 2 X.A; B; 0; 1/ such that for every natural number p, xk .t/ ! x.t/ as k ! 1 uniformly on Œ0; p;

(3.42)

I f .0; p; x; u/  p.f / C M C  :

(3.43)

It follows from (3.43) and Proposition 3.4 that .x; u/ 2 X.A; B; 0; 1/ is an .f ; A; B/good trajectory-control pair and there exists a positive number 0 such that jx.t/  xf j  =4 for all t  0 :

(3.44)

Relation (3.42) implies that there exists an integer k0 > 0 C 8 such that for each integer k  k0 , jxk .t/  x.t/j  =4 for all t 2 Œ0 ; 0 C 4:

(3.45)

Relations (3.44) and (3.45) imply that for every integer k  k0 and every number t 2 Œ0 ; 0 C 4, jxf  xk .t/j  jxf  x.t/j C jx.t/  xk .t/j  =2: This contradicts (3.39). The contradiction we have reached proves Proposition 3.34. Proposition 3.35. Let M;  > 0. Then there exists a natural number L such that for each T  L, each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  T.f / C M

(3.46)

and each number S satisfying ŒS; S C L  Œ0; T the following inequality holds: minfjx.t/  xf j W t 2 ŒS; S C Lg  :

(3.47)

104

3 Linear Control Systems with Nonconvex Integrands

Proof. Proposition 3.2 implies that there exists a positive constant  such that for every positive number T and every trajectory-control pair .x; u/ 2 X.A; B; 0; T/ we have I f .0; T; x; u/  T.f /   : It follows from Proposition 3.34 that there exists an integer L  1 such that for every trajectory-control pair .x; u/ 2 X.A; B; 0; L/ satisfying I f .0; L; x; u/  L.f / C M C 2 the following inequality holds: minfjx.t/  xf j W t 2 Œ0; Lg  :

(3.48)

Assume that T  L, .x; u/ 2 X.A; B; 0; T/ satisfies (3.46) and a real number S satisfies (3.47). In view of the choice of  we have I f .0; S; x; u/  S.f /   ; I f .S C L; T; x; u/  .T  S  L/.f /   : Combined with (3.46) this implies that I f .S; S C L; x; u/ D I f .0; T; x; u/  I f .0; S; x; u/  I f .S C L; T; x; u/  T.f / C M  S.f / C   .T  .S C L//.f / C  L.f / C 2 : By the inequality above and the choice of L (see (3.48)), there exists t 2 ŒS; S C L such that jx.t/  xf j  . Proposition 3.35 is proved. Proposition 3.36. Any .f ; A; B/-overtaking optimal pair .x; u/ 2 X.A; B; 0; 1/ is .f ; A; B/-good. Proof. Let .x; u/ 2 X.A; B; 0; 1/ be .f ; A; B/-overtaking optimal. By Proposition 3.27, there exists .x1 ; u1 / 2 X.A; B; 0; 1/ such that x1 .0/ D x.0/; x1 .t/ D xf ; u1 .t/ D uf ; t 2 Œ1; 1/; I f .0; 1; x1 ; u1 / D .f ; x1 .0/; xf ; 1/:

3.5 Auxiliary Results for Theorems 3.7, 3.9 and 3.10

105

Since the pair .x; u/ is .f ; A; B/-overtaking optimal it follows from the relations above that 0  lim supŒI f .0; T; x; u/  I f .0; T; x1 ; u1 / T!1

D lim supŒI f .0; T; x; u/  .f ; x.0/; xf ; 1/  .T  1/.f /: T!1

Together with Proposition 3.4 this implies that the pair .x; u/ is .f ; A; B/-good. Proposition 3.36 is proved. Proposition 3.37. Let  2 .0; 1/. Then there exists ı > 0 such that for each T  1 and each .x; u/ 2 X.A; B; 0; T/ satisfying jx.0/  xf j; jx.T/  xf j  ı; I .0; T; x; u/  .f ; x.0/; x.T/; T/ C ı f

the inequality jx.t/  xf j   holds for all t 2 Œ0; T. Proof. By Propositions 3.29 and 3.32, for each integer k  1, there is ık 2 .0; 4k /

(3.49)

such that the following properties hold: (i) for each pair of points y; z 2 Rn satisfying jy  xf j; jz  xf j  ık , j .f ; y; z; 1/  .f /j  4k I (ii) for each number T  1 and each pair of points y; z 2 Rn satisfying jy  xf j; jz  xf j  ık , .f ; y; z; T/  T.f / C 4k : We may assume without loss of generality that the sequence fık g1 kD1 is decreasing. Assume that the proposition does not hold. Then for each integer k  1 there exist Tk  1 and a trajectory-control pair .xk ; uk / 2 X.A; B; 0; Tk / such that jxk .0/  xf j  ık ; jxk .Tk /  xf j  ık ;

(3.50)

I f .0; Tk ; xk ; uk /  .f ; xk .0/; xk .Tk /; Tk / C ık ;

(3.51)

supffjxk .t/  xf j W t 2 Œ0; Tk g > :

(3.52)

It follows from property (ii) and (3.49)–(3.51) that for every natural number k we have I f .0; Tk ; xk ; uk /  Tk .f / C 2  4k :

(3.53)

106

3 Linear Control Systems with Nonconvex Integrands

Proposition 3.27 implies that there exists a trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ such that x.t/ D x1 .t/; u.t/ D u1 .t/; t 2 Œ0; T1 ; x.T1 C 1/ D x2 .0/; I f .T1 ; T1 C 1; x; u/ D .f ; x1 .T1 /; x2 .0/; 1/

(3.54) (3.55)

and for every natural number k we have x

X  k .Ti C 1/ C t D xkC1 .t/; iD1

X  k u .Ti C 1/ C t D ukC1 .t/; iD1

I

f

X kC1

 kC1 X .Ti C 1/  1; .Ti C 1/; x; u D .f ; xkC1 .Tk /; xkC2 .0/; 1/:

iD1

(3.56)

iD1

In view of (3.53), (3.56) and property (i), for each integer k  2,  X  X k k I f 0; .Ti C 1/; x; u D .I f .0; Ti ; xi ; ui / C .f ; xi .Ti /; xiC1 .0/; 1// iD1

iD1



k X ŒTi .f / C 2  4i C .f / C 4i  iD1

 .f /

k X

.Ti C 1/ C 6:

iD1

Since the relation above holds for any integer k  2 it follows from Proposition 3.4 that the trajectory-control pair .x; u/ is .f ; A; B/-good and lim x.t/ D xf :

t!1

Thus there exists an integer i0  1 such that for each integer k  i0 and all t 2 Œ0; Tk , jxk .t/  xf j  =2: This contradicts (3.52). The contradiction we have reached proves Proposition 3.37.

3.6 Proof of Theorem 3.7

107

3.6 Proof of Theorem 3.7 By Proposition 3.33, there exists M2 > 0 such that for each   1 and each y; z 2 Rn satisfying jyj; jzj  M0 C M1 , .f ; y; z; /  .f / C M2 :

(3.57)

By Proposition 3.37, there exists ı 2 .0; / such that the following property holds: (i) for each T  1 and each .x; u/ 2 X.A; B; 0; T/ satisfying jx.0/  xf j; jx.T/  xf j  ı; I .0; T; x; u/  .f ; x.0/; x.T/; T/ C ı f

the inequality jx.t/  xf j   holds for all t 2 Œ0; T. By Proposition 3.35, there exists L0 > 0 such that the following property holds: (ii) for each T  L0 , each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  T.f / C M1 C M2 and each number S satisfying ŒS; S C L0   Œ0; T we have minfjx.t/  xf j W t 2 ŒS; S C L0 g  ı: Fix L > 4L0 C 4: Assume that T > 2L, .x; u/ 2 X.A; B; 0; T/, for each S 2 Œ0; T  L, I f .S; S C L; x; u/  .f ; x.S/; x.S C L/; L/ C ı

(3.58)

and that at least one of the conditions (a), (b), and (c) of Theorem 3.7 holds. It follows from conditions (a)–(c) and the choice of M2 (see (3.57)) that I f .0; T; x; u/  T.f / C M2 C M1 :

(3.59) q

It follows from (3.59) and property (ii) that there exists a sequence fSi giD1 such that 0  S1  L0 ; 1  SiC1  Si  2 C L0 for all integers i satisfying 1  i < q;

108

3 Linear Control Systems with Nonconvex Integrands

Tq  Sq  Tq  L0  1;

(3.60)

jx.Si /  xf j  ı; i D 1; : : : ; q:

(3.61)

Clearly, if jx.0/  xf j  ı, then we may assume that S1 D 0 and if jx.T/  xf j  ı, then we may assume that Sq D T. Assume that t 2 ŒS1 ; Sq : Then there is an integer j 2 f1; : : : ; q  1g such that t 2 ŒSj ; SjC1 : In view of (3.60), there exists a number S such that ŒSj ; SjC1   ŒS; S C L  Œ0; T: Combined with (3.58) this implies that I f .Sj ; SjC1 ; x; u/  .f ; x.Sj /; x.SjC1 /; SjC1  Sj / C ı: Together with (3.61) and property (i) this implies that jx.t/  xf j  . Theorem 3.7 is proved.

3.7 Proof of Theorem 3.9 We may assume without loss of generality that M > jxf j C 1;  < 21 : By Theorem 3.7, there exist L > 0, ı > 0 such that the following property holds: (i) for each T > 2L and each .x; u/ 2 X.A; B; 0; T/ which satisfies jx.0/j; jx.T/j  M; I f .0; T; x; u/ D .f ; x.0/; x.T/; T/ we have jx.t/  xf j   for all t 2 ŒL; T  L and if jx.0/  xf j  ı, then jx.t/  xf j   for all t 2 Œ0; T  L Assume that .x; u/ 2 X.A; B; 0; 1/ is an .f ; A; B/-overtaking optimal pair and jx.0/j  M:

(3.62)

3.8 Proof of Theorem 3.10

109

By (A3) and Proposition 3.36, lim x.t/ D xf :

t!1

Thus there exists T0 > 0 such that for all t  T0 , jx.t/j < M:

(3.63)

T > T0 C 2L:

(3.64)

Let

By property (i), (3.62) and (3.64), jx.t/  xf j   for all t 2 ŒL; T  L and if jx.0/  xf j  ı, then jx.t/  xf j   for all t 2 Œ0; T  L: Since T is any number satisfying (3.64) we conclude that Theorem 3.9 is proved.

3.8 Proof of Theorem 3.10 In view of Proposition 3.36, (i) implies (ii). By (A3), (ii) implies (iii). Clearly, (iv) follows from (iii). Assume that (iv) holds. We show that (i) is true. It follows from (iv), .f ; A; B/-minimality of .x; u/ and Theorem 3.7 that lim x.t/ D xf :

t!1

(3.65)

Assume that the pair .x; u/ is not .f ; A; B/-overtaking optimal. By Theorem 3.8, there exists an .f ; A; B/-overtaking optimal pair .Qx; uQ / 2 X.A; B; 0; 1/ satisfying xQ .0/ D x.0/: Clearly, lim supŒI f .0; T; xQ ; uQ /  I f .0; T; x; u/  0: T!1

Since the pair .x; u/ is not .f ; A; B/-overtaking optimal we have lim supŒI f .0; T; x; u/  I f .0; T; xQ ; uQ / > 0: T!1

(3.66)

110

3 Linear Control Systems with Nonconvex Integrands

Thus there exist  > 0 and a strictly increasing sequence of positive numbers Tk ! 1 as k ! 1 such that for all integers k  1, I f .0; Tk ; x; u/  I f .0; Tk ; xQ ; uQ /  2:

(3.67)

Proposition 3.29 and (A2) imply that there exists ı > 0 such that j .f ; z1 ; z2 ; 1/  .f /j  =4

(3.68)

for all z1 ; z2 2 Rn satisfying jzi  xf j  ı, i D 1; 2: It follows from Proposition 3.36 and (A3) that the pair .Qx; uQ / is .f ; A; B/-good and lim xQ .t/ D xf :

t!1

(3.69)

By property (iv), the .f ; A; B/-minimality of .x; u/, Proposition 3.33, and (3.65), lim supŒI f .0; T; x; u/  T.f / < 1: T!1

Thus the pair .x; u/ is .f ; A; B/-good. In view of (3.65) and (3.69) there exists S0 > 0 such that for all t  S0 , jx.t/  xf j; jQx.t/  xf j  ı:

(3.70)

Choose a natural number k such that Tk > S0 . By Proposition 3.27, there exists .x1 ; u1 / 2 X.A; B; 0; Tk C 1/ such that x1 .t/ D xQ .t/; u1 .t/ D uQ .t/; t 2 Œ0; Tk ; x1 .Tk C 1/ D x.Tk C 1/; I f .Tk ; Tk C 1; x1 ; u1 / D .f ; xQ .Tk /; x.Tk C 1/; 1/:

(3.71)

It follows from (3.66) and (3.71) that x1 .0/ D x.0/; x1 .Tk C 1/ D x.Tk C 1/:

(3.72)

By the relation Tk > S0 , (3.68), and (3.70), j .f ; xQ .Tk /; x.Tk C 1/; 1/  .f ; x.Tk /; x.Tk C 1/; 1/j  =2:

(3.73)

In view of (3.67), (3.71), (3.73), and the .f ; A; B/-minimality of .x; u/, I f .0; Tk C 1; x1 ; u1 / D I f .0; Tk ; xQ ; uQ / C .f ; xQ .Tk /; x.Tk C 1/; 1/  I f .0; Tk ; x; u/   C .f ; x.Tk /; x.Tk C 1/; 1/ C =2 D I f .0; TkC1 ; x; u/  =2:

3.9 Proofs of Propositions 3.14, 3.16, 3.17, and 3.22

111

Combined with (3.72) this contradicts the .f ; A; B/-minimality of .x; u/. The contradiction we have reached proves that the pair .x; u/ is .f ; A; B/-overtaking optimal. Theorem 3.10 is proved.

3.9 Proofs of Propositions 3.14, 3.16, 3.17, and 3.22 Proof of Proposition 3.14. Let  > 0. By Proposition 3.29, there exists ı > 0 such that j .f ; z1 ; z2 ; 1/  .f /j  =4

(3.74)

for all z1 ; z2 2 Rn satisfying jzi  xf j  ı; i D 1; 2: By Theorem 3.9, exists  2 .0; ı/ such that for any .f ; A; B/-overtaking optimal trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ satisfying jx.0/  xf j   , we have jx.t/  xf j  ı for all numbers t  0:

(3.75)

Let z1 ; z2 2 Rn ; jzi  xf j  ; i D 1; 2; .x; u/ 2 .z1 /:

(3.76)

By (3.76) and the choice of  (see (3.75)) relation (3.75) is true. It follows from (3.12) that  f .z1 / D lim infŒI f .0; T; x; u/  T.f / T!1

D I f .0; 1; x; u/  .f / C lim infŒI f .1; T; x; u/  .T  1/.f /: T!1

(3.77)

By Proposition 3.27 there exists .x1 ; u1 / 2 X.A; B; 0; 1/ such that x1 .0/ D z2 ; x1 .t/ D x.t/; u1 .t/ D u.t/ for all t  1; I f .0; 1; x1 ; u1 / D .f ; z2 ; x.1/; 1/:

(3.78)

It follows from (3.75), (3.76), and the choice of ı (see (3.74)) that j .f ; z2 ; x.1/; 1/  .f ; z1 ; x.1/; 1/j  =2:

(3.79)

112

3 Linear Control Systems with Nonconvex Integrands

By (3.76)–(3.79) and Proposition 3.11,  f .z2 /  lim infŒI f .0; T; x1 ; u1 /  T.f / T!1

D I f .0; 1; x1 ; u1 /  .f / C lim infŒI f .1; T; x; u/  .T  1/.f / T!1

  .z1 / C .f ; z2 ; x.1/; 1/  .f ; z1 ; x.1/; 1/ f

  f .z1 / C =2: This implies that j f .z1 /   f .z2 /j  =2 for all z1 ; z2 2 Rn satisfying (3.76), Proposition 3.14 is proved. t u Proof of Proposition 3.16. Let M > 0. We show that the set fx 2 Rn W  f .x/  Mg is bounded. Assume the contrary. Then there exists a sequence fzk g1 kD1 such that lim jzk j D 1;

(3.80)

 f .zk /  M; k D 1; 2; : : : :

(3.81)

k!1

By Proposition 3.2, there is  > 0 such that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/, I f .0; T; x; u/  T.f /   :

(3.82)

In view of Theorem 3.8 and (3.12), for each integer k  1 there exists .f ; A; B/overtaking optimal trajectory-control pair .xk ; uk / 2 X.A; B; 0; 1/ such that xk .0/ D zk ;  f .zk / D lim infŒI f .0; T; xk ; uk /  T.f /: T!1

(3.83)

Let k  1 be an integer and T > 0. It follows from (3.81) and (3.83) that there is S > T such that I f .0; S; xk ; uk /  S.f /   f .zk / C 1  M C 1:

(3.84)

Relations (3.82) and (3.84) imply that I f .0; T; xk ; uk /  T.f / D I f .0; S; xk ; uk /  S.f /  I f .T; S; xk ; uk /  .S  T/.f /  M C 1 C  : Thus for each integer k  1 and each T > 0, I f .0; T; xk ; uk /  T.f /  M C 1 C  :

3.9 Proofs of Propositions 3.14, 3.16, 3.17, and 3.22

113

By the inequality above and Proposition 3.25, extracting a subsequence and reindexing if necessary we may assume without loss of generality that there exists .x; u/ 2 X.A; B; 0; 1/ such that for each integer p  1, xk .t/ ! x.t/ as k ! 1 uniformly on Œ0; p; I f .0; p; x; u/  lim inf I f .0; p; xk ; uk /: k!1

Combined with (3.83) this implies that limk!1 zk D x.0/. This contradicts (3.80). The contradiction we have reached proves Proposition 3.16. t u n n Proof of Proposition 3.17. Assume that fzk g1 kD1  R , z 2 R and that

lim zk D z:

k!1

(3.85)

We show that  f .z/  lim inf  f .zk /: k!1

By Proposition 3.2, the sequence f f .zk /g1 kD1 is bounded from below. We may assume without loss of generality there exists limk!1  f .zk / < 1. Clearly,  WD lim  f .zk / k!1

is finite. By Theorem 3.8 and (3.12), for each integer k  1 there exists .f ; A; B/overtaking optimal trajectory-control pair .xk ; uk / 2 X.A; B; 0; 1/ such that xk .0/ D zk ;  f .zk / D lim infŒI f .0; T; xk ; uk /  T.f /: T!1

(3.86)

Proposition 3.12 implies that for each integer k  1 and each T > 0, I f .0; T; xk ; uk /  T.f / D  f .xk .0//   f .xk .T//:

(3.87)

Proposition 3.2 implies that there is  > 0 such that for each T > 0 and each .x; u/ 2 X.A; B; 0; T/, I f .0; T; x; u/  T.f /   :

(3.88)

Let k  1 be an integer and T > 0. It follows from (3.86), (3.88), and the choice of  that there is S > T such that I f .0; S; xk ; uk /  S.f /   C 1:

114

3 Linear Control Systems with Nonconvex Integrands

The relations above and (3.88) imply that I f .0; T; xk ; uk /  T.f / D I f .0; S; xk ; uk /  S.f /  I f .T; S; xk ; uk /  .S  T/.f /   C 1 C  : Thus for each integer k  1 and each T > 0, I f .0; T; xk ; uk /  T.f /   C 1 C  : By the inequality above and Proposition 3.25, extracting a subsequence and reindexing if necessary we may assume without loss of generality that there exists .x; u/ 2 X.A; B; 0; 1/ such that for each integer p  1, xk .t/ ! x.t/ as k ! 1 uniformly on Œ0; p; I f .0; p; x; u/  lim inf I f .0; p; xk ; uk /: k!1

(3.89)

Let  > 0. In view of Proposition 3.14, there exists a positive number ı such that for each  2 Rn satisfying j  xf j  ı, j f ./j  =2:

(3.90)

By (3.85) and Theorem 3.9, there exists L0 > 0 such that for each integer k  1 and t  L0 , jxk .t/  xf j  ı:

(3.91)

Assume that an integer k  1 and T  L0 . In view of (3.90), (3.91) and Proposition 3.12, j f .xk .T//j  =2 and I f .0; T; xk ; uk / D T.f / C  f .xk .0//   f .xk .T//  T.f / C  f .xk .0// C =2: By the relation above, (3.86) and (3.89), for each integer T > L0 , I f .0; T; x; u/  lim inf I f .0; T; xk ; uk / k!1

 T.f / C lim  f .zk / C =2 k!1

3.9 Proofs of Propositions 3.14, 3.16, 3.17, and 3.22

115

and I f .0; T; xk ; uk /  T.f /  lim  f .zk / C =2: k!1

In view of Proposition 3.11, (3.85), (3.86) and (3.89),  f .z/  lim  f .zk / C =2: k!1

Since  is any positive number this completes the proof of Proposition 3.17.

t u

Proof of Proposition 3.22. Assume that .x; u/ 2 X.A; B; 0; 1/ is an .f ; A; B/-good pair. Proposition 3.5 implies that there exists a number M > supfx.t/j W t 2 Œ0; 1/g:

(3.92)

Let  > 0. By Theorem 3.7, there exist L; ı > 0 such that the following property holds: (i) for each number T > 2L and each .y; v/ 2 X.A; B; 0; 1/ which satisfies jy.0/j; jy.T/j  M; I f .0; T; y; v/  .f ; y.0/; y.T/; T/ C ı the inequality jy.t/  xf j   is true for all numbers t 2 ŒL; T  L. In view of Proposition 3.4, there exists T0 > 0 such that for each pair of numbers S2 > S1  T0 , I f .S1 ; S2 ; x; u/   .f ; x.S1 /; x.S2 /; S2  S1 / C ı:

(3.93)

Let T > T0 C 2L. Set y.t/ D x.T  t C T0 /; v.t/ D u.T  t C T0 /; t 2 ŒT0 ; T:

(3.94)

In view of (3.17) and (3.18), .y; v/ 2 X.A; B; T0 ; T/. By (3.92) and (3.94), jy.T0 /j; jy.T/j  M: It follows from (3.93), (3.94), and Proposition 3.20 that I f .T0 ; T; y; v/  .f ; y.T0 /; y.T/; T  T0 / C ı: Together with (3.95) and property (i) this implies that jy.t/  xf j  ; t 2 ŒT0 C L; T  L

(3.95)

116

3 Linear Control Systems with Nonconvex Integrands

and jx.t/  xf j  ; t 2 ŒT0 C L; T  L: Since T is any natural number satisfying T > T0 C 2L we conclude that jx.t/  xf j   for all t  T0 : Since  is any positive number we conclude that limt!1 x.t/ D xf . Proposition 3.22 is proved. t u

3.10 The Basic Lemma for Theorem 3.23 Lemma 3.38. Let S0 > 0,  2 .0; 1/. Then there exists ı 2 .0; / such that for each .x; u/ 2 X.A; B; 0; S0 / which satisfies  f .x.0//  inf. f / C ı; I f .0; S0 ; x; u/  S0 .f /   f .x.0// C  f .x.S0 //  ı there exists an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that  f .x .0// D inf. f /; jx.t/  x .t/j   for all t 2 Œ0; S0 : Proof. Assume that the lemma does not hold. Then there exist a sequence fık g1 kD1  .0; 1 and a sequence f.xk ; uk /g1 kD1  X.A; B; 0; S0 / such that lim ık D 0

(3.96)

 f .xk .0//  inf. f / C ık ;

(3.97)

k!1

and that for all integer k  1,

I f .0; S0 ; xk ; uk /  S0 .f /   f .xk .0// C  f .xk .S0 //  ık

(3.98)

and that the following property holds: (i) for each .f ; A; B/-overtaking optimal pair .y; v/ 2 X.A; B; 0; 1/ satisfying  f .y.0// D inf. f / we have supfjxk .t/  y.t/j W t 2 Œ0; S0 g > :

3.10 The Basic Lemma for Theorem 3.23

117

In view of (3.97) and (3.98) and the boundedness from below of the function  f , the sequence fI f .0; S0 ; xk ; uk /g1 kD1 is bounded. By Proposition 3.25, extracting a subsequence and re-indexing if necessary, we may assume without loss of generality that there exists .x; u/ 2 X.A; B; 0; S0 / such that xk .t/ ! x.t/ as k ! 1 uniformly on Œ0; S0 ;

(3.99)

I f .0; S0 ; x; u/  lim inf I f .0; S0 ; xk ; uk /:

(3.100)

k!1

It follows from (3.96), (3.97), (3.99), and the lower semicontinuity of  f that  f .x.0//  lim inf  f .xk .0// D inf. f /;  f .x.0// D inf. f /: k!1

(3.101)

By (3.99) and the lower semicontinuity of  f ,  f .x.S0 //  lim inf  f .xk .S0 //:

(3.102)

k!1

It follows from (3.96)–(3.98), (3.100)–(3.102) that I f .0; S0 ; x; u/  S0 .f /   f .x.0// C  f .x.S0 //  lim infŒI f .0; S0 ; xk ; uk /  S0 .f /  lim  f .xk .0// C lim  f .xk .S0 // k!1

k!1

k!1

 lim infŒI .0; S0 ; xk ; uk /  S0 .f /   .xk .0// C  .xk .S0 //  0: f

f

f

k!1

In view of the inequality above and Proposition 3.11, I f .0; S0 ; x; u/  S0 .f /   f .x.0// C  f .x.S0 // D 0:

(3.103)

Theorem 3.8 implies that there exists an .f ; A; B/-overtaking optimal pair .Qx; uQ / 2 X.A; B; 0; 1/ such that xQ .0/ D x.S0 /:

(3.104)

x.t/ D xQ .t  S0 /; u.t/ D uQ .t  S0 /:

(3.105)

For all t > S0 set

It is not difficult to see that the pair .x; u/ 2 X.A; B; 0; 1/ is an .f ; A; B/-good pair. By (3.13), (3.105), and Propositions 3.11 and 3.12, I f .0; S; x; u/  S.f /   f .x.0// C  f .x.S// D 0 for all S > 0:

118

3 Linear Control Systems with Nonconvex Integrands

Combined with Proposition 3.18 and (3.10) this implies that .x; u/ 2 X.A; B; 0; 1/ is an .f ; A; B/-overtaking optimal pair satisfying  f .x.0// D inf. f /: By (3.99), for all sufficiently large natural numbers k, jxk .t/  x.t/j  =2 for all t 2 Œ0; S0 : This contradicts the property (i). The contradiction we have reached proves Lemma 3.38. Note that Lemma 3.38 can also be applied for the triplet .f ; A; B/.

3.11 Proofs of Theorems 3.23 and 3.24 Proof of Theorem 3.23. By Lemma 3.38 applied to the triplet .f ; AB/ there exist ı1 2 .0; =4/ such that the following property holds: (P1)

for each .x; u/ 2 X.A; B; 0; L0 / which satisfies f .x.0//  inf.f / C ı1 ; I f .0; L0 ; x; u/  L0 .f /  f .x.0// C f .x.L0 //  ı1

there exists an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that f .x .0// D inf.f /; jx.t/  x .t/j   for all t 2 Œ0; L0 : In view of Propositions 3.13, 3.14, and 3.29, there exists ı2 2 .0; ı1 / such that for each z 2 Rn satisfying jz  xf j  2ı2 , jf .z/j D jf .z/  f .xf /j  ı1 =8I

(3.106)

3.11 Proofs of Theorems 3.23 and 3.24

119

for each y; z 2 Rn satisfying jy  xf j  2ı2 , jz  xf j  2ı2 , j .f ; y; z; 1/  .f /j  ı1 =8:

(3.107)

By Theorem 3.7, there exist l0 > 0, ı3 2 .0; ı2 =8/ such that the following property holds: (P2)

for each T > 2l0 and each .x; u/ 2 X.A; B; 0; T/

such that jx.0/j  M; I f .0; T; x; u/  .f ; x.0/; T/ C ı3 we have jx.t/  xf j  ı2 for all t 2 Œl0 ; T  l0 :

(3.108)

By Theorem 2.8, there exists an .f ; A; B/-overtaking optimal pair .Nx ; uN  / 2 X.A; B; 0; 1/ such that f .Nx .0// D inf.f /:

(3.109)

Proposition 3.36 and (A3) imply that there exists l1 > 0 such that jNx .t/  xf j  ı2 for all t  l1 :

(3.110)

Choose ı > 0 and L1 > 0 such that ı  ı3 =4;

(3.111)

L1 > 2L0 C 2l0 C 2l1 C 8:

(3.112)

T  L1 ; .x; u/ 2 X.A; B; 0; T/

(3.113)

Assume that

and that jx.0/j  M; I f .0; T; x; u/  .f ; x.0/; T/ C ı:

(3.114)

In view of property (P2) and (3.112)–(3.114), relation (3.108) holds. It follows from (3.112) and (3.113) that ŒT  l0  l1  L0  4; T  l0  l1  L0   Œl0 ; T  l0  l1  L0 :

(3.115)

120

3 Linear Control Systems with Nonconvex Integrands

Relations (3.108) and (3.115) imply that jx.t/  xf j  ı2 for all t 2 ŒT  l0  l1  L0  4; T  l0  l1  L0 :

(3.116)

By Proposition 3.27, there exists a trajectory-control pair .x1 ; u1 / 2 X.A; B; 0; T/ such that x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 Œ0; T  l0  l1  L0  4; x1 .t/ D xN  .T  t/; u1 .t/ D uN  .T  t/; t 2 ŒT  l0  l1  L0  3; T; I f .T  l0  l1  L0  4; T  l0  l1  L0  3; x1 ; u1 / D .f ; x.T  l0  l1  L0  4/; xN  .l0 C l1 C L0 C 3/; 1/:

(3.117)

It follows from (3.114) and (3.117) that  ı  I f .0; T; x1 ; u1 /  I f .0; T; x; u/ D I f .T  l0  l1  L0  4; T  l0  l1  L0  3; x1 ; u1 / CI f .T  l0  l1  L0  3; T; x1 ; u1 / I f .T  l0  l1  L0  4; T  l0  l1  L0  3; x; u/ I f .T  l0  l1  L0  3; T; x; u/:

(3.118)

In view of (3.110), (3.116), (3.117), and the choice of ı2 (see (3.107)), I f .T  l0  l1  L0  4; T  l0  l1  L0  3; x1 ; u1 /  .f / C ı1 =8:

(3.119)

By (3.116) and the choice of ı2 (see (3.107)), I f .T  l0  l1  L0  4; T  l0  l1  L0  3; x; u/  .f /  ı1 =8:

(3.120)

It follows from (3.118)–(3.120) that I f .T  l0  l1  L0  3; T; x1 ; u1 /  I f .T  l0  l1  L0  3; T; x; u/  ı  ı1 =4:

(3.121)

3.11 Proofs of Theorems 3.23 and 3.24

121

Since .Nx ; uN  / is an .f ; A; B/-overtaking optimal pair it follows from (3.117) and Proposition 3.12 that I f .T  l0  l1  L0  3; T; x1 ; u1 / D I f .0; l0 C l1 C L0 C 3; xN  ; uN  / D .f /.l0 C l1 C L0 C 3/ C f .Nx .0// f .Nx .l0 C l1 C L0 C 3//:

(3.122)

In view of (3.110) and the choice of ı2 (see (3.106)) we have jf .Nx .l0 C l1 C L0 C 3//j  ı1 =8: Combined with (3.121) and (3.122) this implies that I f .T  l0  l1  L0  3; T; x; u/  .f /.l0 C l1 C L0 C 3/ C f .Nx .0// C ı C 3ı1 =8: (3.123) Set xQ .t/ D x.T  t/; uQ .t/ D u.T  t/; t 2 Œ0; T:

(3.124)

Evidently, .Qx; uQ / 2 X.A; B; 0; T/ and by (3.123) and (3.124) we have I f .0; l0 C l1 C L0 C 3; xQ ; uQ / D I f .T  l0  l1  L0  3; T; x; u/  f .Nx .0// C .f /.l0 C l1 C L0 C 3/ C ı C 3ı1 =8: (3.125) In view of (3.124) and (3.116), jQx.l0 C l1 C L0 C 3/  xf j  ı2 : By the relation above and the choice of ı2 (see (3.106)), jf .Qx.l0 C l1 C L0 C 3//j  ı1 =8:

(3.126)

It follows from (3.125), (3.126), and Proposition 3.11 that f .Qx.0//  f .Nx .0// CI f .0; L0 ; xQ ; uQ /  L0 .f /  f .Qx.0// C f .Qx.L0 //  f .Qx.0//  f .Nx .0// C I f .0; l0 C l1 C L0 C 3; xQ ; uQ / .f /.l0 C l1 C L0 C 3/  f .Qx.0// C f .Qx.l0 C l1 C L0 C 3//

122

3 Linear Control Systems with Nonconvex Integrands

 f .Qx.0//  f .Nx .0// C f .Nx .0// C .f /.l0 C l1 C L0 C 3/ C ı C 3ı1 =8 .f /.l0 C l1 C L0 C 3/  f .Qx.0// C ı1 =8  ı C ı1 =2  ı1 : By the relation above, Proposition 3.11 and the relation f .Nx .0// D inf.f /, f .Qx.0//  inf.f / C ı1 ; I f .0; L0 ; xQ ; uQ /  L0 .f /  f .Qx.0// C f .Qx.L0 //  ı1 : It follows from the two inequalities above and property (P1) that there exists an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that f .x .0// D inf.f /; jQx.t/  x .t/j   for all t 2 Œ0; L0 : Together with (3.124) this implies that jx.T  t/  x .t/j   holds for all t 2 Œ0; L0 : t u

Theorem 3.23 is proved. Theorems 3.7 and 3.23 imply the following result.

Theorem 3.39. Let L0 > 0,  2 .0; 1/. Then there exist ı > 0 and L1 > L0 such that for each T  L1 and each .x; u/ 2 X.A; B; 0; T/ which satisfies I f .0; T; x; u/  .f ; T/ C ı there exists an .f ; A; B/-overtaking optimal pair .Nx ; uN  / 2 X.A; B; 0; 1/ such that f .Nx .0// D inf.f /; jx.T  t/  xN  .t/j   for all t 2 Œ0; L0 : Theorem 3.39 and Proposition 3.20 imply Theorem 3.24.

3.12 Proof of Proposition 3.6

123

3.12 Proof of Proposition 3.6 Since .x ; u / is the unique minimizer of the function L we have .h/  c:

(3.127)

We show that .h/ D c. Assume the contrary. Then .h/ < c and there exist  > 0 and .x; u/ 2 X.A; B; 0; 1/ such that lim inf T 1 I h .0; T; x; u/ < c  : T!1

(3.128)

It is not difficult to see that for each T > 0, Z I h .0; T; x; u/  T.h/ D Z

T

D Z

0 T

D Z

0 T

D 0

T 0

Œh.x.t/; u.t//  .h/dt

L.x.t/; u.t//dt C cT  .h/T C

Z

T

Z L.x.t/; u.t//dt C .c  .h//T C

hl; Ax.t/ C Bu.t/idt

0 T 0

hl; x0 .t/idt

L.x.t/; u.t//dt C .c  .h//T C hl; x.T/  x.0/i:

(3.129)

By (3.129), there exists a sequence of numbers fTk g1 kD1 such that Tk  10; TkC1  Tk  10; k D 1; 2; : : : ; Tk1 I h .0; Tk ; x; u/ < c  ; k D 1; 2; : : : :

(3.130)

In view of (3.2) there exists M0 > 1 such that .M0 / > a0 C 4 C jcj:

(3.131)

It follows from (3.130), (3.131), and (A1) that for each natural number k there exists Sk 2 Œ0; Tk  such that jx.Sk /j  M0 ; jx.t/j > M0 for all t satisfying Sk < t  Tk :

(3.132)

124

3 Linear Control Systems with Nonconvex Integrands

Let k  1 be an integer. By (3.129), (3.131), (3.132), and (A1), I h .0; Tk ; x; u/ D I h .0; Sk ; x; u/ C I h .Sk ; Tk ; x; u/ Z Sk D L.x.t/; u.t//dt C cSk C hl; x.Sk /  x.0/i 0

CI h .Sk ; Tk ; x; u/  cSk  jlj.M0 C jx.0/j/ C .Tk  Sk /. .M0 /  a0 /  cSk  .Tk  Sk /jcj  jlj.M0 C jx.0/j/  cTk  jlj.M0 C jx.0/j/: This implies that lim inf Tk1 I h .0; Tk ; x; u/  c: k!1

This contradicts (3.128). The contradiction we have reached proves that .h/ D c D h.x ; u /: Now it is easy to see that (A2) holds. Let us show that (A3) holds. Let .x; u/ 2 X.A; B; 0; 1/ be .h; A; B/-good. We show that lim x.t/ D x :

t!1

Assume the contrary. Then there exist  > 0 and a sequence of numbers ftk g1 kD1 such that for all integers k  1, tkC1  tk  2; jx.tk /  x j  :

(3.133)

For each integer k  1 set xk .t/ D x.tk C t/; uk .t/ D u.tk C t/; t 2 Œ0; 1:

(3.134)

Clearly, f.xk ; uk /g1 kD1  X.A; B; 0; 1/. (A1) and (3.134) imply that the sequence fI h .0; 1; xk ; uk /g1 kD1 is bounded. By the lower semicontinuity of the integral functionals (see Proposition 4.2 of [52]), extracting a subsequence and re-indexing if necessary, we may assume without loss of generality that there exists .Ox; uO / 2 X.A; B; 0; 1/ such that xk .t/ ! xO .t/ as k ! 1 uniformly in Œ0; 1;

(3.135)

I .0; 1; xO ; uO /  lim inf I .0; 1; xk ; uk /;

(3.136)

h

h

k!1

3.12 Proof of Proposition 3.6

125

In view of the equality .h/ D c, for each integer k  1, I h .0; 1; xk ; uk /  .h/ Z 1 Z 1 L.xk .t/; uk .t//dt C hl; Axk .t/ C Buk .t/idt D Z

0

0

1

D 0

L.xk .t/; uk .t//dt C hl; xk .1/  xk .0/i;

(3.137)

I h .0; 1; xO ; uO /  .h/ Z 1 Z 1 L.Ox.t/; uO .t//dt C hl; AOx.t/ C BOu.t/idt D Z

0

0

1

D 0

L.Ox.t/; uO .t//dt C hl; xO .1/  xO .0/i:

(3.138)

By (3.135)–(3.138), Z 0

1

Z L.Ox.t/; uO .t//dt  lim inf k!1

0

1

L.xk .t/; uk .t//dt:

(3.139)

Proposition 3.5 implies that the function x is bounded. Together with (3.129) this implies that Z

T

sup 0

 L.x.t/; u.t//dt W T > 0 < 1:

Combined with (3.134) and (3.139) this implies that Z 0

1

L.Ox.t/; uO .t//dt D 0;

x.t/ D x for all t 2 Œ0; 1 and lim x.tk / D lim xk .0/ D xO .0/ D x :

k!1

k!1

This contradicts (3.133). The contradiction we have reached proves that (A3) holds and completes the proof of Proposition 3.6. t u

Chapter 4

Stability Properties

In this chapter we continue to study the structure of optimal trajectories of linear control systems with autonomous nonconvex integrands on large intervals. We show that the turnpike property and the convergence of solutions in regions close to the endpoints of the time intervals, which was established in Chap. 3, are stable under small perturbations of objective functions (integrands).

4.1 Preliminaries and Main Results We use the notation, definitions, and assumptions introduced in Sects. 3.1–3.3. Recall that a0 > 0 and W Œ0; 1/ ! Œ0; 1/ is an increasing function such that lim

t!1

.t/ D 1:

We continue to study the structure of optimal trajectories of the controllable linear control system x0 D Ax C Bu; where A and B are given matrices of dimensions nn and nm, with the continuous integrand f W Rn  Rm ! R1 which satisfy assumptions (A1)–(A3) and (3.10). Denote by M the set of all borelian functions g W RnCmC1 ! R1 which satisfy g.t; x; u/  maxf .jxj/;

.juj/;

.ŒjAx C Buj  a0 jxjC /ŒjAx C Buj  a0 jxjC g  a0

(4.1)

for each .t; x; u/ 2 RnCmC1 . © Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6_4

127

128

4 Stability Properties

We equip the set M with the uniformity which is determined by the following base: E.N; ; / D f.f ; g/ 2 M  M W jf .t; x; u/  g.t; x; u/j   for each .t; x; u/ 2 RnCmC1 satisfying jxj; juj  Ng \f.f ; g/ 2 M  M W .jf .t; x; u/j C 1/.jg.t; x; u/j C 1/1 2 Œ1 ;  for each .t; x; u/ 2 RnCmC1 satisfying jxj  Ng; where N > 0,  > 0 and  > 1. Clearly, the uniform space M is Hausdorff and has a countable base. Therefore M is metrizable. It is not difficult to show that the uniform space M is complete. Denote by Mb the set of all functions g 2 M which are bounded on bounded subsets of RnCmC1 . Clearly, Mb is a closed subset of M. We consider the topological subspace Mb  M equipped with the relative topology. For each pair of numbers T1 2 R1 , T2 > T1 , each .x; u/ 2 X.A; B; T1 ; T2 / and each borelian bounded from below function g W ŒT1 ; T2   Rn  Rm set Z I g .T1 ; T2 ; x; u/ D

T2

g.t; x.t/; u.t//dt: T1

We consider the following optimal control problems I g .T1 ; T2 ; x; u/ ! min; .x; u/ 2 X.A; B; T1 ; T2 / such that x.T1 / D y; x.T2 / D z; I g .T1 ; T2 ; x; u/ ! min; .x; u/ 2 X.A; B; T1 ; T2 / such that x.T1 / D y; I g .T1 ; T2 ; x; u/ ! min; .x; u/ 2 X.A; B; T1 ; T2 /; where y; z 2 Rn , 1 > T2 > T1 > 1 and g 2 M. Let y; z 2 Rn , T1 2 R1 , T2 > T1 and g W ŒT1 ; T2   Rn  Rm be a borelian bounded from below function. Set .g; y; z; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T1 / D y; x.T2 / D zg;

(4.2)

.g; y; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T1 / D yg;

(4.3)

4.1 Preliminaries and Main Results

129

O .g; z; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T2 / D zg; .g; T1 ; T2 / D inffI .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 /g: g

(4.4) (4.5)

Recall that f W Rn  Rm ! R1 is a continuous function which satisfies (3.10) and assumptions (A1)–(A3). For each .t; x; u/ 2 RnCmC1 set F.t; x; u/ D f .x; u/:

(4.6)

In this chapter we prove the following three stability results. They show that the turnpike phenomenon, for approximate solutions on large intervals, is stable under small perturbations of the objective function (integrand) f . Theorem 4.1. Let ; M > 0. Then there exist L0  1 and ı0 > 0 such that for each L1  L0 there exists a neighborhood U of F in Mb such that the following assertion holds. Assume that T > 2L1 , g 2 U , .x; u/ 2 X.A; B; 0; T/ and that a finite sequence of q numbers fSi giD0 satisfy S0 D 0; Si C1  Si 2 ŒL0 ; L1 ; i D 0; : : : ; q  1; Sq 2 .T  L1 ; T; I g .Si ; Si C1 ; x; u/  .Si C1  Si /.f / C M for each integer i 2 Œ0; q  1, I g .Si ; SiC2 ; x; u/  .g; x.Si /; x.SiC2 /; Si ; SiC2 / C ı0 for each nonnegative integer i  q  2 and I g .Sq2 ; T; x; u/  .g; x.Sq2 /; x.T/; Sq2 ; T/ C ı0 : The there exist p1 ; p2 2 Œ0; T such that p1  p2 , p1  2L0 , p2 > T  2L1 and that jx.t/  xf j   for all t 2 Œp1 ; p2 : Moreover if jx.0/  xf j  ı, then p1 D 0 and if jx.T/  xf j  ı, then p2 D T. Theorem 4.2. Let  2 .0; 1/; M0 ; M1 > 0. Then there exist L > 0, ı 2 .0; / and a neighborhood U of F in Mb such that for each T > 2L, each g 2 U and each .x; u/ 2 X.A; B; 0; T/ which satisfies for each S 2 Œ0; T  L, I g .S; S C L; x; u/  .g; x.S/; x.S C L/; S; S C L/ C ı and satisfies at least one of the following conditions:

130

4 Stability Properties

(a) jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C M1 I (b) jx.0/j  M0 ; I g .0; T; x; u/  .g; x.0/; 0; T/ C M1 I (c) I g .0; T; x; u/  .g; 0; T/ C M1 there exist p1 2 Œ0; L, p2 2 ŒT  L; T such that jx.t/  xf j   for all t 2 Œp1 ; p2 : Moreover if jx.0/  xf j  ı, then p1 D 0 and if jx.T/  xf j  ı, then p2 D T. Denote by Card.A/ the cardinality of the set A. Theorem 4.3. Let  2 .0; 1/; M0 ; M1 > 0. Then there exist l > 0, an integer Q  1 and a neighborhood U of F in Mb such that for each T > lQ, each g 2 U and each .x; u/ 2 X.A; B; 0; T/ which satisfies and satisfies at least one of the following conditions: (a) jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C M1 I (b) jx.0/j  M0 ; I g .0; T; x; u/  .g; x.0/; 0; T/ C M1 I (c) I g .0; T; x; u/  .g; 0; T/ C M1 q

q

there exist strictly increasing sequences of numbers fai giD1 , fbi giD1  Œ0; T such that q  Q, for all i D 1; : : : ; q, 0  bi  ai  l; bi  aiC1 for all integers i satisfying 1  i < q and that q

jx.t/  xf j   for all t 2 Œ0; T n [iD1 Œai ; bi : In this chapter we also prove the following three stability results. They show that the convergence of approximate solutions on large intervals, in the regions close to the end points, is stable under small perturbations of the objective function (integrand) f . Theorem 4.4. Let L0 > 0,  2 .0; 1/; M > 0. Then there exist ı > 0, a neighborhood U of F in Mb and L1 > L0 such that for each T  L1 , each g 2 U , and each .x; u/ 2 X.A; B; 0; T/ which satisfies jx.0/j  M; I g .0; T; x; u/  .g; x.0/; 0; T/ C ı

4.1 Preliminaries and Main Results

131

there exists an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that f .x .0// D inf.f /; jx.T  t/  x .t/j   for all t 2 Œ0; L0 : Theorem 4.5. Let L0 > 0,  2 .0; 1/. Then there exist ı > 0, a neighborhood U of F in Mb and L1 > L0 such that for each T  L1 , each g 2 U , and each .x; u/ 2 X.A; B; 0; T/ which satisfies I g .0; T; x; u/  .g; 0; T/ C ı there exist an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ and an .f ; A; B/-overtaking optimal pair .Nx ; uN  / 2 X.A; B; 0; 1/ such that  f .x .0// D inf. f /; f .Nx .0// D inf.f / and for all t 2 Œ0; L0 , jx.t/  x .t/j  ; jx.T  t/  xN  .t/j  : Theorem 4.6. Let y; z 2 Rn , L0 > 0,  2 .0; 1/. Then there exist ı > 0, a neighborhood U of F in Mb and L1 > L0 such that for each T  L1 , each g 2 U , and each .x; u/ 2 X.A; B; 0; T/ which satisfies x.0/ D y; x.T/ D z; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C ı there exist an .f ; A; B/-overtaking optimal pair .x1 ; u1 / 2 X.A; B; 0; 1/ and an .f ; A; B/-overtaking optimal pair .x2 ; u2 / 2 X.A; B; 0; 1/ such that x1 .0/ D y; x2 .0/ D z and for all t 2 Œ0; L0 , jx.t/  x1 .t/j  ; jx.T  t/  x2 .t/j  : This chapter is organized as follows. Section 4.2 contains two auxiliary results and the proof of Theorem 4.1. A basic lemma for Theorem 4.2 is proved in Sect. 4.3

132

4 Stability Properties

while Theorem 4.2 itself is proved in Sect. 4.4. Section 4.5 contains the proof of Theorem 4.3. Auxiliary results for Theorem 4.4 are given in Sect. 4.6. Theorems 4.4 and 4.5 are proved in Sect. 4.7. Section 4.8 contains the proof of Theorem 4.6.

4.2 Two Auxiliary Results and Proof of Theorem 4.1 In the sequel we use the following auxiliary results. Proposition 4.7 (Proposition 2.7 of [52]). Let M1 > 0 and 0 < 0 < 1 . Then there exists M2 > 0 such that for each g 2 M, each T1 2 R1 , each T2 2 ŒT1 C 0 ; T1 C 1  and each .x; u/ 2 X.A; B; T1 ; T2 / which satisfies I g .T1 ; T2 ; x; u/  M1 the inequality jx.t/j  M2 holds for all t 2 ŒT1 ; T2 : Proposition 4.8 (Proposition 2.9 of [52]). Let g 2 M, 0 < c1 < c2 and D;  > 0. Then there exists a neighborhood U of g in M such that for each h 2 U , each T1 2 R1 , each T2 2 ŒT1 C c1 ; T1 C c2 , and each trajectory-control pair .x; u/ 2 X.A; B; T1 ; T2 / which satisfies minfI g .T1 ; T2 ; x; u/; I h .T1 ; T2 ; x; u/g  D the inequality jI g .T1 ; T2 ; x; u/  I h .T1 ; T2 ; x; u/j   holds. Proof of Theorem 4.1. For every z 2 R1 set bzc D maxfj W j is an integer and j  zg: By Proposition 3.37, there exists ı0 2 .0; 1=8/ such that the following property holds: (P1)

for each T  1, each .x; u/ 2 X.A; B; 0; T/ satisfying jx.0/  xf /j; jx.T/  xf j  4ı0 ; I .0; T; x; u/  .f ; x.0/; x.T/; T/ C 4ı0 f

we have jx.t/  xf j   for all t 2 Œ0; T:

4.2 Two Auxiliary Results and Proof of Theorem 4.1

133

By Proposition 3.35, there exists L0  5 such that the following property holds: (P2) for each T  .L0  4/, each .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  T.f / C M C 4 and each number S satisfying ŒS; S C L0  4  Œ0; T we have maxfjx.t/  xf j W t 2 ŒS; S C L0  4g  ı0 : Let L1  L0 :

(4.7)

In view of Proposition 3.33, there exists M0 > 0 such that for each   1 and each pair of points y; z 2 Rn satisfying jyj; jzj  jxf j C 4 we have .f ; y; z; /  .f / C M0 :

(4.8)

By Proposition 4.8, there exists a neighborhood U of F in M such that the following property holds: (P3) for each g 2 U , each T1 2 R1 , each T2 2 ŒT1 C 1; T1 C 4L1  and each trajectory-control pair .x; u/ 2 X.A; B; T1 ; T2 / satisfying minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  4L1 j.f /j C M C M0 C 1 we have jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  ı0 : Assume that T > 2L1 ; g 2 U ; .x; u/ 2 X.A; B; 0; T/; q fSi giD0

 Œ0; T; S0 D 0;

SiC1  Si 2 ŒL0 ; L1 ; i D 0; : : : ; q  1; Sq 2 .T  L1 ; T; I .Si ; SiC1 ; x; u/  .SiC1  Si /.f / C M g

(4.9) (4.10) (4.11) (4.12)

for each integer i 2 Œ0; q  1, I g .Si ; SiC2 ; x; u/  .g; x.Si /; x.SiC2 /; Si ; SiC2 / C ı0

(4.13)

for each nonnegative integer i  q  2 and I g .Sq2 ; T; x; u/  .g; x.Sq2 /; x.T/; Sq2 ; T/ C ı0 :

(4.14)

134

4 Stability Properties

Let i 2 Œ0; q  1 be an integer. It follows from (4.11), (4.12) and the choice of U (see property (P3)) that I f .Si ; SiC1 ; x; u/  I g .Si ; SiC1 ; x; u/ C ı0  .SiC1  Si /.f / C M C 1: By the inequality above, (4.11) and property (P2), there exists a number i such that i 2 ŒSi C 3; Si C L0 ; jx.i /  xf j  ı0 :

(4.15)

Let a nonnegative integer i  q  2. Relations (4.11) and (4.15) imply that i ; iC1 2 ŒSi C 3; SiC2 ; 3  iC1  i  2L1 :

(4.16)

In view of (4.13) and (4.16), I g .i ; iC1 ; x; u/  .g; x.i /; x.iC1 /; i ; iC1 / C ı0 :

(4.17)

Thus we have shown that there exists a strictly increasing sequence of numbers fi gkiD0 where k is a natural number such that 0  L0 ; k > T  2L1 ;

(4.18)

jx.i /  xf j  ı0 ; i D 0; : : : ; k;

(4.19)

3  iC1  i  2L1 ; i D 0; : : : ; k  1

(4.20)

and (4.17) holds for all i D 0; : : : ; k  1. Clearly, if jx.0/  xf j  ı0 , then we may assume that 0 D 0 and if jx.T/  xf j  ı0 , then we may assume that k D T. Let i 2 f0; : : : ; k  1g. By (4.19), (4.20), and the choice of M0 (see (4.8)), .f ; x.i /; x.iC1 /; iC1  i /  .f /.iC1  i / C M0 :

(4.21)

By the relation above, property (P3), (4.9), and (4.20), j .f ; x.i /; x.iC1 /; iC1  i /  .g; x.i /; x.iC1 /; i ; iC1 /j  ı0 : It follows from (4.17), (4.21), and (4.22) that I g .i ; iC1 ; x; u/  .f ; x.i /; x.iC1 /; iC1  i / C 2ı0  .f /.iC1  i / C M0 C 1: By the relation above, (P3), (4.9), and (4.10), I f .i ; iC1 ; x; u/  I g .i ; iC1 ; x; u/ C ı0  .f ; x.i /; x.iC1 /; iC1  i / C 3ı0 :

(4.22)

4.3 Basic Lemma for Theorem 4.2

135

It follows from the relation above, property (P1), (4.10), and (4.19) that for all i D 0; : : : ; k  1, jx.t/  xf j  ; t 2 Œi ; iC1 : This completes the proof of Theorem 4.1.

t u

4.3 Basic Lemma for Theorem 4.2 Lemma 4.9. Let  2 .0; 1/; M0 ; M1 > 0. Then there exist L > 0 and a neighborhood U of F in Mb such that the following assertion holds. Assume that T > L, g 2 U , 0  S1  S2  L; ŒS1 ; S2   Œ0; T

(4.23)

and .x; u/ 2 X.A; B; 0; T/ satisfies at least one of the following conditions: (a) jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C M1 I (b) jx.0/j  M0 ; I g .0; T; x; u/  .g; x.0/; 0; T/ C M1 I (c) I g .0; T; x; u/  .g; 0; T/ C M1 : Then minfjx.t/  xf j W t 2 ŒS1 ; S2 g  :

(4.24)

Proof. By Proposition 3.35 there exists an integer L0 > 0 such that the following property holds: (P4)

for each T  L0 , each trajectory-control pair .x; u/ 2 X.A; B; 0; T/ satisfying I f .0; T; x; u/  T.f / C 16.1 C a0 /

and each number S satisfying ŒS; S C L0   Œ0; T we have minfjx.t/  xf j W t 2 ŒS; S C L0 g  =2: We may assume without loss of generality that M0 > jxf j C 4

(4.25)

136

4 Stability Properties

and that M1 > supfj .f ; z1 ; z2 ; 1/j W z1 ; z2 2 Rn ; jz1 j; jz2 j  M0 g

(4.26)

(see Proposition 3.28). Proposition 3.33 implies that there exists M2 > M1 C M0 such that for each S  1 and each pair of points y; z 2 Rn satisfying jyj; jzj  M0 , .f ; y; z; S/  S.f / C M2 :

(4.27)

Fix an integer l  1 such that l > 28 C M1 C j.f /j.2L0 C M2 C 18/ C4.2L0 C 18/.1 C a0 / C 4M2 C a0 .L0 C 8/

(4.28)

and set L D .L0 C 1/l:

(4.29)

By Proposition 4.8, there exists a neighborhood U of F in Mb such that the following property holds: (P5) for each g 2 U , each T1  0, each T2 2 ŒT1 C1; T1 C4L and each trajectorycontrol pair .x; u/ 2 X.A; B; T1 ; T2 / which satisfies minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  .4L C 2/j.f /j C 4M2 C 4 C 16.1 C a0 / we have jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  .8L/1 : Assume that T > L; g 2 U ;

(4.30)

numbers S1 ; S2 satisfy (4.23) and a trajectory-control pair .x; u/ 2 X.A; B; 0; T/ satisfies at least one of the conditions (a), (b), (c). We claim that (4.24) holds. Assume the contrary. Then jx.t/  xf j >  for all t 2 ŒS1 ; S2 :

(4.31)

4.3 Basic Lemma for Theorem 4.2

137

We may assume without loss of generality that at least one of the following conditions hold: S1  1; S2  T  1I

(4.32)

S1 > 1; there is SO 1 2 ŒS1  1; S1  such that jx.SO 1 /  xf j D ; S2  T  1I (4.33) S1  1; S2 < T  1; there is SO 2 2 ŒS2 ; S2 C 1 such that jx.SO 2 /  xf j D ; (4.34) S1 > 1; S2 < T  1; there are SO 1 2 ŒS1  1; S1 ; SO 2 2 ŒS2 ; S2 C 1 such that jx.SO i /  xf j D ; i D 1; 2:

(4.35)

In view of (4.1), (4.23), and (4.29), I g .S1 ; S2 ; x; u/ D I g .S1 ; S1 C L0 b.S2  S1 /L01 c; x; u/ CI g .S1 C L0 b.S2  S1 /L01 c; S2 ; x; u/  I g .S1 ; S1 C L0 b.S2  S1 /L01 c; x; u/  L0 a0 b.S2 S1 /L01 c1

X

D

iD0

I g .S1 C iL0 ; S1 C .i C 1/L0 ; x; u/  a0 L0 : (4.36)

Let j 2 f0; : : : ; b.S2  S1 /L01 c  1g:

(4.37)

It follows from (4.37), (4.31), and property (P4) that I f .S1 C jL0 ; S1 C .j C 1/L0 ; x; u/ > L0 .f / C 16.1 C a0 /:

(4.38)

We claim that I g .S1 C jL0 ; S1 C .j C 1/L0 ; x; u/  L0 .f / C 16.1 C a0 /  1: Assume the contrary. Then I g .S1 C jL0 ; S1 C .j C 1/L0 ; x; u/ < L0 .f / C 16.1 C a0 /  1:

(4.39)

138

4 Stability Properties

Combined with (P5), (4.30), and (4.39) this implies that I f .S1 C jL0 ; S1 C .j C 1/L0 ; x; u/  1 C I g .S1 C jL0 ; S1 C .j C 1/L0 ; x; u/ < L0 .f / C 16.1 C a0 /: This contradicts (4.38). The contradiction we have reached proves (4.39). Thus (4.39) holds for all j 2 f0; : : : ; b.S2  S1 /L01 c  1g. Put z0 D x.0/ if jx.0/j  M0 ; z0 D 0 if jx.0/j > M0 ; I z1 D x.T/ if jx.T/j  M0 ; z1 D 0 if jx.T/j > M0 :

(4.40)

It is not difficult to see that there exists .x1 ; u1 / 2 X.A; B; 0; T/ such that: if (4.32) holds, then x1 .0/ D z0 ; x1 .t/ D xf ; u1 .t/ D uf ; t 2 Œ1; T  1; x1 .T/ D z1 ; I f .0; 1; x1 ; u1 /  .f ; z0 ; xf ; 1/ C 1; I f .T  1; T; x1 ; u1 /  .f ; xf ; z1 ; 1/ C 1I if (4.33) holds, then x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 Œ0; SO 1 ; x1 .t/ D xf ; u1 .t/ D uf ; t 2 ŒSO 1 C 1; T  1; I f .SO 1 ; SO 1 C 1; x1 ; u1 /  .f ; x.SO 1 /; xf ; 1/ C 1; x1 .T/ D z1 ; I f .T  1; T; x1 ; u1 /  .f ; xf ; z1 ; 1/ C 1I if (4.34) holds, then x1 .0/ D z0 ; x1 .t/ D xf ; u1 .t/ D uf ; t 2 Œ1; SO 2  1; x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 ŒSO 2 ; T; I f .0; 1; x1 ; u1 /  .f ; z0 ; xf ; 1/ C 1; I f .SO 2  1; SO 2 ; x1 ; u1 /  .f ; xf ; x.SO 2 /; 1/ C 1I if (4.45) holds, then x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 Œ0; SO 1  [ ŒSO 2 ; T; x1 .t/ D xf ; u1 .t/ D uf ; t 2 ŒSO 1 C 1; SO 2  1; I f .SO 1 ; SO 1 C 1; x1 ; u1 /  .f ; x.SO 1 /; xf ; 1/ C 1; I f .SO 2  1; SO 2 ; x1 ; u1 /  .f ; xf ; x.SO 2 /; 1/ C 1:

4.3 Basic Lemma for Theorem 4.2

139

It follows from conditions (a)–(c) and the choice of .x1 ; u1 / that I g .0; T; x; u/  I g .0; T; x1 ; u1 / C M:

(4.41)

We consider the cases (4.32)–(4.35) separately and obtain a lower bound for I g .0; T; x; u/  I g .0; T; x1 ; u1 /. Assume that (4.32) holds. In view of (4.1), (4.32), and (4.39), I g .0; T; x; u/  I g .S1 ; S2 ; x; u/  2a0 b.S2 S1 /L01 c1

  2a0  a0 L0 C

X

I g .S1 C iL0 ; S1 C .i C 1/L0 ; x; u/

iD0

  a0 .2 C L0 / C b.S2  S1 /L01 c.L0 .f / C 16.1 C a0 /  1/  .S2  S1 /.f /  L0 j.f /j  a0 .2 C L0 / C ..S2  S1 /L01  1/.16.a0 C 1/  1/:

(4.42)

By (4.25), (4.32), (4.40), and the choice of M2 (see (4.27)), I f .0; 1; x1 ; u1 /  .f / C M2 ; I f .T  1; T; x1 ; u1 /  .f / C M2 C 1: It follows from these inequalities, (4.30), and property (P5) that I g .0; 1; x1 ; u1 /; I g .T  1; T; x1 ; u1 /  .f / C M2 C 5=4:

(4.43)

In view of (4.1), Z I g .1; T  1; x1 ; u1 / 

bTc

1

g.t; xf ; uf /dt C a0 :

(4.44)

By (4.30) and property (P5), for each i 2 f1; : : : ; bT  1cg, Z

iC1

g.t; xf ; uf /dt  .f / C .8L/1 :

i

Together with (4.1) and (4.44) this implies that I g .1; T  1; x1 ; u1 /  a0 C bT  1c..f / C .8L/1 /: By (4.29), (4.30), (4.32), (4.41)–(4.43), and (4.45), M1  I g .0; T; x; u/  I g .0; T; x1 ; u1 /  T.f /  2j.f /j  L0 j.f /j  a0 .2 C L0 /

(4.45)

140

4 Stability Properties

C.16.a0 C 1/  1/.TL01  4/  2.f /  2M2  5=2  a0 bT  1c.f /  bT  1c.8L/1  T.L01 .16.a0 C 1/  1/  .8L/1 /  64.a0 C 1/ 6j.f /j  L0 j.f /j  a0 .2 C L0 /  2M2  3  a0  l  1  64.a0 C 1/  j.f /j.L0 C 6/  a0 .3 C L0 /  2M2  3: This contradicts (4.28). Thus if (4.32) holds we have reached a contradiction. Assume that (4.33) holds. In view of (4.33) and the choice of .x1 ; u1 /, I g .0; T; x; u/  I g .0; T; x1 ; u1 / D I g .SO 1 ; T; x; u/  I g .SO 1 ; T; x1 ; u1 /:

(4.46)

By (4.1), (4.33), and (4.39), I g .SO 1 ; T; x; u/  I g .S1 ; T; x; u/  a0 b.S2 S1 /L01 c1



X

I g .S1 C jL0 ; S1 C .j C 1/L0 ; x; u/  a0 .L0 C 2/

jD0

  a0 .L0 C 2/ C b.S2  S1 /L01 c.L0 .f / C 16.1 C a0 /  1/   a0 .L0 C 2/  .S2  S1 /.f /  j.f /.1 C L0 / C .b.S2  S1 /L01 c  1/.16a0 C 15/:

(4.47)

By (4.25), (4.33), (4.40), and the choice of M2 (see (4.27)) and .x1 ; u1 /, I f .SO 1 ; SO 1 C 1; x1 ; u1 /  .f / C M2 C 1; I f .T  1; T; x1 ; u1 /  .f / C M2 C 1: In view of these inequalities, (4.30) and property (P5), I g .SO 1 ; SO 1 C 1; x1 ; u1 /; I g .T  1; T; x1 ; u1 /  .f / C M2 C 3=2:

(4.48)

It follows from (4.1) that I g .SO 1 C 1; T  1; x1 ; u1 /  I g .SO 1 C 1; bT  SO 1  1c C SO 1 C 1; x1 ; u1 / C a0 D a0 C

bT SO 1 2c X

I g .SO 1 C i C 1; SO 1 C i C 2; x1 ; u1 /:

iD0

(4.49)

4.3 Basic Lemma for Theorem 4.2

141

By (4.30), (4.33), the choice of .x1 ; u1 / and property (P5), for each i 2 f0; : : : ; bT  SO 1  3cg, I g .SO 1 C i C 1; SO 1 C i C 2; x1 ; u1 /  I f .SO 1 C i C 1; SO 1 C i C 2; x1 ; u1 / C .8L/1 D .f / C .8L/1 ;

(4.50)

I g .T  2; T  1; x1 ; u1 /  .f / C .8L/1 :

(4.51)

It follows from (4.1), (4.33), (4.48)–(4.51), and the choice of .x1 ; u1 / that I g .SO 1 C 1; T  1; x1 ; u1 /  a0 C bT  SO 1  2c..f / C .8L/1 / CI g .SO 1 C 1 C bT  SO 1  2c; S1 C 2 C bT  SO 1  2c; x1 ; u1 /  a0 C bT  SO 1  2c..f / C .8L/1 / C I g .T  2; T; x1 ; u1 / C 2a0  a0 C bT  SO 1  2c..f / C .8L/1 / C M2 C .f / C 3=2 CI g .T  2; T  1; x1 ; u1 / C 2a0  3a0 C M2 C .f / C 3=2 C ..f / C .8L/1 /bT  SO 1  1c: By the relation above, (4.23), (4.29), (4.41), and (4.46)–(4.48), M1  I g .0; T; x; u/  I g .0; T; x1 ; u1 / D I g .SO 1 ; T; x; u/  I g .SO 1 ; T; x1 ; u1 /  a0 .2 C L0 / C .S2  S1 /.f /  j.f /j.1 C L0 / C..S2  S1 /L01  2/.16a0 C 15/  2M2  2j.f /j  3  3a0 .T  SO 1  2/..f / C .8L/1 /  2j.f /j  3  M2  a0 .L0 C 5/  j.f /j.11 C L0 /  3M2  6 C.S2  S1 /.L01 .16a0 C 15/  .8L/1 /  2.16a0 C 15/  1  l  a0 .L0 C 5/  j.f /j.11 C L0 /  3M2  8  2.16a0 C 15/: This contradicts (4.28). Thus if (4.33) holds we have reached a contradiction. Assume that (4.34) holds. In view of (4.34) and the choice of .x1 ; u1 /, I g .0; T; x; u/  I g .0; T; x1 ; u1 / D I g .0; SO 2 ; x; u/  I g .0; SO 2 ; x1 ; u1 /:

(4.52)

142

4 Stability Properties

By (4.1), (4.34), and (4.39), I g .0; SO 2 ; x; u/  I g .S1 ; S2 ; x; u/  2a0 b.S2 S1 /L01 c1

X

D

I g .S1 C jL0 ; S1 C .j C 1/L0 ; x; u/  a0 .L0 C 2/

jD0

 a0 .L0 C 2/ C .b.S2  S1 /L01 c  1/.L0 .f / C 16a0 C 15/  a0 .L0 C 2/  2L0 j.f /j  32a0  30 C.S2  S1 /.f / C .S2  S1 /L01 .16a0 C 15/:

(4.53)

By (4.34), (4.25), and the choice of M2 (see (4.27)) and .x1 ; u1 /, I f .0; 1; x1 ; u1 /  .f / C M2 C 1; I f .SO 2  1; SO 2 ; x1 ; u1 /  .f / C M2 C 1: In view of these inequalities, (4.30) and property (P5), I g .0; 1; x1 ; u1 /; I g .SO 2  1; SO 2 ; x1 ; u1 /  .f / C M2 C 5=4:

(4.54)

It follows from (4.1), (4.34), and the choice of .x1 ; u1 / that I g .1; SO 2  1; x1 ; u1 / D

Z Z

SO 2 1 1 bSO 2 c

 1

g.t; xf ; uf /dt

g.t; xf ; uf /dt C a0 :

(4.55)

By (4.30) and property (P5), for each j 2 f1; : : : ; bSO 2  1cg, Z

jC1

g.t; xf ; uf /dt  .f / C .8L/1 :

j

Together with (4.55) this implies that I g .1; SO 2  1; x1 ; u1 /  a0 C bSO 2  1c..f / C .8L/1 /: By the relation above, (4.23), (4.29), (4.34), (4.41), (4.52)–(4.54), and the choice of .x1 ; u1 /, M1  I g .0; T; x; u/  I g .0; T; x1 ; u1 / D I g .0; SO 2 ; x; u/  I g .0; SO 2 ; x1 ; u1 /   a0 .2 C L0 / C 32a0  30  2L0 j.f /j C .S2  S1 /.f /

4.3 Basic Lemma for Theorem 4.2

143

C.S2  S1 /L01 .16a0 C 15/  2M2  2j.f /j  3  a0 bSO 2  1c..f / C .8L/1 /  2j.f /j C .S2  S1 /.L01 .16a0 C 15/  .8L/1 /  5 a0 .L0 C 3/  32a0  30  .2 C 2L0 /j.f /j  2M2  l  6  j.f /j.4 C 2L0 /  a0 .L0 C 3/  32a0  30  2M2 : This contradicts (4.28). Thus if (4.34) holds we have reached a contradiction. Assume that (4.35) holds. By (4.35) and the choice of .x1 ; u1 /, I g .0; T; x; u/  I g .0; T; x1 ; u1 / D I g .SO 1 ; SO 2 ; x; u/  I g .SO 1 ; SO 2 ; x1 ; u1 /:

(4.56)

It follows from (4.1), (4.35), (4.39), and the choice of .x1 ; u1 / that I g .SO 1 ; SO 2 ; x; u/  I g .S1 ; S2 ; x; u/  2a0 b.S2 S1 /L01 c1



X

I g .S1 C iL0 ; S1 C .i C 1/L0 ; x; u/  a0 .L0 C 2/

iD0

 a0 .L0 C 2/ C b.S2  S1 /L01 c.L0 .f / C 16a0 C 15/:

(4.57)

By (4.35), (4.25), and the choice of M2 (see (4.27)) and .x1 ; u1 /, I f .SO 1 ; SO 1 C 1; x1 ; u1 /  .f / C M2 C 1; I f .SO 2  1; SO 2 ; x1 ; u1 /  .f / C M2 C 1: In view of these inequalities, (4.30) and property (P5), I g .SO 1 ; SO 1 C 1; x1 ; u1 /; I g .SO 2  1; SO 2 ; x1 ; u1 /  .f / C M2 C 5=4:

(4.58)

It follows from (4.1) that I .SO 1 C 1; SO 2  1; x1 ; u1 / D

Z

g

Z  Z 

SO 2 1 SO 1 C1 S2

g.t; xf ; uf /dt

g.t; xf ; uf /dt C 2a0

S1 S1 CbS2 S1 cC1 S1

g.t; xf ; uf /dt C 3a0 :

(4.59)

144

4 Stability Properties

By (4.30) and property (P5), for each i 2 f0; : : : ; bS2  S1 cg, Z

iC1

g.t; xf ; uf /dt  .f / C .8L/1 :

i

Together with (4.58) and (4.59) this implies that I g .SO 1 ; SO 2 ; x1 ; u1 /  2.f / C 2M2 C 3 C 3a0 C bS2  S1 C 1c..f / C .8L/1 /: By the relation above, (4.23), (4.29), (4.41), (4.56), and (4.57), M1  I g .SO 1 ; SO 2 ; x; u/  I g .SO 1 ; SO 2 ; x1 ; u1 /  .S2  S1 /.f /  L0 j.f /j C .S2  S1 /L01 .16a0 C 15/ 16a0  15  a0 .2 C L0 /  bS2  S1 c..f / C .8L/1 / 4j.f /j  2M2  4  3a0  l  .5 C L0 /j.f /j  19a0  19  a0 .L0 C 2/  2M2 : This contradicts (4.28). Thus if (4.35) holds we have reached a contradiction. Thus in all the cases we have reached a contradiction which proves (4.24) and Lemma 4.9 itself.

4.4 Proof of Theorem 4.2 By Proposition 3.27, there exists ı0 2 .0; / such that the following property holds: (P6) for each   1 and each trajectory-control pair .x; u/ 2 X.A; B; 0;  / satisfying jx.0/  xf j; jx. /  xf j  4ı0 ; I f .0; ; x; u/  .f ; x.0/; x. /; / C 4ı0 we have jx.t/  xf /j  ; t 2 Œ0;  : By Lemma 4.9, there exist L0 > 0 and a neighborhood U0 of F in Mb such that the following property holds: (P7) for each T > L0 , each g 2 U0 , each pair of numbers S1 ; S2 satisfying 0  S1  S2  L0 ; S2  T and each trajectory-control pair .x; u/ 2 X.A; B; 0; T/ for which at least one of the conditions (a), (b), (c) of Theorem 4.2 holds, we have minfjx.t/  xf j W t 2 ŒS1 ; S2 g  ı0 :

4.4 Proof of Theorem 4.2

145

Fix L  4.L0 C 1/; ı 2 .0; 41 ı0 /:

(4.60)

In view of assumption (A1) and Proposition 3.33, there exists a number M2 > 0 such that the following property holds: (P8) for each S 2 Œ1; L and each pair of points y; z 2 Rn satisfying jyj; jzj < 4 C jxf j we have j .f ; y; z; S/j < M2 . By Proposition 4.8, there exists a neighborhood U of F in Mb such that U  U0 and that the following property holds: (P9) for each g 2 U , each T1 2 R1 , each T2 2 ŒT1 C 1; T1 C 4L and each trajectory-control pair .x; u/ 2 X.A; B; T1 ; T2 / which satisfies minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  M2 C 4 we have jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  ı: Assume that T > 2L; g 2 U ; .x; u/ 2 X.A; B; 0; T/;

(4.61)

for each S 2 Œ0; T  L, I g .S; S C L; x; u/  .g; x.S/; x.S C L/; S; S C L/ C ı

(4.62)

and at least one of the following conditions below holds: (a) jx.0/j; jx.T/j  M0 ; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C M1 I (b) jx.0/j  M0 ; I g .0; T; x; u/  .g; x.0/; 0; T/ C M1 I (c) I g .0; T; x; u/  .g; 0; T/ C M1 : By conditions (a)–(c) and property (P7), there exists a finite strictly increasing sequence of integers Si 2 Œ0; T, i D 1; : : : ; q such that 0  S1  L0 ; Sq  T  .1 C L0 /; 1  SiC1  Si  L0 C 2; i D 1; : : : ; q  1; jx.Si /  xf j  ı0 ; i D 1; : : : ; q:

(4.63)

146

4 Stability Properties

We may assume without loss of generality that if jx.0/  xf j  ı; then S1 D 0 and if jx.T/  xf j  ı; then Sq D T: Assume that a number  2 ŒS1 ; Sq :

(4.64)

Then there exists a natural number k 2 f1; : : : ; q  1g such that Sk   < SkC1 :

(4.65)

In view of (4.60), (4.61), and (4.63), there is S 2 Œ0; T  L such that ŒSk ; SkC1   ŒS; S C L: Together with (4.62) this implies that I g .Sk ; SkC1 ; x; u/  .g; x.Sk /; x.SkC1 /; Sk ; SkC1 / C ı:

(4.66)

In view of (4.63) and property (P8), .f ; x.Sk /; x.SkC1 /; SkC1  Sk / < M2 : By the relation above, (4.61), (4.63), (4.66), and property (P9), .g; x.Sk /; x.SkC1 /; Sk ; SkC1 /  .f ; x.Sk /; x.SkC1 /; SkC1  Sk / C ı

(4.67)

and I g .Sk ; SkC1 ; x; u/  M2 C 2: It follows from the relation above, (4.61), (4.63), (4.66), (4.67), and property (P9) that I f .Sk ; SkC1 ; x; u/  I g .Sk ; SkC1 ; x; u/ C ı  .f ; x.Sk /; x.SkC1 /; SkC1  Sk / C 3ı: Combined with (4.63), (4.65), (4.64), and property (P6) this implies that jx./  u t xf j   for all number  2 ŒS1 ; Sq . Theorem 4.2 is proved.

4.5 Proof of Theorem 4.3

147

4.5 Proof of Theorem 4.3 By Proposition 3.37, there exists ı0 2 .0; / such that the following property holds: (P10)

for each   1, each .x; u/ 2 X.A; B; 0; / satisfying jx.0/  xf j; jx. /  xf j  4ı0 ; I f .0; ; x; u/  .f ; x.0/; x. /; / C 4ı0

we have jx.t/  xf j  ; t 2 Œ0;  : By Lemma 4.9, there exist a number L0 > 0 and a neighborhood U0 of F in Mb such that the following property holds: (P11) for each T > L0 , each g 2 U0 , each pair of numbers S1 ; S2 satisfying 0  S1  S2  L0 , S2  T and each trajectory-control pair .x; u/ 2 X.A; B; 0; T/ which satisfies at least one of the conditions (a), (b), and (c) of Theorem 4.3, we have minfjx.t/  xf j W t 2 ŒS1 ; S2 g  ı0 : Fix numbers l  4.L0 C 1/; ı 2 .0; 41 ı0 /

(4.68)

Q > 2ı01 M1 C 6:

(4.69)

and an integer

By (A1) and Proposition 3.33, there is M2 > 0 such that the following property holds: (P12) for each S 2 Œ1; l and each y; z 2 Rn satisfying jyj; jzj  4 C jxf j we have j .f ; y; z; S/j < M2 : By Proposition 4.8, there exists a neighborhood U of F in Mb such that U  U0 and that the following property holds: (P13) for each g 2 U , each T1 2 R1 , each T2 2 ŒT1 C 1; T1 C 4l and each trajectory-control pair .x; u/ 2 X.A; B; T1 ; T2 / which satisfies minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  M2 C 4

148

4 Stability Properties

we have jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  ı: Assume that T > lQ; g 2 U ; .x; u/ 2 X.A; B; 0; T/

(4.70)

and that at least one of the conditions (a), (b), (c) of Theorem 4.3 holds. By conditions (a)–(c), (4.70), and property (P11), there exists a finite strictly increasing sequence of numbers Si 2 Œ0; T, i D 1; : : : ; q such that 0  S1  L0 ; Sq  T  .1 C L0 /; 1  SiC1  Si  L0 C 1; i D 1; : : : ; q  1; (4.71) jx.Si /  xf j  ı0 ; i D 1; : : : ; q:

(4.72)

Define by induction a finite strictly increasing sequence of natural numbers i1 ; : : : ; ik 2 f1; : : : ; qg. Set i1 D 1:

(4.73)

Assume that p  1 is an integer and that we defined integers i1 < : : : < ip belonging to f1; : : : ; qg such that for each natural number m < p the following properties hold: (i) I g .Sim ; SimC1 ; x; u/ > .g; x.Sim /; x.SimC1 /; Sim ; SimC1 / C ı0 I

(4.74)

(ii) if imC1 > im C 1, then I g .Sim ; SimC1 1 ; x; u/  .g; x.Sim /; x.SimC1 1 /; Sim ; SimC1 1 / C ı0 :

(4.75)

(Note that by (4.73) our assumption holds for p D 1.) Let us define ipC1 . If ip D q, then our construction is completed, k D p, ik D q and for each natural number m < p D k, properties (i) and (ii) hold. Assume that ip < q. There are two cases: I g .Sip ; Sq ; x; u/  .g; x.Sip /; x.Sq /; Sip ; Sq / C ı0 I

(4.76)

I g .Sip ; Sq ; x; u/ > .g; x.Sip /; x.Sq /; Sip ; Sq / C ı0 :

(4.77)

4.5 Proof of Theorem 4.3

149

Assume that (4.76) holds. Then we set k D p C 1, ik D q the construction is completed, for each natural number m < k  1, (4.74) is true and for each natural number m < k, property (ii) holds. Assume that (4.77) holds. Then we set ipC1 D minfj > Sip W j is an integer and I g .Sip ; Sj ; x; u/ > .g; x.Sip /; x.Sj /; Sip ; Sj / C ı0 g:

(4.78)

It is easy to see that the assumption made for p also holds for p C 1. As a result we obtain a finite strictly increasing sequence of integers i1 ; : : : ; ik 2 f1; : : : ; qg such that ik D q, for all integers m satisfying 1  m < k1, (4.74) holds and each natural number m < k, property (ii) holds. It follows from conditions (a)–(c) and (4.74) that M1  I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ X  fI g .Sij ; SijC1 ; x; u/  .g; x.Sij /; x.SijC1 /; Sij ; SijC1 / W j is an integer satisfying 1  j < k  1g  ı0 .k  2/; k  ı01 M1 C 2:

(4.79)

Define A D fj 2 f1; : : : ; kg W j < k and SijC1  Sij  4.L0 C 1/g:

(4.80)

Let j 2 A:

(4.81)

Relations (4.81), (4.80), (4.71), (4.75), and property (ii) imply that I g .Sij ; SijC1 1 ; x; u/  .g; x.Sij /; x.SijC1 1 /; Sij ; SijC1 1 / C ı0 ;

(4.82)

ijC1 > ij C 3:

(4.83)

p 2 fij ; : : : ; ijC1  2g:

(4.84)

Let

In view of (4.84), fSp ; SpC1 g  fSij ; : : : ; SijC1 1 g

150

4 Stability Properties

and by (4.82), I g .Sp ; SpC1 ; x; u/  .g; x.Sp /; x.SpC1 /; Sp ; SpC1 / C ı0 :

(4.85)

It follows from the choice of M2 (see (P12)), (4.71), (4.68), and (4.72) that .f ; x.Sp /; x.SpC1 /; Sp ; SpC1 /  M2 : Combined with (4.68), (4.70), (4.71), and property (P13) this implies that .g; x.Sp /; x.SpC1 /; Sp ; SpC1 /  .f ; x.Sp /; x.SpC1 /; SpC1  Sp / C ı0  M2 C ı0 : It follows from (4.70), (4.71), (4.84), (4.85), the inequality above, and property (P13) that I f .Sp ; SpC1 ; x; u/  .f ; x.Sp /; x.SpC1 /; SpC1  Sp / C 3ı0 : By the inequality above, (4.71), (4.72), and property (P10), for all integers p 2 fij ; : : : ; ijC1  2g, jx.t/  xf j  ; t 2 ŒSp ; SpC1 : Since j is an any integer belonging to A we conclude that ft 2 Œ0; T W jx.t/  xf j > g  Œ0; S1  [ ŒSq ; T [fŒSij ; SijC1  W j 2 f1; : : : ; k  1g; SijC1  Sij < 4.L0 C 1/g [fŒSijC1 2 ; SijC1  W j 2 Ag: It is not difficult to see that the right-hand side of the inclusion is a finite union of closed intervals, by (4.68) and (4.71) their lengths does not exceed 4.L0 C 1/  l and by (4.69) and (4.79) their number does not exceed 2 C 2k  2ı01 M1 C 6 < Q: Theorem 4.3 is proved.

t u

4.6 Auxiliary Results for Theorem 4.4

151

4.6 Auxiliary Results for Theorem 4.4 Assume that S1 2 R1 , S2 > S1 and g 2 M. For each .x; u/ 2 Rn  Rm and each t 2 ŒS1 ; S2  set LS1 ;S2 .g/.t; x; u/ D g.S2  t C S1 ; x; u/;

(4.86)

for each .t; x; u/ 2 .1; S1 /  Rn  Rm set LS1 ;S2 .g/.t; x; u/ D LS1 ;S2 .g/.S1 ; x; u/ and for each .t; x; u/ 2 .S2 ; 1/  Rn  Rm set LS1 ;S2 .g/.t; x; u/ D LS1 ;S2 .g/.S2 ; x; u/: It is clear that LS1 ;S2 .g/ 2 M, if g 2 Mb , then LS1 ;S2 .g/ 2 Mb and that LS1 ;S2 is a self-mapping of M and of Mb . It is not difficult to see that the following result holds. Proposition 4.10. Let V be a neighborhood of F in M. Then there exists a neighborhood U of F in M such that LS1 ;S2 .g/ 2 V for each g 2 U , each S1 2 R1 and each S2 > S1 . Let S1 2 R1 , S2 > S1 , g 2 M and .x; u/ 2 X.A; B; S1 ; S2 / .X.A; B; S1 ; S2 / respectively/: Recall that (see (3.16)) xN .t/ D x.S2  t C S1 /; uN .t/ D u.S2  t C S1 /; t 2 ŒS1 ; S2 :

(4.87)

In view of (4.86) and (4.87), Z

S2 S1

Z LS1 ;S2 .g/.t; xN .t/; uN .t//dt D Z D

S2

g.S2  tCS1 ; x.S2 t C S1 /; u.S2  t C S1 //dt

S1 S2

g.t; x.t/; u.t//dt:

(4.88)

S1

Let T2 > T1 be a pair of real numbers, y; z 2 Rn and g 2 Mb . For each .x; u/ 2 X.A; B; T1 ; T2 / set Z I .T1 ; T2 ; x; u/ D g

T2

g.t; x.t/; u.t//dt T1

152

4 Stability Properties

and set  .g; y; z; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T1 / D y; x.T2 / D zg;

(4.89)

 .g; y; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T1 / D yg;

(4.90)

O  .g; z; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 / and x.T2 / D zg;  .g; T1 ; T2 / D inffI g .T1 ; T2 ; x; u/ W .x; u/ 2 X.A; B; T1 ; T2 /g:

(4.91) (4.92)

Relations (4.88) implies the following result. Proposition 4.11. Let S2 > S1 be real numbers, M  0, g 2 Mb and let .xi ; ui / 2 X.A; B; S1 ; S2 /, i D 1; 2. Then I g .S1 ; S2 ; x1 ; u1 /  I g .S1 ; S2 ; x2 ; u2 /  M if and only if I gN .S1 ; S2 ; xN 1 ; uN 1 /  I gN .S1 ; S2 ; xN 2 ; uN 2 /  M; where gN D LS1 ;S2 .g/. Proposition 4.11 implies the following result. Proposition 4.12. Let S2 > S1 be real numbers, M  0, g 2 Mb , gN D LS1 ;S2 .g/ and .x; u/ 2 X.A; B; S1 ; S2 /: Then the following assertions hold: I g .S1 ; S2 ; x; u/  .g; S1 ; S2 / C M if and only if I gN .S1 ; S2 ; xN ; uN /   .Ng; S1 ; S2 / C MI I g .S1 ; S2 ; x; u/  .g; x.S1 /; x.S2 /; S1 ; S2 / C M if and only if I gN .S1 ; S2 ; xN ; uN /   .Ng; xN .S1 /; xN .S2 /; S1 ; S2 / C MI I g .S1 ; S2 ; x; u/  O .g; x.S2 /; S1 ; S2 / C M if and only if I gN .S1 ; S2 ; xN ; uN /   .Ng; xN .S1 /; S1 ; S2 / C MI I g .S1 ; S2 ; x; u/  .g; x.S1 /; S1 ; S2 / C M if and only if I gN .S1 ; S2 ; xN ; uN /  O  .Ng; xN .S2 /; S1 ; S2 / C M:

4.7 Proofs of Theorems 4.4 and 4.5

153

4.7 Proofs of Theorems 4.4 and 4.5 Proof of Theorem 4.4. By Lemma 3.38 applied to the triplet .f ; A  B/ there exist ı1 2 .0; =4/ such that the following property holds: (P14)

for each .x; u/ 2 X.A; B; 0; L0 / which satisfies f .x.0//  inf.f / C ı1 ; I f .0; L0 ; x; u/  L0 .f /  f .x.0// C f .x.L0 //  ı1

there exists an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that f .x .0// D inf.f /; jx.t/  x .t/j   holds for all t 2 Œ0; L0 : In view of Propositions 3.13, 3.14, and 3.24, there exists ı2 2 .0; ı1 / such that: for each z 2 Rn satisfying jz  xf j  2ı2 , jf .z/j D jf .z/  f .xf /j  ı1 =8I

(4.93)

for each y; z 2 Rn satisfying jy  xf j  2ı2 ; jz  xf j  2ı2 we have j .f ; y; z; 1/  .f /j  ı1 =8:

(4.94)

By Theorem 4.2, there exist l0 > 0, ı3 2 .0; ı2 =8/ and a neighborhood U1 of F in Mb such that the following property holds: (P15) for each T > 2l0 , each g 2 U1 and each .x; u/ 2 X.A; B; 0; T/

154

4 Stability Properties

such that jx.0/j  M; I .0; T; x; u/  .g; x.0/; 0; T/ C ı3 g

we have jx.t/  xf j  ı2 for all t 2 Œl0 ; T  l0 :

(4.95)

By Theorem 3.8, there exists an .f ; A; B/-overtaking optimal pair .Nx ; uN  / 2 X.A; B; 0; 1/ such that f .Nx .0// D inf.f /:

(4.96)

Assumption (A3) implies that there exists l1 > 0 such that jNx .t/  xf j  ı2 for all t  l1 :

(4.97)

By Proposition 4.8, there exists a neighborhood U  U1 of F in Mb such that the following property holds: (P16) for each g 2 U , each T1 2 R1 , each T2 2 ŒT1 C 1; T1 C 2L0 C 2l0 C 2l1 C 4 and each trajectory-control pair .x; u/ 2 X.A; B; T1 ; T2 / which satisfies minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  .j.f /j C 2/.2L0 C 2l0 C 2l1 C 4/ C 2 C jf .Nx .0//j the inequality jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  ı3 =8 holds. Choose ı > 0 and L1 > 0 such that ı  ı3 =4; L1  2L0 C 2l0 C 2l1 C 4:

(4.98)

T  L1 ; g 2 U ; .x; u/ 2 X.A; B; 0; T/;

(4.99)

jx.0/j  M; I .0; T; x; u/  .g; x.0/; 0; T/ C ı:

(4.100)

Assume that

g

It follows from property (P15) and (4.98)–(4.100) that relation (4.95) is true. By (4.98) and (4.99), ŒT  l0  l1  L0  4; T  l0  l1  L0   Œl0 ; T  l0  l1  L0 :

(4.101)

4.7 Proofs of Theorems 4.4 and 4.5

155

Relations (4.95) and (4.101) imply that jx.t/  xf j  ı2 for all t 2 ŒT  l0  l1  L0  4; T  l0  l1  L0 :

(4.102)

By Proposition 3.27, there exists a trajectory-control pair .x1 ; u1 / 2 X.A; B; 0; T/ such that x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 Œ0; T  l0  l1  L0  4; x1 .t/ D xN  .T  t/; u1 .t/ D uN  .T  t/; t 2 ŒT  l0  l1  L0  3; T; I f .T  l0  l1  L0  4; T  l0  l1  L0  3; x1 ; u1 / D .f ; x.T  l0  l1  L0  4/; xN  .l0 C l1 C L0 C 3//:

(4.103)

It follows from (4.103) and (4.100) that  ı  I g .0; T; x1 ; u1 /  I g .0; T; x; u/ D I g .T  l0  l1  L0  4; T  l0  l1  L0  3; x1 ; u1 / CI g .T  l0  l1  L0  3; T; x1 ; u1 / I g .T  l0  l1  L0  4; T  l0  l1  L0  3; x; u/ I g .T  l0  l1  L0  3; T; x; u/:

(4.104)

It follows from (4.94), (4.97), (4.102), and (4.103) that I f .T  l0  l1  L0  4; T  l0  l1  L0  3; x1 ; u1 /  .f / C ı1 =8: Together with (4.99) and property (P16) this implies that I g .T  l0  l1  L0  4; T  l0  l1  L0  3; x1 ; u1 /  .f / C ı1 =8 C ı3 =8:

(4.105)

By (4.102) and the choice of ı2 (see (4.94)), I f .T  l0  l1  L0  4; T  l0  l1  L0  3; x; u/  .f /  ı1 =8: If I g .T  l0  l1  L0  4; T  l0  l1  L0  3; x; u/ < .f /  ı1 =2; then by property (P16) and (4.99),

(4.106)

156

4 Stability Properties

I f .T  l0  l1  L0  4; T  l0  l1  L0  3; x; u/ < .f /  ı1 =2 C ı3 =8 < .f /  3ı1 =8 and this contradicts (4.106). Hence I g .T  l0  l1  L0  4; T  l0  l1  L0  3; x; u/  .f /  ı1 =2: Together with (4.104) and (4.105) this implies that I g .T  l0  l1  L0  3; T; x1 ; u1 /  I g .T  l0  l1  L0  3; T; x; u/  ı  ı1 =8  ı3 =8  ı1 =2:

(4.107)

Since .Nx ; uN  / is an .fN ; A; B/-overtaking optimal pair it follows from (3.18), (4.103), and Proposition 3.12 that I f .T  l0  l1  L0  3; T; x1 ; u1 / D I f .0; l0 C l1 C L0 C 3; xN  ; uN  / D .f /.l0 C l1 C L0 C 3/ C f .Nx .0// f .Nx .l0 C l1 C L0 C 3//:

(4.108)

In view of (4.97) and the choice of ı2 (see (4.93), jf .Nx .l0 C l1 C L0 C 3//j  ı1 =8: Combined with (4.108) this implies that I f .T  l0  l1  L0  3; T; x1 ; u1 /  f .Nx .0// C .f /.l0 C l1 C L0 C 3/ C ı1 =8: (4.109) By (P16), (4.99), and (4.109), I g .T  l0  l1  L0  3; T; x1 ; u1 /  f .Nx .0// C .f /.l0 C l1 C L0 C 3/ C ı1 =8 C ı3 =8:

(4.110)

It follows from (4.107) and (4.110) that I g .T  l0  l1  L0  3; T; x; u/  f .Nx .0// C .f /.l0 C l1 C L0 C 3/ C ı C 3ı1 =4 C ı3 =4: (4.111)

4.7 Proofs of Theorems 4.4 and 4.5

157

By property (P16), (4.99), and (4.111), I f .T  l0  l1  L0  3; T; x; u/  f .Nx .0// C .f /.l0 C l1 C L0 C 3/ C ı C 3ı1 =4 C 3ı3 =8: (4.112) Define xQ .t/ D x.T  t/; uQ .t/ D u.T  t/; t 2 Œ0; T:

(4.113)

It is clear that .Qx; uQ / 2 X.A; B; 0; T/ and by (4.112), (4.113), I f .0; l0 C l1 C L0 C 3; xQ ; uQ / D I f .T  l0  l1  L0  3; T; x; u/  f .Nx .0// C .f /.l0 C l1 C L0 C 3/ Cı C 3ı1 =4 C 3ı3 =8:

(4.114)

In view of (4.102) and (4.113), jQx.l0 C l1 C L0 C 3/  xf j  ı2 : By the relation above and the choice of ı2 (see (4.93)), jf .Qx.l0 C l1 C L0 C 3//j  ı1 =8:

(4.115)

By Propositions 3.11 and 3.12, (4.114), (4.115), and .f ; A; B/-overtaking optimality of .Nx ; uN  /,  f .Qx.0//  f .Nx .0// C I f .0; L0 ; xQ ; uQ /  L0 .f /  f .Qx.0// C f .Qx.L0 //  f .Qx.0//  f .Nx .0// C I f .0; l0 C l1 C L0 C 3; xQ ; uQ /  .f /.l0 C l1 C L0 C 3/  f .Qx.0// C f .Qx.l0 C l1 C L0 C 3//  f .Qx.0//  f .Nx .0// C .f /.l0 C l1 C L0 C 3/ C ı C 3ı3 =8 C 3ı1 =4 C f .Nx .0//  .f /.l0 C l1 C L0 C 3/  f .Qx.0// C ı1 =8  ı C 3ı1 =4 C ı3 =2  ı1 : It follows from the relation above, (4.96), and Proposition 3.11 that f .Qx.0//  inf.f / C ı1 ; I f .0; L0 ; xQ ; uQ /  L0 .f / C f .Qx.0// C f .Qx.L0 //  ı1 :

158

4 Stability Properties

By the inequalities above and property (P14), there exists an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that f .x .0// D inf.f /; jQx.t/  x .t/j   holds for all t 2 Œ0; L0 : Combined with (4.113) this implies that jx.T  t/  xN  .t/j   holds for all t 2 Œ0; L0 : t u

Theorem 4.4 is proved. Proof of Theorem 4.5. Theorems 4.2 and 4.4 imply the following result.

Theorem 4.13. Let L0 > 0,  2 .0; 1/. Then there exist ı > 0, a neighborhood U of F in Mb and L1 > L0 such that for each T  L1 , each g 2 U , and each .x; u/ 2 X.A; B; 0; T/ which satisfies I g .0; T; x; u/  .g; 0; T/ C ı there exists an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that  f .x .0// D inf.f / and for all t 2 Œ0; L0  jx.T  t/  x .t/j  : Theorem 4.5 follows from Propositions 4.10, 4.12, and Theorem 4.13.

4.8 Proof of Theorem 4.6 Theorem 4.6 follows from Propositions 4.10 and 4.12 and the next result. Theorem 4.14. Let y 2 Rn , L0 > 0,  2 .0; 1/; M > 0. Then there exist ı > 0, a neighborhood U of F in Mb and L1 > L0 such that for each T  L1 , each g 2 U

4.8 Proof of Theorem 4.6

159

and each .x; u/ 2 X.A; B; 0; T/ which satisfies x.0/ D y; jx.T/j  M; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C ı there exists an .f ; A; B/-overtaking optimal pair .Qx; uQ / 2 X.A; B; 0; 1/ such that xQ .0/ D y and for all t 2 Œ0; L0 , jx.t/  xQ .t/j  : Proof. Denote by d the metric of the space M. Assume that Theorem 4.14 does not hold. Then there exist a sequence fık g1 kD1  .0; 1/ such that ık < 4k ; k D 1; 2; : : : ;

(4.116)

Tk > L0 C 2k; k D 1; 2; : : :

(4.117)

a sequence

a sequence fgk g1 kD1  Mb such that d.gk ; f /  k1 ; k D 1; 2; : : :

(4.118)

and a sequence .xk ; uk / 2 X.A; B; 0; Tk /, k D 1; 2; : : : such that for each natural number k, xk .0/ D y; jxk .Tk /j  M; I .0; Tk ; xk ; uk /  .gk ; xk .0/; xk .Tk /; 0; Tk / C ık gk

(4.119) (4.120)

and that the following property holds: (i) for each .f ; A; B/-overtaking optimal pair .; v/ 2 X.A; B; 0; 1/ satisfying .0/ D y we have maxfjxk .t/  .t/j W t 2 Œ0; L0 g > :

(4.121)

In view of (4.116)–(4.120) and Theorem 4.2, the following property holds: (ii) for each  > 0 there exists l > 0 and an integer k  1 and each integer k  k C 2l , jxk .t/  xf j  ; t 2 Œl ; Tk  l :

160

4 Stability Properties

Let S  l1 . By Proposition 3.28 and (4.119), the sequence f .f ; y; xk .S/; 0; S/g1 kDk1 is bounded. Proposition 4.8 and (4.118) imply that the sequence f .gk ; y; xk .S/; 0; S/g1 kDk1 is bounded and lim j .gk ; y; xk .S/; 0; S/  .f ; y; xk .S/; S/j D 0:

k!1

(4.122)

Together with (4.116) and (4.120) this implies that the sequence fI gk .0; S; xk ; uk /g1 kDk1 is bounded. Combined with Proposition 4.8 and (4.118) this implies that the sequence fI f .0; S; xk ; uk /g1 kDk1 is bounded and that lim jI gk .0; S; xk ; uk /  I f .0; S; xk ; uk /j D 0:

k!1

(4.123)

In view of (4.116), (4.120), (4.122) and (4.123), lim jI f .0; S; xk ; uk /  .f ; y; xk .S/; 0; S/j D 0:

k!1

(4.124)

Since the sequence f .f ; y; xk .S/; 0; S/g1 kDk1 is bounded, the sequence fI f .0; S; xk ; uk /g1 kDk1 is bounded too for any integer S  l1 . By Proposition 3.25, extracting subsequences and re-indexing we may assume without loss of generality that there exists .x; u/ 2 X.A; B; 0; 1/ such that for each integer S  l1 , xk .t/ ! x.t/ as k ! 1 uniformly on Œ0; S;

(4.125)

I f .0; S; x; u/  lim inf I f .0; S; xk ; uk /:

(4.126)

k!1

By (4.125) and property (ii) the following property holds: (iii) Let  > 0 and let l > 0 be as guaranteed by property (ii). Then jx.t/  xf j   for all t  l :

4.8 Proof of Theorem 4.6

161

We show that the pair .x; u/ is .f ; A; B/-overtaking optimal. It view of property (iii) and Theorem 3.10, it is sufficient to show that the pair .x; u/ is .f ; A; B/-minimal. Assume the contrary. Then there exist  > 0, Q0 > l1 and .Qx; uQ / 2 X.A; B; 0; Q0 / such that I f .0; Q0 ; xQ ; uQ / < I f .0; Q0 ; x; u/  ; xQ .0/ D y; xQ .Q0 / D x.Q0 /:

(4.127)

By Proposition 3.29, there exists  > 0 such that for each z1 ; z2 2 Rn satisfying jzi  xf j  2 , i D 1; 2, we have j .f ; z1 ; z2 ; 1/  .f /j  =16:

(4.128)

Let l > 0, k  1 be as guaranteed by property (ii). By (4.124) and (4.126) there exists a natural number k > 2k C 2k1 C 2l C 2l1 C 2Q0

(4.129)

such that jI f .0; Q0 C l C 1; xk ; uk /  .f ; y; xk .Q0 C l C 1/; Q0 C l C 1/j  =16; (4.130) I f .0; Q0 C l ; x; u/  I f .0; Q0 C l ; xk ; uk / C =16:

(4.131)

It follows from (4.120) and property (ii) that jxk .t/  xf j  ; t 2 Œl ; Tk  l :

(4.132)

Property (iii) implies that jx.t/  xf j  ; t 2 Œl ; 1/:

(4.133)

Œl C Q0 ; l C Q0 C 1  Œl ; Tk  l :

(4.134)

By (4.117) and (4.129),

Proposition 3.27 and (4.127) imply that there exists .Ox; uO / 2 X.A; B; 0; Q0 C l C 1/

162

4 Stability Properties

such that xO .t/ D xQ .t/; uO .t/ D uQ .t/; t 2 Œ0; Q0 ; xO .t/ D x.t/; uO .t/ D u.t/; t 2 ŒQ0 ; Q0 C l ; xO .Q0 C l C 1/ D xk .Q0 C l C 1/; I f .Q0 C l ; Q0 C l C 1; xO ; uO / D .f ; x.Q0 C l /; xk .Q0 C l C 1/; 1/:

(4.135)

It follows from (4.127), (4.129), (4.130), and (4.135) that I f .0; Q0 C l C 1; xO ; uO /  .f ; y; xk .Q0 C l C 1/; Q0 C l C 1/  I f .0; Q0 C l C 1; xk ; uk /  =16:

(4.136)

By (4.127), (4.129), (4.131)–(4.133), (4.135), and the choice of  (see (4.128)), I f .0; Q0 C l C 1; xO ; uO /  I f .0; Q0 C l C 1; xk ; uk / I f .0; Q0 C l ; xO ; uO / C I f .Q0 C l ; Q0 C l C 1; xO ; uO / I f .0; Q0 C l ; xk ; uk /  I f .Q0 C l ; Q0 C l C 1; xk ; uk /  I f .0; Q0 C l ; xO ; uO /  I f .0; Q0 C l ; xk ; uk / C .f ; x.Q0 C l /; xk .Q0 C l C 1/; 1/  .f ; xk .Q0 C l /; xk .Q0 C l C 1/; 1/  I f .0; Q0 C l ; xO ; uO /  I f .0; Q0 C l ; xk ; uk / C =8 D I f .0; Q0 C l ; xO ; uO /  I f .0; Q0 C l ; x; u/ CI f .0; Q0 C l ; x; u/  I f .0; Q0 C l ; xk ; uk / C =8  I f .0; Q0 ; xQ ; uQ /  I f .0; Q0 ; x; u/ C 3=16 <  C 3=16:

This contradicts (4.136). The contradiction we have reached proves that the pair .x; u/ is .f ; A; B/-overtaking optimal. By (4.125), there exists a natural number p0 such that for all integers p  p0 , jxp .t/  x.t/j  =2; t 2 Œ0; L0 : This contradicts property (i). The contradiction we have reached proves Theorem 4.14.

Chapter 5

Linear Control Systems with Discounting

In this chapter we extend the stability results obtained in Chap. 4 to the linear control systems with discounting. We establish the turnpike property of approximate solutions on large intervals and the convergence of approximate solutions in regions close to the endpoints of the time intervals. We also show that this convergence as well as the turnpike property are stable under small perturbations of objective functions (integrands).

5.1 Preliminaries and Main Results We use the notation, definitions, and assumptions introduced in Sects. 3.1–3.3 of Chap. 3 and in Sects. 4.1 and 4.6 of Chap. 4.. Recall that a0 > 0 and W Œ0; 1/ ! Œ0; 1/ is an increasing function such that lim

t!1

.t/ D 1:

We continue to study the structure of optimal trajectories of the controllable linear control system x0 D Ax C Bu where A and B are given matrices of the dimensions n  n and n  m and with the continuous integrand f W Rn  Rm ! R1 which satisfies assumptions (A1)–(A3) and (3.10).

© Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6_5

163

164

5 Linear Control Systems with Discounting

We continue to consider the set M of all borelian functions g W RnCmC1 ! R1 which satisfy the growth condition g.t; x; u/  maxf .jxj/;

.juj/;

.ŒjAx C Buj  a0 jxjC /ŒjAx C Buj  a0 jxjC g  a0

(5.1)

for each .t; x; u/ 2 RnCmC1 : The set M is equipped with the uniformity which is determined by the following base: E.N; ; / D f.f ; g/ 2 M  M W jf .t; x; u/  g.t; x; u/j   for each .t; x; u/ 2 RnCmC1 satisfying jxj; juj  Ng \f.f ; g/ 2 M  M W .jf .t; x; u/j C 1/.jg.t; x; u/j C 1/1 2 Œ1 ;  for each .t; x; u/ 2 RnCmC1 satisfying jxj  Ng;

(5.2)

where N > 0,  > 0 and  > 1. Remind that the uniform space M is metrizable and complete. Denote by Mb the set of all functions g 2 M which are bounded on bounded subsets of RnCmC1 . Note that Mb is a closed subset of M. We consider the topological subspace Mb  M equipped with the relative topology. Recall that f W Rn  Rm ! R1 is a continuous function which satisfies assumptions (3.10), (A1)–(A3). For each .t; x; u/ 2 RnCmC1 set F.t; x; u/ D f .x; u/:

(5.3)

For each g 2 M, each borelian function ˛ W Œ0; 1/ ! R1 define a function ˛g W RnCmC1 ! R1 by ˛g.t; x; u/ D ˛.t/g.t; x; u/; .t; x; u/ 2 Œ0; 1/  Rn  Rm ; ˛g.t; x; u/ D 0; .t; x; u/ 2 .1; 0/  Rn  Rm

(5.4)

and for each T1 2 R1 , each T2 > T1 and each .x; u/ 2 X.A; B; T1 ; T2 / set I ˛g .T1 ; T2 ; x; u/ D

Z

T2

˛.t/g.t; x.t/; u.t//dt:

T1

In this chapter we prove the following two stability results. The first of them (Theorem 5.1) establishes the stability of the turnpike property of approximate solutions on large intervals for our linear control systems with discounting. The second one (Theorem 5.2) establishes the stability of the convergence of approximate solutions in regions close to the endpoints of the time intervals in the case of discounting.

5.1 Preliminaries and Main Results

165

Theorem 5.1. Let ; M > 0. Then there exist L > 0, ı 2 .0; /,  2 .0; 1/, and a neighborhood U of F in Mb such that for each T > 2L, each g 2 U , each borelian function ˛ W Œ0; 1/ ! .0; 1 which satisfies ˛.t1 /˛.t2 /1   for each t1 ; t2 2 Œ0; T satisfying jt1  t2 j  L and each .x; u/ 2 X.A; B; 0; T/ which satisfies at least one of the following conditions: (a) jx.0/j  M; I ˛g .0; T; x; u/  .˛g; x.0/; 0; T/ C ı inff˛.t/ W t 2 Œ0; TgI (b) jx.0/j; jx.T/j  M; I ˛g .0; T; x; u/  .˛g; x.0/; x.T/; 0; T/ C ı inff˛.t/ W t 2 Œ0; Tg there exist p1 2 Œ0; L, p2 2 ŒT  L; T such that jx.t/  xf j   for all t 2 Œp1 ; p2 : Moreover if jx.0/  xf j  ı, then p1 D 0 and if jx.T/  xf j  ı, then p2 D T. Theorem 5.2. Let 0  1,  > 0; M > 0. Then there exist ı 2 .0; /,  2 .0; 1/, T0  0 and a neighborhood U of F in Mb such that for each T  T0 , each g 2 U , each borelian function ˛ W Œ0; 1/ ! .0; 1 which satisfies ˛.t1 /˛.t2 /1   for each t1 ; t2 2 Œ0; T satisfying jt1  t2 j  T0 and each .x; u/ 2 X.A; B; 0; T/ which satisfies jx.0/j  M; I ˛g .0; T; x; u/  .˛g; x.0/; 0; T/ C ı inff˛.t/ W t 2 Œ0; Tg there exists an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that f .x .0// D inf.f /; jx.T  t/  x .t/j   for all t 2 Œ0; 0 : This chapter is organized as follows. Section 5.2 contains auxiliary results. Theorem 5.1 is proved in Sect. 5.3. Section 5.4 contains the proof of Theorem 5.2.

166

5 Linear Control Systems with Discounting

5.2 Auxiliary Results for Theorems 5.1 and 5.2 Let g 2 M, L > 0,  2 .0; 1/ and let a borelian function ˛ W Œ0; 1/ ! .0; 1 satisfy ˛.t1 /˛.t2 /1   for each t1 ; t2  0 satisfying jt1  t2 j  L: Let T  T2  T1  0; T2  T1  L: Define a function g˛;T1 ;T2 W RnCmC1 ! R1 as follows. For every .x; u/ 2 Rn  Rm set g˛;T1 ;T2 .t; x; u/ D ˛.t/g.t; x; u/ inff˛.s/ W s 2 ŒT1 ; T2 g1 C a0 .1  1/

(5.5)

for all t 2 ŒT1 ; T2 ; g˛;T1 ;T2 .t; x; u/ D g˛;T1 ;T2 .T1 ; x; u/ for all t 2 .1; T1 /; g˛;T1 ;T2 .t; x; u/ D g˛;T1 ;T2 .T2 ; x; u/ for all t 2 .T2 ; 1/:

(5.6) (5.7)

Clearly, the function g˛;T1 ;T2 is borelian and belongs to the space M and if g 2 Mb , then g˛;T1 ;T2 2 Mb . Proposition 5.3. Let M;  > 0. Then there exists M1 > 0 such that for each yQ ; zQ 2 Rn satisfying jQyj; jQzj  M there is .x; u/ 2 X.A; B; 0;  / such that x.0/ D yQ , x./ D zQ and that jx.t/j; ju.t/j  M1 for all t 2 Œ0;  : Proof. It follows from Proposition 3.26 that for each yQ , zQ 2 Rn there exists a unique solution x./, y./ of the following system .x0 ; y0 /t D C..x; y/t /

(5.8)

with the boundary constraints x.0/ D yQ , x. / D zQ and C..x; y/t / D .Ax C BBt y; x  At y/t : (Here Bt denotes the transpose of B.) For any initial value .x0 ; y0 / 2 Rn  Rn there exists a unique solution of (5.8) x./; y./ and .x.s/; y.s//t D esC .x0 ; y0 /t ; s 2 R1 :

5.2 Auxiliary Results for Theorems 5.1 and 5.2

167

Clearly, for each yQ ; zQ 2 Rn there exists a unique of vector D.Qy; zQ/ 2 Rn such that the function .x.s/; y.s// D .esC ..Qy; D.Qy; zQ//t //t ; s 2 R1 satisfies (5.8) with the boundary constraints x.0/ D yQ ; x. / D zQ: It is easy to see that D W Rn  Rn ! Rn is a linear operator. Now the validity of the proposition follows from the boundedness of the set fjesC ..Qy; D.Qy; zQ//t /j W yQ ; zQ 2 Rn ; jQyj; jQzj  M; s 2 Œ0;  g: We suppose that the sum over empty set is zero. Lemma 5.4. Let M > 0. Then there exist L > 0,  2 .0; 1/, M0 > 0, and a neighborhood U of F in Mb such that for each T > L, each g 2 U , each borelian function ˛ W Œ0; 1/ ! .0; 1 which satisfies ˛.t1 /˛.t2 /1   for each t1 ; t2 2 Œ0; T satisfying jt1  t2 j  L

(5.9)

and each .x; u/ 2 X.A; B; 0; T/ which satisfies at least one of the following conditions: (a) jx.0/j  M; I ˛g .0; T; x; u/  .˛g; x.0/; 0; T/ C inff˛.t/ W t 2 Œ0; TgI (b) jx.0/j; jx.T/j  M; ˛g

I .0; T; x; u/  .˛g; x.0/; x.T/; 0; T/ C inff˛.t/ W t 2 Œ0; Tg the inequality jx.t/j  M0 holds for all t 2 Œ0; T. Proof. We may assume that M > jxf j C juf j C 4:

(5.10)

By Proposition 4.7 there exists M1 > M C 2 such that the following property holds: (i) for each g 2 M, each T1 2 R1 , each T2 2 ŒT1 C 81 ; T1 C 8 and each .x; u/ 2 X.A; B; T1 ; T2 / which satisfies I g .T1 ; T2 ; x; u/  64.jf .xf ; uf /j C 4 C a0 / we have jx.t/j  M1 for all t 2 ŒT1 ; T2 :

168

5 Linear Control Systems with Discounting

By Proposition 5.3 there exists M2 > M1 such that the following property holds: (ii) for each z1 ; z2 2 Rn satisfying jzi j  M1 ; i D 1; 2 there is .; / 2 X.A; B; 0; 1/ such that .0/ D z1 ; .1/ D z2 ; j.t/j; j .t/j  M2 for all t 2 Œ0; 1: In view of the continuity of the function f , there exists M3 > 16 such that jf .z1 ; z2 /j  M3 for each .z1 ; z2 / 2 RnCm satisfying jzi j  M2 C 2; i D 1; 2: (5.11) Choose numbers M ; L;  such that M > 64.a0 C 2/ C 8  128M3 ; L > 8M3 C 8;  2 .21 ; 1/:

(5.12) (5.13)

By Proposition 4.7 there exists M0 > 0 such that the following property holds: (iii) for each g 2 M, each T1 2 R1 , each T2 2 ŒT1 C 81 ; T1 C 8 and each .x; u/ 2 X.A; B; T1 ; T2 / which satisfies I g .T1 ; T2 ; x; u/  M2 C M1 C M C 64.a0 C 2/ C M we have jx.t/j  M0 for all t 2 ŒT1 ; T2 : There exists a neighborhood U of F in Mb such that U  fg 2 Mb W jF.t; z; v/  g.t; z; v/j  1 for all .t; z; v/ 2 RnCmC1 satisfying jzj; jvj  M2 C 1g:

(5.14)

Assume that T  L; g 2 U ;

(5.15)

a borelian function ˛ W Œ0; 1/ ! .0; 1 satisfies (5.9), .x; u/ 2 X.A; B; 0; T/ satisfies at least one of the conditions (a), (b). We show that jx.t/j  M0 ; t 2 Œ0; T:

(5.16)

5.2 Auxiliary Results for Theorems 5.1 and 5.2

169

Assume the contrary. Then there is a number 0 such that 0 2 .0; T/; jx.0 /j > M0 :

(5.17)

Clearly, there is 1 2 .0; T  2/ such that 1 < 0 < 1 C 2:

(5.18)

S1 D maxft 2 Œ0; 1  W jx.t/  M1 g

(5.19)

S2 D minft 2 Œ1 C 2; T W jx.t/  M1 g

(5.20)

Set

and set

if ft 2 Œ1 C 2; T W jx.t/  M1 g 6D ;; otherwise put S2 D T. Recall that for any z 2 R1 , Œz D minfi 2 R1 W i is an integer and i  zg: Set k1 D Œ1  S1 ; k2 D ŒS2  1  2:

(5.21)

By (5.10), (5.19), (5.20), property (ii), and conditions (a) and (b), there exists .Qx; uQ / 2 X.A; B; 0; T/ such that xQ .t/ D x.t/; uQ .t/ D u.t/; t 2 Œ0; S1  [ .ŒS2 ; T n fTg/;

(5.22)

xQ .S1 C 1/ D xf ; jQx.t/j; jQu.t/j  M2 ; t 2 ŒS1 ; S1 C 1;

(5.23)

xQ .S2  1/ D xf ; jQx.t/j; jQu.t/j  M2 ; t 2 ŒS2  1; S2 ;

(5.24)

xQ .S2 / D x.S2 / if jx.S2 /j  M1 ; otherwise xQ .S2 / D 0;

(5.25)

xQ .t/ D xf ; uQ .t/ D uf ; t 2 ŒS1 C 1; S2  1:

(5.26)

By (5.22)–(5.26) and conditions (a), (b), inff˛.t/ W t 2 Œ0; Tg  I ˛g .0; T; x; u/  I ˛g .0; T; xQ ; uQ /  I ˛g .S1 ; S2 ; x; u/  I ˛g .S1 ; S2 ; xQ ; uQ /:

(5.27)

170

5 Linear Control Systems with Discounting

It is easy to see that I ˛g .S1 ; S2 ; x; u/ D I ˛g .S1 C k1 ; S2  k2 ; x; u/ X C fI ˛g .S1 C i; S1 C i C 1; x; u/ W i is an integer, 0  i < k1 g X C fI ˛g .S2  i  1; S2  i; x; u/ W i is an integer, 0  i < k2 g (5.28) and that I ˛g .S1 ; S2 ; xQ ; uQ / D I ˛g .S1 ; S1 C 1; xQ ; uQ / C I ˛g .S2  1; S2 ; xQ ; uQ / C I ˛g .S1 C 1; S2  1; xQ ; uQ /: (5.29) We estimate I ˛g .S1 C k1 ; S2  k2 ; x; u/. In view of (5.21), .S2  k2 /  .S1 C k1 /  2; .S2  k2 /  .S1 C k1 /  .S2  .S2  1  3//  .S1 C 1  S1  1/  4:

(5.30) (5.31)

It follows from (5.17), (5.18), and (5.21) that 0 2 ŒS1 C k1 ; S2  k2 ; jx.0 /j > M0 :

(5.32)

gO D g˛;S1 Ck1 ;S2 k2 :

(5.33)

Let

Clearly, g 2 Mb . In view of (5.30)–(5.33) and property (iii), I gO .S1 C k1 ; S2  k2 ; x; u/ > M :

(5.34)

By (5.5), (5.33), and (5.34), M < I gO .S1 C k1 ; S2  k2 ; x; u/ D .S2  S1  k2  k1 /a0 .1  1/ C inff˛.t/ W t 2 ŒS1 C k1 ; S2  k2 g1 I ˛g .S1 C k1 ; S2  k2 ; x; u/ and I ˛g .S1 C k1 ; S2  k2 ; x; u/ > .M  .S2  S1  k2  k1 /a0 .1  1// inff˛.t/ W t 2 ŒS1 C k1 ; S2  k2 g: (5.35)

5.2 Auxiliary Results for Theorems 5.1 and 5.2

171

If k1 > 0, then by (5.19) and (5.21), jx.S1 C i/j > M1 ; i D 1; : : : ; S1 C k1 :

(5.36)

Assume that k1 > 0; i 2 f0; : : : ; k1  1g: We estimate I ˛g .S1 C i; S1 C i C 1; x; u/: Let gO D g˛;S1 Ci;S1 CiC1 :

(5.37)

In view of (5.36), (5.37), and property (i), I gO .S1 C i; S1 C i C 1; x; u/ > 64.jf .xf ; uf /j C 4 C 2a0 /:

(5.38)

By (5.5), (5.33), and (5.38), 64.jf .xf ; uf /j C 4 C 2a0 / < I gO .S1 C i; S1 C i C 1; x; u/ D a0 .1  1/ C inff˛.t/ W t 2 ŒS1 C i; S1 C i C 1g1 I ˛g .S1 C i; S1 C i C 1; x; u/

and I ˛g .S1 C i; S1 C i C 1; x; u/ > .64.jf .xf ; uf /j C 4 C 2a0 /  a0 .1  1// inff˛.t/ W t 2 ŒS1 C i; S1 C i C 1g; i D 0; : : : ; k1  1: Combined with (5.9) and (5.13) this implies that I ˛g .S1 ; S1 C k1 ; x; u/  .64.jf .xf ; uf /j C 4 C a0 //

X f˛.S1 C i/ W i is an integer, 0  i < k1 g: (5.39)

If k2 > 0, then by (5.20) and (5.21), jx.S2  i/j > M1 ; i D 1; : : : ; k2 : Assume that k2 > 0; i 2 f0; : : : ; k2  1g:

(5.40)

172

5 Linear Control Systems with Discounting

We estimate I ˛g .S2  i  1; S2  i; x; u/: Let gO D g˛;S2 i1;S2 i :

(5.41)

In view of (5.40), (5.41), and property (i), I gO .S2  i  1; S2  i; x; u/ > 64.jf .xf ; uf /j C 4 C 2a0 /: Together with (5.5) this implies that 64.jf .xf ; uf /j C 4 C 2a0 / < I gO .S2  i  1; S2  i; x; u/ D a0 .1  1/ C inff˛.t/ W t 2 ŒS2  i  1; S2  ig1 I ˛g .S2  i  1; S2  i; x; u/

and I ˛g .S2  i  1; S2  i; x; u/ > .64.jf .xf ; uf /j C 4 C 2a0 /  a0 .1  1// inff˛.t/ W t 2 ŒS2  i  1; S2  ig; i D 0; : : : ; k2  1: Combined with (5.9) and (5.13) this implies that I ˛g .S2  k2 ; S2 ; x; u/  64.jf .xf ; uf /j C 4 C a0 /

X f˛.S2  i/ W i is an integer, 0  i < k2 g:

(5.42)

It follows from (5.11), (5.13), (5.28), (5.31), (5.35), (5.39), and (5.42) that I ˛g .S1 ; S2 ; x; u/  21 M inff˛.t/ W t 2 ŒS1 C k1 ; S2  k2 g X C 64.jf .xf ; uf /j C 4 C a0 / f˛.S1 C i/ W i is an integer, 0  i < k1 g X f˛.S2  i/ W i is an integer, 0  i < k2 g: (5.43) C 64.jf .xf ; uf /j C 4 C a0 /

By (5.14), (5.15), (5.23)–(5.25), and the choice of M3 (see (5.11)), for all t 2 ŒS1 ; S1 C 1 [ ŒS2  1; S2 , jg.t; xQ .t/; uQ .t//j  1 C jF.t; xQ .t/; uQ .t//j  M3 C 1:

(5.44)

5.2 Auxiliary Results for Theorems 5.1 and 5.2

173

In view of (5.10), (5.14), (5.15), and (5.26), for all t 2 ŒS1 C 1; S2  1, jg.t; xQ .t/; uQ .t//j  1 C jF.t; xQ .t/; uQ .t//j  jf .xf ; uf /j C 1:

(5.45)

Relations (5.9), (5.12), (5.31), (5.44), and (5.45) imply that I ˛g .S1 ; S2 ; xQ ; uQ / D I ˛g .S1 ; S1 C 1; xQ ; uQ / C I ˛g .S2  1; S2 ; xQ ; uQ / C

Z

S2 1

S1 C1

˛.t/g.t; xQ .t/; uQ .t//dt

 .M3 C 1/˛.S1 /1 C .M3 C 1/˛.S2  1/1 Z S2 1 C ˛.t/jf .xf ; uf j C 1/dt S1 C1

 .M3 C 1/1 .˛.S1 / C ˛.S2  1// X 1 C .jf .xf ; uf j C 1/ f˛.S1 C i/ W i is an integer, 0  i < k1 g C

X

 f˛.S2  i/ W i is an integer, 0  i < k2 g

C.S2  S1  k2  k1 /1 inff˛.t/ W t 2 ŒS1 C k1 ; S2  k2 g: By (5.9), (5.13), (5.27), (5.31), (5.43), and the relation above, inff˛.t/ W t 2 Œ0; Tg  I ˛g .S1 ; S2 ; x; u/  I ˛g .S1 ; S2 ; xQ; uQ /  inff˛.t/ W t 2 ŒS1 C k1 ; S2  k2 g.21 M   41 / C16.jf .xf ; uf /j C 4 C a0 /  2.jf .xf ; uf /j C 1/ X  f˛.S1 C i/ W i is an integer, 0  i < k1 g C

X

 f˛.S2  i/ W i is an integer, 0  i < k2 g

.M3 C 1/1 .˛.S1 / C ˛.S2  1//  81 M inff˛.t/ W t 2 ŒS1 C k1 ; S2  k2 g X C12 f˛.S1 C i/ W i is an integer, 0  i < k1 g X C12 f˛.S2  i/ W i is an integer, 0  i < k2 g 4M3 .˛.S1 / C ˛.S2  1//

174

5 Linear Control Systems with Discounting  161 M ˛.S1 C k1 / X C12 f˛.S1 C i/ W i is an integer, 0  i < k1 g  4M3 ˛.S1 / C161 M ˛.S2  k2 / X C12 f˛.S2  i/ W i is an integer, 0  i < k2 g 4M3 ˛.S2  1/:

(5.46)

There are two cases: k1  L; k1 > L. If k1  L, then in view of (5.9), ˛.S1 C k1 /  ˛.S1 / and by (5.11) and (5.13), 161 M ˛.S1 C k1 /  4M3 ˛.S1 /  161 M 2 ˛.S1 /  4M3 ˛.S1 /  ˛.S1 /.641 M  M3 /  1281 M ˛.S1 / > 8M3 ˛.S1 /: If k1 > L, then (5.9), (5.12), and (5.13) imply that 12

X

f˛.S1 C i/ W i is an integer, 0  i < k1 g  4M3 ˛.S1 / X  12 f˛.S1 C i/ W i D 0; : : : ; Lg  4M3 ˛.S1 /  12L˛.S1 /  4M3 ˛.S1 /  .6L  4M3 /˛.S1 /  8˛.S1 /:

In the both cases 161 M ˛.S1 C k1 / X C12 f˛.S1 C i/ W i is an integer, 0  i < k1 g  4M3 ˛.S1 /  8˛.S1 /:

(5.47)

There are two cases: k2  L; k2 > L. If k2  L, then in view of (5.9), ˛.S2  k2 /  ˛.S2  1/ and by (5.11)–(5.13), 161 M ˛.S2  k2 /  4M3 ˛.S2  1/  161 M 2 ˛.S2  1/  4M3 ˛.S2  1/  ˛.S2  1/.641 M  4M3 /  2561 M ˛.S2  1/ > 8˛.S2  1/:

5.3 Proof of Theorem 5.1

175

If k2 > L, then (5.9), (5.12), and (5.13) imply that 12

X

f˛.S2  i/ W i is an integer, 0  i < k2 g  4M3 ˛.S2  1/ X  12 f˛.S2  i/ W i D 0; : : : ; Lg  4M3 ˛.S2  1/  12L˛.S2  1/  4M3 ˛.S2  1/  .6L  4M3 /˛.S2  1/  8˛.S2  1/:

In the both cases 161 M ˛.S2  k2 / X C12 f˛.S2  i/ W i is an integer, 0  i < k2 g 4M3 ˛.S2  1/  8˛.S2  1/:

(5.48)

By (5.46)–(5.48), inff˛.t/ W t 2 Œ0; Tg  8˛.S1 / C 8˛.S2  1/; a contradiction. The contradiction we have reached proves Lemma 5.4 itself.

5.3 Proof of Theorem 5.1 By Lemma 5.4, there exist L0 > 0, 0 2 .0; 1/, M0 > 0 and a neighborhood U0 of F in Mb such that the following property holds: (i) for each T > L0 , each g 2 U0 , each borelian function ˛ W Œ0; 1/ ! .0; 1 which satisfies ˛.t1 /˛.t2 /1  0 for each t1 ; t2 2 Œ0; T satisfying jt1  t2 j  L0 and each .x; u/ 2 X.A; B; 0; T/ which satisfies at least one of the conditions (a), (b) of Theorem 5.1 we have jx.t/j  M0 ; t 2 Œ0; T: By Proposition 3.33, there exists 0 > 0 such that .f ; y; z; T/  T.f / C 0 for all y; z 2 Rn satisfying jyj; jzj  M0 C 1 and all T  1.

(5.49)

176

5 Linear Control Systems with Discounting

By Theorem 4.1, there exist L1  1 and ı1 2 .0; 1=8/ and a neighborhood U1  U0 of F in Mb such that the following property holds: (ii) for each T > 4L1 , each g 2 U1 , each .x; u/ 2 X.A; B; 0; T/ and each finite q sequence of numbers fSi giD0 which satisfies S0 D 0; SiC1  Si 2 ŒL1 ; 2L1 ; i D 0; : : : ; q  1; Sq 2 ŒT  2L1 ; T; I g .Si ; SiC1 ; x; u/  .SiC1  Si /.f / C 0 C 8 for each integer i 2 Œ0; q  1, I g .Si ; SiC2 ; x; u/  .g; x.Si /; x.SiC2 /; Si ; SiC2 / C ı1 for each nonnegative integer i  q  2 and I g .Sq2 ; T; x; u/  .g; x.Sq2 /; x.T/; Sq2 ; T/ C ı1 ; there exist p1 ; p2 2 Œ0; T such that p1  p2 , p1  2L1 , p2 > T  4L1 and that jx.t/  xf j   for all t 2 Œp1 ; p2 : Moreover if jx.0/  xf j  ı1 , then p1 D 0 and if jx.T/  xf j  ı1 , then p2 D T. Choose positive numbers L; ı such that L > 8.L0 C L1 C 1/; ı < minf; ı1 ; 1g=64:

(5.50)

By Proposition 4.8, there exists a neighborhood U2  U1 of F in Mb such that the following property holds: (iii) for each g 2 U2 , each T1 2 R1 , each T2 2 ŒT1 C 81 ; T1 C 8L and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  160 C 16 C 16Lj.f /j C 16L we have jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  ı1 =16: By (5.2) there exist 1 2 .0; minf; 1g/, 1 > 1 and N1 > 1 such that fg 2 Mb W .F; g/ 2 E.N1 ; 1 ; 1 /g  U2 :

(5.51)

5.3 Proof of Theorem 5.1

177

Set 1 D supfjf .z; /j W .z; / 2 RnCm ; jzj; jj  N1 g;

(5.52)

U D fg 2 Mb W .g; F/ 2 E.N1 ; 1 =4; 21 .1 C 1 //g:

(5.53)

Choose a number  such that 0 <  < 1; .1  1/.1 C 1 C a0 / < 1 =4; 1

2 .1 C 1 /Œ

1

1

C ja0  1j.

 1/ < 1 :

(5.54) (5.55)

Assume that T > 2L; g 2 U ;

(5.56)

a borelian function ˛ W Œ0; 1/ ! .0; 1 satisfies ˛.t1 /˛.t2 /1   for each t1 ; t2 2 Œ0; T satisfying jt1  t2 j  L;

(5.57)

.x; u/ 2 X.A; B; 0; T/ and at least one of the conditions (a), (b) of Theorem 5.1 holds. By property (i), (5.50), (5.56), and (5.57), jx.t/j  M0 ; t 2 Œ0; T:

(5.58)

Set q D bT=L1 c D maxfi 2 R1 W i  T=L1 is an integerg;

(5.59)

S0 D 0; Si D iL1 ; i D 0; : : : ; bT=L1 c  1; Sq D T:

(5.60)

Let j; i 2 f0; : : : ; q  1g; j  i 2 f1; 2g:

(5.61)

We estimate I f .Si ; Sj ; x; u/. By conditions (a) and (b), I ˛g .Si ; Sj ; x; u/  .˛g; x.Si /; x.Sj /; Si ; Sj / C ı inff˛.t/ W t 2 Œ0; Tg:

(5.62)

Set gO D g˛;Si ;Sj : Clearly, gO 2 Mb . In view of (5.5)–(5.7) and (5.63), for each .y; v/ 2 X.A; B; Si ; Sj /

(5.63)

178

5 Linear Control Systems with Discounting

we have I gO .Si ; Sj ; y; v/ D a0 .1  1/.Sj  Si / C inff˛.t/ W t 2 ŒSi ; Sj g1 I ˛g .Si ; Sj ; y; v/:

(5.64)

It follows from (5.62) and (5.64) that I gO .Si ; Sj ; x; u/  .Og; x.Si /; x.Sj /; Si ; Sj / C ı:

(5.65)

.F; gO / 2 E.N1 ; 1 ; 1 /; gO 2 U2 :

(5.66)

We show that

Let .t; z; / 2 RnCmC1 satisfy jzj; jj  N1 :

(5.67)

jOg.t; z; /  f .z; /j  1 :

(5.68)

We claim that

In view of (5.5)–(5.7) and (5.63), we may assume without loss of generality that t 2 ŒSi ; Sj :

(5.69)

jg.t; z; /  f .z; /j  1 =4:

(5.70)

By (5.53), (5.56), and (5.67),

Relations (5.52), (5.67), and (5.70) imply that jg.t; z; /j  1 C 1 =4:

(5.71)

It follows from (5.5), (5.50), (5.54), (5.57), (5.61), (5.63), (5.69), and (5.71) that jOg.t; z; /  g.t; z; /j  a0 .1  1/ C jg.t; z; /jj˛.t/ inff˛.s/ W s 2 ŒSi ; Sj g1  1j  a0 .1  1/ C .1 C 1/.1  1/ < 1 =4: Together with (5.70) this implies (5.68). Let .t; z; / 2 RnCmC1 satisfy jzj  N1 :

(5.72)

5.3 Proof of Theorem 5.1

179

We claim that .jOg.t; z; /j C 1/.jf .z; /j C 1/1 2 Π1 ; 1 :

(5.73)

In view of (5.5)–(5.7) and (5.63), we may assume without loss of generality that t 2 ŒSi ; Sj : By (5.53), (5.56), and (5.72), .jf .z; /j C 1/.jg.t; z; /j C 1/1 2 Œ.21 .1 C 1//1 ; 21 .1 C 1/:

(5.74)

In view of (5.5), (5.63), and the inclusion t 2 ŒSi ; Sj , gO .t; z; / D ˛.t/ inff˛.s/ W s 2 ŒSi ; Sj g1 g.t; z; / C a0 .1  1/:

(5.75)

The inclusion t 2 ŒSi ; Sj , (5.5), (5.50), (5.57), and (5.63) imply that jOg.t; z; /j C 1  ˛.t/ inff˛.s/ W s 2 ŒSi ; Sj g1 jg.t; z; /j C a0 .1  1/ C 1  1 jg.t; z; /j C a0 .1  1/ C 1  1 .jg.t; z; /j C 1/ C .a0  1/.1  1/ and .jOg.t; z; /j C 1/.jg.t; z; /j C 1/1  1 C ja0  1j.1  1/:

(5.76)

It follows from (5.5), (5.63), and the inclusion t 2 ŒSi ; Sj  that g.t; z; / D .Og.t; z; /  a0 .1  1//˛.t/1 inff˛.s/ W s 2 ŒSi ; Sj g: Together with (5.56) and (5.57) this implies that jg.t; z; /j C 1  jOg.t; z; /j1 C a0 .1  1/1 C 1  1 .jOg.t; z; /j C 1/ C .1  1/.a0  1/ and .jg.t; z; /j C 1/.jOg.t; z; /j C 1/1  1 C ja0  1j.1  1/: By (5.55), (5.56), (5.74), (5.76), (5.77), and the inclusion t 2 ŒSi ; Sj , .jf .z; /j C 1/.jOg.t; z; /j C 1/1 ; .jOg.t; z; /j C 1/.jf .z; /j C 1/1  21 .1 C 1 /.1 C ja0  1j.1  1// < 1 and (5.73) holds.

(5.77)

180

5 Linear Control Systems with Discounting

It follows from (5.51), (5.68), and (5.73) that relation (5.66) holds. In view of (5.49), (5.58), (5.59), and (5.61), .f ; x.Si /; x.Sj /; Sj  Si /  .Sj  Si /.f / C 0 :

(5.78)

Property (iii), (5.66), and (5.78) imply that .Og; x.Si /; x.Sj /; Si ; Sj /  .Sj  Si /.f / C 0 C ı1 =16

(5.79)

j .f ; x.Si /; x.Sj /; Sj  Si /  .Og; x.Si /; x.Sj /; Si ; Sj /j  ı1 =16:

(5.80)

and

By (5.65), (5.79), and property (iii), jI gO .Si ; Sj ; x; u/  I f .Si ; Sj ; x; u/j  ı1 =16:

(5.81)

It follows from (5.65), (5.78), (5.80), and (5.81) that I f .Si ; Sj ; x; u/  .f ; x.Si /; x.Sj /; Sj  Si / C ı1 =4  .Sj  Si /.f / C 0 C 1:

(5.82)

Therefore (5.82) holds for all integers i; j satisfying (5.61). Together with (5.50), (5.56), (5.59), (5.60), (5.82), and property (ii) this implies that there exist p1 ; p2 2 Œ0; T such that p1  p2 , p1  2L1 , p2 > T  4L1 and that jx.t/  xf j   for all t 2 Œp1 ; p2 : Moreover if jx.0/  xf j  ı1 , then p1 D 0 and if jx.T/  xf j  ı1 , then p2 D T. Theorem 5.1 is proved. u t

5.4 Proof of Theorem 5.2 By Lemma 3.38 applied to the triplet .f ; A  B/ there exist ı1 2 .0; =4/ such that the following property holds: (P1)

for each .x; u/ 2 X.A; B; 0; 0 / which satisfies f .x.0//  inf.f / C 64ı1 ;

5.4 Proof of Theorem 5.2

181

I f .0; 0 ; x; u/  0 .f /  f .x.0// C f .x.0 //  64ı1 there exists an .f ; A; B/-overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that f .x .0// D inf.f /; jx.t/  x .t/j   for all t 2 Œ0; 0 : In view of Propositions 3.13, 3.14, and 3.29, there exists ı2 2 .0; ı1 / such that: for each z 2 Rn satisfying jz  xf j  2ı2 , jf .z/j D jf .z/  f .xf /j  ı1 =8I

(5.83)

for each y; z 2 Rn satisfying jy  xf j  2ı2 ; jz  xf j  2ı2 we have j .f ; y; z; 1/  .f /j  ı1 =8:

(5.84)

By Theorem 5.1 and Lemma 5.4, there exist l0 > 0, ı3 2 .0; ı2 =8/, 0 2 .0; 1/ and a neighborhood U0 of F in Mb such that the following property holds: (P2) for each T > 2l0 , each g 2 U0 , each borelian function ˛ W Œ0; 1/ ! .0; 1 which satisfies ˛.t1 /˛.t2 /1  0 for each t1 ; t2 2 Œ0; T satisfying jt1  t2 j  l0 and each .x; u/ 2 X.A; B; 0; T/ which satisfies jx.0/j  M; I ˛g .0; T; x; u/  .˛g; x.0/; 0; T/ C ı3 inff˛.t/ W t 2 Œ0; Tg we have jx.t/  xf j  ı2 for all t 2 Œl0 ; T  l0 ; jx.t/j  M0 for all t 2 Œ0; T: By Theorem 3.8, there exists an .f ; A; B/-overtaking optimal pair .Nx ; uN  / 2 X.A; B; 0; 1/

182

5 Linear Control Systems with Discounting

such that f .Nx .0// D inf.f /:

(5.85)

(A3) implies that there exists l1 > 0 such that jNx .t/  xf j  ı2 for all t  l1 :

(5.86)

L > 8.l0 C l1 C 0 / C 8;

(5.87)

T0  2l1 C 2l0 C 20 C 8 C 8L:

(5.88)

Choose numbers

By Proposition 4.8, there exists a neighborhood U1  U0 of F in Mb such that the following property holds: (P3) for each g 2 U1 , each T1 2 R1 , each T2 2 ŒT1 C 81 ; T1 C 8T0  and each .x; u/ 2 X.A; B; T1 ; T2 / satisfying minfI f .T1 ; T2 ; x; u/; I g .T1 ; T2 ; x; u/g  .j.f /j C 2/8T0 C 8 C jf .Nx .0//j we have jI f .T1 ; T2 ; x; u/  I g .T1 ; T2 ; x; u/j  ı3 =8: By (5.2), there exist 1 2 .0; minf; 1g/, 1 > 1 and N1 > 1 such that fg 2 Mb W .F; g/ 2 E.N1 ; 1 ; 1 /g  U1 :

(5.89)

Set U D fg 2 Mb W .g; F/ 2 E.N1 ; 1 =4; 21 .1 C 1 //g; 1 D supfjf .z; /j W .z; / 2 R

nCm

; jzj; jj  N1 g:

(5.90) (5.91)

Choose a number  2 .0; 1/ such that  > 8=9; 2T0 a0 .1  1/ < ı3 =8;

(5.92)

0 <  < 1; 16.2  1/.1 C 1 C j.f /j C a0 / < 1 =4;

(5.93)

21 .1 C 1 /Œ1 C ja0  1j.1  1/ C ja0 1  1j.1  1/ < 1

(5.94)

5.4 Proof of Theorem 5.2

183

and a positive number ı < minfı1 ; ı2 ; ı3 g=8:

(5.95)

T  T0 ; g 2 U ;

(5.96)

Assume that

a borelian function ˛ W Œ0; 1/ ! .0; 1 satisfies ˛.t1 /˛.t2 /1   for each t1 ; t2 2 Œ0; T satisfying jt1  t2 j  T0

(5.97)

and .x; u/ 2 X.A; B; 0; T/ satisfies jx.0/j  M; I ˛g .0; T; x; u/  .˛g; x.0/; 0; T/ C ı inff˛.t/ W t 2 Œ0; Tg:

(5.98)

By property (P2), (5.88), (5.93), and (5.96)–(5.98), jx.t/j  M0 for all t 2 Œ0; T;

(5.99)

jx.t/  xf j  ı2 for all t 2 Œl0 ; T  l0 :

(5.100)

In view of (5.88), ŒT  l0  l1  0  4; T  l0  l1  0   Œl0 ; T  l0  l1  0 :

(5.101)

Relations (5.100) and (5.101) imply that jx.t/  xf j  ı2 for all t 2 ŒT  l0  l1  0  4; T  l0  l1  0 :

(5.102)

Proposition 3.27 implies that there exists .x1 ; u1 / 2 X.A; B; 0; T/ such that x1 .t/ D x.t/; u1 .t/ D u.t/; t 2 Œ0; T  l0  l1  0  4;

(5.103)

x1 .t/ D xN  .T  t/; u1 .t/ D uN  .T  t/; t 2 ŒT  l0  l1  0  3; T;

(5.104)

I f .T  l0  l1  0  4; T  l0  l1  0  3; x1 ; u1 / D .f ; x.T  l0  l1  0  4/; xN  .l0 C l1 C 0 C 3//:

(5.105)

184

5 Linear Control Systems with Discounting

By (5.103)–(5.105) and (5.98),  ı inff˛.t/ W t 2 Œ0; Tg  I ˛g .0; T; x1 ; u1 /  I ˛g .0; T; x; u/ D I ˛g .T  l0  l1  0  4; T  l0  l1  0  3; x1 ; u1 / C I ˛g .T  l0  l1  0  3; T; x1 ; u1 /  I ˛g .T  l0  l1  0  4; T  l0  l1  0  3; x; u/  I ˛g .T  l0  l1  0  3; T; x; u/:

(5.106)

Let S1 ; S2 2 Œ0; T satisfy S2  S1 2 Œ81 ; L:

(5.107)

.F; g˛;S1 ;S2 / 2 E.N1 ; 1 ; 1 /; g˛;S1 ;S2 2 U1 :

(5.108)

gO D g˛;S1 ;S2 :

(5.109)

jzj; jj  N1 :

(5.110)

jOg.t; z; /  f .z; /j  1 :

(5.111)

We show that

Set

Let .t; z; / 2 RnCmC1 satisfy

We claim that

In view of (5.5)–(5.7) and (5.109), we may assume without loss of generality that t 2 ŒS1 ; S2 : By (5.90), (5.96), and (5.110), jg.t; z; /  f .z; /j  1 =4:

(5.112)

Relations (5.91), (5.110), and (5.112) imply that jg.t; z; /j  1 C 1 =4:

(5.113)

It follows from (5.5), (5.88), (5.93), (5.97), (5.107), (5.109), and (5.113) that jOg.t; z; /  g.t; z; /j  a0 .1  1/ C jg.t; z; /jj˛.t/ inff˛.s/ W s 2 ŒS1 ; S2 g1  1j  a0 .1  1/ C .1 C 1/.1  1/ < 1 =4:

5.4 Proof of Theorem 5.2

185

Together with (5.112) this implies (5.111). Let .t; z; / 2 RnCmC1 satisfy jzj  N1 :

(5.114)

We claim that .jOg.t; z; /j C 1/.jf .z; /j C 1/1 2 Π1 ; 1 :

(5.115)

In view of (5.5)–(5.7) and (5.109), we may assume without loss of generality that t 2 ŒS1 ; S2 : By (5.90), (5.96), and (5.114), .jf .z; /j C 1/.jg.t; z; /j C 1/1 2 Œ.21 .1 C 1//1 ; 21 .1 C 1/:

(5.116)

In view of (5.5), (5.88), (5.97), (5.107), and (5.109), gO .t; z; / D ˛.t/ inff˛.s/ W s 2 ŒS1 ; S2 g1 g.t; z; / C a0 .1  1/;

(5.117)

jOg.t; z; /j C 1  ˛.t/ inff˛.s/ W s 2 ŒS1 ; S2 g1 jg.t; z; /j C a0 .1  1/ C 1  1 jg.t; z; /j C a0 .1  1/ C 1  1 .jg.t; z; /j C 1/ C .a0  1/.1  1/ and .jOg.t; z; /j C 1/.jg.t; z; /j C 1/1  1 C ja0  1j.1  1/: It follows from (5.117) that g.t; z; / D .Og.t; z; /  a0 .1  1//˛.t/1 inff˛.s/ W s 2 ŒS1 ; S2 g: Together with (5.88), (5.97), and (5.107) this implies that jg.t; z; /j C 1  jOg.t; z; /j1 C a0 .1  1/1 C 1  1 .jOg.t; z; /j C 1/ C .1  1/ja0 1  1j: By the relation above, .jg.t; z; /j C 1/.jOg.t; z; /j C 1/1  1 C ja0 1  1j.1  1/:

(5.118)

186

5 Linear Control Systems with Discounting

Together with (5.94), (5.116), and (5.118) this implies that .jf .z; /j C 1/.jOg.t; z; /j C 1/1 ; .jOg.t; z; /j C 1/.jf .z; /j C 1/1  21 .1 C 1 /.1 C ja0  1j.1  1/ C ja0 =  1j.1  1// < 1 and (5.115) holds. Thus the following property holds: (P4)

Relation (5.108) holds for all S1 ; S2 2 Œ0; T satisfying S2  S1 2 Œ1=8; L.

It follows from (5.84), (5.86), (5.102), and (5.105) that I f .T  l0  l1  0  4; T  l0  l1  0  3; x1 ; u1 /  .f / C ı1 =8:

(5.119)

Set gO D g˛;Tl0 l1 0 4;Tl0 l1 0 3 :

(5.120)

By (5.108), (5.119), (5.120), and properties (P3) and (P4), I gO .T  l0  l1  0  4; T  l0  l1  0  3; x1 ; u1 /  .f / C ı1 =8 C ı3 =8:

(5.121)

It follows from (5.102) and the choice of ı2 (see (5.84)) that I f .T  l0  l1  0  4; T  l0  l1  0  3; x; u/  .f /  ı1 =8:

(5.122)

If I gO .T  l0  l1  0  4; T  l0  l1  0  3; x; u/ < .f /  ı1 =2; then by (5.108), (5.120), and properties (P3) and (P4), I f .T  l0  l1  0  4; T  l0  l1  0  3; x; u/ < .f /  ı1 =2 C ı3 =8 < .f /  3ı1 =8 and this contradicts (5.122). Thus I gO .T  l0  l1  0  4; T  l0  l1  0  3; x; u/  .f /  ı1 =2: By (5.5), (5.120), and (5.121), I ˛g .T  l0  l1  0  4; T  l0  l1  0  3; x1 ; u1 /  inff˛.s/ W s 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g

(5.123)

5.4 Proof of Theorem 5.2

187

 I gO .T  l0  l1  0  4; T  l0  l1  0  3; x1 ; u1 /  ..f / C ı1 =8 C ı3 =8/ inff˛.s/ W s 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g:

(5.124) In view of (5.5), (5.120), and (5.123), I ˛g .T  l0  l1  0  4; T  l0  l1  0  3; x; u/  inff˛.s/ W s 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g  .I gO .T  l0  l1  0  4; T  l0  l1  0  3; x; u/  a0 .1  1//  ..f /  ı1 =2  a0 .1  1// inff˛.s/ W s 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g:

(5.125) It follows from (5.92), (5.124), and (5.125) that I ˛g .T  l0  l1  0  4; T  l0  l1  0  3; x1 ; u1 /  I ˛g .T  l0  l1  0  4; T  l0  l1  0  3; x; u/  inff˛.s/ W s 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g.5ı1 =8 C ı3 =8 C a0 .1  1//  inff˛.s/ W s 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g.5ı1 =8 C ı3 =4/:

(5.126) Relations (5.106) and (5.126) imply that I ˛g .T  l0  l1  0  3; T; x1 ; u1 /  I ˛g .T  l0  l1  0  3; T; x; u/   ı inff˛.t/ W t 2 Œ0; Tg  inff˛.t/ W t 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g.5ı1 =8 C ı3 =4/   inff˛.t/ W t 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g.5ı1 =8 C 3ı3 =8/:

(5.127) N Since .Nx ; uN  / is an .f ; A; B/-overtaking optimal pair it follows from (5.104) and Proposition 3.12 that I f .T  l0  l1  0  3; T; x1 ; u1 / D I f .0; l0 C l1 C 0 C 3; xN  ; uN  / D .f /.l0 C l1 C 0 C 3/ C f .Nx .0// f .Nx .l0 C l1 C 0 C 3//:

(5.128)

By (5.86) and the choice of ı2 (see (5.83)), f .Nx .l0 C l1 C 0 C 3//  ı1 =8: By (5.128) and the relation above, I f .T  l0  l1  0  3; T; x1 ; u1 /  f .Nx .0// C .f /.l0 C l1 C 0 C 3/ C ı1 =8: (5.129)

188

5 Linear Control Systems with Discounting

Set gO D g˛;Tl0 l1 0 3;T :

(5.130)

By (5.87), (5.129), (5.130), and properties (P3) and (P4), I gO .T  l0  l1  0  3; T; x1 ; u1 /  f .Nx .0// C .f /.l0 C l1 C 0 C 3/ C ı1 =8 C ı3 =8: (5.131) It follows from (5.5), (5.130), and (5.131) that I ˛g .T  l0  l1  0  3; T; x1 ; u1 /  inff˛.s/ W s 2 ŒT  l0  l1  0  3; TgI gO .T  l0  l1  0  3; T; x1 ; u1 /  inff˛.s/ W s 2 ŒT  l0  l1  0  3; Tg  .f .Nx .0// C .f /.l0 C l1 C 0 C 3/ C 3ı1 =8/:

(5.132)

In view of (5.127) and (5.132), I ˛g .T  l0  l1  0  3; T; x; u/  inff˛.t/ W t 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g.5ı1 =8 C 3ı3 =8/ C inff˛.s/ W s 2 ŒT  l0  l1  0  3; Tg  .f .Nx .0// C .f /.l0 C l1 C 0 C 3/ C 3ı1 =8/:

(5.133)

Relations (5.5), (5.87), (5.92), (5.97), (5.130), and (5.133) imply that I gO .T  l0  l1  0  3; T; x; u/  inff˛.s/ W s 2 ŒT  l0  l1  0  3; Tg1  I ˛g .T  l0  l1  0  3; T; x; u/ C a0 .1  1/.l0 C l1 C 0 C 3/  inff˛.s/ W s 2 ŒT  l0  l1  0  3; Tg1  inff˛.t/ W t 2 ŒT  l0  l1  0  4; T  l0  l1  0  3g.5ı1 =8 C 3ı3 =8/ C f .Nx .0// C .f /.l0 C l1 C 0 C 3/ C 3ı1 =8 C a0 .1  1/.l0 C l1 C 0 C 3/  1 .5ı1 =8 C 3ı3 =8/ C a0 .1  1/.l0 C l1 C 0 C 3/ C f .Nx .0// C .f /.l0 C l1 C 0 C 3/ C 3ı1 =8  f .Nx .0// C .f /.l0 C l1 C 0 C 3/ C 9.5ı1 C 3ı3 / C 3ı1 =8 C ı3 =8: (5.134)

5.4 Proof of Theorem 5.2

189

It follows from properties (P3) and (P4), (5.88), (5.130), and (5.134) that I f .T  l0  l1  0  3; T; x; u/  f .Nx .0// C .f /.l0 C l1 C 0 C 3/ C9.5ı1 C 3ı3 / C 3ı1 =8 C ı3 =2:

(5.135)

xQ .t/ D x.T  t/; uQ .t/ D u.T  t/; t 2 Œ0; T:

(5.136)

Set

Clearly, .Qx; uQ / 2 X.A; B; 0; T/ and in view of (5.135), (5.136), I f .0; l0 C l1 C 0 C 3; xQ; uQ / D I f .T  l0  l1  0  3; T; x; u/ f   .Nx .0// C .f /.l0 C l1 C 0 C 3/ C 60ı1 :

(5.137)

It follows from (5.102) and (5.136) that jQx.l0 C l1 C 0 C 3/  xf j  ı2 : By the relation above and the choice of ı2 (see (5.83)), jf .Qx.l0 C l1 C 0 C 3//j  ı1 =8:

(5.138)

By Propositions 3.11, (5.137), and (5.138), f .Qx.0//  f .Nx .0// C I f .0; 0 ; xQ ; uQ /  0 .f /  f .Qx.0// C f .Qx.0 //  f .Qx.0//  f .Nx .0// C I f .0; l0 C l1 C 0 C 3; xQ ; uQ / .f /.l0 C l1 C 0 C 3/  f .Qx.0// C f .Qx.l0 C l1 C 0 C 3//  f .Qx.0//  f .Nx .0// C .f /.l0 C l1 C 0 C 3/ C 60ı1 C f .Nx .0//  .f /.l0 C l1 C 0 C 3/  f .Qx.0// C ı1 =8  60ı1 C ı1 =8  61ı1 :

(5.139)

By (5.139), (5.85), and Proposition 3.11, f .Qx.0//  inf.f / C 61ı1 ; I f .0; 0 ; xQ ; uQ /  0 .f /  f .Qx.0// C f .Qx.0 //  61ı1 :

(5.140) (5.141)

190

5 Linear Control Systems with Discounting

It follows from (5.140), (5.141), and property (P1) that there exists an .f ; A; B/overtaking optimal pair .x ; u / 2 X.A; B; 0; 1/ such that f .x .0// D inf.f /; jx.T  t/  x .t/j D jQx.t/  x .t/j   for all t 2 Œ0; 0 : Theorem 5.2 is proved.

t u

Chapter 6

Dynamic Zero-Sum Games with Linear Constraints

In this chapter we study the existence and turnpike properties of approximate solutions for a class of dynamic continuous-time two-player zero-sum games without using convexity-concavity assumptions. We describe the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals and show that approximate solutions are determined mainly by the objective function, and are essentially independent of the choice of interval and endpoint conditions.

6.1 Preliminaries and Main Results We use the notation and definitions of Sects. 3.1 and 3.2 of Chap. 3. Let n1 ; n2 ; m1 ; m2 be natural numbers, f W Rn1  Rm1  Rn2  Rm2 ! R1 be a Borel measurable function, A1 ; B1 ; A2 ; B2 be given matrices of dimensions n1  n1 ; n1  m1 ; n2  n2 ; n2  m2 , respectively. Linear control controllable systems are described by x0 .t/ D A1 x.t/ C B1 u.t/;

(6.1)

y0 .t/ D A2 y.t/ C B2 v.t/;

(6.2)

for almost every (a. e.) t 2 I, where I is either R1 or ŒT1 ; 1/ or ŒT1 ; T2  (here 1 < T1 < T2 < 1), x W I ! Rn1 , y W I ! Rn2 are a. c. functions and the control functions u W I ! Rm1 , v W I ! Rm2 are Lebesgue measurable functions. Given z1 ; z2 2 Rn1 , 1 ; 2 2 Rn2 , and a positive number T we consider a continuous-time two-player zero-sum game over the interval Œ0; T denoted by  .z1 ; z2 ; 1 ; 2 ; T/. For this game the set of strategies for the first player is the set of all pairs .x; u/ 2 X.A1 ; B1 ; 0; T/ satisfying x.0/ D z1 and x.T/ D z2 , the set © Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6_6

191

192

6 Dynamic Zero-Sum Games with Linear Constraints

of strategies for the second player is the set of all pairs .y; v/ 2 X.A2 ; B2 ; 0; T/ satisfying y.0/ D 1 and y.T/ D 2 , and the objective function for the first player associated with the strategies .x; u/ 2 X.A1 ; B1 ; 0; T/ and .y; v/ 2 X.A2 ; B2 ; 0; T/ is given by Z

T 0

f .x.t/; u.t/; y.t/; v.t//dt

if this integrand is well defined in the sense which is explained below. We recall that a0 > 0 and W Œ0; 1/ ! Œ0; 1/ be an increasing function such that lim

t!1

.t/ D 1:

Suppose that xf 2 Rn1 ; uf 2 Rm1 ; yf 2 Rn2 ; vf 2 Rm2 and A1 xf C B1 uf D 0; A2 yf C B2 vf D 0:

(6.3)

We suppose that the following assumption holds. (C1) (i) the function f is bounded on all bounded subsets of Rn1  Rm1  Rn2  Rm2 ; (ii) the function f .; ; yf ; vf / W Rn1  Rm1 ! R1 is continuous; (iii) for each .x; u/ 2 Rn1  Rm1 , f .x; u; yf ; vf /  maxf .jxj/;

.juj/;

.ŒjA1 x C B1 uj  a0 jxjC /ŒjA1 x C B1 uj  a0 jxjC g  a0 I (iv) the function f .xf ; uf ; / W Rn2  Rm2 ! R1 is continuous; (v) for each .y; v/ 2 Rn2  Rm2 , f .xf ; uf ; y; v/   maxf .jyj/;

.jvj/;

.ŒjA2 y C B2 vj  a0 jyjC /ŒjA2 y C B2 vj  a0 jyjC g C a0 I (vi) for each x 2 Rn1 the function f .x; ; yf ; vf / W Rm1 ! R1 is convex and for each y 2 Rn2 the function f .xf ; uf ; y; / W Rm2 ! R1 is concave;

6.1 Preliminaries and Main Results

193

(vii) for each M;  > 0 there exist ; ı > 0 such that jf .x1 ; u1 ; yf ; vf /  f .x2 ; u2 ; yf ; vf /j   maxff .x1 ; u1 ; yf ; vf /; f .x2 ; u2 ; yf ; vf /g for each u1 ; u2 2 Rm1 and each x1 ; x2 2 Rn1 which satisfy jxi j  M; jui j  ; i D 1; 2; maxfjx1  x2 j; ju1  u2 jg  ıI (viii) for each M;  > 0 there exist ; ı > 0 such that jf .xf ; uf ; y1 ; v1 /  f .xf ; uf ; y2 ; v2 /j   maxff .xf ; uf ; y1 ; v1 /; f .xf ; uf ; y2 ; v2 /g for each y1 ; y2 2 Rn2 and each v1 ; v2 2 Rm2 which satisfy jyi j  M; jvi j  ; i D 1; 2; maxfjy1  y2 j; jv1  v2 jg  ıI (ix) for each K > 0 there exist a constant aK > 0 and an increasing function K

W Œ0; 1/ ! Œ0; 1/

such that K .t/

! 1 as t ! 1

and f .x; u; yf ; uf / 

K .juj/juj

 aK

for each u 2 Rm1 and each x 2 Rn1 satisfying jxj  K, and f .xf ; uf ; y; v/ 

K .jvj/jvj

 aK

for each v 2 Rm2 and each y 2 Rn2 satisfying jyj  K; (x) for each M > 0, there is a number cM > 0 such that f .x; u; y; v/  cM

194

6 Dynamic Zero-Sum Games with Linear Constraints

for each x 2 Rn1 , each u 2 Rm1 , each y 2 Rn2 , and each v 2 Rm2 satisfying jyj; jvj  M and f .x; u; y; v/  cM for each x 2 Rn1 , each u 2 Rm1 , each y 2 Rn2 , and each v 2 Rm2 satisfying jxj; juj  M. Define f .1/ .x; u/ D f .x; u; yf ; vf /; .x; u/ 2 Rn1  Rm1 ;

(6.4)

f .2/ .y; v/ D f .xf ; uf ; y; v/; .y; v/ 2 Rn2  Rm2 :

(6.5)

It is not difficult to see that (A1) holds for f D f .i/ , i D 1; 2. For f D f .i/ , i D 1; 2, we defined .f .i/ / by (3.5): .i/

.f .i/ / D infflim inf T 1 I f .0; T; x; u/ W .x; u/ 2 X.Ai ; Bi ; 0; 1/g: T!1

(6.6)

Propositions 3.1 and 3.3 applied to the triplets .f .i/ ; Ai ; Bi /, i D 1; 2 imply the following results. Proposition 6.1. 1. .f .1/ / D f .xf ; uf ; yf ; vf / if and only if there is c > 0 such that .1/

I f .0; T; x; u/  Tf .xf ; uf ; yf ; vf /  c for each T > 0 and each .x; u/ 2 X.A1 ; B1 ; 0; 1/. 2. .f .2/ / D f .xf ; uf ; yf ; vf / if and only if there is c > 0 such that .2/

I f .0; T; x; u/  Tf .xf ; uf ; yf ; vf /  c for each T > 0 and each .x; u/ 2 X.A2 ; B2 ; 0; 1/. We suppose that the following assumption holds.

6.1 Preliminaries and Main Results

(C2)

195

There is c > 0 such that for each T > 0 and each .x; u/ 2 X.A1 ; B1 ; 0; 1/, Z

T

0

f .x.t/; u.t/; yf ; vf /dt  Tf .xf ; uf ; yf ; vf /  c

and for each T > 0 and each .y; v/ 2 X.A2 ; B2 ; 0; 1/, Z

T 0

f .xf ; uf ; y.t/; v.t//dt  Tf .xf ; uf ; yf ; vf / C c :

By (C2) and Proposition 6.1, .f .1/ / D .f .2/ / D f .xf ; uf ; yf ; vf /:

(6.7)

We suppose that the following assumption holds. (C3)

If .x; u/ 2 Rn1  Rm1 satisfies A1 x C B1 u D 0; .f .1/ / D f .x; u; yf ; uf /;

then x D xf ; If .x; u/ 2 Rn2  Rm2 satisfies A2 x C B2 u D 0; .f .2/ / D f .xf ; uf ; x; u/; then x D yf . Clearly, (A2) holds for f D f .i/ ; i D 1; 2. Let RN D R1 [ f1; 1g. We suppose that for all real numbers ,  C 1 D 1;   1 D 1; 1 <  < 1; maxf; 1g D 1; maxf; 1g D ; minf; 1g D ; minf; 1g D 1: For each z 2 R1 set zC D maxfz; 0g; z D maxfz; 0g: Define f C .x1 ; x2 ; y1 ; y2 / D maxff .x1 ; x2 ; y1 ; y2 /; 0g; f  .x1 ; x2 ; y1 ; y2 / D maxff .x1 ; x2 ; y1 ; y2 /; 0g for all x1 2 Rn1 , x2 2 Rm1 , y1 2 Rn2 and y2 2 Rm2 . Let 1 < T1 < T2 < 1, .x; u/ 2 X.A1 ; B1 ; T1 ; T2 /; .y; v/ 2 X.A2 ; B2 ; T1 ; T2 /:

196

6 Dynamic Zero-Sum Games with Linear Constraints

The pair ..x; u/; .y; v// is called admissible if at least one of the integrals Z

T2

Z

f C .x.t/; u.t/; y.t/; v.t//dt;

T1

T2

f  .x.t/; u.t/; y.t/; v.t//dt

T1

is finite. If ..x; u/; .y; v/// is admissible, then we set Z

T2

I f .T1 ; T2 ; x; u; y; v/ WD D

T1 Z T2

f .x.t/; u.t/; y.t/; v.t//dt f C .x.t/; u.t/; y.t/; v.t//dt

T1

Z



T2

f  .x.t/; u.t/; y.t/; v.t//dt:

T1

We can apply the results of Chap. 3 for f D f .i/ , i D 1; 2. Let us now define approximate solutions (saddle points) of games  .z1 ; z2 ; 1 ; 2 ; T/ with z1 ; z2 2 Rn1 , 1 ; 2 2 Rn2 and a positive constant T. Let M  0, 1 < T1 < T2 < 1, .x; u/ 2 X.A1 ; B1 ; T1 ; T2 /, .y; v/ 2 X.A2 ; B2 ; T1 ; T2 / be such that the pair ..x; u/; .y; v// is admissible. The pair ..x; u/; .y; v// is called .M/-good if the integral I f .T1 ; T2 ; x; u; y; v/ is finite and the following properties hold: for each .z; / 2 X.A1 ; B1 ; T1 ; T2 / such that the pair ..z; /; .y; v// is admissible and z.Ti / D x.Ti /, i D 1; 2, I f .T1 ; T2 ; z; ; y; v/  I f .T1 ; T2 ; x; u; y; v/  MI for each .z; / 2 X.A2 ; B2 ; T1 ; T2 / such that the pair ..x; u/; .z; // is admissible and z.Ti / D y.Ti /, i D 1; 2, I f .T1 ; T2 ; x; u; z; /  I f .T1 ; T2 ; x; u; y; v/ C M: If ..x; u/; .y; v// is .0/-good, then ..x; u/; .y; v// is called a saddle point of the game  .x.T1 /; x.T2 /; y.T1 /; y.T2 /; T2  T1 /. Note that the existence of a saddle point of the game  .z1 ; z2 ; 1 ; 2 ; T/ with z1 ; z2 2 Rn1 , 1 ; 2 2 Rn2 and T > 0 is not guaranteed. Nevertheless, the next result which is proved in Sect. 6.2 holds. Theorem 6.2. Let M > 0. Then there exists M > 0 such that for each T1 2 R1 , each T2 > T1 C 2, each z1 ; z2 2 Rn1 satisfying jzi j  M, i D 1; 2 and each 1 ; 2 2 Rn2 satisfying ji j  M, i D 1; 2 there exists an .M /-good pair .x; u/ 2 X.A1 ; B1 ; T1 ; T2 /; .y; v/ 2 X.A2 ; B2 ; T1 ; T2 /

6.1 Preliminaries and Main Results

197

such that x.Ti / D zi ; y.Ti / D i ; i D 1; 2; jx.t/j; ju.t/j; jy.t/j; jv.t/j  M ; t 2 ŒT1 ; T1 C 1 [ ŒT2  1; T2 ; x.t/ D xf ; u.t/ D uf ; y.t/ D yf ; v.t/ D vf ; t 2 ŒT1 C 1; T2  1: It should be mentioned that in Theorem 6.2 the constant M does not depend on the length of the interval T2  T1 . In this chapter we establish a turnpike property of good solutions of our dynamic games which means that they spend most of the time in a small neighborhood of the pair .xf ; yf /. It is known in the optimal control theory that turnpike properties of approximately optimal solutions are deduced from an asymptotic turnpike property of solutions of corresponding infinite horizon optimal control problems [44, 53]. We say that f possesses the asymptotic turnpike property (or briefly ATP), if the following properties hold: for each .x; u/ 2 X.A1 ; B1 ; 0; 1/ such that Z sup 0

T

 f .x.t/; u.t/; yf ; uf /dt  Tf .xf ; uf ; yf ; vf / W T 2 .0; 1/ < 1

we have limt!1 jx.t/  xf j D 0; for each .y; v/ 2 X.A2 ; B2 ; 0; 1/ such that Z inf 0

T

 f .xf ; uf ; y.t/; v.t//dt  Tf .xf ; uf ; yf ; vf / W T 2 .0; 1/ > 1

we have limt!1 jy.t/  yf j D 0. Clearly, f has (ATP) if and only if f ..i// , i D 1; 2 satisfy (A3). The following theorem is the mail result of this chapter which will be proved in Sect. 6.3. Theorem 6.3. Let f possess (ATP) and M;  > 0. Then there exist l > 0 and an integer Q  1 such that for each T > Ql and each .M/-good admissible pair .x; u/ 2 X.A1 ; B1 ; 0; T/; .y; v/ 2 X.A2 ; B2 ; 0; T/ such that jx.0/j; jx.T/j; jy.0/j; jy.T/j  M

198

6 Dynamic Zero-Sum Games with Linear Constraints

there exist a natural number q  Q and a sequence of closed intervals Œai ; bi   Œ0; T, i D 1; : : : ; q such that 0  bi  ai  l; i D 1; : : : ; q; jx.t/  xf j  ; jy.t/  yf j   q

for all t 2 Œ0; T n [iD1 Œai ; bi :

6.2 Proof of Theorem 6.2 Proposition 5.3 implies the following result. Proposition 6.4. Let ; M > 0. Then there exist M1 > 0 such that for each z1 ; z2 2 Rn1 satisfying jzi j  M, i D 1; 2 and each 1 ; 2 2 Rn2 satisfying ji j  M, i D 1; 2 there exist .x; u/ 2 X.A1 ; B1 ; 0; /; .y; v/ 2 X.A2 ; B2 ; 0;  / such that x.0/ D z1 ; x. / D z2 ; y.0/ D 1 ; y./ D 2 ; jx.t/j; ju.t/j; jy.t/j; jv.t/j  M1 for all t 2 Œ0;  : Proof of Theorem 6.2. We may assume that M > jxf j C juf j C jyf j C jvf j:

(6.8)

By Proposition 6.4, there exist M1 > M such that the following property holds: (P1) for each z1 ; z2 2 Rn1 satisfying jzi j  M, i D 1; 2 and each 1 ; 2 2 Rn2 satisfying ji j  M, i D 1; 2 there exist .x; u/ 2 X.A1 ; B1 ; 0; 1/; .y; v/ 2 X.A2 ; B2 ; 0; 1/ such that x.0/ D z1 ; x.1/ D z2 ; y.0/ D 1 ; y.1/ D 2 ; jx.t/j; ju.t/j; jy.t/j; jv.t/j  M1 for all t 2 Œ0; 1:

6.2 Proof of Theorem 6.2

199

By (C1)(i) and (C1)(x) there exists M2 > 0 such that jf .z1 ; z2 ; 1 ; 2 /j  M2 for each z1 2 R ; z2 2 Rm1 ; each 1 2 Rn2 ; 2 2 Rm2 n1

satisfying jzi j; ji j  M1 ; i D 1; 2;

(6.9)

f .z1 ; z2 ; 1 ; 2 /  M2 for each pair z1 2 Rn1 ; z2 2 Rm1 ; each pair 1 2 Rn2 ; 2 2 Rm2 satisfying ji j  M1 ; i D 1; 2;

(6.10)

f .z1 ; z2 ; 1 ; 2 /  M2 for each pair z1 2 Rn1 ; z2 2 Rm1 ; each pair 1 2 Rn2 ; 2 2 Rm2 satisfying jzi j  M1 ; i D 1; 2:

(6.11)

M D 8M2 C c C M1 :

(6.12)

Set

Let T1 2 R1 ; T2 > T1 C 2; z1 ; z2 2 Rn1 ; 1 ; 2 2 Rn2 ; jzi j; ji j  M; i D 1; 2:

(6.13)

By property (P1), (6.8), and (6.13), there exist .x; u/ 2 X.A1 ; B1 ; T1 ; T2 /; .y; v/ 2 X.A2 ; B2 ; T1 ; T2 / such that x.T1 / D z1 ; x.t/ D xf ; u.t/ D uf ; t 2 ŒT1 C 1; T2  1; x.T2 / D z2 ;

(6.14)

jx.t/j; ju.t/j  M1 ; t 2 ŒT1 ; T1 C 1 [ ŒT2  1; T2 ;

(6.15)

y.T1 / D 1 ; y.t/ D yf ; v.t/ D vf ; t 2 ŒT1 C 1; T2  1; y.T2 / D 2 ;

(6.16)

jy.t/j; jv.t/j  M1 ; t 2 ŒT1 ; T1 C 1 [ ŒT2  1; T2 ;

(6.17)

200

6 Dynamic Zero-Sum Games with Linear Constraints

It follows from (6.8), (6.9), and (6.14)–(6.17) that jf .x.t/; u.t/; y.t/; v.t//j  M2 ; t 2 ŒT1 ; T2 :

(6.18)

By (6.14)–(6.18), ˇ Z T2 ˇ ˇ ˇ ˇ ˇ f .x.t/; u.t/; y.t/; v.t//dt  .T  T /f .x ; u ; y ; v / 2 1 f f f f ˇ ˇ T1

ˇZ ˇ  ˇˇ

T1 C1

T1

Z

T2

C

T2 1

f .x.t/; u.t/; y.t/; v.t//dt ˇ ˇ f .x.t/; u.t/; y.t/; v.t//dt  2f .xf ; uf ; yf ; vf /ˇˇ  4M2 : (6.19)

We show that the pair ..x; u/; .y; v// is M -good. Let .; / 2 X.A1 ; B1 ; T1 ; T2 /; .Ti / D x.Ti /; i D 1; 2: In view of (6.16), (6.17), and (C1)(x), the pair ..; /; .y; v// is admissible. It follows from (6.8)–(6.10), (6.16), (6.17), (6.19), and (C2) that Z

T2

T1

f ..t/; .t/; y.t/; v.t//dt Z

T1 C1

D

Z f ..t/; .t/; y.t/; v.t//dt C

T1

Z

T2

C

T2 1

T2 1

T1 C1

f ..t/; .t/; y.t/; v.t//dt

f ..t/; .t/; y.t/; v.t//dt

 2M2 C

Z

T2 1

T1 C1

f ..t/; .t/; yf ; vf /dt

 2M2 C .T2  T1  2/f .xf ; uf ; yf ; vf /  c Z T2  f .x.t/; u.t/; y.t/; v.t//dt  6M2  c  2jf .xf ; uf ; yf ; vf /j T1

Z 

T2

f .x.t/; u.t/; y.t/; v.t//dt  M :

T1

Let .; / 2 X.A2 ; B2 ; T1 ; T2 /; .Ti / D y.Ti /; i D 1; 2:

6.3 Proof of Theorem 6.3

201

In view of (6.14), (6.15), and (C1)(x), the pair ..x; u/; .; // is admissible. It follows from (6.8), (6.11)–(6.15), (6.19), and (C2) that Z

T2

f .x.t/; u.t/; .t/; .t//dt

T1

Z

T1 C1

D

Z f .x.t/; u.t/; .t/; .t//dt C

T1

Z

T2

C

T2 1

T1 C1

f .x.t/; u.t/; .t/; .t//dt

f .x.t/; u.t/; .t/; .t//dt

Z

 2M2 C

T2 1

T2 1 T1 C1

f .xf ; uf ; .t/; .t//dt

 2M2 C .T2  T1  2/f .xf ; uf ; yf ; vf / C c Z T2  f .x.t/; u.t/; y.t/; v.t//dt C 6M2 C c Z 

T1 T2

f .x.t/; u.t/; y.t/; v.t//dt C M :

T1

t u

Theorem 6.2 is proved.

6.3 Proof of Theorem 6.3 By Theorem 6.2, there exists M1 > 0 such that the following property holds: (P2) for each T1 2 R1 , each T2 > T1 C 2, each z1 ; z2 2 Rn1 satisfying jzi j  M, i D 1; 2 and each 1 ; 2 2 Rn2 satisfying ji j  M, i D 1; 2 there exists an .M1 /-good pair Q 2 X.A2 ; B2 ; T1 ; T2 / .Qx; uQ / 2 X.A1 ; B1 ; T1 ; T2 /; .Qy; v/ such that xQ .Ti / D zi ; yQ .Ti / D i ; i D 1; 2; jQx.t/j; jQu.t/j; jQy.t/j; jv.t/j Q  M1 ; t 2 ŒT1 ; T1 C 1 [ ŒT2  1; T2 ; Q D vf ; t 2 ŒT1 C 1; T2  1: xQ .t/ D xf ; uQ .t/ D uf ; yQ .t/ D yf ; v.t/ By (C1)(x), there exists M2 > 0 such that f .z1 ; z2 ; 1 ; 2 /  M2 for each z1 2 Rn1 ; z2 2 Rm1 ; each 1 2 Rn2 ; 2 2 Rm2

202

6 Dynamic Zero-Sum Games with Linear Constraints

satisfying ji j  M1 ; i D 1; 2;

(6.20)

f .z1 ; z2 ; 1 ; 2 /  M2 for each z1 2 Rn1 ; z2 2 Rm1 ; each 1 2 Rn2 ; 2 2 Rm2 satisfying jzi j  M1 ; i D 1; 2:

(6.21)

By Theorem 4.3 and Proposition 3.2, there exist l1 ; l2 > 0 and integers Q1 ; Q2  1 such that the following properties hold: (P3)

for each T > l1 Q1 and each .x; u/ 2 X.A1 ; B1 ; 0; T/ which satisfies Z

T 0

f .x.t/; u.t/; yf ; vf /dt  2M C 4M2 C c C Tf .xf ; uf ; yf ; vf / q

q

there exist strictly increasing sequences of numbers fai giD1 , fbi giD1  Œ0; T such that q  Q1 , for all i D 1; : : : ; q, 0  bi  ai  l1 ; bi  aiC1 for all integers i satisfying 1  i < q and that q

jx.t/  xf j   for all t 2 Œ0; T n [i D1 Œai ; bi I (P4)

for each T > l2 Q2 and each .y; v/ 2 X.A2 ; B2 ; 0; T/ which satisfies Z 0

T

f .xf ; uf ; y.t/; v.t///dt  2M  4M2  c C Tf .xf ; uf ; yf ; vf / q

q

there exist strictly increasing sequences of numbers fai gi D1 , fbi gi D1  Œ0; T such that q  Q2 , for all i D 1; : : : ; q, 0  bi  ai  l2 ; bi  aiC1 for all integers i satisfying 1  i < q and that q

jy.t/  xf j   for all t 2 Œ0; T n [iD1 Œai ; bi : Set Q D Q1 C Q2 C 4; l D maxfl1 ; l2 g C 4:

(6.22)

6.3 Proof of Theorem 6.3

203

Assume that T > Ql and .x; u/ 2 X.A1 ; B1 ; 0; T/; .y; v/ 2 X.A2 ; B2 ; 0; T/ is an .M/-good admissible pair such that jx.0/j; jx.T/j; jy.0/j; jy.T/j  M:

(6.23)

Q such that By property (P2) and (6.23) there exists an .M1 /-good pair ..Qx; uQ /; .Qy; v// .Qx; uQ / 2 X.A1 ; B1 ; 0; T/; xQ .0/ D x.0/; xQ .T/ D x.T/;

(6.24)

.Qy; v/ Q 2 X.A2 ; B2 ; 0; T/; yQ .0/ D y.0/; yQ .T/ D y.T/;

(6.25)

jQx.t/j; jQu.t/j; jQy.t/j; jv.t/j Q  M1 ; t 2 Œ0; 1 [ ŒT  1; T;

(6.26)

xQ .t/ D xf ; uQ .t/ D uf ; yQ .t/ D yf ; v.t/ Q D vf ; t 2 Œ1; T  1:

(6.27)

In view of (C2), Z Z

T1

f .x.t/; u.t/; yf ; vf /dt  .T  2/f .xf ; uf ; yf ; vf /  c ;

(6.28)

f .xf ; uf ; y.t/; v.t//dt  .T  2/f .xf ; uf ; yf ; vf / C c :

(6.29)

1 T1 1

Since the pair ..x; u/; .y; v// is .M/-good, it follows from (6.24)–(6.27) that Z

1

M C Z C

0 T1

1

Z 0

Z

f .x.t/; u.t/; yQ .t/; v.t//dt Q

T

f .x.t/; u.t/; yQ .t/; v.t//dt Q

f .x.t/; u.t/; y.t/; v.t//dt Z

T

MC 0

Z

1

DMC Z C

T

T1

0 T

Z

f .x.t/; u.t/; yQ .t/; v.t//dt Q C

D M C 

f .x.t/; u.t/; yQ .t/; v.t//dt Q

0 T

T1

f .Qx.t/; uQ .t/; y.t/; v.t//dt Z f .Qx.t/; uQ .t/; y.t/; v.t//dt C

f .Qx.t/; uQ .t/; y.t/; v.t//dt:

1

T1

f .Qx.t/; uQ .t/; y.t/; v.t//dt (6.30)

204

6 Dynamic Zero-Sum Games with Linear Constraints

By (6.21), (6.20), and (6.26)–(6.30), M  2M2 C .T  2/f .xf ; uf ; yf ; vf /  c Z T1  M  2M2 C f .x.t/; u.t/; yf ; vf //dt 1

Z

1

 M C Z C

0 T

f .x.t/; u.t/; yQ .t/; v.t//dt Q Z

f .x.t/; u.t/; yQ .t/; v.t//dt Q C

T1

Z

 M C 2M2 C

T1

1

T1 1

f .x.t/; u.t/; yf ; vf /dt

f .xf ; uf ; y.t/; v.t//dt

 M C 2M2 C c C .T  2/f .xf ; uf ; yf ; vf /:

(6.31)

It follows from (6.28) and (6.31) that Z T1 f .x.t/; u.t/; yf ; vf /dt  2M C 4M2 C c C .T  2/f .xf ; uf ; yf ; vf /; Z

1

1

T1

f .xf ; uf ; y.t/; v.t//dt  2M  4M2  c C .T  2/f .xf ; uf ; yf ; vf /:

By the inequalities above, the relation T > Ql, and properties (P3) and (P4), there exist .1/ q

.1/ q

.2/ q

.2/ q

1 1 fai giD1 ; fbi giD1  Œ1; T  1; 2 2 ; fbi giD1  Œ1; T  1; fai giD1

such that qi  Qi , i D 1; 2, for all i D 1; : : : ; q1 , .1/

.1/

0  bi  ai

 l1 ;

for all i D 1; : : : ; q2 , .2/

.2/

0  bi  ai

 l2 ; q

.1/

.1/

q

.2/

.2/

1 jx.t/  xf j   for all t 2 Œ1; T  1 n [iD1 Œai ; bi ; 2 jy.t/  yf j   for all t 2 Œ1; T  1 n [iD1 Œai ; bi :

This completes the proof of Theorem 6.3.

t u

6.4 Examples

205

6.4 Examples We use the notation and definitions of Sect. 6.1. Example 6.5. Assume that f W Rn1  Rm1  Rn2  Rm2 ! R1 is a Borel measurable function which is bounded on all bounded subsets of Rn1  Rm1  Rn2  Rm2 and satisfies (C1)(x). Let xf 2 Rn1 ; uf 2 Rm1 ; yf 2 Rn2 ; vf 2 Rm2 satisfy (6.3). Assume that a1 > 0, l1 2 Rm1 , l2 2 Rm2 , function such that lim

t!1

0 .t/

0

W Œ0; 1/ ! Œ0; 1/ is an increasing

D1

and that L1 W Rn1  Rm1 ! Œ0; 1/ and L2 W Rn2  Rm2 ! Œ0; 1/ are continuous functions such that for all .x; u/ 2 Rn1  Rm1 ; L1 .x; u/ D 0 if and only if x D xf ; u D uf ; L1 .x; u/  maxf

0 .jxj/;

0 .juj/jujg

 a1 C jl1 jjA1 x C B1 uj;

for all .y; v/ 2 Rn2  Rm2 ; L2 .y; v/ D 0 if and only if y D yf ; v D vf ; L2 .y; v/  maxf

0 .jyj/;

0 .jvj/jvjg

 a1 C jl2 jjA2 y C B2 vj;

for each M;  > 0 there exist ; ı > 0 such that for i D 1; 2, jLi .x1 ; u1 /  Li .x2 ; u2 /j   maxfLi .x1 ; u1 /; Li .x2 ; u2 /g for each x1 ; x2 2 Rni and each u1 ; u2 2 Rmi which satisfy jx1 j; jx2 j  M; ju1 j; ju2 j  ; jx1  x2 j; ju1  u2 j  ı and that for i D 1; 2 and each x 2 Rni , the function Li .x; / W Rmi ! R1 is convex. Assume that for each .x; u/ 2 Rn1  Rm1 ; f .x; u; yf ; vf / D f .xf ; uf ; yf ; vf / C L1 .x; u/ C hl1 ; A1 x C B1 ui and that for each .y; v/ 2 Rn2  Rm2 ; f .xf ; uf ; y; v/ D f .xf ; uf ; yf ; vf /  L2 .y; v/  hl2 ; A2 y C B2 vi:

206

6 Dynamic Zero-Sum Games with Linear Constraints

We show that assumptions (C1)–(C3) and (ATP) hold for the integrand f . It is clear that (C1)(i), (C1)(ii), (C1)(iv), (C1)(vi), (C1)(vii), (C1)(viii), and (C1)(ix) hold. It is not difficult to see that (C1)(iii) and (C1)(v) hold under the appropriate choice of a0 > 0, . Thus assumption (C1) holds. Set f .1/ .x; u/ D f .x; u; yf ; vf / D f .xf ; uf ; yf ; vf / C L1 .x; u/ C hl1 ; A1 x C B1 ui for each .x; u/ 2 Rn1  Rm1 ; f .2/ .y; v/ D f .xf ; uf ; y; v/ D f .xf ; uf ; yf ; vf / C L2 .y; v/ C hl2 ; A2 y C B2 vi for each .y; v/ 2 Rn2  Rm2 : In Sect. 3.1 (see also Example 1.12) it was explained that for the triplets .fi ; Ai ; Bi /, i D 1; 2 assumptions (A1) holds under the appropriate choice of a0 > 0, and in view of Proposition 3.6, (A2) holds for the triplets .fi ; Ai ; Bi /, i D 1; 2, .f .1/ / D f .xf ; uf ; yf ; vf /; .f .2/ / D f .xf ; uf ; yf ; vf / and the following properties hold: for every .f1 ; A1 ; B1 /-good trajectory-control pair .x; u/ 2 X.A1 ; B1 ; 0; 1/, lim x.t/ D xf I

t!1

for every .f2 ; A2 ; B2 /-good trajectory-control pair .y; v/ 2 X.A2 ; B2 ; 0; 1/, lim y.t/ D yf :

t!1

Combined with Proposition 6.1 this implies that (C2) and (C3) hold and that the integrand f possesses (ATP). Therefore Theorems 6.2 and 6.3 hold for the integrand f . Example 6.6. Assume that f W Rn1  Rm1  Rn2  Rm2 ! R1 is a Borel measurable function which is bounded on all bounded subsets of Rn1  Rm1  Rn2  Rm2 and satisfies (C1)(x). Let xf 2 Rn1 ; uf 2 Rm1 ; yf 2 Rn2 ; vf 2 Rm2 satisfy (6.3).

6.4 Examples

207

Set f .1/ .x; u/ D f .x; u; yf ; vf / for each .x; u/ 2 Rn1  Rm1 ; f .2/ .y; v/ D f .xf ; uf ; y; v/ for each .y; v/ 2 Rn2  Rm2 : Suppose that f .1/ ; f .2/ are continuous strictly convex functions, assumption (A1) holds for triplets .fi ; Ai ; Bi /, i D 1; 2 and for i D 1; 2, f .i/ .x; u/=juj ! 1 as juj ! 1 uniformly in x 2 Rni : As it was mentioned in Sect. 3.1 (see also Example 1.11) Corollary 2.11 of Chap. 2 implies that assumptions (A2) and (A3) hold for the triplets .fi ; Ai ; Bi /, i D 1; 2, Combined with Proposition 6.1 this implies that (C1)–(C3) and (ATP) hold for the integrand f . Therefore Theorems 6.2 and 6.3 hold for the integrand f .

Chapter 7

Genericity Results

In this chapter we continue to study the class of optimal control problems studied in Chaps. 3 and 4. There we established the convergence of approximate solutions on large intervals in the regions close to the end points. In this chapter we show that for a typical (in the sense of Baire category) integrand the values of approximate solutions at the end points converge to the limit which is a unique solution of the corresponding minimization problem associated with the integrand.

7.1 Preliminaries and Main Results We use the notation, definitions, and assumptions introduced in Sects. 3.1–3.3 of Chap. 3 and in Sects. 4.1 and 4.6 of Chap. 4. Recall that a0 > 0 and W Œ0; 1/ ! Œ0; 1/ is an increasing function such that lim

t!1

.t/ D 1:

We continue to study the structure of optimal trajectories of the controllable linear control system x0 D Ax C Bu where A and B are given matrices of dimensions n  n and n  m. We consider the space M of all borelian functions g W Rn C m C 1 ! R1 which satisfy the growth assumption (5.1). The space M is equipped with the uniformity which is determined by the base (5.2). This uniform space is metrizable and complete. In Chap. 5 we considered the set Mb of all functions g 2 M which are

© Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6_7

209

210

7 Genericity Results

bounded on bounded subsets of RnCmC1 . Note that Mb is a closed subset of M. We consider the topological subspace Mb  M equipped with the relative topology. Denote by Mc the set of all continuous functions g 2 M which satisfy assumption (A1) and which do not depend on the variable t. By Proposition 4.10 of [52], Mc is a closed subset of M. We consider the topological subspace Mc  M equipped with the relative topology. In this chapter any element g 2 Mc will be considered as a function g W RnCm ! R1 . Let f 2 Mc . By Proposition 4.9 of [52], there exists an .A; B/-trajectory-control pair xQ f W R1 ! Rn ; uQ f W R1 ! Rm whose restriction to any finite interval ŒT1 ; T2  belongs to X.A; B; T1 ; T2 / and a number bQ f > 0 such that the following assumption holds: (B1)

(i) .f ; xQ f .T1 /; xQ f .T2 /; T1 ; T2 / D I f .T1 ; T2 ; xQ f ; uQ f / for each T1 2 R1 and each T2 > T1 ;

(ii) supfI f .j; j C 1; xQ f ; uQ f / W j D 0; ˙1; ˙2; : : :g < 1I (iii) for each S1 > 0 there exist S2 > 0 and c > 0 such that I f .T1 ; T2 ; xQ f ; uQ f /  I f .T1 ; T2 ; x; u/ C S2 for each T1 2 R1 , each T2  T1 C c and each .x; u/ 2 X.A; B; T1 ; T2 / which satisfies jx.T1 /j; jx.T2 /j  S1 ; (iv) for each  > 0 there exists ı > 0 such that for each .T; z/ 2 RnC1 which satisfies jz  xQ f .T/j  ı there are 1 2 .T; T C bQ f  and 2 2 ŒT  bQ f ; T/; and trajectory-control pairs .x1 ; u1 / 2 X.A; B; T; 1 /; .x2 ; u2 / 2 X.A; B; 2 ; T/

7.1 Preliminaries and Main Results

211

which satisfy x1 .T/ D x2 .T/ D z; xi .i / D xQ f .i /; i D 1; 2; jx1 .t/  xQ f .t/j   for all t 2 ŒT; 1 ; jx2 .t/  xQ f .t/j   for all t 2 Œ2 ; T; I f .T; 1 ; x1 ; u1 /  I f .T; 1 ; xQ f ; uQ f / C ; I f .2 ; T; x2 ; u2 /  I f .2 ; T; xQ f ; uQ f / C : Note that assumption (B1) means that the trajectory-control pair xQ f W R1 ! Rn ; uQ f W R1 ! Rm is a solution of the corresponding infinite horizon optimal control problem associated with the integrand f and that certain controllability properties hold near this trajectory-control pair. Let f 2 Mc . Set .f / WD infflim inf T 1 I f .0; T; x; u/ 2 X.A; B; 0; 1/g: T!1

(7.1)

By (5.1), the value .f / is well defined and finite and f .0; 0/  .f /  a0 : Denote by M the set of all f 2 Mc for which there exists .xf ; uf / 2 Rn  Rm such that Axf C Buf D 0; .f / D f .xf ; uf /:

(7.2)

In Sect. 7.2 we prove the following result. Proposition 7.1. M is a closed subset of the space Mc . We consider the topological subspace M  Mc equipped with the relative topology. For every f 2 M let .xf ; uf / 2 Rn  Rm satisfy (7.2). Let f 2 M . By (7.2), Propositions 3.1 and 3.2 (see Chap. 3), and Proposition 4.6 of [52] (see also Proposition 7.7), assumption (B1) holds with the constant trajectory-control pair xQ f .t/ D xf ; uQ f .t/ D uf for all t 2 R1 :

212

7 Genericity Results

For each r 2 .0; 1/ set fr .x; u/ D f .x; u/ C r minfjx  xf j; 1g; .x; u/ 2 Rn  Rm :

(7.3)

It is not difficult to see that fr 2 M for all r 2 .0; 1/. Theorem 2.3 and Corollary 2.4 of [52] imply the following result. Theorem 7.2. There exists a set F0  M which is a countable intersection of open everywhere dense subsets of M such that for each f 2 F0 the following property holds. For each S;  > 0 there exist real numbers  > 0, ı 2 .0; / such that for each T1 2 R1 , each T2  T1 C 2 and each .x; u/ 2 X.A; B; T1 ; T2 / which satisfies I f .T1 ; T2 ; x; u/  .f ; T2  T1 / C S and I f .T1 ; T2 ; x; u/  .f ; x.T1 /; x.T2 /; T2  T1 / C ı the inequality jx.t/  xf j   holds for all t 2 ŒT1 C ; T2  : Let F0  M be as guaranteed by Theorem 7.2. Propositions 3.1, 3.2, and Theorem 7.2 imply the following result. Proposition 7.3. Every f 2 F0 satisfies assumptions (A2) and (A3). Denote by FQ the set of all f 2 M which satisfy assumptions (A2) and (A3). The following theorem is the main result of this chapter. Theorem 7.4. There exists a set F  FQ which is a countable intersection of open everywhere dense subsets of M such that for each f 2 F the following properties hold:. f

f

(i) there exists a unique point xC 2 Rn such that  f .xC / D inf. f /; f f (ii) there exists a unique point x 2 Rn such that f .x / D inf.f /. Theorem 7.4 follows from Propositions 3.21 and 3.22 and the next result which is proved in Sect. 7.4. Theorem 7.5. There exists a set F  FQ which is a countable intersection of open everywhere dense subsets of M such that for each f 2 F there exists a unique point xf 2 Rn such that  f .xf / D inf. f /. Let f 2 M . Denote by L.f / the set of all pairs .x; u/ 2 Rn  Rm such that Ax C Bu D 0; .f / D f .x; u/:

7.2 Proof of Proposition 7.1

213

Q Clearly, L.f / is a nonempty set which is not necessarily a singleton and if f 2 F, then by (A2), fx 2 Rn W there is u 2 Rm such that .x; u/ 2 L.f /g is a singleton denoted by fxf g, where xf 2 Rn . Chapter 7 is organized as follows. Proposition 7.1 is proved in Sect. 7.2. Section 7.3 contains auxiliary results for Theorem 7.5 which is proved in Sect. 7.4.

7.2 Proof of Proposition 7.1 Assumptions (A1) and (B1) and Proposition 2.7 of [52] imply the following result. Lemma 7.6. Let f 2 Mc and xQ f ./; uQ f ./ be as guaranteed by (B1). Then the following assertions hold. 1. supfjQx.t/j W t 2 R1 g < 1: 2. Let S0 > 0. Then there exist S > 0, c  1 such that for each T1 2 R1 , each T2  T1 C c and each trajectory-control pair .x; u/ 2 X.A; B; T1 ; T2 / satisfying jx.T1 /j  S0 the inequality I f .T1 ; T2 ; xQ f ; uQ f /  I f .T1 ; T2 ; x; u/ C S holds. Proposition 7.7 (Proposition 4.6 of [52]). Let f 2 Mc , M; ;  > 0. Then there exists a number ı > 0 such that: 1. for each T 2 R1 , each y1 ; y2 ; z1 ; z2 2 Rn satisfying jyi j; jzi j  M; i D 1; 2; jy1  y2 j; jz1  z2 j  ı the following relation holds: j .f ; y1 ; z1 ; T; T C /  .f ; y2 ; z2 ; T; T C  /j  : 2. for each T 2 R1 , each y1 ; y2 ; z1 :z2 2 Rn satisfying jyi j; jzi j  M; i D 1; 2; jy1  y2 j; jz1  z2 j  ı

214

7 Genericity Results

and each trajectory-control pair .x1 ; u1 / 2 X.A; B; T; T C  / which satisfies x1 .T/ D y1 ; x1 .T C / D z1 ; I f .T; T C ; x1 ; u1 / D .f ; y1 ; z1 ; T; T C  / there exists a trajectory-control pair .x2 ; u2 / 2 X.A; B; T; T C  / such that x2 .T/ D y2 ; x2 .T C / D z2 ; jI .T; T C ; x2 ; u2 /  I f .T; T C ; x1 ; u1 /j  ; f

jx1 .t/  x2 .t/j  ; t 2 ŒT; T C  : Proof of Proposition 7.1. Let ffi g1 iD1  M ; lim fi D f i!1

(7.4)

in Mc . We show that f 2 M . Let an .A; B/-trajectory-control pair .Qxf ; uQ f / be as guaranteed by (B1). Clearly, for each integer i  1 there exists .xfi ; ufi / 2 Rn  Rm such that Axfi C Bufi D 0; .fi / D fi .xfi ; ufi /:

(7.5)

.f / D lim inf T 1 I f .0; T; xQ f ; uQ f /:

(7.6)

Lemma 7.6 implies that T!1

Let  2 .0; 1/. By Lemma 7.6, supfjQxf .t/j W t 2 R1 g < 1: Proposition 7.7 implies that there exists a number M0 > supfj .f ; z1 ; z2 ; 0; 1/j W z1 ; z2 2 Rn ; jz1 j; jz2 j  supfjQxf .t/j W t 2 R1 g C 4g:

(7.7)

7.2 Proof of Proposition 7.1

215

In view of (7.6), there exists a number T0 such that T0  40 C .2M0 C 2/.4/1 ; 2T01 j.f /j < =4;

(7.8)

j.f /  T01 I f .0; T0 ; xQ f ; uQ f /j  T01 :

(7.9)

Clearly, there exists .x; u/ 2 X.A; B; 0; T0 C 2/ such that x.0/ D 0; x.T0 C 2/ D 0; x.t/ D xQ f .t  1/; u.t/ D uQ f .t  1/; t 2 Œ1; T0 C 1; I f .0; 1; x; u/  .f ; 0; xQ f .0/; 0; 1/ C 1; I f .T0 C 1; T0 C 2; x; u/  .f ; xQ f .T0 /; 0; T0 C 1; T0 C 2/ C 1:

(7.10)

By (7.7), (7.9), and (7.10), I f .0; T0 C 2; x; u/  2M0 C I f .0; T0 ; xQ f ; uQ f /  2M0 C T0 .f / C :

(7.11)

It follows from (7.4), (7.11), and Proposition 4.8 that there exists a natural number k0 such that for each integer k  k0 , I fk .0; T0 C 2; x; u/  2M0 C T0 .f / C 2:

(7.12)

In view of (7.4), (7.8), (7.10), and (7.12), for each integer k  k0 , fk .xfk ; ufk / D .fk /  .T0 C 2/1 I fk .0; T0 C 2; x; u/  .T0 C 2/1 .2M0 C 2/ C T0 .T0 C 2/1 .f /  .f / C =2: (7.13) (A1) and (7.13) imply that the sequence f.xfk ; ufk /g1 kD1 is bounded. Extracting a subsequence and re-indexing we may assume that there exists  D lim xfk ; D lim ufk : k!1

k!1

(7.14)

By (7.4), (7.13), and (7.14), f .; / D lim fk .xfk ; ufk / D lim .fk /  .f / C =2: k!1

k!1

Since  is any number belonging to the interval .0; 1/ we conclude that f .; /  .f /: Evidently, A C B D 0. This implies that .f / D f .; /, f 2 M and completes the proof of Proposition 7.1. t u

216

7 Genericity Results

7.3 Auxiliary Results for Theorem 7.5 In the sequel we need the following result (see Proposition 3.7.1 of [44]). Lemma 7.8. Let ˝ be a closed subset of Rs . Then there exists a bounded nonnegative function  2 C1 .Rs / such that ˝ D fx 2 Rs W .x/ D 0g and for each p sequence of nonnegative integers p1 ; : : : ; ps , the function @jpj =@x11 : : : @xsps W Rs ! P s R1 is bounded, where jpj D iD1 pi . Q ffk g1  M , Lemma 7.9. Let f 2 F, kD1 lim fk D f ;

k!1

.xk ; uk / 2 L.fk /; k D 1; 2; : : : : Then the sequence f.xk ; uk /g1 kD1 is bounded, any its limit point belongs to L.f /, there exists limk!1 xk and for every .x; u/ 2 L.f /, f .x; u/ D lim f .xk ; uk / D lim fk .xk ; uk /: k!1

k!1

Proof. Let .x; u/ 2 L.f /. It is clear that for all integers k  1, fk .xk ; uk /  fk .x; u/ ! f .x; u/ as k ! 1:

(7.15)

By (A1) and (7.15) the sequence f.xk ; uk /g1 kD1 is bounded. Let .; / be its limit point. Clearly, A C Bu D 0: In view of (7.15), f .; /  lim sup f .xk ; uk / D lim sup fk .xk ; uk / D f .x; u/ D .f /: k!1

k!1

This implies that f .; / D .f / and  D x. This completes the proof of Lemma 7.9. Lemma 7.10. Let f 2 FQ and  2 .0; 1/. Then there exist ı > 0 and a neighborhood U of f in M such that for each g 2 U \ FQ and each z 2 Rn satisfying jz  xf j  ı, j g .z/j  : Proof. By Lemma 7.9, there exists a neighborhood U0 of f in M such that for each g 2 U0 \ M , jxf  xg j  41 :

(7.16)

7.3 Auxiliary Results for Theorem 7.5

217

By Proposition 7.7, there exists  2 .0; =2/ such that j .f ; z1 ; z2 ; 0; 1/  .f /j  =16 for each z1 ; z2 2 Rn satisfying jzi  xf j  2; i D 1; 2:

(7.17)

By Proposition 4.8, there exists a neighborhood U1 of f in M such that U1  U0 and the following property holds: (i) for each g 2 U1 , each S 2 R1 , each and each trajectory-control pair .x; u/ 2 X.A; B; S; S C 1/ which satisfies minfI g .S; S C 1; x; u/; I f .S; S C 1; x; u/g  j.f /j C 4 the inequality jI g .S; S C 1; x; u/  I f .S; S C 1; x; u/j  =16 holds. By Theorem 4.2, there exist ı 2 .0; =4/, l0 > 0 and a neighborhood U2 of f in M such that U2  U1 and the following property holds: (ii) for each T > 2l0 , each g 2 U2 and each .x; u/ 2 X.A; B; 0; T/ which satisfies jx.0/  xf j  ı; jx.T/j  jxf j C 1; I g .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C ı we have jx.t/  xf j   for all t 2 Œ0; T  l0 : By Lemma 7.9, there exists a neighborhood U of f in M such that U  U2 and the following property holds: Q (iii) for each g 2 U \ F, jxf  xg j  ı: Assume that Q z1 ; z2 2 Rn ; jzi  xf j  ı; i D 1; 2: g 2 U \ F;

(7.18)

Theorem 3.8 and (7.18) imply that there exists an .g; A; B/-overtaking optimal pair .x; u/ 2 X.A; B; 0; 1/ satisfying x.0/ D z1 :

(7.19)

218

7 Genericity Results

In view of (A3) and (7.18), lim x.t/ D xg :

t!1

(7.20)

It is clear that (7.16) holds. By (7.16) and (7.20), for all sufficiently large numbers t, jx.t/j  jxf j C 1: Together with (7.18) and property (ii) this implies that jx.t/  xf j   for all t 2 Œ0; 1/:

(7.21)

In view of (7.19),  g .z1 / D lim infŒI g .0; T; x; u/  T.g/ T!1

D I g .0; 1; x; u/  .g/ C lim infŒI g .1; T; x; u/  .T  1/.g/: T!1

(7.22) By Proposition 3.27, there exists .x1 ; u1 / 2 X.A; B; 0; 1/ such that x1 .0/ D z2 ; x1 .t/ D x.t/; u1 .t/ D u.t/ for all t  1; I g .0; 1; x1 ; u1 / D .g; z2 ; x.1/; 0; 1/:

(7.23)

Proposition 3.11, (7.19), (7.22), and (7.23) imply that  g .z2 /   g .z1 /  .g; z2 ; x.1/; 0; 1/  .g; z1 ; x.1/; 0; 1/:

(7.24)

It follows from (7.17), (7.18), and (7.21) that j .f ; zi ; x.1/; 0; 1/  .f /j  =16; i D 1; 2:

(7.25)

In view of (7.25) and property (i), for i D 1; 2, j .f ; zi ; x.1/; 0; 1/  .g; zi ; x.1/; 0; 1/j  =16: By (7.25) and (7.26), j .g; z1 ; x.1/; 0; 1/  .g; z2 ; x.1/; 0; 1/j  =4:

(7.26)

7.3 Auxiliary Results for Theorem 7.5

219

Together with (7.24) this implies that  g .z2 /   g .z1 /  =4: Thus we have shown that j g .z2 /   g .z1 /j  =4 for all z1 ; z2 2 Rn satisfying jzi  xf j  ı; i D 1; 2. Combined with property (iii) and the equality  g .xg / D 0 this completes the proof of Lemma 7.10. Lemma 7.11. Let f 2 FQ and M0 > 0. Then there exist M1 > 0 and a neighborhood U of f in M such that for each g 2 U \ FQ and each z 2 Rn satisfying jzj  M0 ,  g .z/  M1 : Proof. By Lemma 7.9, there exists a neighborhood U0 of f in M such that for each g 2 U0 \ FQ  , jxf  xg j  1; j.f /  .g/j  1:

(7.27)

By Proposition 3.28, there exists M2 > 0 such that j .f ; z1 ; z2 ; 1/j  M2 for each z1 ; z2 2 Rn satisfying jzi j  jxf j C M0 C 1; i D 1; 2:

(7.28)

By Proposition 4.8, there exists a neighborhood U of f in M such that U  U0 and the following property holds: (iv) for each g 2 U and each trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ which satisfies minfI g .0; 1; x; u/; I f .0; 1; x; u/g  M2 C 1 the inequality jI g .0; 1; x; u/  I f .0; 1; x; u/j  1 holds. Set M1 D M2 C 2 C j.f /j:

(7.29)

220

7 Genericity Results

Assume that Q z 2 Rn ; jzj  M0 : g 2 U \ F;

(7.30)

In view of (7.30) and the choice of U0 (see (7.27)), (7.27) is true. By Proposition 3.27, there exists .x; u/ 2 X.A; B; 0; 1/ such that x.0/ D z; x.t/ D xg ; u.t/ D ug for all t  1; I f .0; 1; x; u/ D .f ; z; xg ; 1/:

(7.31)

In view of (7.31) and Proposition 3.11,  g .z/  lim infŒI g .0; T; x; u/  T.g/ T!1

D I g .0; 1; x; u/  .g/:

(7.32)

It follows from (7.27), (7.28), (7.30), and (7.31) that I f .0; 1; x; u/ D .f ; z; xg ; 0; 1/  M2 : Together with (7.30) and property (iv) this implies that I g .0; 1; x; u/  I f .0; 1; x; u/ C 1  M2 C 1: Combined with (7.27) and (7.32) this implies that  g .z/  M2 C 2 C j.f /j D M1 : Lemma 7.11 is proved. Lemma 7.12. Let f 2 FQ and M0 > 0. Then there exist M1 > 0 and a neighborhood U of f in M such that for each g 2 U \ FQ and each z 2 Rn satisfying  g .z/  inf. g / C M0 the inequality jzj  M1 holds. Proof. By Lemma 7.9, there exists a neighborhood U of f in M such that for each Q g 2 U \ F, j.f /  .g/j  1:

(7.33)

By Proposition 4.7, there exists M1 > 0 such that for each g 2 M and each trajectory-control pair .x; u/ 2 X.A; B; 0; 1/ which satisfies I g .0; 1; x; u/  M0 C 2 C j.f /j

7.3 Auxiliary Results for Theorem 7.5

221

we have jx.t/j  M1 ; t 2 Œ0; 1:

(7.34)

Q z 2 Rn ;  g .z/  inf. g / C M0 : g 2 U \ F;

(7.35)

Assume that

Evidently, (7.33) holds. By Theorem 3.8 and (7.35), there exists .g; A; B/-overtaking optimal pair .x; u/ 2 X.A; B; 0; 1/ such that x.0/ D z: Proposition 3.12 implies that  g .z/ D I g .0; 1; x; u/  .g/ C  g .x.1//:

(7.36)

By (7.35) and (7.36), I g .0; 1; x; u/  .g/ D  g .z/   g .x.1//   g .z/  inf. g /  M0 :

(7.37)

In view of (7.33) and (7.37), I f .0; 1; x; u/  M0 C 2 C j.f /j: By the relation above and the choice of M1 (see (7.34)), jzj D jx.0/j  M1 : Lemma 7.12 is proved. Q  2 .0; 1/, L0 > 0. Then there exist a neighborhood U Lemma 7.13. Let f 2 F, of f in M and ı > 0 such that for each g 2 U \ FQ and each .g; A; B/-overtaking optimal pair .x; u/ 2 X.A; B; 0; 1/ satisfying  g .x.0//  inf. g / C ı there exists an .f ; A; B/-overtaking optimal pair .; / 2 X.A; B; 0; 1/ such that  f ..0// D inf. f /; j.t/  x.t/j   for all t 2 Œ0; L0 :

222

7 Genericity Results

Proof. By Lemma 3.38, there exists 0 2 .0; =2/ such that the following property holds: (v) for each .x; u/ 2 X.A; B; 0; L0 / which satisfies  f .x.0//  inf. f / C 0 ; I f .0; L0 ; x; u/  L0 .f /   f .x.0// C  f .x.L0 //  0 there exists an .f ; A; B/-overtaking optimal pair .; / 2 X.A; B; 0; 1/ such that  f ..0// D inf. f /; jx.t/  .t/j   for all t 2 Œ0; L0 : By Proposition 3.29, there exists 1 2 .0; 0 / such that j .f ; z1 ; z2 ; 1/  .f /j  0 =64 for each z1 ; z2 2 Rn satisfying jzi  xf j  1 ; i D 1; 2:

(7.38)

By Lemmas 7.9, 7.11, and 7.12, exist a neighborhood U0 of f in M , M0 > 0, M1 > 0 such that the following properties hold: Q (vi) for each g 2 U0 \ F, jxf  xg j  1; j.f /  .g/j  1I (vii) for each g 2 U0 \ FQ and each z 2 Rn satisfying  g .z/  inf. g / C 4, we have jzj  M0 I (viii) for each g 2 U0 \ FQ and each z 2 Rn satisfying jzj  M0 C 1 we have  g .z/  M1 : We may assume without loss of generality that M0 > jxf j C 2:

(7.39)

Q Properties (vii) and (viii) imply that for each g 2 U0 \ F, inf. g /  M1 :

(7.40)

It follows from Lemma 7.10 that there exist ı 2 .0; 1 =4/ and a neighborhood U1 of f in M such that the following property holds:

7.3 Auxiliary Results for Theorem 7.5

223

(ix) for each g 2 U1 \ FQ and each z 2 Rn satisfying jz  xf j  ı, j g .z/j  0 =32: By Lemma 7.9, there exists a neighborhood U2 of f in M such that U2  U1 and Q for each g 2 U2 \ F, jxf  xg j  ı:

(7.41)

By Theorem 4.2, there exist ı 2 .0; ı/, L1 > 4 C L0 and a neighborhood U3 of f in M such that U3  U2 and the following property holds: (x) for each T > 2L1 , each g 2 U3 and each .x; u/ 2 X.A; B; 0; T/ which satisfies jx.0/j; jx.T/j  M0 C 1; I .0; T; x; u/  .g; x.0/; x.T/; 0; T/ C ı g

we have jx.t/  xf j  ı for all t 2 ŒL1 ; T  L1 : By Proposition 4.8, there exists a neighborhood U4 of f in M such that and the following property holds: (xi) for each g 2 U4 , each T 2 Œ1; L1 C 2 each and each trajectory-control pair .x; u/ 2 X.A; B; 0; T/ which satisfies minfI g .0; T; x; u/; I f .0; T; x; u/g  4 C M1 C 2L1 .j.f /j C 2/ the inequality jI g .0; T; x; u/  I f .0; T; x; u/j  0 =64 holds. By Lemma 7.9, there exists a neighborhood U5 of f in M such that for each Q g 2 U \ F, j.g/  .f /j  .L1 C 2/1 0 =64:

(7.42)

U D \5iD1 Ui :

(7.43)

Set

224

7 Genericity Results

Assume that Q g1 ; g2 2 U \ F;

(7.44)

z 2 R ;  .z/  inf. / C 1 n

g1

g1

(7.45)

and .x; u/ 2 X.A; B; 0; 1/ is a .g1 ; A; B/-overtaking optimal pair satisfying x.0/ D z:

(7.46)

Proposition 3.15 and (7.46) imply that  g1 .z/ D lim ŒI g1 .0; T; x; u/  T.g1 /:

(7.47)

jzj  M0 :

(7.48)

lim x.t/ D xg1 :

(7.49)

T!1

By (7.43)–(7.45),

In view of (7.44) and (A3), t!1

It follows from (7.43), (7.44), and property (vi) that jxf  xg1 j  1:

(7.50)

By (7.39), (7.49), and (7.50), for all sufficiently large numbers t, jx.t/  xf j  jx.t/  xg1 j C jxg1  xf j  5=4; jx.t/j  jxf j C 2 < M0 :

(7.51)

Since the pair .x; u/ is .g1 ; A; B/-overtaking optimal it follows from (7.44), (7.46), (7.48), (7.51), and property (x) that jx.t/  xf j  ı for all t  L1 :

(7.52)

Since the pair .x; u/ is .g1 ; A; B/-overtaking optimal it follows from (7.46) and Proposition 3.12 that  g1 .z/ D I g1 .0; L1 ; x; u/  L1 .g1 / C  g1 .x.L1 //:

(7.53)

In view of (7.43), (7.44), (7.48), and property (viii),  g1 .z/  M1 :

(7.54)

7.3 Auxiliary Results for Theorem 7.5

225

Property (ix), (7.43), (7.44), and (7.52) imply that j g1 .x.L1 //j  321 0 :

(7.55)

By (7.53)–(7.55) and property (vi), I g1 .0; L1 ; x; u/  2 C M1 C L1 j.g1 /j  2 C M1 C L1 .j.f /j C 1/:

(7.56)

It follows from (7.43), (7.44), (7.56), and property (xi) that jI g1 .0; L1 ; x; u/  I f .0; L1 ; x; u/j  0 =64; jI f .0; L1 ; x; u/  I g2 .0; L1 ; x; u/j  0 =64:

(7.57)

jI g1 .0; L1 ; x; u/  I g2 .0; L1 ; x; u/j  0 =32:

(7.58)

In view of (7.57),

Proposition 3.27 and (7.44) imply that there exists .x1 ; u1 / 2 X.A; B; 0; 1/ such that x1 .t/ D x.t/; u1 .t/ D u.t/ for all t 2 Œ0; L1 ; x1 .t/ D xg2 ; u1 .t/ D ug2 for all t 2 ŒL1 C 1; 1/; I g2 .L1 ; L1 C 1; x1 ; u1 / D .g2 ; x.L1 /; xg2 ; 0; 1/:

(7.59)

By (7.46), (7.59), and Proposition 3.11,  g2 .z/  lim infŒI g2 .0; T; x1 ; u1 /  T.g2 / T!1

D I g2 .0; L1 ; x; u/  L1 .g2 / C .g2 ; x.L1 /; xg2 ; 0; 1/  .g2 /: (7.60) In view of (7.42)–(7.44), jL1 .g2 /  L1 .g1 /j  0 =32:

(7.61)

It follows from (7.38), (7.41), (7.43), (7.44), and (7.52) that j .f ; x.L1 /; xg2 ; 0; 1/  .f /j  0 =64:

(7.62)

Property (xi) and (7.43), (7.44), (7.62) imply that j .f ; x.L1 /; xg2 ; 0; 1/  .gi ; x.L1 /; xg2 ; 0; 1/j  0 =64; i D 1; 2:

(7.63)

226

7 Genericity Results

It follows from (7.53), (7.55), and (7.58)–(7.63) that  g2 .z/  I g1 .0; L1 ; x; u/ C 0 =32  L1 .g1 / C 0 =32 C 0 =8   g1 .z/ C 70 =32:

(7.64)

By (7.43), (7.44), (7.52), (7.58), (7.61), and property (ix), jŒI g2 .0; L1 ; x; u/  L1 .g2 /   g2 .x.L1 // ŒI g1 .0; L1 ; x; u/  L1 .g1 /   g1 .x.L1 //j  0 =16 C 0 =16: Therefore, in view of (7.44) and the relation above, we have shown that the following property holds: (C)

for each Q g1 ; g2 2 U \ F;

each z 2 Rn satisfying  g1 .z/  inf. g1 / C 1 and each .g1 ; A; B/-overtaking optimal pair .x; u/ 2 X.A; B; 0; 1/ satisfying x.0/ D z we have  g2 .z/   g1 .z/ C 0 =4; jŒI g2 .0; L1 ; x; u/  L1 .g2 /   g2 .x.L1 // ŒI g1 .0; L1 ; x; u/  L1 .g1 /   g1 .x.L1 //j  0 =8: Property (C) implies that for each Q g1 ; g2 2 U \ F; we have j inf. g2 /  inf. g1 /j  0 =4:

(7.65)

Q g 2 U \ F;

(7.66)

Assume that

and .x; u/ 2 X.A; B; 0; 1/ is a .g; A; B/-overtaking optimal pair satisfying  g .x.0//  inf. g / C ı:

(7.67)

7.4 Proof of Theorem 7.5

227

By Property (C) and (7.65)–(7.67),  f .x.0//   g .x.0//C0 =4  inf. g /Cı C0 =4  inf. f /Cı C0 =2:

(7.68)

Since .x; u/ 2 X.A; B; 0; 1/ is a .g; A; B/-overtaking optimal pair it follows from (7.65)–(7.77), Proposition 3.12, and property (C) that I f .0; L1 ; x; u/  L1 .f /   f .x.L1 // C  f .x.0//  0 =8 C ŒI g .0; L1 ; x; u/  L1 .g/   g .x.L1 // C  g .x.0//   g .x.0// C  f .x.0//  0 =8   g .x.0// C  f .x.0//  0 =8 C 0 =4:

(7.69)

By (7.68), (7.69), the relation L1 > L0 , property (v), and Proposition 3.11, there exists an .f ; A; B/-overtaking optimal pair .; / 2 X.A; B; 0; 1/ such that  f ..0// D inf. f /; jx.t/  .t/j   for all t 2 Œ0; L0 : Lemma 7.13 is proved.

7.4 Proof of Theorem 7.5 Denote by E the set of all f 2 FQ for which there exists a unique point zf 2 Rn satisfying  f .zf / D inf. f /: Lemma 7.14. The set E is an everywhere dense subset of M . Q In order to prove the lemma it is sufficient to show that for every Proof. Let f 2 F. neighborhood V of f in M we have V \ E 6D ;. There are two cases:  f .xf / > inf. f /I

(7.70)

 .xf / D inf. /:

(7.71)

f

f

Assume that (7.70) holds. There exists z0 2 Rn such that  f .z0 / D inf. f /:

(7.72)

By Theorem 3.8, there exists an .f ; A; B/-overtaking optimal pair .y; v/ 2 X.A; B; 0; 1/ such that y.0/ D z0 :

(7.73)

228

7 Genericity Results

(A3) implies that lim y.t/ D xf :

t!1

(7.74)

Together with (7.70) and (7.72) this implies that there exists  > 0 such that for all sufficiently large numbers t,  f .z0 / C  <  f .y.t//: Therefore there exists a number 0  0 such that  f .y.0 // D  f .z0 /;  f .y.t// >  f .z0 / for all t > 0 : We may assume without loss of generality that 0 D 0. Then  f .y.t// >  f .z0 / for all t > 0:

(7.75)

Since .y; v/ 2 X.A; B; 0; 1/ is a .f ; A; B/-overtaking optimal pair it follows from Propositions 3.12 and 3.15 that  f .z0 / D lim ŒI f .0; T; y; v/  T.f / D lim Œ f .y.0//   f .y.T//: T!1

T!1

(7.76)

By Lemma 7.8 and (7.74), there exists a bounded nonnegative function  2 C1 .Rn / such that for each sequence of nonnegative integers P p1 ; : : : ; pn , the function p @jpj =@x11 : : : @xnpn W Rn ! R1 is bounded, where jpj D niD1 pi and fx 2 Rn W .x/ D 0g D fxf g [ fy.t/ W t 2 Œ0; 1/g:

(7.77)

For any r 2 .0; 1/ define a function fr W Rn  Rm ! R1 by fr .x; y/ D f .x; y/ C r.x/; .x; y/ 2 RnCm :

(7.78)

Let r 2 .0; 1/. Clearly, fr 2 Mc . In view of (7.78), .fr /  .f /: Since .y; v/ 2 X.A; B; 0; 1/ is a .f ; A; B/-good pair it follows from (7.77) and (7.78) that .fr /  lim inf T 1 I fr .0; L1 ; y; v/ T!1

D lim inf T 1 I f .0; L1 ; y; v/ D .f /: T!1

7.4 Proof of Theorem 7.5

229

Thus .fr / D .f / D f .xf ; uf / D fr .xf ; uf /

(7.79)

where .xf ; uf / 2 L.f /. It is easy now to see that fr satisfies assumption (A2). Proposition 3.4, (A3) which holds for the function f , and (7.79) imply that if .x; u/ 2 X.A; B; 0; 1/ is an .fr ; A; B/-good pair, then it is also an .f ; A; B/-good pair and limt!1 x.t/ D xf . Thus fr satisfies assumption (A3). Therefore we have shown that for each r 2 .0; 1/, Q .fr / D .f /; L.fr / D L.f /; fr 2 F:

(7.80)

Let r 2 .0; 1/. We show that fr 2 E. By Proposition 3.11, (3.12), (7.73), (7.77), (7.78), and (7.80),  fr .z0 /  lim infŒI fr .0; T; y; v/  T.fr / T!1

D lim infŒI f .0; T; y; v/  T.f / D  f .z0 /: T!1

(7.81)

Proposition 3.11, (3.12), (7.78), (7.80), and (7.81) imply that  fr .z/   f .z/ for all z 2 Rn ;  fr .z0 / D  f .z0 /:

(7.82)

Let z 2 Rn n fz0 g. We show that  fr .z/ >  fr .z0 /. If z 2 fy.t/ W t 2 Œ0; 1/g [ fxf g; then in view of (7.70), (7.72), (7.75), and (7.82),  fr .z0 / D  f .z0 / <  f .z/   fr .z/: Assume that z 62 fy.t/ W t 2 Œ0; 1/g [ fxf g:

(7.83)

By Theorem 3.8 and (7.80), there exists an .fr ; A; B/-overtaking optimal pair .x; u/ 2 X.A; B; 0; 1/ such that x.0/ D z:

(7.84)

Since .x; u/ is an .fr ; A; B/-overtaking optimal pair it follows from Propositions 3.11 and 3.15, (7.72), (7.77), (7.78), (7.80), (7.82), (7.83), and (7.84) that

230

7 Genericity Results

 fr .z/ D lim ŒI fr .0; T; x; u/  T.f / T!1  Z D lim I f .0; T; x; u/  T.f / C r T!1

  f .z/ C r

Z

1

0

T 0

 .u.t//dt

.u.t//dt >  f .z/   fr .z0 /:

Thus we have shown that  fr .z/ >  fr .z0 / for all z 2 Rn n fz0 g; fr 2 E for all r 2 .0; 1/ and for any neighborhood V of f in M we have V \ E 6D ;. Assume that (7.71) holds. By Lemma 7.8, there exists a bounded nonnegative function  2 C1 .Rn / such that for each sequence of nonnegative integers p1 jpj pn n 1 pP 1 ; : : : ; pn , the function @ =@x1 : : : @xn W R ! R is bounded, where jpj D n iD1 pi and fx 2 Rn W .x/ D 0g D fxf g:

(7.85)

For any r 2 .0; 1/ define a function fr W Rn  Rm ! R1 by fr .x; y/ D f .x; y/ C r.x/; .x; y/ 2 RnCm :

(7.86)

Let r 2 .0; 1/. Clearly, fr 2 Mc and .fr / D .f / D f .xf ; uf / D fr .xf ; uf /

(7.87)

where .xf ; uf / 2 L.f / D L.fr /. Assume that .x; u/ 2 X.A; B; 0; 1/ is an .fr ; A; B/-good pair. Then by (7.87), Proposition 3.4, and (A3) which holds for f , .x; u/ is also an .f ; A; B/-good pair Q It and limt!1 x.t/ D xf . Thus fr satisfies assumptions (A2) and (A3) and fr 2 F. follows from Propositions 3.11 and 3.13, (3.12) and (7.86) that  fr .xf / D  f .xf / D 0;  fr .z/   f .z/ for all z 2 Rn :

(7.88)

Let z 2 Rn n fxf g. By Theorem 3.8, there exists an .fr ; A; B/-overtaking optimal pair .x; u/ 2 X.A; B; 0; 1/ such that x.0/ D z: It follows from the relation above, Propositions 3.11 and 3.15, (7.71), and (7.85)– (7.87) that

7.4 Proof of Theorem 7.5

231

 fr .z/ D lim ŒI fr .0; T; x; u/  T.f / T!1

D lim

T!1



Z I .0; T; x; u/  T.f / C r

T

f

Z   .z/ C r f

0

1

0

 .u.t//dt

.u.t//dt >  f .z/   fr .xf /:

Thus we have shown that fr 2 E for all r 2 .0; 1/ and for any neighborhood V of f in M we have V \ E 6D ;. This completes the proof of Lemma 7.14. Completion of the Proof of Theorem 7.5. By Lemma 7.14, the set E is everywhere dense. For each f 2 E there exist xf ; zf 2 Rn , uf 2 Rm such that Axf C Buf D 0; f .xf ; uf / D .f /;  f .zf / D inf. f /; fzf g D fz 2 Rn W  f .z/ D inf. f /g:

(7.89)

Let f 2 E and k  1 be an integer. By (7.13), there exists an open neighborhood U .f ; k/ of f in M such that the following property holds: (a) for each g 2 U .f ; k/ \ FQ and each z 2 Rn satisfying  g .z/ D inf. g / we have jz  zf j  2k . Set Q F D \1 kD1 [ fU .f ; k/ W f 2 Eg \ F:

(7.90)

Clearly, F is a countable intersection of open everywhere dense subsets of M and Q F  F. Let g 2 F, z1 ; z2 2 Rn and  g .zi / D inf. g /; i D 1; 2:

(7.91)

Let k  1 be an integer. In view of (7.90), there exist fk 2 E such that g 2 U .fk ; k/: By the relation above, (7.91), and property (a), jzi  zfk j  2k ; i D 1; 2 and jz1  z2 j  2kC1 for any integer k  1. This implies that z1 D z2 . Theorem 7.5 is proved. t u

Chapter 8

Variational Problems with Extended-Valued Integrands

In this chapter we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous extended-valued integrand. In our recent research we showed that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. In this chapter our goal is to study the structure of approximate solutions in regions close to the endpoints of the time intervals.

8.1 Existence of Solutions and Their Turnpike Properties In this chapter we consider the following variational problems Z

T

0

f .v.t/; v 0 .t//dt ! min;

(P1 )

v W Œ0; T ! Rn is an absolutely continuous (a.c.) function such that v.0/ D x; v.T/ D y; Z 0

T

f .v.t/; v 0 .t//dt ! min;

(P2 )

v W Œ0; T ! Rn is an a. c. function such that v.0/ D x

© Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6_8

233

234

8 Variational Problems with Extended-Valued Integrands

and Z 0

T

f .v.t/; v 0 .t//dt ! min;

(P3 )

v W Œ0; T ! Rn is an a. c. function, where x; y 2 Rn . Here Rn is the n-dimensional Euclidean space with the Euclidean norm j  j and f W Rn  Rn ! R1 [ f1g is an extended-valued integrand. In [46, 49, 53] we studied the problems (P1 ) and (P2 ) and showed under certain assumptions that the turnpike property holds and that the turnpike xN is a unique solution of the minimization problem f .x; 0/ ! min, x 2 Rn . In this chapter which is based on [55] we study the structure of approximate solutions of the problems (P2 ) and (P3 ) in regions close to the endpoints of the time intervals. We show that in regions close to the right endpoint T of the time interval these approximate solutions are determined only by the integrand, and are essentially independent of the choice of interval and endpoint value x. For the problems (P3 ), approximate solutions are determined only by the integrand function also in regions close to the left endpoint 0 of the time interval. More precisely, we define fN .x; y/ D f .x; y/ for all x; y 2 Rn and consider the set P.fN / of all solutions of a corresponding infinite horizon variational problem associated with the integrand fN . For a given pair of real positive numbers , , we show that if T is large enough and v W Œ0; T ! Rn is an approximate solution of the problem (P2 ), then jv.T  t/  y.t/j   for all t 2 Œ0;  , where y./ 2 P.fN /. The prototype of our result was obtained in [47] under an additional assumption that the function f W Rn  Rn ! R1 [ f1g is strictly convex. This assumption played a crucial role there. In this chapter the integrand f is a nonconvex function. We denote by mes.E/ the Lebesgue measure of a Lebesgue measurable set E  R1 , denote by j  j the Euclidean norm of the space Rn and by h; i the inner product of Rn . For each function f W X ! R1 [ f1g, where X is a nonempty, set dom.f / D fx 2 X W f .x/ < 1g: Let a be a real positive number, such that lim

W Œ0; 1/ ! Œ0; 1/ be an increasing function

t!1

.t/ D 1

(8.1)

and let f W Rn  Rn ! R1 [ f1g be a lower semicontinuous function such that the set dom.f / D f.x; y/ 2 Rn  Rn W f .x; y/ < 1g

(8.2)

8.1 Existence of Solutions and Their Turnpike Properties

235

is nonempty, convex, and closed and that .jyj/jyjg  a for each x; y 2 Rn :

f .x; y/  maxf .jxj/;

(8.3)

Recall that a function v defined on an infinite subinterval of R1 with values in Rn is called absolutely continuous (a. c.) if v is absolutely continuous on any finite subinterval of its domain. For each x; y 2 Rn and each T > 0 define Z .f ; T; x/ D inf

T

0

f .v.t/; v 0 .t//dt W v W Œ0; T ! Rn

 is an a. c. function satisfying v.0/ D x ; Z .f ; T; x; y/ D inf

T

0

f .v.t/; v 0 .t//dt W v W Œ0; T ! Rn

 is an a. c. function satisfying v.0/ D x; v.T/ D y ; Z .f ; T/ D inf

T

0

(8.5)

f .v.t/; v 0 .t//dt W v W Œ0; T ! Rn

 is an a.c. function ; Z O .f ; T; y/ D inf

(8.4)

T 0

(8.6)

f .v.t/; v 0 .t//dt W v W Œ0; T ! Rn

 is an a. c. function satisfying v.T/ D y :

(8.7)

(Here we assume that infimum over an empty set is infinity.) We suppose that there exists a point xN 2 Rn such that f .Nx; 0/  f .x; 0/ for each x 2 Rn

(8.8)

and that the following assumptions hold: (A1) .Nx; 0/ is an interior point of the set dom.f / and the function f is continuous at the point .Nx; 0/; (A2) for each M > 0 there exists cM > 0 such that .f ; T; x/  Tf .Nx; 0/  cM for each x 2 Rn satisfying jxj  M and each real number T > 0;

236

(A3)

8 Variational Problems with Extended-Valued Integrands

for each x 2 Rn the function f .x; / W Rn ! R1 [ f1g is convex.

Assumption (A2) implies that for each a.c. function v W Œ0; 1/ ! Rn the function Z

T

T! 0

f .v.t/; v 0 .t//dt  Tf .Nx; 0/; T 2 .0; 1/

is bounded from below. It should be mentioned that inequality (8.8) and assumptions (A1)–(A3) are common in the literature and hold for many infinite horizon optimal control problems. In particular, we need inequality (8.8) and assumption (A2) in the cases when the problems (P1 ) and (P2 ) possess the turnpike property and the point xN is its turnpike. Assumption (A2) means that the constant function v.t/ N D xN , t 2 Œ0; 1/ is an approximate solution of the infinite horizon variational problem with the integrand f related to the problems (P1 ) and (P2 ). We say that an a. c. function v W Œ0; 1/ ! Rn is .f /-good [44, 51] if ˇ Z ˇ sup ˇˇ

T 0

ˇ  ˇ f .v.t/; v .t//dt  Tf .Nx; 0/ˇˇ W T 2 .0; 1/ < 1: 0

The following result was obtained in [46]. Proposition 8.1. Let v W Œ0; 1/ ! Rn be an a.c. function. Then either the function v is .f /-good or Z

T 0

f .v.t/; v 0 .t//dt  Tf .Nx; 0/ ! 1 as T ! 1:

Moreover, if the function v is .f /-good, then supfjv.t/j W t 2 Œ0; 1/g < 1. For each pair of numbers T1 2 R1 , T2 > T1 and each a.c. function v W ŒT1 ; T2  ! R put n

Z I f .T1 ; T2 ; v/ D

T2

f .v.t/; v 0 .t//dt

(8.9)

T1

and for any T 2 ŒT1 ; T2  set I f .T; T; v/ D 0. For each M > 0 denote by XM;f the set of all x 2 Rn such that jxj  M and there exists an a.c. function v W Œ0; 1/ ! Rn which satisfies v.0/ D x; I f .0; T; v/  Tf .Nx; 0/  M for each T 2 .0; 1/:

(8.10)

It is clear that [fXM;f W M 2 .0; 1/g is the set of all points x 2 X for which there exists an .f /-good function v W Œ0; 1/ ! Rn such that v.0/ D x.

8.1 Existence of Solutions and Their Turnpike Properties

237

We suppose that the following assumption holds: (A4) (the asymptotic turnpike property) for each .f /-good function v W Œ0; 1/ ! Rn , limt!1 jv.t/  xN j D 0. Examples of integrands f which satisfy assumptions (A1)–(A4) are considered in [46, 53]. The following turnpike result for the problem (P2 ) was established in [46]. Theorem 8.2. Let ; M be positive numbers. Then there exist an integer L  1 and a real number ı > 0 such that for each real number T > 2L and each a.c. function v W Œ0; T ! Rn which satisfies v.0/ 2 XM;f and I f .0; T; v/  .f ; T; v.0// C ı there exist a pair of numbers 1 2 Œ0; L and 2 2 ŒT  L; T such that jv.t/  xN j   for all t 2 Œ1 ; 2  and if jv.0/  xN j  ı, then 1 D 0. Let M > 0. Denote by YM;f the set of all points x 2 Rn for which there exist a number T 2 .0; M and an a. c. function v W Œ0; T ! Rn such that v.0/ D xN , v.T/ D x and I f .0; T; v/  M. The following turnpike results for the problems (P1 ) were established in [49]. Theorem 8.3. Let ; M0 ; M1 > 0. Then there exist numbers L; ı > 0 such that for each number T > 2L, each point z0 2 XM0 ;f and each point z1 2 YM1 ;f , the value .f ; T; z0 ; z1 / is finite and for each a.c. function v W Œ0; T ! Rn which satisfies v.0/ D z0 ; v.T/ D z1 ; I f .0; T; v/  .f ; T; z0 ; z1 / C ı there exists a pair of numbers 1 2 Œ0; L; 2 2 ŒT  L; T such that jv.t/  xN j  ; t 2 Œ1 ; 2 : Moreover if jv.0/  xN j  ı, then 1 D 0 and if jv.T/  xN j  ı, then 2 D T. In the sequel we use a notion of an overtaking optimal function [44, 53]. An a.c. function v W Œ0; 1/ ! Rn is called .f /-overtaking optimal if for each a.c. function u W Œ0; 1/ ! Rn satisfying u.0/ D v.0/ the inequality lim supŒI f .0; T; v/  I f .0; T; u/  0 T!1

holds. The following result which establishes the existence of an overtaking optimal function was obtained in [46].

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8 Variational Problems with Extended-Valued Integrands

Theorem 8.4. Assume that x 2 Rn and that there exists an .f /-good function v W Œ0; 1/ ! Rn satisfying v.0/ D x. Then there exists an .f /-overtaking optimal function u W Œ0; 1/ ! Rn such that u .0/ D x. Assumption (A1) implies that there exists a number rN 2 .0; 1/ such that: ˝0 WD f.x; y/ 2 Rn  Rn W jx  xN j  rN and jyj  rN g  dom.f /I 0 WD supfjf .z1 ; z2 /j W .z1 ; z2 / 2 ˝0 g < 1:

(8.11) (8.12)

It is easy to see that the value .f ; T; x; y/ is finite for each number T  1 and each pair of points x; y 2 Rn such that jx  xN j; jy  xN j  rN =2: Let M > 0. Denote by YN M;f the set of all points x 2 Rn such that jxj  M and for which there exist a number T 2 .0; M and an a. c. function v W Œ0; T ! Rn such that v.0/ D x, v.T/ D xN and I f .0; T; v/  M. It is easy to see that the following result holds. Proposition 8.5. For each M > 0 there exists M0 > 0 such that YN M;f  XM0 ;f . The next result follows from Lemma 8.27 and (A1). Proposition 8.6. For each M > 0 there exists M0 > 0 such that XM;f  YN M0 ;f . An a. c. function v W Œ0; 1/ ! Rn is called .f /-minimal [5, 44] if for each T1  0, each T2 > T1 and each a.c. function u W ŒT1 ; T2  ! Rn satisfying u.Ti / D v.Ti /, i D 1; 2, we have Z

T2

T1

f .v.t/; v 0 .t//dt 

Z

T2

f .u.t/; u0 .t//dt:

T1

Following theorem obtained in [48] shows that the optimality notions introduced above are equivalent. Theorem 8.7. Assume that x 2 Rn and that there exists an .f /-good function vQ W Œ0; 1/ ! Rn satisfying v.0/ Q D x. Let v W Œ0; 1/ ! Rn be an a.c. function such that v.0/ D x. Then the following conditions are equivalent: (i) the function v is .f /-overtaking optimal; (ii) the function v is .f /-good and .f /-minimal; (iii) the function v is .f /-minimal and limt!1 v.t/ D xN ; (iv) the function v is .f /-minimal and lim inft!1 jv.t/  xN j D 0. The following two theorems obtained in [48] describe the asymptotic behavior of overtaking optimal functions. Theorem 8.8. Let  be a positive number. Then there exists a positive number ı such that: (i) For each point x 2 Rn satisfying jx  xN j  ı there exists an .f /-overtaking optimal and .f /-good function v W Œ0; 1/ ! Rn such that v.0/ D x.

8.2 Preliminaries

239

(ii) For each .f /-overtaking optimal function v W Œ0; 1/ ! Rn satisfying jv.0/  xN j  ı, the inequality jv.t/  xN j   holds for all numbers t 2 Œ0; 1/. Theorem 8.9. Let ; M > 0. Then there exists L > 0 such that for each x 2 XM;f and each .f /-overtaking optimal function v W Œ0; 1/ ! Rn satisfying v.0/ D x the following inequality holds: jv.t/  xN j   for all t 2 ŒL; 1/: In Sect. 8.5 we prove the following turnpike result for approximate solutions of the problems of the type (P3 ). Theorem 8.10. Let  > 0. Then there exist numbers L; ı > 0 such that for each number T > 2L and each a. c. function v W Œ0; T ! Rn which satisfies I f .0; T; v/  .f ; T/ C ı there exists a pair of numbers 1 2 Œ0; L; 2 2 ŒT  L; T such that jv.t/  xN j  ; t 2 Œ1 ; 2 : Moreover if jv.0/  xN j  ı, then 1 D 0 and if jv.T/  xN j  ı; then 2 D T.

8.2 Preliminaries We use the notation, definitions, and assumptions introduced in Sect. 8.1. We define a function  f .x/, x 2 Rn which plays an important role in our study. For all x 2 Rn n [fXM;f W M 2 .0; 1/g set  f .x/ D 1: Let x 2 [fXM;f W M 2 .0; 1/g:

(8.13)

Denote by .f ; x/ the set of all .f /-overtaking optimal functions v W Œ0; 1/ W Rn satisfying v.0/ D x. By (8.13) and Theorem 8.4, the set .f ; x/ is nonempty. In view of (8.13), any element of .f ; x/ is an .f /-good function. Define  f .x/ D lim infŒI f .0; T; v/  Tf .Nx; 0/; T!1

(8.14)

240

8 Variational Problems with Extended-Valued Integrands

where v 2 .f ; x/. Clearly,  f .x/ does not depend on the choice of v. In view of (A2) and (8.13),  f .x/ is finite. Definition (8.14) and the definition of overtaking optimal functions imply the following result. Proposition 8.11. 1. Let v W Œ0; 1/ ! Rn be an .f /-good function. Then  f .v.0//  lim infŒI f .0; T; v/  Tf .Nx; 0/ T!1

and for each T  0 and each S > T,  f .v.T//  I f .T; S; v/  .S  T/f .Nx; 0/ C  f .v.S//:

(8.15)

2. Let S > T  0 and v W Œ0; S ! Rn be an a. c. function such that  f .v.T//;  f .v.S// < 1. Then (8.15) holds. The next result follows from definition (8.14). Proposition 8.12. Let v W Œ0; 1/ ! Rn be an .f /-overtaking optimal and .f /-good function. Then for each T  0 and each S > T,  f .v.T// D I f .T; S; v/  .S  T/f .Nx; 0/ C  f .v.S//: Proposition 8.13.  f .Nx/ D 0. Proof. Set v.t/ D xN for all t  0. By Theorem 8.7 and (A2), the function v is a .f /-overtaking optimal. In view of (8.14),  f .Nx/ D 0. The following two results are proved in Sect. 8.6. Proposition 8.14. The function  f is finite in a neighborhood of xN and continuous at xN . Proposition 8.15. For each M > 0 the set fx 2 Rn W  f .x/  Mg is bounded. (Here we assume that an empty set is bounded.) Set inf. f / D inff f .z/ W z 2 Rn g:

(8.16)

By (A2) and Proposition 8.15, inf. f / is finite. Set Xf D fx 2 Rn W  f .x/  inf. f / C 1g:

(8.17)

Proposition 8.16. Assume that x 2 [fXM;f W M 2 .0; 1/g and v 2 .f ; x/. Then  f .x/ D lim ŒI f .0; T; v/  Tf .Nx; 0/: T!1

8.2 Preliminaries

241

Proof. It follows from (A4) and Propositions 8.12–8.14 that  f .x/ D lim . f .v.0//   f .v.T/// D lim ŒI f .0; T; v/  Tf .Nx; 0/: T!1

T!1

Proposition 8.16 is proved. The next result is proved in Sect. 8.6. Proposition 8.17. There exists M > 0 such that Xf  XM;f . Propositions 8.6 and 8.17 imply the following result. Proposition 8.18. There exists L > 0 such that Xf  YN L;f . The following result is proved in Sect. 8.6. Proposition 8.19. The function  f W Rn ! R1 [ f1g is lower semicontinuous. Set D.f / D fx 2 Rn W  f .x/ D inf. f /g:

(8.18)

By Propositions 8.15 and 8.19, the set D.f / is nonempty bounded and closed subset of Rn . The following proposition is proved in Sect. 8.6. Proposition 8.20. Let v W Œ0; 1/ ! Rn be an .f /-good function such that for all T > 0, I f .0; T; v/  Tf .Nx; 0/ D  f .v.0//   f .v.T//:

(8.19)

Then v is an .f /-overtaking optimal function. The next result easily follows from (8.17), (8.18), Proposition 8.17, and Theorem 8.9. Proposition 8.21. For each  > 0 there exists T > 0 such that for each z 2 D.f / and each v 2 .f ; z/ the inequality jv.t/  xN j   holds for all t  T . In order to study the structure of solutions of the problems (P2 ) and (P3 ) we introduce the following notation and definitions. Define a function fN W Rn  Rn ! R1 [ f1g by fN .x; y/ D f .x; y/ for all x; y 2 Rn :

(8.20)

Clearly, dom.fN / D f.x; y/ 2 Rn  Rn W .x; y/ 2 dom.f /g;

(8.21)

242

8 Variational Problems with Extended-Valued Integrands

dom.fN / is a nonempty closed convex set, fN is a lower semicontinuous function satisfying fN .x; y/  maxf .jxj/;

.jyj/jyjg  a for each x; y 2 Rn :

(8.22)

The notation introduced for the function f is also used for the function fN . Namely, for each pair of numbers T1 2 R1 , T2 > T1 , and each a.c. function v W ŒT1 ; T2  ! Rn put Z

fN

T2

I .T1 ; T2 ; v/ D

fN .v.t/; v 0 .t//dt

T1

and for each x; y 2 Rn and each T > 0 define N .fN ; T; x/ D inffI f .0; T; v/ W v W Œ0; T ! Rn

is an a. c. function satisfying v.0/ D xg; N

.fN ; T; x; y/ D inffI f .0; T; v/ W v W Œ0; T ! Rn is an a. c. function satisfying v.0/ D x; v.T/ D yg; N

.fN ; T/ D inffI f .0; T; v/ W v W Œ0; T ! Rn is an a.c. functiong; N

O .fN ; T; y/ D inffI f .0; T; v/ W v W Œ0; T ! Rn is an a. c. function satisfying v.T/ D yg: Let v W Œ0; T ! Rn be an a.c. function. Set v.t/ N D v.T  t/; t 2 Œ0; T: It is easy to see that Z

T 0

fN .v.t/; N vN 0 .t//dt D

Z Z

T 0 T

D Z

0 T

D 0

f .v.t/; N vN 0 .t//dt f .v.T  t/; v 0 .T  t//dt f .v.t/; v 0 .t//dt:

(8.23)

Clearly, for all x 2 Rn , fN .Nx; 0/ D f .Nx; 0/  f .x; 0/ D fN .x; 0/;

(8.24)

8.3 The Main Results

243

.Nx; 0/ is an interior point of the set dom.fN / and the function fN is continuous at the point .Nx; 0/. Thus (A1) holds for the function fN . Clearly, for each x 2 Rn the function fN .x; 0/ W Rn ! R1 [ f1g is convex. Thus (A3) holds for fN . The next result easily follows from (8.23). Proposition 8.22. Let T > 0, M  0, and vi W Œ0; T ! Rn , i D 1; 2 be a. c. functions. Then I f .0; T; v1 /  I f .0; T; v2 /  M if and only if N

N

I f .0; T; vN 1 /  I f .0; T; vN 2 /  M: Proposition 8.22 implies the following result. Proposition 8.23. Let T > 0 and v W Œ0; T ! Rn be an a. c. function. Then the following assertions hold: I f .0; T; v/  .f ; T/ C M if and only if N N  .fN ; T/ C MI I f .0; T; v/

I f .0; T; v/  .f ; T; v.0/; v.T// C M if and only if N N  .fN ; T; v.0/; N v.T// N C MI I f .0; T; v/ N N  .fN ; T; v.0// N C MI I f .0; T; v/  O .f ; T; v.T// C M if and only if I f .0; T; v/ N

N  O .fN ; T; v.T// N C M: I f .0; T; v/  .f ; T; v.0// C M if and only if I f .0; T; v/ The next result is proved in Sect. 8.6. Proposition 8.24. 1. For each M > 0 there exists cM > 0 such that .fN ; T; x/  T fN .Nx; 0/  cM for each x 2 Rn satisfying jxj  M and each T > 0. 2. For each .fN /-good function v W Œ0; 1/ ! Rn , limt!1 v.t/ D xN : In view of Proposition 8.24 the function fN satisfies assumptions (A2) and (A4). We have already mentioned that assumptions (A1) and (A3) hold for the function fN . Therefore all the results stated above for the function f are also true for the function fN .

8.3 The Main Results We use the notation, definitions, and assumptions introduced in Sects. 8.1 and 8.2. Recall that f W Rn  Rn ! R1 [ f1g is a lower semicontinuous function with a nonempty closed convex dom.f / satisfying (8.3), xN 2 Rn satisfies (8.8), and that (A1)–(A4) hold.

244

8 Variational Problems with Extended-Valued Integrands

The following two theorems which describe the structure of approximate solutions of the problems (P1 ) and (P2 ) are proved in Sects. 8.8 and 8.9, respectively. Theorem 8.25. Let L0 > 0, 0 > 0 and  > 0. Then there exist ı > 0 and T0 > 0 such that for each T  T0 and each a. c. function v W Œ0; T ! Rn which satisfies v.0/ 2 YN L0 ;f ; I f .0; T; v/  .f ; T; v.0// C ı there exists an .fN /-overtaking optimal function v  W Œ0; 1/ ! Rn such that v  .0/ 2 D.fN / and jv.T  t/  v  .t/j   for all t 2 Œ0; 0 . Theorem 8.26. Let 0 > 0 and  > 0. Then there exist ı > 0 and T0 > 0 such that for each T  T0 and each a. c. function v W Œ0; T ! Rn which satisfies I f .0; T; v/  .f ; T/ C ı there exist an .f /-overtaking optimal function u W Œ0; 1/ ! Rn and an .fN /overtaking optimal function v  W Œ0; 1/ ! Rn such that u .0/ 2 D.f /, v  .0/ 2 D.fN / and that for all t 2 Œ0; 0 , jv.t/  u .t/j   and jv.T  t/  v  .t/j  : The results of this section were obtained in [55]. Note that in [47] analogous results were obtained for strictly convex integrands.

8.4 Auxiliary Results This section contains several auxiliary results which will be used in the paper. Lemma 8.27 ([46]). Let M;  > 0. Then there exists L0 > 0 such that for each number T  L0 , each a.c. function v W Œ0; T ! Rn satisfying jv.0/j  M; I f .0; T; v/  Tf .Nx; 0/ C M and each number s 2 Œ0; T  L0  the inequality minfjv.t/  xN j W t 2 Œs; s C L0 g   holds. Lemma 8.28 ([46]). Let  > 0. Then there exists a number ı 2 .0; rN =2/ such that for each number T  2 and each a.c. function v W Œ0; T ! Rn which satisfies

8.4 Auxiliary Results

245

jv.0/  xN j; jv.T/  xN j  ı; I f .0; T; v/  .f ; T; v.0/; v.T// C ı the inequality jv.t/  xN j   is true for all numbers t 2 Œ0; T. Lemma 8.29 ([46]). Let M0 ; M1 > 0. Then there exists M2 > 0 such that for each T > 0 and each a.c. function v W Œ0; T ! Rn which satisfies jv.0/j  M0 ; I f .0; T; v/  Tf .Nx; 0/ C M1 the following inequality holds: jv.t/j  M2 for all t 2 Œ0; T: Proposition 8.30 ([46]). Let T > 0 and let vk W Œ0; T ! Rn , k D 1; 2; : : : be a sequence of a.c. functions such that the sequence fI f .0; T; vk /g1 kD1 is bounded and that the sequence fvk .0/g1 is bounded. Then there exist a strictly increasing kD1 n sequence of natural numbers fki g1 and an a.c. function v W Œ0; T ! R such that iD1 vki .t/ ! v.t/ as i ! 1 uniformly on Œ0; T; I f .0; T; v/  lim inf I f .0; T; vki /: i!1

Lemma 8.31 ([46]). Let  > 0. Then there exists ı > 0 such that for each a.c. function v W Œ0; 1 ! Rn satisfying jv.0/  xN j, jv.1/  xN j  ı, I f .0; 1; v/  f .Nx; 0/  : Lemma 8.32. Let 0 < L0 < L1 and M0 > 0. Then there exists M1 > 0 such that for each T 2 ŒL0 ; L1  and each a. c. function v W Œ0; T ! Rn satisfying I f .0; T; v/  M0 the inequality jv.t/j  M1 holds for all t 2 Œ0; T. Proof. By (8.1), there exists c0 > 0 such that that for all t  0,

.t/  4 for all t  c0 . This implies

.t/t  4t  4c0 :

(8.25)

In view of (8.1), choose M1 > 0 such that .M1 =2/ > L01 M0 C aL1 L01 C 1; M1 > 2.M0 C aL1 C 4c0 L1 /:

(8.26)

Let T 2 ŒL0 ; L1  and an a. c. function v W Œ0; T ! Rn satisfy I f .0; T; v/  M0 . Together with (8.3) this implies that

246

8 Variational Problems with Extended-Valued Integrands

Z M0 C aL1 

Z

T

.jv.t/j/dt;

0

T

0

.jv 0 .t/j/jv 0 .t/jdt:

(8.27)

By (8.25) and (8.27), Z M0 C aL1  4 Z 0

T 0

T

Z

0

.jv .t/j  c0 /dt  4c0 L1 C 4

T 0

jv 0 .t/jdt;

.jv 0 .t/j/dt  M0 C aL1 C 4c0 L1 :

Together with (8.26) this implies that supfjv.t2 /  v.t1 /j W t1 ; t2 2 Œ0; Tg  M0 C aL1 C 4c0 L1 < M1 =2:

(8.28)

Assume that maxfjv.t/j W t 2 Œ0; Tg > M1 : In view of (8.27) and (8.28), jv.t/j  M1 =2 for all t 2 Œ0; T; Z T .M1 =2/L0  .jv.t/j/dt  M0 C aL1 ; 0

.M1 =2/  L01 M0 C aL1 L01 : This contradicts (8.26). The contradiction we have reached proves Lemma 8.32.

8.5 Proof of Theorem 8.10 Theorem 8.10 easily follows from Lemma 8.28 and the following result. Lemma 8.33. Let M;  > 0. Then there exists L0 > 0 such that for each number T  L0 , each a.c. function v W Œ0; T ! Rn satisfying I f .0; T; v/  Tf .Nx; 0/ C M and each number s 2 Œ0; T  L0  the inequality minfjv.t/  xN j W t 2 Œs; s C L0 g   holds.

(8.29)

8.5 Proof of Theorem 8.10

247

Proof. In view of (8.1), there exists a number M0 > 0 such that .M0 / > jf .Nx; 0/j C a C 2:

(8.30)

By (A2), there exists c0 > 0 such that .f ; T; x/  Tf .Nx; 0/  c0 for each x 2 Rn satisfying jxj  M0 and each T > 0:

(8.31)

By Lemma 8.27, there exists L1 > 0 such that the following property holds: (Pi)

for each number T  L1 , each a.c. function u W Œ0; T ! Rn satisfying ju.0/j  M0 ; I f .0; T; u/  Tf .Nx; 0/ C M

and each number s 2 Œ0; T  L1 , minfju.t/  xN j W t 2 Œs; s C L1 g  : Choose L0 > 4L1 C M C c0 :

(8.32)

Assume that T  L0 and an a. c. function v W Œ0; T ! Rn satisfies (8.29). Let us show that minfjv.t/j W t 2 Œ0; Tg  M0 :

(8.33)

Assume the contrary. Then jv.t/j > M0 for all t 2 Œ0; T. Together with (8.3), (8.29), and (8.30) this implies that f .v.t/; v 0 .t//  f .Nx; 0/ C 2 for all t 2 Œ0; T and M  I f .0; T; v/  Tf .Nx; 0/  2T  L0 : This contradicts the choice of L0 (see (8.32)). Thus (8.33) holds. Set 0 D minft 2 Œ0; T W jv.t/j  M0 g:

(8.34)

In view of (8.29) and (8.34), M C Tf .Nx; 0/  I f .0; T; v/ D I f .0; 0 ; v/ C I f .0 ; T; v/:

(8.35)

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8 Variational Problems with Extended-Valued Integrands

By (8.3), (8.30), and (8.34), I f .0; 0 ; v/  0 . .M0 /  a/  0 .f .Nx; 0/ C 2/:

(8.36)

It follows from (8.31) and (8.34) that I f .0 ; T; v/  .T  0 /f .Nx; 0/  c0 :

(8.37)

By (8.35)–(8.37), M C Tf .Nx; 0/  0 .f .Nx; 0/ C 2/ C .T  0 /f .Nx; 0/  c0 ; 20  M C c0 :

(8.38)

In view of (8.29) and (8.36), I f .0 ; T; v/ D I f .0; T; v/  I f .0; 0 ; v/  Tf .Nx; 0/ C M  0 f .Nx; 0/  .T  0 /f .Nx; 0/ C M:

(8.39)

By (8.32) and (8.38), T  0  L0  M  c0 > 4L1 : It follows from (8.34), (8.39), the inequality above, and property (Pi) that for each number S satisfying ŒS; S C L1   Œ0 ; T, minfjv.t/  xN j W t 2 ŒS; S C L1 g  :

(8.40)

Let a number S satisfy ŒS; S C L0   Œ0; T: By (8.32), (8.38), and the relation above, E WD ŒS; S C L0  \ Œ0 ; T is a closed interval, mes.E/  L0  0  4L1 . Together with (8.40) this implies that minfjv.t/  xN j W t 2 ŒS; S C L0 g  minfjv.t/  xN j W t 2 Eg  : Lemma 8.33 is proved.

8.6 Proofs of Propositions 8.14, 8.15, 8.17, 8.19, 8.20 and 8.24

249

8.6 Proofs of Propositions 8.14, 8.15, 8.17, 8.19, 8.20 and 8.24 Proof of Proposition 8.14. By (8.14) and Theorem 8.8, there exists ı0 > 0 such that  f .x/ is finite for any x 2 Rn satisfying jx  xN j  ı0 . Let us show that  f is continuous at xN . Let  > 0. By (A1) there exists ı 2 .0; ı0 / such that for each .y; z/ 2 Rn  Rn satisfying jy  xN j C jzj  6ı we have .y; z/ 2 dom.f /; jf .y; z/  f .Nx; 0/j  :

(8.41)

jxi  xN j  ı; i D 1; 2:

(8.42)

Let x1 ; x2 2 Rn satisfy

In order to complete the proof of proposition it is sufficient to show that  f .x2 /   f .x1 / C . By the choice of ı0 and (8.14), there exists an .f /-overtaking optimal function v0 W Œ0; 1/ ! Rn such that v0 .0/ D x1 ;

(8.43)

 f .x1 / D lim infŒI f .0; T; v0 /  Tf .Nx; 0/:

(8.44)

T!1

Define v.t/ D x2 C t.x1  x2 /; t 2 Œ0; 1; v.t/ D v0 .t  1/; t 2 .1; 1:

(8.45)

In view of (8.43) and (8.45), v W Œ0; 1/ ! Rn is an a. c. function. By (8.42) and (8.45), for all t 2 .0; 1/, jv 0 .t/j D jx1  x2 j  2ı; jv.t/  xN j  3ı:

(8.46)

By (8.46) and the choice of ı (see (8.41)), for all t 2 .0; 1/, jf .v.t/; v 0 .t//  f .Nx; 0/j  :

(8.47)

It follows from (8.44), (8.45), (8.47), and Proposition 8.11 that  f .x2 /  lim infŒI f .0; T; v/  Tf .Nx; 0/ T!1

D I f .0; 1; v/  f .Nx; 0/ C lim infŒI f .0; T; v0 /  Tf .Nx; 0/   C  f .x1 /: T!1

250

8 Variational Problems with Extended-Valued Integrands

t u

This completes the proof of Proposition 8.14.

Proof of Proposition 8.15. Let M > 0 and suppose that the set fx 2 R W  .x/  Mg is nonempty. By Propositions 8.13 and 8.14, there exists ı > 0 such that for each x 2 Rn satisfying jx  xN j  ı,  f .x/ is finite and n

f

j f .x/j  1:

(8.48)

By Lemma 8.33, there exists L0 > 0 such that the following property holds: (a) for each T  L0 , each a.c. function v W Œ0; T ! Rn satisfying I f .0; T; v/  Tf .Nx; 0/ C M C 1 and each S 2 Œ0; T  L0 , minfjv.t/  xN j W t 2 ŒS; S C L0 g  ı: By Lemma 8.32, there exists M1 > 0 such that the following property holds: (b) for each S 2 Œ1; L0 C 1 and each a. c. function v W Œ0; S ! Rn satisfying I f .0; S; v/  .1 C L0 /jf .Nx; 0/j C M C 1 the inequality jv.t/j  M1 holds for all t 2 Œ0; S. Assume that x 2 Rn satisfies  f .x/  M:

(8.49)

In view of (8.14) and (8.49),  f .x/ D lim infŒI f .0; T; v/  Tf .Nx; 0/  M;

(8.50)

v 2 .f ; x/:

(8.51)

T!1

where

By (8.50) and property (a), there exists t0 2 Œ1; L0 C 1

(8.52)

jv.t0 /  xN j  ı:

(8.53)

such that

8.6 Proofs of Propositions 8.14, 8.15, 8.17, 8.19, 8.20 and 8.24

251

It follows from (8.53) and the choice of ı (see (8.48)) that j f .v.t0 //j  1:

(8.54)

By (8.49), (8.51), (8.54), and Proposition 8.12, I f .0; t0 ; v/  t0 f .Nx; 0/ D  f .v.0//   f .v.t0 //  M C 1:

(8.55)

In view of (8.52), (8.55), and property (b), jv.t/j  M1 for all t 2 Œ0; t0  and in view of (8.51), jxj D jv.0/j  M1 . Proposition 8.15 is proved. t u Proof of Proposition 8.17. By Proposition 8.15, there exists M0 > 0 such that Xf  fz 2 Rn W jzj  M0 g:

(8.56)

By Lemma 8.29, there exists M1 > 0 such that for each T > 0 and each a.c. function v W Œ0; T ! Rn which satisfies jv.0/j  M0 ; I f .0; T; v/  Tf .Nx; 0/ C inf. f / C 2 we have jv.t/j  M1 for all t 2 Œ0; T:

(8.57)

By (A2), there exists c1 > 0 such that for each T > 0 and each z 2 Rn satisfying jzj  M1 we have .f ; T; z/  Tf .Nx; 0/  c1 :

(8.58)

 f .x/  inf. f / C 1:

(8.59)

Let x 2 Xf . Then

By (8.59) and Proposition 8.16,  f .x/ D lim ŒI f .0; T; v/  Tf .Nx; 0/;

(8.60)

v 2 .f ; x/:

(8.61)

T!1

where

By (8.59) and (8.60), there exists a number T0 > 0 such that for all T  T0 , I f .0; T; v/  Tf .Nx; 0/  inf. f / C 2:

(8.62)

252

8 Variational Problems with Extended-Valued Integrands

By (8.56), (8.59), (8.61), (8.62), and the choice of M1 (see (8.57)), jv.t/j  M1 ; t 2 Œ0; 1/:

(8.63)

It follows from (8.62), (8.63), and the choice of c1 (see (8.58)) that for any T 2 .0; T0 /, I f .0; T; v/  Tf .Nx; 0/ D I f .0; T0 ; v/  T0 f .Nx; 0/ .I f .T; T0 ; v/  .T0  T/f .Nx; 0//  inf. f / C 2 C c1 :

(8.64)

In view of (8.61) and (8.63), x D v.0/ 2 XM;f , where M D j inf. f /j C 2 C c1 C M1 : t u

Proposition 8.17 is proved. Proof of Proposition 8.19. Assume that

fxk g1 kD1

 R , x 2 R and that n

n

lim xk D x:

k!1

(8.65)

We show that  f .x/  lim infk!1  f .xk /. We may assume that there exists a finite M0 WD lim  f .xk / k!1

(8.66)

and that  f .xk / is finite for all integers k  1. By Proposition 8.16, for each integer k  1,  f .xk / D lim ŒI f .0; T; vk /  Tf .Nx; 0/;

(8.67)

vk 2 .f ; xk /:

(8.68)

T!1

where

We may assume without loss of generality that for all integers k  1,  f .xk /  M0 C 1:

(8.69)

By (8.65), (8.67)–(8.69), and Lemma 8.29, there exists M1 > 0 such that for each integer k  1, jvk .t/j  M1 ; t 2 Œ0; 1/:

(8.70)

8.6 Proofs of Propositions 8.14, 8.15, 8.17, 8.19, 8.20 and 8.24

253

By (A2) there exists c1 > 0 such that for each T > 0 and each z 2 Rn satisfying jzj  M1 , .f ; T; z/  Tf .Nx; 0/  c1 :

(8.71)

Let k  1 be an integer and S > 0. By (8.67) and (8.69), there exists T > S such that I f .0; T; vk /  Tf .Nx; 0/  M0 C 2:

(8.72)

In view of the choice of c1 (see (8.71)) and (8.70), I f .S; T; vk /  .T  S/f .Nx; 0/  c1 :

(8.73)

Relations (8.72) and (8.73) imply that I f .0; S; vk /  Sf .Nx; 0/ D I f .0; T; vk /  Tf .Nx; 0/ .I f .S; T; vk /  .T  S/f .Nx; 0//  M0 C c1 C 2: Thus for any integer k  1 and any S > 0, I f .0; S; vk /  Sf .Nx; 0/  M2 ;

(8.74)

M2 D M0 C M1 C c1 C 2:

(8.75)

where

It follows from (8.68), (8.70), and (8.74) that for any integer k  1, xk D vk .0/ 2 XM2 ;f :

(8.76)

By (8.70), (8.74), and Proposition 8.30 using diagonalization process and reindexing if necessary we may assume without loss of generality that there exists an a.c. function v W Œ0; 1/ ! Rn such that for each natural number p, vk .t/ ! v.t/ as k ! 1 uniformly on Œ0; p;

(8.77)

I f .0; p; v/  lim inf I f .0; p; vk /:

(8.78)

k!1

Let  > 0. In view of Propositions 8.13 and 8.14, there exists a positive number ı < minf41 ; 41 rN g

(8.79)

254

8 Variational Problems with Extended-Valued Integrands

such that for each y 2 Rn satisfying jy  xN j  ı, j f .y/j D j f .y/   f .Nx/j  =4:

(8.80)

By (8.69), (8.76), and Theorem 8.9, there exists L > 0 such that for each integer k  1, jvk .t/  xN j  ı; t 2 ŒL; 1/:

(8.81)

In view of (8.80) and (8.81), for all integers k  1, jvk .L/  xN j  ı; j f .vk .L//j  =4:

(8.82)

It follows from (8.77), (8.80), and (8.82) that jv.L/  xN j  ı; j f .v.L//j  =4:

(8.83)

Let k  1 be an integer. By (8.68), (8.82), and Proposition 8.12, I f .0; L; vk / D  f .vk .0//   f .vk .L// C Lf .Nx; 0/   f .vk .0// C Lf .Nx; 0/ C =4: Together with (8.66), (8.76), and (8.78), this implies that I f .0; L; v/  lim infΠf .vk .0// C Lf .Nx; 0/ C =4 D lim  f .xk / C Lf .Nx; 0/ C =4: k!1

k!1

(8.84) By (8.65), (8.66), (8.76), (8.77), (8.83), (8.84), and Proposition 8.11,  f .x/ D  f .v.0//  I f .0; L; v/  Lf .Nx; 0/ C  f .v.L//  lim  f .xk / C =2: k!1

Since  is any positive number this completes the proof of Proposition 8.19.

t u

Proof of Proposition 8.20. In view of (A4), lim v.t/ D xN :

t!1

(8.85)

Theorem 8.4 implies that there exists an .f /-overtaking optimal function u W Œ0; 1/ ! Rn such that u.0/ D v.0/:

(8.86)

 f .v.0// D lim ŒI f .0; T; u/  Tf .Nx; 0/:

(8.87)

By (8.86) and Proposition 8.16, T!1

8.6 Proofs of Propositions 8.14, 8.15, 8.17, 8.19, 8.20 and 8.24

255

On the other hand, it follows from (8.19), (8.85), and Propositions 8.13 and 8.14 that for any T > 0, I f .0; T; v/  Tf .Nx; 0/ D  f .v.0//   f .v.T// !  f .v.0// as T ! 1:

(8.88)

By (8.87) and (8.88), lim ŒI f .0; T; u/  I f .0; T; v/ D 0:

T!1

This implies that v is an .f /-overtaking optimal function. Proposition 8.20 is proved. t u Proof of Proposition 8.24. First we prove assertion 1. Let M > 0. In view of (8.1) we may assume without loss of generality that .M/ > jf .Nx; 0/j C a C 2:

(8.89)

By (A2), there exists c0 > 0 such that for each S > 0 and each a. c. function u W Œ0; S ! Rn satisfying ju.0/j  M; I f .0; S; u/  Sf .Nx; 0/  c0 :

(8.90)

Assume that T > 0 and that v W Œ0; T ! Rn is an a. c. function satisfying jv.0/j  M:

(8.91)

In order to prove assertion 1 it is sufficient to show that N

I f .0; T; v/  Tf .Nx; 0/  c0 : Let u.t/ D v.t/ N D v.T  t/; t 2 Œ0; T:

(8.92)

In view of (8.23) and (8.92), N

I f .0; T; u/ D I f .0; T; v/:

(8.93)

ju.T/j  M:

(8.94)

0 D minft 2 Œ0; T W ju.t/j  Mg:

(8.95)

By (8.91) and (8.92),

Let

256

8 Variational Problems with Extended-Valued Integrands

In view of (8.94), 0 is well defined. By (8.95) and the choice of c0 (see (8.90)), I f .0 ; T; u/  .T  0 /f .Nx; 0/  c0 :

(8.96)

By (8.3), (8.89), and (8.95), I f .0; 0 ; u/  0 . .M/  a/  0 f .Nx; 0/: Together with (8.93) and (8.96) this implies that N

I f .0; T; v/ D I f .0; 0 ; u/ C I f .0 ; T; u/  T fN .Nx; 0/  c0 : Thus assertion 1 holds. Let us prove assertion 2. Note that the integrand fN satisfies (A1)–(A3). Let v W Œ0; 1/ ! Rn be an .fN /-good function. Then there exists M1 > 0 such that for each pair of integers T2 > T1  0, ˇZ ˇ ˇ ˇ

T2

T1

ˇ ˇ 0 N Œf .v.t/; v .t//  f .Nx; 0/dtˇˇ  M1 :

(8.97)

In order to complete the proof of assertion 2 it is sufficient to show that lim v.t/ D xN :

t!1

Let  > 0. By Proposition 8.1 applied to the function fN , there exists M2 > 0 such that jv.t/j  M2 for all t 2 Œ0; 1/:

(8.98)

By Lemma 8.28 and Proposition 8.23, there exists ı 2 .0; rN =2/ such that the following property holds: (i) for each number T  2 and each a.c. function w W Œ0; T ! Rn which satisfies jw.0/  xN j; jw.T/  xN j  ı; N

I f .0; T; w/  .fN ; T; w.0/; w.T// C ı the inequality jw.t/  xN j   is true for all numbers t 2 Œ0; T. By Lemma 8.27 and (8.23), there exists L0 > 0 such that the following property holds: (ii) for each a.c. function w W Œ0; L0  ! Rn satisfying N

jw.t/j  M1 C M2 ; t 2 Œ0; L0 ; I f .0; L0 ; w/  L0 f .Nx; 0/ C M1 C M2

8.7 The Basic Lemma for Theorem 8.25

257

we have minfjw.t/  xN j W t 2 Œ0; L0 g  ı: Since the function v is .fN /-good there exists 0 > 0 such that for each pair of numbers T2 > T1  0 , N I f .T1 ; T2 ; v/  .fN ; T2  T1 ; v.T1 /; v.T2 // C ı:

(8.99)

It follows from (8.97), (8.98), and property (ii) that there exists a strictly increasing sequence of numbers fSk g1 kD1 such that S1  0 ; SkC1  Sk C 2 for all natural numbers k; jv.Sk /  xN j  ı for all integers k  1:

(8.100)

By (8.99), (8.100), and property (i), for each integer k  1, jv.t/  xN j  ; t 2 ŒSk ; SkC1 : Therefore jv.t/  xN j   for all t  S1 and limt!1 v.t/ D xN . Proposition 8.24 is proved. u t

8.7 The Basic Lemma for Theorem 8.25 Lemma 8.34. Let T0 > 0 and  2 .0; 1/. Then there exists ı 2 .0; / such that for each a. c. function v W Œ0; T0  ! Rn which satisfies  f .v.0//  inf. f / C ı; I .0; T0 ; v/  T0 f .Nx; 0/   .v.0// C  .v.T0 //  ı f

f

f

(8.101) (8.102)

there exists an .f /-overtaking optimal function u W Œ0; 1/ ! Rn which satisfies  f .u.0// D inf. f /;

(8.103)

ju.t/  v.t/j   for all t 2 Œ0; T0 : Proof. Assume the contrary. Then there exist a sequence fık g1 kD1  .0; 1 and a sequence of a. c. functions vk W Œ0; T0  ! Rn , k D 1; 2; : : : such that lim ık D 0

k!1

(8.104)

258

8 Variational Problems with Extended-Valued Integrands

and that for each integer k  1 and each .f /-overtaking optimal function u W Œ0; 1/ ! Rn satisfying (8.103),  f .vk .0//  inf. f / C ık ;

(8.105)

I f .0; T0 ; vk /  T0 f .Nx; 0/   f .vk .0// C  f .vk .T0 //  ık ; supfju.t/  vk .t/j W t 2 Œ0; T0 g > :

(8.106) (8.107)

By (8.105) and Proposition 8.15, the sequence fvk .0/g1 kD1 is bounded. In view of (8.105) and (8.106), for each integer k  1,  f .vk .0// and  f .vk .T0 // are finite and the sequence fI f .0; T0 ; vk /g1 kD1 is bounded. By Proposition 8.30 and 1 f the boundedness of the sequences fvk .0/g1 kD1 and fI .0; T0 ; vk /gkD1 , extracting a subsequence and re-indexing if necessary, we may assume without loss of generality that there exists an a.c. function v W Œ0; T0  ! Rn such that vk .t/ ! v.t/ as k ! 1 uniformly on Œ0; T0 ;

(8.108)

I f .0; T0 ; v/  lim inf I f .0; T0 ; vk /:

(8.109)

k!1

By (8.105), (8.108), and the lower semicontinuity of  f (see Proposition 8.19),  f .v.0//  lim inf  f .vk .0// D inf. f / k!1

and  f .v.0// D inf. f /:

(8.110)

In view of (8.108) and Proposition 8.19,  f .v.T0 //  lim inf  f .vk .T0 //:

(8.111)

k!1

By (8.104)–(8.106), (8.109)–(8.111), I f .0; T0 ; v/  T0 f .Nx; 0/   f .v.0// C  f .v.T0 //  lim inf I f .0; T0 ; vk /  T0 f .Nx; 0/  lim  f .vk .0// C lim inf  f .vk .T0 // k!1

k!1

k!1

 lim infŒI .0; T0 ; vk /  T0 f .Nx; 0/   .vk .0// C  .vk .T0 //  0: (8.112) f

f

f

k!1

Relations (8.109), (8.110), and (8.112) imply that  f .v.T0 // is finite. Together with (8.110) and Proposition 8.11 this implies that I f .0; T0 ; v/  T0 f .Nx; 0/   f .v.0// C  f .v.T0 //  0:

8.8 Proof of Theorem 8.25

259

By the inequality above and (8.112), I f .0; T0 ; v/  T0 f .Nx; 0/   f .v.0// C  f .v.T0 // D 0:

(8.113)

Since  f .v.T0 // is finite, Theorem 8.4 implies that there is an .f /-overtaking optimal and .f /-good function w W Œ0; 1/ ! Rn such that w.0/ D v.T0 /:

(8.114)

v.t/ D w.t  T0 /:

(8.115)

For all t > T0 set

Clearly, v W Œ0; 1/ ! Rn is an a. c. function. Since the function w is .f /-good and .f /-overtaking it follows from Propositions 8.11 and 8.12 and (8.113) that for all T > 0, I f .0; T; v/  Tf .Nx; 0/   f .v.0// C  f .v.T// D 0:

(8.116)

It is clear that the function v is .f /-good. In view of (8.116) and Proposition 8.20, v is an .f /-overtaking optimal function satisfying (8.110). By (8.108), for all sufficiently large natural numbers k, jvk .t/  v.t/j  =2 for all t 2 Œ0; T0 . This contradicts (8.107). The contradiction we have reached proves Lemma 8.34. Since the function fN satisfies the same assumptions as the function f Lemma 8.34 can be applied with the function fN .

8.8 Proof of Theorem 8.25 Recall (see (8.11) and (8.12)) that rN 2 .0; 1/ and ˝0 D f.x; y/ 2 Rn  Rn W jx  xN j  rN and jyj  rN g  dom.f /:

(8.117)

By Lemma 8.34 applied to the function fN , there exists ı1 2 .0; minf; rN =2g/ such that the following property holds: (Pii)

for each a. c. function v W Œ0; 0  ! Rn which satisfies N

N

 f .v.0//  inf. f / C ı1 ; N

N

N

I f .0; 0 ; v/  0 f .Nx; 0/   f .v.0// C  f .v.0 //  ı1

(8.118) (8.119)

260

8 Variational Problems with Extended-Valued Integrands

there exists an .fN /-overtaking optimal function u W Œ0; 1/ ! Rn such that N

N

 f .u.0// D inf. f /;

(8.120)

ju.t/  v.t/j   for all t 2 Œ0; 0 :

(8.121)

By Propositions 8.13 and 8.14, (A1) and Lemma 8.31, there exists ı2 2 .0; ı1 / such that for each z 2 Rn satisfying jz  xN j  2ı2 , N

N

N

j f .z/j D j f .z/   f .Nx/j  ı1 =8;

(8.122)

for each .x; y/ 2 Rn  Rn satisfying jx  xN j  4ı2 , jyj  4ı2 , jf .x; y/  f .Nx; 0/j  ı1 =8

(8.123)

and that the following property holds: (Piii)

for each a.c. function v W Œ0; 1 ! Rn satisfying jv.0/  xN j, jv.1/  xN j  ı2 , I f .0; 1; v/  f .Nx; 0/  ı1 =8:

By Theorem 8.2 and Proposition 8.5, there exist L  1 and ı3 > 0 such that the following property holds: (Piv)

for each T > 2L and each a.c. function v W Œ0; T ! Rn which satisfies v.0/ 2 YN L0 ;f and I f .0; T; v/  .f ; T; v.0// C ı3

(8.124)

we have jv.t/  xN j  ı2 for all t 2 ŒL; T  L:

(8.125)

By Proposition 8.21 applied to the function fN there exists 1 > 0 such that for each .fN /-overtaking optimal function u W Œ0; 1/ ! Rn satisfying N

N

 f .u.0// D inf. f /

(8.126)

ju.t/  xN j  ı2 for all t  1 :

(8.127)

we have

Choose ı > 0 and T0 > 0 such that ı < .16.L C 1 C 0 C 6//1 minfı1 ; ı2 ; ı3 g; T0 > 2L C 20 C 21 C 4:

(8.128) (8.129)

8.8 Proof of Theorem 8.25

261

Assume that T  T0 and that an a. c. function v W Œ0; T ! Rn satisfies v.0/ 2 YN L0 ;f and I f .0; T; v/  .f ; T; v.0// C ı:

(8.130)

By (8.128)–(8.130) and property (Piv), relation (8.125) holds. In view of (8.129), ŒT  L  0  1  4; T  L  0  1   ŒL; T  L  0  1 :

(8.131)

Relations (8.125) and (8.131) imply that jv.t/  xN j  ı2 for all t 2 ŒT  L  0  1  4; T  L  0  1 :

(8.132)

By Theorem 8.4, there exists an .fN /-overtaking optimal function u W Œ0; 1/ ! Rn satisfying (8.120). Then (8.127) holds. Define v.t/ Q D v.t/; t 2 Œ0; T  L  0  1  4; v.t/ Q D u.T  t/; t 2 ŒT  L  0  1  3; T; v.t Q C T  L  0  1  4/ D v.T  L  0  1  4/ C tŒu.L C 0 C 1 C 3/  v.T  L  0  1  4/; t 2 .0; 1/:

(8.133)

Clearly, vQ W Œ0; T ! Rn is an a. c. function. In view of (8.127) and (8.133), jv.T Q  L  0  1  3/  xN j D ju.L C 0 C 1 C 3/  xN j  ı2 :

(8.134)

It follows from (8.127), (8.132), and (8.133) that for all t 2 .T  L  0  1  4, T  L  0  1  3/, jvQ 0 .t/j  ju.L C 0 C 1 C 3/  v.T  L  0  1  4/j  2ı2 ; jv.t/ Q  xN j  3ı2 :

(8.135)

By (8.123) and (8.135), for all t 2 .T  L  0  1  4; T  L  0  1  3/, jf .v.t/; Q vQ 0 .t//  f .Nx; 0/j  ı1 =8: It follows from (8.130) and (8.133) that Q ı  I f .0; T; v/  I f .0; T; v/ D I f .T  L  0  1  4; T; v/  I f .T  L  0  1  4; T; v/: Q

(8.136)

262

8 Variational Problems with Extended-Valued Integrands

Together with (8.136) this implies that I f .T  L  0  1  4; T; v/  ı C I f .T  L  0  1  4; T  L  0  1  3; v/ Q C I f .T  L  0  1  3; T; v/ Q N

 ı C ı1 =8 C f .Nx; 0/ C I f .0; L C 0 C 1 C 3; u/:

(8.137)

By (8.132) and property (Piii), I f .T  L  0  1  4; T  L  0  1  3; v/  fN .Nx; 0/  ı1 =8: Together with (8.137) this implies that N

I f .T  L  0  1  3; T; v/  ı C ı1 =4 C I f .0; L C 0 C 1 C 3; u/:

(8.138)

Set y.t/ D v.T  t/; t 2 Œ0; L C 0 C 1 C 3:

(8.139)

It follows from (8.23), (8.138), and (8.139) that N

I f .0; L C 0 C 1 C 3; y/ D I f .T  L  0  1  3; T; v/ N

 ı C ı1 =4 C I f .0; L C 0 C 1 C 3; u/:

(8.140)

By (8.120), (8.128)–(8.130), (8.132), (8.140), Propositions 8.11 and 8.12, and .fN /overtaking optimality of u, N

N

N

N

N

 f .y.0//  inf. f / C I f .0; 0 ; y/  0 f .Nx; 0/   f .y.0// C  f .y.0 // N

N

N

  f .y.0//   f .u.0// C I f .0; L C 0 C 1 C 3; y/ N

N

 .L C 0 C 1 C 3/f .Nx; 0/   f .y.0// C  f .y.L C 0 C 1 C 3// N

N

N

  f .y.0//   f .u.0// C 3ı1 =8 C I f .0; L C 0 C 1 C 3; u/ N

N

 .L C 0 C 1 C 3/f .Nx; 0/   f .y.0// C  f .y.L C 0 C 1 C 3// N

N

N

N

D  f .y.0//   f .u.0// C  f .u.0//   f .u.L C 0 C 1 C 3// C 3ı1 =8 N

N

  f .y.0// C  f .y.L C 0 C 1 C 3// N

N

D  f .y.L C 0 C 1 C 3//   f .u.L C 0 C 1 C 3// C 3ı1 =8:

(8.141)

8.9 Proof of Theorem 8.26

263

By (8.132) and (8.139), jy.L C 0 C 1 C 3/  xN j  ı2 . Together with (8.134) and the choice of ı2 (see (8.122)) this implies that N

N

j f .y.L C 0 C 1 C 3//j; j f .u.L C 0 C 1 C 3//j  ı1 =8: By the inequalities above and (8.141), N

N

N

N

N

 f .y.0//  inf. f / C I f .0; 0 ; y/  0 f .Nx; 0/   f .y.0// C  f .y.0 //  ı1 : Combined with Proposition 8.11, (8.130), (8.132), and (8.139) the inequality above implies that N

N

 f .y.0//  inf. f /  ı1 ; N

N

N

I f .0; 0 ; y/  0 f .Nx; 0/   f .y.0// C  f .y.0 //  ı1 : By the inequalities above and property (Pii) there exists an .fN /-overtaking optimal function w W Œ0; 1/ ! Rn such that w.0/ 2 D.fN / and for all t 2 Œ0; 0 ,   jy.t/  w.t/j D jv.T  t/  w.t/j: t u

Theorem 8.25 is proved.

8.9 Proof of Theorem 8.26 Theorems 8.10 and 8.25 and Lemma 8.32 imply the following result. Theorem 8.35. Let 0 > 0 and  > 0. Then there exist ı > 0 and T0 > 0 such that for each T  T0 and each a. c. function v W Œ0; T ! Rn which satisfies I f .0; T; v/  .f ; T/ C ı there exists an .fN /-overtaking optimal function v  W Œ0; 1/ ! Rn such that v  .0/ 2 D.fN / and that for all t 2 Œ0; 0 , jv.T  t/  v  .t/j  : Theorem 8.26 follows from Theorem 8.35 applied to the functions f and fN .

264

8 Variational Problems with Extended-Valued Integrands

8.10 Structure of Solutions of Problem (P1 ) We use the notation, definitions, and assumptions introduced in Sects. 8.1 and 8.2. The following theorem describes the structure of solutions of problems (P1 ) in the regions closed to the endpoints of their domains. Theorem 8.36. Let 0 > 0,  > 0, x 2 [fYN L;f W L > 0g, and y 2 [fYL;f W L > 0g. Then there exist ı > 0 and T0 > 0 such that for each T  T0 and each a. c. function v W Œ0; T ! Rn which satisfies v.0/ D x; v.T/ D y; I f .0; T; v/  .f ; T; x; y/ C ı there exist an .f /-overtaking optimal function  W Œ0; 1/ ! Rn and an .fN /overtaking optimal function W Œ0; 1/ ! Rn such that .0/ D x, .0/ D y and j.t/  v.t/j  ; jv.T  t/  .t/j   for all t 2 Œ0; 0 . Theorem 8.36 follows from the next result applied to the functions f and fN . Theorem 8.37. Let 0 > 0,  > 0, L0 > 0, and x 2 YN L0 ;f . Then there exist ı > 0 and T0 > 0 such that for each T  T0 and each a. c. function v W Œ0; T ! Rn which satisfies v.0/ D x; v.T/ 2 YL0 ;f ; I f .0; T; v/  .f ; T; x; v.T// C ı there exists an .f /-overtaking optimal function  W Œ0; 1/ ! Rn such that .0/ D x and j.t/  v.t/j   for all t 2 Œ0; 0 . Theorem 8.37 is proved in Sect. 8.11. Theorems 8.2 and 8.37 and Proposition 8.5 imply the following result. Theorem 8.38. Let 0 > 0,  > 0 and x 2 [fYN L;f W L > 0g. Then there exist ı > 0 and T0 > 0 such that for each T  T0 and each a. c. function v W Œ0; T ! Rn which satisfies v.0/ D x; I f .0; T; v/  .f ; T; x/ C ı there exists an .f /-overtaking optimal function  W Œ0; 1/ ! Rn such that .0/ D x and j.t/  v.t/j   for all t 2 Œ0; 0 .

8.11 Proof of Theorem 8.37

265

8.11 Proof of Theorem 8.37 Assume that Theorem 8.37 does not hold. Then for each natural number k there exist Tk  0 C k C 2L0

(8.142)

and an a. c. function vk W Œ0; Tk  ! Rn such that vk .0/ D x; vk .Tk / 2 YL0 ;f ;

(8.143)

I f .0; Tk ; vk /  .f ; Tk ; x; vk .Tk // C 1=k

(8.144)

and that for each a. c. function  2 .f ; x/

(8.145)

supfj.t/  vk .t/j W t 2 Œ0; 0 g > :

(8.146)

we have

By (8.142)–(8.144), Proposition 8.5, Theorem 8.3, and the inclusion x 2 YN L0 ;f

(8.147)

the following property holds: (Pv) for each  > 0 there exist L./ > 0 and a natural number k./ such that for each integer k  k./ and each Tk > 2L./, jvk .t/  xN j  ; t 2 ŒL./; Tk  L./: Relations (8.142), (8.143), and (8.147) imply that for each integer k  1, .f ; Tk ; x; vk .Tk //  Tk f .Nx; 0/ C 2L0 C 2L0 jf .Nx; 0/j:

(8.148)

By (8.143), (8.144), (8.148), and Lemma 8.29, there exists M0 > 0 such that for all integers k  1, jvk .t/j  M0 for all t 2 Œ0; Tk :

(8.149)

(A2) implies that there exists c0 > 0 such that for each T > 0 and each z 2 Rn satisfying jzj  M0 , .f ; T; z/  Tf .Nx; 0/  c0 :

(8.150)

266

8 Variational Problems with Extended-Valued Integrands

Let p  1 be an integer. In view of (8.142), (8.144), (8.148)–(8.150), for each integer k  p, I f .0; p; vk / D I f .0; Tk ; vk /  I f .p; Tk ; vk /  Tk f .Nx; 0/ C 2L0 C 2L0 jf .Nx; 0/j C 1  .Tk  p/f .Nx; 0/ C c0  pf .Nx; 0/ C 2L0 .1 C jf .Nx; 0/j/ C c0 C 1:

(8.151)

By (8.143), (8.151), and Proposition 8.30, extracting subsequences and re-indexing if necessary, we may assume without loss of generality that there exists an a. c. function v W Œ0; 1/ ! Rn such that for any integer p  1, vk .t/ ! v.t/ as k ! 1 uniformly on Œ0; p;

(8.152)

I f .0; p; v/  lim inf I f .0; p; vk /:

(8.153)

k!1

In view of (8.151) and (8.153), for any integer p  1, I f .0; p; v/  pf .Nx; 0/ C 2L0 .1 C jf .Nx; 0/j/ C c0 C 1:

(8.154)

We show that for each T > 0, I f .0; T; v/ D .f ; T; v.0/; v.T//: Assume the contrary. Then there exist S > 0 and  > 0 such that I f .0; S; v/ > .f ; S; v.0/; v.S// C 7: Clearly, there exists an a. c. function vQ W Œ0; S ! Rn such that Q < I f .0; S; v/  6: v.0/ Q D x; v.S/ Q D v.S/; I f .0; S; v/

(8.155)

By (A1) and Lemma 8.31, there exists ı 2 .0; rN =4/ such that for each .x; y/ 2 Rn Rn satisfying jx  xN j  2ı, jyj  2ı, jf .x; y/  f .Nx; 0/j  =2

(8.156)

and that for each a.c. function v W Œ0; 1 ! Rn satisfying jv.0/  xN j, jv.1/  xN j  2ı, I f .0; 1; v/  f .Nx; 0/  =2:

(8.157)

By (Pv) and (8.142), there exist a number L1 > 0 and an integer k1 > 2L1 such that for each integer k  k1 , jvk .t/  xN j  ı; t 2 ŒL1 ; Tk  L1 :

(8.158)

8.11 Proof of Theorem 8.37

267

It follows from (8.142), (8.152), and (8.158) that jv.t/  xN j  ı for all t  L1 :

(8.159)

k2 > k1 C S C 2L1 C 9:

(8.160)

Choose a natural number

In view of (8.153), I f .0; k1 C S C L1 C 8; v/  lim inf I f .0; k1 C S C L1 C 8; vk /: k!1

(8.161)

For each integer k  k2 define Q t 2 Œ0; S; vO k .t/ D v.t/; t 2 .S; k1 C S C L1 C 8; vO k .t/ D v.t/; vO k .k1 C S C L1 C 8 C t/ D v.k1 C S C L1 C 8/ C tŒvk .k1 C S C L1 C 9/  v.k1 C S C L1 C 8/; t 2 .0; 1:

(8.162)

In view of (8.155) and (8.162), for each integer k  k2 , vO k W Œ0; k1 CSCL1 C9 ! Rn is an a. c. function which satisfies vO k .0/ D x; vO k .k1 C S C L1 C 9/ D vk .k1 C S C L1 C 9/:

(8.163)

By (8.143), (8.144), and (8.163), for each integer k  k2 , I f .0; k1 C S C L1 C 9; vk /  I f .0; k1 C S C L1 C 9; vO k / C 1=k:

(8.164)

In view of (8.162), there exists an integer k  k2 such that I f .0; k1 C S C L1 C 8; v/  I f .0; k1 C S C L1 C 8; vk / C ; k1 .k1 C S C L1 C 10/ < :

(8.165) (8.166)

It follows from (8.142) and (8.160) that Tk  L1  k C 2L0  L1 > k1 C S C 2L0 C L1 C 9:

(8.167)

268

8 Variational Problems with Extended-Valued Integrands

Relations (8.158), (8.159), (8.162), and (8.167) imply that for all t 2 .k1 CSCL1 C8; k1 C S C L1 C 9/; jvO k0 .t/j  jvk .k1 C S C L1 C 9/  xN j C jNx  v.k1 C S C L1 C 8/j  2ı; jvO k .t/  xN j  jvk .k1 C S C L1 C 9/  xN j C jNx  v.k1 C S C L1 C 8/j  2ı:

(8.168)

In view of (8.156) and (8.168), jI f .k1 C S C L1 C 8; k1 C S C L1 C 9; vO k /  f .Nx; 0/j  =2:

(8.169)

By the choice of ı (see (8.157)), (8.158), and (8.167), I f .k1 C S C L1 C 8; k1 C S C L1 C 9; vk /  f .Nx; 0/  =2:

(8.170)

By (8.155), (8.162), (8.164), (8.165), (8.169), and (8.170), 1=k  I f .0; k1 C S C L1 C 9; vO k /  I f .0; k1 C S C L1 C 9; vk / D I f .0; k1 C S C L1 C 8; vO k / C I f .k1 C S C L1 C 8; k1 C S C L1 C 9; vO k /  I f .0; k1 C S C L1 C 8; vk /  I f .k1 C S C L1 C 8; k1 C S C L1 C 9; vk /  I f .0; S; v/ Q C I f .S; k1 C S C L1 C 8; v/  I f .0; k1 C S C L1 C 8; vk / C f .Nx; 0/ C =2  f .Nx; 0/ C =2  I f .0; S; v/  6 C I f .S; k1 C S C L1 C 8; v/  I f .0; k1 C S C L1 C 8; vk / C   4;  < 1=k: This contradicts (8.166). The contradiction we have reached proves that I f .0; T; v/ D .f ; T; v.0/; v.T// for all T > 0. Together with (8.154) and Theorem 8.7 this implies that v is an .f /overtaking optimal function satisfying v.0/ D x. By (8.152) for all sufficiently large natural numbers k, jvk .t/  v.t/j  =2; t 2 Œ0; 0 : This contradicts (8.146). The contradiction we have reached proves Theorem 8.37. t u

Chapter 9

Dynamic Games with Extended-Valued Integrands

In this chapter we study a class of dynamic continuous-time two-player zerosum unconstrained games with extended-valued integrands. We do not assume convexity-concavity assumptions and establish the existence and the turnpike property of approximate solutions.

9.1 Preliminaries and Main Results In this chapter we continue to use the notation and definitions introduced in Chap. 8. We denote by mes.E/ the Lebesgue measure of a Lebesgue measurable set E  R1 , denote by j  j the Euclidean norm of the space Rq and by h; i the inner product of Rq where q is a natural number. For each function f W X ! RN WD R1 [ f1; 1g, where the set X is nonempty put dom.f / D fx 2 X W 1 < f .x/ < 1g: We suppose that a > 0,

W Œ0; 1/ ! Œ0; 1/ is an increasing function such that lim

t!1

.t/ D 1;

n; m be natural numbers and that f W Rn  Rn  Rm  Rm ! RN be a Borel measurable function. We suppose that for all real numbers ,  C 1 D 1;   1 D 1; 1 <  < 1:

© Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6_9

269

270

9 Dynamic Games with Extended-Valued Integrands

For each z 2 R1 set zC D maxfz; 0g; z D maxfz; 0g: Given z1 ; z2 2 Rn , 1 ; 2 2 Rm , and a positive number T we consider a continuous-time two-player zero-sum game over the interval Œ0; T denoted by  .z1 ; z2 ; 1 ; 2 ; T/. For this game the set of strategies for the first player is the set of all a. c. functions x W Œ0; T ! Rn satisfying x.0/ D z1 and x.T/ D z2 , the set of strategies for the second player is the set of all a. c. functions y W Œ0; T ! Rm satisfying y.0/ D 1 and y.T/ D 2 , and the objective function for the first player associated with the strategies x W Œ0; T ! Rn , y W Œ0; T ! Rm is given by RT 0 0 0 f .x.t/; x .t/; y.t/; y .t//dt if this integrand is well defined in the sense which is explained below. Let xf 2 Rn ; yf 2 Rm : Set f .1/ .x; y/ D f .x; y; yf ; 0/; .x; y/ 2 Rn  Rn ; f .2/ .x; y/ D f .xf ; 0; x; y/; .x; y/ 2 Rm  Rm : We assume that f .1/ .x; y/ > 1 for all .x; y/ 2 Rn  Rn ; f .2/ .x; y/ > 1 for all .x; y/ 2 Rm  Rm ; the functions f .1/ W Rn  Rn ! R1 [ f1g, f .2/ W Rm  Rm ! R1 [ f1g are lower semicontinuous functions, the sets dom.f .1/ / D f.x; y/ 2 Rn  Rn W f .x; y; yf ; 0/ < 1g; dom.f .2/ / D f.x; y/ 2 Rm  Rm W f .xf ; 0; x; y/ > 1g are nonempty, convex, and closed and f .x; y; yf ; 0/ D f .1/ .x; y/  maxf .jxj/; .jyj/jyjg  a for each x; y 2 Rn ;

(9.1)

f .xf ; 0; x; y/ D f .2/ .x; y/   maxf .jxj/; .jyj/jyjg C a for each x; y 2 Rm :

(9.2)

9.1 Preliminaries and Main Results

271

For each T 1 2 R1 , each T2 > T1 , and each pair of functions v W ŒT1 ; T2  ! Rn , u W ŒT1 ; T2  ! Rm set I

f .1/

Z .T1 ; T2 ; v/ D

T2

f .1/ .v.t/; v 0 .t//dt;

T1 .2/

Z

I f .T1 ; T2 ; v/ D 

T2

f .2/ .u.t/; u0 .t//dt:

T1

We suppose that dom.f / D dom.f .1/ /  dom.f .2/ /;

(9.3)

the function f is bounded on all bounded subsets of dom.f /, f .xf ; 0; yf ; 0/  f .x; 0; yf ; 0/ for each x 2 Rn ;

(9.4)

f .xf ; 0; yf ; 0/  f .xf ; 0; y; 0/ for each y 2 Rm

(9.5)

and that the following assumptions hold: (C1) .xf ; 0/ is an interior point of dom.f .1/ / and f .1/ is continuous at the point .xf ; 0/; .yf ; 0/ is an interior point of dom.f .2/ / and f .2/ is continuous at the point .yf ; 0/; (C2) for each M > 0 there is a number cM > 0 such that f .x; u; y; v/  cM for each .x; u/ 2 Rn  Rn and each .y; v/ 2 dom.f .2/ / satisfying jyj; jvj  M, and f .x; u; y; v/  cM for each .y; v/ 2 Rm  Rm and each .x; u/ 2 dom.f .1/ / satisfying jxj; juj  M; (C3) for each M > 0 there exists cM > 0 such that for each T > 0 and each a.c. function x W Œ0; T ! Rn satisfying jx.0/j  M, Z

T 0

f .x.t/; x0 .t/; yf ; 0/dt  Tf .xf ; 0; yf ; 0/  cM

and for each T > 0 and each a.c. function y W Œ0; T ! Rm satisfying jy.0/j  M, Z 0

T

f .xf ; 0; y.t/; y0 .t//dt  Tf .xf ; 0; yf ; 0/ C cM I

(C4) for each x 2 Rn the function f .x; ; yf ; 0/ W Rn ! R1 [ f1g is convex and for each y 2 Rm the function f .xf ; 0; y; / W Rm ! R1 [ f1g is concave. By (9.4) and (9.5), the pair .xf ; yf / is a saddle point for the function fN .x; y/ WD f .x; 0; y; 0/, x 2 Rn , y 2 Rm .

272

9 Dynamic Games with Extended-Valued Integrands

Assumption (C3) imply that for each a. c. function v W Œ0; 1/ ! Rn the function Z

T

T! 0

f .v.t/; v 0 .t/; yf ; 0/dt  Tf .xf ; 0; yf ; 0/; T 2 .0; 1/

is bounded from below and that for each a. c. function u W Œ0; 1/ ! Rm the function Z

T

T! 0

f .xf ; 0; u.t/; u0 .t//dt  Tf .xf ; 0; yf ; 0/; T 2 .0; 1/

is bounded from above. For all x1 ; x2 2 Rn , y1 ; y2 2 Rm define f C .x1 ; x2 ; y1 ; y2 / D maxff .x1 ; x2 ; y1 ; y2 /; 0g; f  .x1 ; x2 ; y1 ; y2 / D maxff .x1 ; x2 ; y1 ; y2 /; 0g: Let 1 < T1 < T2 < 1, x W ŒT1 ; T2  ! Rn , y W ŒT1 ; T2  ! Rm be a. c. functions. The pair .x; y/ is called admissible if at least one of the integrals Z

T2

C

0

Z

0

f .x.t/; x .t/; y.t/; y .t//dt;

T1

T2

f  .x.t/; x0 .t/; y.t/; y0 .t//dt

T1

is finite. If .x; y/ is admissible, then we set Z

T2

I .T1 ; T2 ; x; y/ D f

Z

f .x.t/; x0 .t/; y.t/; y0 .t//dt

T1 T2

D

f C .x.t/; x0 .t/; y.t/; y0 .t//dt

T1

Z



T2

f  .x.t/; x0 .t/; y.t/; y0 .t//dt:

T1

It should be mentioned that analogs of assumption (C3) are used in the infinite horizon optimal control and they are usually posed, when one obtains a turnpike result where the turnpike is a singleton. See, for example, [53]. Let us now define approximate solutions (saddle points) of games  .z1 ; z2 ; 1 ; 2 ; T/ with z1 ; z2 2 Rn , 1 ; 2 2 Rm and a positive constant T. Let M  0, 1 < T1 < T2 < 1, x W ŒT1 ; T2  ! Rn , y W ŒT1 ; T2  ! Rm be a.c. functions such that the pair .x; y/ is admissible. The pair .x; y/ is called .M/-good if RT the integral T12 f .x.t/; x0 .t/; y.t/; y0 .t//dt is finite and the following properties hold:

9.1 Preliminaries and Main Results

273

for each a. c. function z W ŒT1 ; T2  ! Rn such that the pair .z; y/ is admissible and z.Ti / D x.Ti /, i D 1; 2, Z

T2

0

0

Z

T2

f .z.t/; z .t/; y.t/; y .t//dt 

T1

f .x.t/; x0 .t/; y.t/; y0 .t//dt  MI

T1

for each a. c. function  W ŒT1 ; T2  ! Rm such that the pair .x; / is admissible and .Ti / D y.Ti /, i D 1; 2, Z

T2

0

0

f .x.t/; x .t/; .t/;  .t//dt 

T1

Z

T2

f .x.t/; x0 .t/; y.t/; y0 .t//dt C M:

T1

If .x; y/ is .0/-good, then .x; y/ is called a saddle point of the game  .x.T1 /; x.T2 /; y.T1 /; y.T2 /; T2  T1 /. Let L > 0. Denote by XL;1 the set of all points x 2 Rn for which there exist a number T 2 .0; L and an a. c. function v W Œ0; T ! Rn such that v.0/ D x; v.T/ D xf ; jv.t/j; jv 0 .t/j  L; t 2 Œ0; L .1/

and I f .0; T; v/ < 1 and by XL;2 the set of all points y 2 Rm for which there exist a number T 2 .0; L and an a. c. function v W Œ0; T ! Rm such that v.0/ D y; v.T/ D yf ; jv.t/j; jv 0 .t/j  L; t 2 Œ0; L .2/

and I f .0; T; v/ < 1: Denote by XN L;1 the set of all points x 2 Rn for which there exist a number T 2 .0; L and an a. c. function v W Œ0; T ! Rn such that v.0/ D xf ; v.T/ D x; jv.t/j; jv 0 .t/j  L; t 2 Œ0; L .1/ and I f .0; T; v/ < 1 and by XN L;2 the set of all points y 2 Rm for which there exist a number T 2 .0; L and an a. c. function v W Œ0; T ! Rm such that

v.0/ D yf ; v.T/ D y; jv.t/j; jv 0 .t/j  L; t 2 Œ0; L .2/

and I f .0; T; v/ < 1.

274

9 Dynamic Games with Extended-Valued Integrands

Note that the existence of a saddle point of the game  .z1 ; z2 ; 1 ; 2 ; T/ with z1 ; z2 2 X, 1 ; 2 2 Y and T > 0 is not guaranteed. Nevertheless, the next result is proved in Sect. 9.3. Theorem 9.1. Let M > 0. Then there exists M > 0 such that for each T1 2 R1 , each T2 > T1 C 2M, each z1 2 XM;1 , each z2 2 XN M;1 , each 1 2 XM;2 and each 2 2 XN M;2 there exists an .M /-good pair of a.c. functions x W ŒT1 ; T2  ! Rn , y W ŒT1 ; T2  ! Rm such that x.Ti / D zi ; y.Ti / D i ; i D 1; 2; x.t/ D xf ; y.t/ D yf ; t 2 ŒT1 C M; T2  M; jx.t/j; jx0 .t/j; jy.t/j; jy0 .t/j  M; t 2 ŒT1 ; T1 C M [ ŒT2  M; T2  and .1/

.1/

I f .T1 ; T1 C M; x/; I f .T2  M; T2 ; x/; .2/

.2/

I f .T1 ; T1 C M; y/; I f .T2  M; T2 ; y/ < 1: It should be mentioned that in Theorem 9.1 the constant M does not depend on the length of the interval T2  T1 . In this chapter we establish a turnpike property of good pairs of a. c. functions which means that they spend most of the time in a small neighborhood of the pair .x ; y /. It is known in the optimal control theory that turnpike properties of approximately optimal solutions are deduced from an asymptotic turnpike property of solutions of corresponding infinite horizon optimal control problems [53]. We say that f possesses the asymptotic turnpike property (or briefly ATP), if the following properties hold: for each a. c. function x W Œ0; 1/ ! Rn such that  Z T f .x.t/; x0 .t/; yf ; 0/dt  Tf .xf ; 0; yf ; 0/ W T 2 .0; 1/ < 1 sup 0

we have limt!1 jx.t/  xf j D 0; for each a. c. function y W Œ0; 1/ ! Rm such that  Z T 0 f .xf ; 0; y.t/; y .t//dt  Tf .xf ; 0; yf ; 0/ W T 2 .0; 1/ > 1 inf 0

we have limt!1 jy.t/  yf j D 0. The following theorem is our main result of this chapter. Theorem 9.2. Let f possess (ATP) and M; M1 ;  > 0. Then there exist l > 0 and an integer Q  1 such that for each T > Ql and each .M1 /-good admissible pair of a.c. functions x W Œ0; T ! Rn , y W Œ0; T ! Rm such that x.0/ 2 XM;1 ; x.T/ 2 XN M;1 ; y.0/ 2 XM;2 ; y.T/ 2 XN M;2

9.2 An Auxiliary Result

275

there exist a natural number q  Q and a sequence of closed intervals Œai ; bi   Œ0; T, i D 1; : : : ; q such that 0  bi  ai  l; i D 1; : : : ; q; jx.t/  xf j  ; jy.t/  yf j   q

for all t 2 Œ0; T n [iD1 Œai ; bi : Chapter 9 is organized as follows. Section 9.2 contains an auxiliary result. Theorem 9.1 is proved in Sect. 9.3. Auxiliary results for Theorem 9.2 are given in Sect. 9.4. Theorem 9.2 is proved in Sect. 9.5.

9.2 An Auxiliary Result Lemma 9.3. Let fg; s; zg g be either ff .1/ ; n; xf g or ff .2/ ; m; yf g. Then there exists c > 0 such that for each T > 0 and each a.c. function x W Œ0; T ! Rs , Z

T

0

g.x.t/; x0 .t//dt  Tg.zg ; 0/  c :

(9.6)

Proof. Clearly, there is M > 0 such that .M/ > a C 4 C jf .xf ; 0; yf ; 0/j:

(9.7)

By (C3), there exists c > 0 such that for each T > 0 and each a.c. function x W Œ0; T ! Rs satisfying jx.0/j  M, Z 0

T

g.x.t/; x0 .t//dt  Tg.zg ; 0/  c :

(9.8)

Let T > 0 and x W Œ0; T ! Rs be an a. c. function. We show that (9.6) holds. If jx.t/j > M for all t 2 Œ0; T, then relations (9.1), (9.2), and (9.7) imply (9.6). Assume that ft 2 Œ0; T W jx.t/j  Mg 6D ;: Set 0 D minft 2 Œ0; T W jx.t/j  Mg:

276

9 Dynamic Games with Extended-Valued Integrands

It is clear that jx.0 /j  M;

(9.9)

jx.t/j > M for all t satisfying 0  t < 0 :

(9.10)

By (9.1), (9.2), (9.7)–(9.10), and the choice of c , Z

T 0

g.x.t/; x0 .t//dt  Tg.zg ; 0/ Z

 . .M/  a/0 C

T

0

g.x.t/; x0 .t//dt  Tg.zg ; 0/

.M/  a  jf .xf ; 0; yf ; 0/j0  c  c :

 Lemma 9.3 is proved.

9.3 Proof of Theorem 9.1 We may assume that M > 4 C jxf j C jyf j: Since the function f is bounded on bounded subsets of dom.f .1/ /dom.f .2/ / there is a constant M0 > 0 such that jf .z1 ; z2 ; 1 ; 2 /j  M0 for all .z1 ; z2 / 2 dom.f .1/ / and all .1 ; 2 / 2 dom .f .2/ / satisfying jzi j; ji j  M; i D 1; 2:

(9.11)

By Lemma 9.3, there exists c > 0 such that: for each T > 0 and each a.c. function v W Œ0; T ! Rn , Z

T 0

f .x.t/; x0 .t/; yf ; 0/dt  Tf .xf ; 0; yf ; 0/  c I

(9.12)

for each T > 0 and each a.c. function u W Œ0; T ! Rm , Z

T 0

f .xf ; 0; u.t/; u0 .t//dt  Tf .xf ; 0; yf ; 0/ C c :

(9.13)

9.3 Proof of Theorem 9.1

277

By (C2), there is a number M1 > 0 such that f .x; u; y; v/  M1 for each .x; u/ 2 Rn  Rn and each .y; v/ 2 dom.f .2/ / satisfying jyj; jvj  MI

(9.14)

f .x; u; y; v/  M1 for each .y; v/ 2 Rm  Rm and each .x; u/ 2 dom.f .1/ / satisfying jxj; juj  M:

(9.15)

M  2M1 M C c C 1 C 2MM0 C 4Mjf .xf ; 0; yf ; 0/j:

(9.16)

Fix

Let T1 2 R1 , T2 > T1 C 2M, z1 2 XM;1 ; z2 2 XN M;1 ; 1 2 XM;2 ; 2 2 XN M;2 :

(9.17)

In view of (8.17), there exist a. c. functions x W ŒT1 ; T2  ! Rn , y W ŒT1 ; T2  ! Rm such that x.T1 / D z1 ; x.t/ D xf ; t 2 ŒT1 C M; T2  M; x.T2 / D z2 ; 0

(9.18)

jx.t/j; jx .t/j  M; t 2 ŒT1 ; T1 C M [ ŒT2  M; T2 ;

(9.19)

y.T1 / D 1 ; y.t/ D yf ; t 2 ŒT1 C M; T2  M; y.T2 / D 2 ;

(9.20)

0

jy.t/j; jy .t/j  M; t 2 ŒT1 ; T1 C M [ ŒT2  M; T2 ; .1/

(9.21)

.1/

I f .T1 ; T1 C M; x/; I f .T2  M; T2 ; x/; .2/

.2/

I f .T1 ; T1 C M; y/; I f .T2  M; T2 ; y/ < 1:

(9.22)

It follows from (9.11), (9.19), (9.21), and (9.22) that for a. e. t 2 ŒT1 ; T1 C M [ ŒT2  M; T2 , .x.t/; x0 .t// 2 dom.f .1/ /; .y.t/; y0 .t// 2 dom.f .2/ /; jf .x.t/; x0 .t/; y.t/; y0 .t//j  M0 :

(9.23) (9.24)

By (9.18), (9.20), and (9.24), ˇZ ˇ ˇ ˇ

T2

T1

ˇ ˇ f .x.t/; x .t/; y.t/; y .t//dt  .T2  T1 /f .xf ; 0; yf ; 0/ˇˇ 0

0

 2MM0 C 2Mjf .xf ; 0; yf ; 0/j:

(9.25)

278

9 Dynamic Games with Extended-Valued Integrands

We show that .x; y/ is an .M /-good pair of a.c. functions. Assume that  W ŒT1 ; T2  ! Rn is an a.c. function. By (9.12), (9.16), (9.20), (9.21), (9.23)–(9.25), and (C2) Z T2 f ..t/;  0 .t/; y.t/; y0 .t//dt T1

Z

T1 CM

D

0

Z

0

f ..t/;  .t/; y.t/; y .t//dt C

T1 CM

T1

Z

T2

C

T2 M

T2 M

f ..t/;  0 .t/; yf ; 0/dt

f ..t/;  0 .t/; y.t/; y0 .t//dt

 2M1 M  c C .T2  T1  2M/f .xf ; 0; yf ; 0/ Z T2  f .x.t/; x0 .t/; y.t/; y0 .t//dt  2MM0  4Mjf .xf ; 0; yf ; 0/j  2M1 M  c Z

T1 T2



f .x.t/; x0 .t/; y.t/; y0 .t//dt  M :

T1

Assume that  W ŒT1 ; T2  ! Rm is an a.c. function. By (9.10), (9.13), (9.15), (9.18), (9.19), (9.23), (9.25), and (C2), Z T2 f .x.t/; x0 .t/; .t/;  0 .t//dt T1

Z

T1 CM

D

f .x.t/; x0 .t/; .t/;  0 .t//dt C

Z

T1

Z

T2

C

T2 M

T2 M

T1 CM

f .xf ; 0; .t/;  0 .t//dt

f ..t/;  0 .t/; x.t/; x0 .t//dt

 2M1 M C c C .T2  T1  2M/f .xf ; 0; yf ; 0/ Z T2  f .x.t/; x0 .t/; y.t/; y0 .t//dt C 2MM0 C 4Mjf .xf ; 0; yf ; 0/j C 2M1 M C c Z 

T1 T2

f .x.t/; x0 .t/; y.t/; y0 .t//dt C M :

T1

t u

Theorem 9.1 is proved.

9.4 Auxiliary Results for Theorem 9.2 Let .g; s; xN / 2 f.f .1/ ; n; xf /; .f .2/ ; m; yf /g:

9.4 Auxiliary Results for Theorem 9.2

279

It is not difficult to see that the triplet .g; s; xN / satisfies all the assumptions of Sect. 3.1 of Chap. 3 of [53] and the results stated there are true for this triplet. Clearly, there is r 2 .0; 1/ such that D WD f.x; y/ 2 Rs  Rs W jx  xN j  rN and jyj  rN g  dom.g/; jg.x; y/j  g.Nx; 0/j  1 for all .x; y/ 2 D: For each pair of points x; y 2 Rn and each positive number T define Z .g; T; x; y/ D inf

T 0

g.v.t/; v 0 .t//dt W v W Œ0; T ! Rs

 is an a. c. function satisfying v.0/ D x; v.T/ D y : For each pair of number T1 2 R1 , T2 > T1 and each a.c. function v W ŒT1 ; T2  ! R put n

Z I g .T1 ; T2 ; v/ D

T2

g.v.t/; v 0 .t//dt:

T1

Theorem 3.4 of [53] implies the following result. Proposition 9.4. Let ; M > 0. Then there exist an integer Q  1 and a positive number l such that for each number T > lQ and each a.c. function v W Œ0; T ! Rn which satisfies jv.0/  xN j; jv.T/  xN j  r; I .0; T; v/  .g; T; v.0/; v.T// C M g

there exists a finite sequence of closed intervals Œai ; bi   Œ0; T, i D 1; : : : ; q such that q  Q; bi  ai  l; i D 1; : : : ; q; q

jv.t/  xN j  ; t 2 Œ0; T n [iD1 Œai ; bi : In this chapter we use the following Lemma 3.10 of [53] (see also Lemma 5.1 of Chap. 8). Lemma 9.5. Let M;  be positive numbers. Then there exists a positive number L0 such that for each number T  L0 , each a.c. function v W Œ0; T ! Rs satisfying jv.0/j  M; I g .0; T; v/  Tg.Nx; 0/ C M

280

9 Dynamic Games with Extended-Valued Integrands

and each number s 2 Œ0; T  L0  the inequality minfjv.t/  xN j W t 2 Œs; s C L0 g   holds. Lemma 9.6. Let M be a positive number. Then there exists a positive number M0 such that for each number T  1, each a.c. function v W Œ0; T ! Rs satisfying I g .0; T; v/  Tg.Nx; 0/ C M the inequality minfjv.t/j W t 2 Œ0; 1g  M0 holds. Proof. By Lemma 9.3, there exists c > 0 such that for each T > 0 and each a.c. function x W Œ0; T ! Rs , I g .0; T; x/  Tg.Nx; 0/  c :

(9.26)

Choose a number M0 > 0 such that .M0 / > jaj C jg.Nx; 0/j C 2M C 2c C 1:

(9.27)

Assume that T  1 and an a.c. function v W Œ0; T ! Rs satisfies I g .0; T; v/  Tg.Nx; 0/ C M:

(9.28)

If jv.t/j > M0 for all t 2 Œ0; T, then by (9.1) and (9.2), for a. e. t 2 Œ0; T, g.v.t/; v 0 .t// 

.M0 /  a:

Together with (9.27) and the inequality T  1 this implies that I g .0; T; v/  Tg.Nx; 0/  T. .M0 /  a  g.Nx; 0// 

.M0 /  a  jg.Nx; 0/j > M C 1:

This contradicts (9.28). The contradiction we have reached proves ft 2 Œ0; T W jv.t/j  M0 g 6D ;: Set 0 D minft 2 Œ0; T W jv.t/j  M0 g:

(9.29)

9.4 Auxiliary Results for Theorem 9.2

281

By (9.1), (9.2), (9.26), (9.28), and (9.29), M  I g .0; T; v/  Tg.Nx; 0/  0 .M0 / C I g .0 ; T; v/  Tg.Nx; 0/  0 a  0 . .M0 /  g.Nx; 0/  a/ C I g .0 ; T; v/  .T  0 /g.Nx; 0/  0 . .M0 /  g.Nx; 0/  a/  c and 0 . .M0 /  jg.Nx; 0/j  a/  c C M: Together with (9.27) this implies that 0  1. Lemma 9.6 is proved. Lemma 9.7. Let M;  be positive numbers. Then there exists a positive number L0 such that for each number T  L0 , each a.c. function v W Œ0; T ! Rn satisfying I g .0; T; v/  Tg.Nx; 0/ C M and each number s 2 Œ0; T  L0  the inequality minfjv.t/  xN j W t 2 Œs; s C L0 g   holds. Proof. By Lemma 9.6, there exists a positive number M0 such that the following property holds: (i) for each number T  1, each a.c. function v W Œ0; T ! Rs satisfying I g .0; T; v/  Tg.Nx; 0/ C M we have minfjv.t/j W t 2 Œ0; 1g  M0 : By Lemma 9.5, there exists a number L0 > 1 such that the following property holds: (ii) for each number T  L0  1, each a.c. function v W Œ0; T ! Rn satisfying jv.0/j  M0 ; I .0; T; v/  Tg.Nx; 0/ C M C a C jg.Nx; 0/j g

282

9 Dynamic Games with Extended-Valued Integrands

and each S 2 Œ0; T  L0 C 1, we have minfjv.t/  xN j W t 2 ŒS; S C L0  1g  : Assume that T  L0 and an a. c. function v W Œ0; T ! Rn satisfies I g .0; T; v/  Tg.Nx; 0/ C M:

(9.30)

In view of (9.30) and property (i), there is 0 2 Œ0; 1

(9.31)

jv.0 /j  M0 :

(9.32)

such that

It follows from (9.1), (9.2), (9.30), and (9.31) that I g .0 ; T; v/ D I g .0; T; v/  I g .0; 0 ; v/  Tg.Nx; 0/ C M C a  .T  0 /g.Nx; 0/ C M C a C jg.Nx; 0/j:

(9.33)

By (9.31)–(9.33), property (ii), and the relation T  L0 > 1, for each number S satisfying ŒS; S C L0  1  Œ0 ; T we have minfjv.t/  xN j W t 2 ŒS; S C L0  1g  : This implies that for each number S satisfying ŒS; S C L0   Œ0; T we have ŒS C 1; .S C 1/ C L0  1  Œ0; T; minfjv.t/  xN j W t 2 ŒS; S C L0 g  : Lemma 9.7 is proved. Lemma 9.8. Let M > 0. Then there exist L0 ; M0 > 0 such that for each number T  2L0 and each a.c. function v W Œ0; T ! Rs which satisfies

9.4 Auxiliary Results for Theorem 9.2

283

I g .0; T; v/  Tg.Nx; 0/ C M there are 1 2 Œ0; L0 ; 2 2 ŒT  L0 ; T such that jv.i /  xN j  r; i D 1; 2

(9.34)

and I g .1 ; 2 ; v/  .g; v.1 /; v.2 /; 2  1 / C M0 : Proof. Let L0 > 0 be as guaranteed by Lemma 9.7 with  D r. By Lemma 9.3, there exists c > 0 such that for each T > 0 and each a.c. function v W Œ0; T ! Rs , I g .0; T; v/  Tg.Nx; 0/  c :

(9.35)

M0  c C 2 C 2L0 C M C 2L0 jg.Nx; 0/j C 2L0 a:

(9.36)

Choose a number

Assume that T  2L0 and an a.c. function v W Œ0; T ! Rs satisfies I g .0; T; v/  Tg.Nx; 0/ C M:

(9.37)

It follows from (9.37), the choice of L0 , and Lemma 9.7 that there are 1 2 Œ0; L0 ; 2 2 ŒT  L0 ; T such that (9.34) holds. By (9.35) and (9.37), .g; v.1 /; v.2 /; 2  1 /  .2  1 /g.Nx; 0/  c :

(9.38)

By (9.1), (9.2), (9.36), (9.37), and (9.38), I g .1 ; 2 ; v/ D I g .0; T; v/  I g .0; 1 ; v/  I g .2 ; T; v/  Tg.Nx; 0/ C M C 2L0 a  .2  1 /g.Nx; 0/ C M C 2L0 a C 2L0 jg.Nx; 0/j  .g; v.1 /; v.2 /; 2  1 / C c C M C 2L0 a C 2L0 jg.Nx; 0/j  .g; v.1 /; v.2 /; 2  1 / C M0 :

284

9 Dynamic Games with Extended-Valued Integrands

Lemma 9.8 is proved. Proposition 9.4 and Lemma 9.8 imply the following result. Proposition 9.9. Let ; M > 0. Then there exist an integer Q  1 and a positive number l such that for each number T > lQ and each a.c. function v W Œ0; T ! Rs which satisfies I g .0; T; v/  Tg.Nx; 0/ C M there exists a finite sequence of closed intervals Œai ; bi   Œ0; T, i D 1; : : : ; q such that q  Q; 0  bi  ai  l; i D 1; : : : ; q; q

jv.t/  xN j  ; t 2 Œ0; T n [iD1 Œai ; bi :

9.5 Proof of Theorem 9.2 By Theorem 9.1, there exists M2 > 0 such that the following property holds: (iii) for each T1 2 R1 , each T2 > T1 C 2M, each z1 2 XM;1 , each z2 2 XN M;1 , each 1 2 XM;2 and each 2 2 XN M;2 there exists an .M2 /-good pair of a.c. functions x W ŒT1 ; T2  ! Rn , y W ŒT1 ; T2  ! Rm such that x.Ti / D zi ; y.Ti / D i ; i D 1; 2; x.t/ D xf ; y.t/ D yf ; t 2 ŒT1 C M; T2  M; jx.t/j; jx0 .t/j; jy.t/j; jy.t/j  M; t 2 ŒT1 ; T1 C M [ ŒT2  M; T2  and .1/

.1/

I f .T1 ; T1 C M; x/; I f .T2  M; T2 ; x/; .2/

.2/

I f .T1 ; T1 C M; y/; I f .T2  M; T2 ; y/ < 1: By (C2), there exists M3 > 0 such that: f .z1 ; z2 ; 1 ; 2 /  M3 for each .z1 ; z2 / 2 Rn  Rn and each .1 ; 2 / 2 dom.f .2/ / satisfying j1 j; j2 j  MI

(9.39)

f .z1 ; z2 ; 1 ; 2 /  M3 for each .1 ; 2 / 2 Rm  Rm and each .z1 ; z2 / 2 dom.f .1/ / satisfying jz1 j; jz2 j  M:

(9.40)

9.5 Proof of Theorem 9.2

285

By Lemma 9.3, there exists c > 0 such that for each T > 0 and each pair of a. c. functions v W Œ0; T ! Rn and u W Œ0; T ! Rm , Z Z

T 0 T

0

f .v.t/; v 0 .t/; yf ; 0/dt  Tf .xf ; 0; yf ; 0/  c ;

(9.41)

f .xf ; 0; u.t/; u0 .t//dt  Tf .xf ; 0; yf ; 0/ C c :

(9.42)

By Proposition 9.9 applied to the functions f .i/ ; i D 1; 2, there exist integer Q1 ; Q2  1 and positive numbers l1 ; l2 such that the following properties hold: (iv) for each number T > l1 Q1 and each a. c. function u W Œ0; T ! Rn which satisfies .1/

I f .0; T; u/  Tf .1/ .xf ; 0/ C 2M1 C c C 4MM3 C 8M there exists a finite sequence of closed intervals Œai ; bi   Œ0; T, i D 1; : : : ; q such that q  Q1 ; bi  ai  l1 ; i D 1; : : : ; q; q

ju.t/  xf j  ; t 2 Œ0; T n [iD1 Œai ; bi I (v) for each number T > l2 Q2 and each a. c. function u W Œ0; T ! Rm which satisfies .2/

I f .0; T; u/  Tf .2/ .yf ; 0/ C 2M1 C c C 4MM3 C 8M there exists a finite sequence of closed intervals Œai ; bi   Œ0; T, i D 1; : : : ; q such that q  Q2 ; bi  ai  l2 ; i D 1; : : : ; q; q

ju.t/  yf j  ; t 2 Œ0; T n [iD1 Œai ; bi : Set Q D Q1 C Q2 C 4; l D maxfl1 ; l2 ; Mg: Assume that T > Ql and x W Œ0; T ! Rn and y W Œ0; T ! Rm is an .M1 /-good admissible pair such that x.0/ 2 XM;1 ; x.T/ 2 XN M;1 ; y.0/ 2 XM;2 ; y.T/ 2 XN M;2 :

(9.43)

286

9 Dynamic Games with Extended-Valued Integrands

By (9.43) and property (ii), there exists an .M2 /-good pair of a.c. functions xQ W Œ0; T ! Rn , yQ W Œ0; T ! Rm such that xQ .0/ D x.0/; xQ .T/ D x.T/; yQ .0/ D y.0/; yQ .T/ D y.T/; xQ .t/ D xf ; yQ .t/ D yf ; t 2 ŒM; T  M; jQx.t/j; jQx0 .t/j; jQy.t/j; jQy0 .t/j  M; t 2 Œ0; M [ ŒT  M; T

(9.44) (9.45) (9.46)

and .1/

.1/

I f .0; M; xQ /; I f .T  M; T; xQ /; .2/

.2/

I f .0; M; yQ /; I f .T  M; T2 ; yQ / < 1:

(9.47)

In view of (9.45)–(9.47), the pairs .x; yQ / and .Qx; y/ are admissible. Since the pair .x; y/ is .M1 /-good it follows from (9.44) and (9.45) that Z M1 C Z C

M

0

f .x.t/; x0 .t/; yQ .t/; yQ 0 .t//dt

TM M

f .x.t/; x .t/; yf ; 0/dt C

Z

T 0

T

0

Z

T 0

Z D M1 C Z

f .x.t/; x0 .t/; yQ .t/; yQ 0 .t//dt

f .x.t/; x0 .t/; yQ .t/; yQ 0 .t//dt

f .x.t/; x0 .t/; y.t/; y0 .t//dt

 M1 C

C

T

TM

Z

D M1 C 

Z

0

T

f .Qx.t/; xQ 0 .t/; y.t/; y0 .t//dt

M 0

f .Qx.t/; xQ 0 .t/; y.t/; y0 .t//dt C

Z

TM

f .xf ; 0; y.t/; y0 .t//dt

M

f .Qx.t/; xQ 0 .t/; y.t/; y0 .t//dt:

TM

By (9.39)–(9.42), (9.46), and (9.48), M1  2MM3  c C .T  2M/f .xf ; 0; yf ; 0/ Z TM  M1  2MM3 C f .x.t/; x0 .t/; yf ; 0/dt M

Z

T

 0

f .x.t/; x0 .t/; y.t/; y0 .t//dt

(9.48)

9.6 Examples

287

Z  M1 C 2MM3 C

TM

f .xf ; 0; y.t/; y0 .t//dt

M

 M1 C 2MM3 C c C .T  2M/f .xf ; 0; yf ; 0/: The relation above implies that Z

TM

f .xf ; 0; y.t/; y0 .t//dt  2M1  4MM3  c C .T  2M/f .xf ; 0; yf ; 0/;

M

(9.49) Z

TM

f .x.t/; x0 .t/; yf ; 0/dt  2M1 C 4MM3 C c C .T  2M/f .xf ; 0; yf ; 0/:

M

(9.50) By (9.49), (9.50), and properties (iv) and (v), there exist finite sequence of closed intervals Œai;1 ; bi;1   ŒM; T  M; i D 1; : : : ; q1 and Œai;2 ; bi;2   ŒM; T  M; i D 1; : : : ; q2 such that q1  Q1 ; q2  Q2 ; 0  bi;1  ai;1  l1 ; i D 1; : : : ; q1 ; 0  bi;2  ai;2  l2 ; i D 1; : : : ; q2 ; q

1 jx.t/  xf j  ; t 2 ŒM; T  M n [iD1 Œai;1 ; bi;1 ;

q

2 jy.t/  yf j  ; t 2 ŒM; T  M n [iD1 Œai;2 ; bi;2 :

This completes the proof of Theorem 9.2.

t u

9.6 Examples Example 9.10. Assume that f W Rn  Rn  Rm  Rm ! R1 is a Borel measurable function which is bounded on all bounded subsets of Rn  Rn  Rm  Rm and satisfies (C2). Let xf 2 Rn ; yf 2 Rm

288

9 Dynamic Games with Extended-Valued Integrands

and set f .1/ .x; y/ D f .x; y; yf ; 0/; .x; y/ 2 Rn  Rn ; f .2/ .x; y/ D f .xf ; 0; x; y/; .x; y/ 2 Rm  Rm : Let a0 be a positive number, satisfying

0

W Œ0; 1/ ! Œ0; 1/ be an increasing function

lim

t!1

0 .t/

D 1;

l1 2 Rn , l2 2 Rm and let L1 W Rn  Rn ! Œ0; 1/ and L2 W Rm  Rm ! Œ0; 1/ be lower semicontinuous functions such that L1 .x; y/  maxf

0 .jxj/;

0 .jyj/jyjg

 a0 C jl1 jjyj for each x; y 2 Rn ;

L2 .x; y/  maxf

0 .jxj/;

0 .jyj/jyjg

 a0 C jl2 jjyj for each x; y 2 Rm ;

for each x; y 2 Rn ; L1 .x; y/ D 0 if and only if .x; y/ D .xf ; 0/; for each x; y 2 Rm ; L2 .x; y/ D 0 if and only if .x; y/ D .yf ; 0/; the function L1 is continuous at .xf ; 0/, the function L2 is continuous at .yf ; 0/, for each point x 2 Rn the function L1 .x; / W Rn ! R1 is convex, for each point x 2 Rm the function L2 .x; / W Rm ! R1 is convex, and that for each x; y 2 Rn ; f .1/ .x; y/ D f .x; y; yf ; 0/ D L1 .x; y/ C f .xf ; 0; yf ; 0/ C hl1 ; yi; for each x; y 2 Rm ; f .2/ .x; y/ D f .xf ; 0; x; y/ D L2 .x; y/ C f .xf ; 0; yf ; 0/  hl2 ; yi: We show that f satisfies all the assumptions made in Sect. 9.1. It is not difficult to see that (9.1) and (9.2) hold under the appropriate choice of a > 0, . Clearly, (9.3)– (9.5), (C1), and (C4) hold. Now we need only to show that (C3) and (ATP) hold. Evidently, all the assumptions made in Example 1.14 hold for the functions f .1/ , .2/ f . In view of Example 1.14, all the assumptions made in Sect. 1.4 (including (A2) and (A4)) hold for the functions f .1/ , f .2/ . This implies that (C3) and (ATP) hold for the function f . Therefore Theorems 9.1 and 9.2 are true for f .

9.6 Examples

289

Example 9.11. Assume that f W Rn  Rn  Rm  Rm ! R1 is a Borel measurable function which is bounded on all bounded subsets of Rn  Rn  Rm  Rm and satisfies (C2). Let xf 2 Rn ; yf 2 Rm and set f .1/ .x; y/ D f .x; y; yf ; 0/; .x; y/ 2 Rn  Rn ; f .2/ .x; y/ D f .xf ; 0; x; y/; .x; y/ 2 Rm  Rm : Assume that f .xf ; 0; y; 0/  f .xf ; 0; yf ; 0/  f .x; 0; yf ; 0/ for each x 2 Rn and each y 2 Rm . Let a be a positive number, satisfying

W Œ0; 1/ ! Œ0; 1/ be an increasing function

lim

t!1

.t/ D 1:

Assume that (9.1) and (9.2) hold and the functions f .1/ , f .2/ are continuous and strictly convex. We claim that f satisfies all the assumptions made in Sect. 9.1. Clearly, (9.3)–(9.5), (C1), and (C4) hold. Now we need only to show that (C3) and (ATP) hold. Evidently, all the assumptions made in Example 1.15 hold for the functions f .1/ , f .2/ . In view of Example 1.15, all the assumptions made in Sect. 1.4 (including (A2) and (A4)) hold for the functions f .1/ , f .2/ . This implies that (C3) and (ATP) hold for the function f . Therefore Theorems 9.1 and 9.2 are true for f .

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Index

A Absolutely continuous function, 2–4, 16, 22, 85, 87, 94, 233, 235 Admissible pair, 196, 197, 200, 201, 203, 272–274, 285, 286 Approximate solution, 5, 8–10, 15–19, 23, 25, 87–88, 129, 130, 163, 164, 191, 196, 197, 209, 233, 234, 236, 239, 244, 269, 272, 274 Asymptotic turnpike property, 18, 197, 237, 274 Autonomous variational problem, 16, 233

E Euclidean norm, 1, 6, 16, 21, 85, 234, 269 Euclidean space, 1, 6, 16, 21, 85, 234 Extended-valued integrand, 16–19, 233–289

B Baire category, 15, 209 Borelian function, 127, 128, 164–168, 175, 177, 181, 183, 209 Borel measurable function, 5, 21, 27, 32, 191, 205, 206, 269, 287, 289

I Increasing function, 7, 11, 12, 14, 17–19, 27, 86, 87, 90, 127, 163, 192, 193, 205, 209, 234, 269, 288, 289 Infinite horizon, 6, 10, 15–18, 85, 87, 234, 236 Infinite horizon optimal control problem, 8, 10, 15, 17, 23, 24, 85, 87, 197, 211, 236, 272, 274 Inner product, 1, 6, 21, 85, 234, 269 Integral functional, 6, 13, 22, 87, 124 Integrand, 5–11, 14–19, 21–125, 127, 129, 130, 163, 192, 206, 207, 209, 211, 233–289 Interior point, 17–19, 235, 243, 271

C Cardinality of a set, 29, 130 Complete uniform space, 164, 209 Control system, 6, 10, 13, 15, 22, 23, 85, 87 Convex function, 1, 7, 11, 14, 17, 89 Convex set, 18, 242

D Differentiable function, 1, 2 Differential equation, 85

G Good function, 18, 236–241, 243, 256, 259, 274, 278, 284, 286 Good trajectory-control pair, 9, 14, 15, 24, 89–91, 96, 103, 206 Growth condition, 2, 3, 164

L Lebesgue measurable function, 5, 11, 21, 22, 27, 85, 87, 94, 191 Lebesgue measurable set, 4, 234, 269

© Springer International Publishing Switzerland 2015 A.J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications 104, DOI 10.1007/978-3-319-19141-6

295

296 Lebesgue measure, 4, 5, 234, 269 Linear control system, 5, 10, 11, 15, 21–125, 127, 163–191, 209 Lower semicontinuous function, 17–19, 93, 234, 241–243, 288 Lower semicontinuous integrand, 16, 233

M Minimal function, 26, 46, 47, 88, 91, 92, 109–111, 161, 238 Minimization problem, 3, 8, 15, 16, 23, 26, 34, 47, 209, 234 Minimizer, 2, 5, 123

O Optimal control problem, 1, 8, 10–15, 17, 22–24, 85, 87, 128, 209, 211, 236, 274 Optimal trajectory, 5, 10, 11, 15, 21–24, 85, 87, 91, 127, 163, 209 Optimization problem, 2, 3 Overtaking optimal function, 237–241, 244, 249, 254, 255, 257–261, 263, 264, 268 Overtaking optimal trajectory-control, 9, 13, 24, 91, 111–113

P Pair, 6, 17, 19, 22, 27, 29–35, 43, 45, 63, 66, 73–75, 78, 81, 82, 84, 87, 92–97, 104, 105, 108–111, 115–119, 121, 122, 128, 131, 133, 136, 144, 145, 147, 151, 153, 154, 156, 158, 159, 161, 162, 165, 181, 187, 190–192, 196, 197, 199–201, 203, 212, 217, 221, 222, 224, 226–230, 234, 236–239, 242, 256, 257, 271–274, 278, 279, 284–286 Periodic convex integrand, 5–10, 21–84

Index S Strategy, 191–192, 270 Strict convexity, 2, 74 Strictly convex function, 1, 2, 6, 7, 11, 14, 22, 30, 36, 89, 207, 234, 244, 289

T Topological subspace, 128, 164, 210, 211 Trajectory-control pair, 8, 9, 13–15, 22–24, 87, 89–91, 94, 96, 102–106, 111–113, 120, 132, 133, 135, 136, 144, 145, 147, 154, 155, 206, 210, 211, 213, 214, 217, 219, 220, 223 Turnpike, 5, 9–11, 16, 17, 24, 25, 234, 236, 272 phenomenon, 1–5, 10, 15, 21–23, 27–29, 85, 87, 129 property, 5, 6, 8–11, 15–19, 21–25, 87, 127, 163, 164, 191, 197, 233–239, 269, 274 result, 1, 21–26, 29, 90–92, 237, 239, 272

U Uniformity, 6, 14, 22, 27, 35, 42, 74, 82, 83, 89, 97, 103, 113, 114, 117, 124, 128, 160, 164, 207, 209, 245, 253, 258, 266 Uniform space, 27, 128, 164, 209

V Variational problem, 1, 2, 16–19, 233–268

W Well-posed optimization problem, 3

Z Zero-sum game, 15, 19, 191–207, 269, 270