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______________________________________ TURNPIKE PROPERTIES IN THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL

Nonconvex Optimization and Its Applications VOLUME 80 Managing Editor: Panos Pardalos University of Florida, U.S.A.

Advisory Board: J. R. Birge University of Chicago, U.S.A. Ding-Zhu Du University of Minnesota, U.S.A. C. A. Floudas Princeton University, U.S.A. J. Mockus Lithuanian Academy of Sciences, Lithuania H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A. G. Stavroulakis Technical University Braunschweig, Germany H. Tuy National Centre for Natural Science and Technology, Vietnam

______________________________________ TURNPIKE PROPERTIES IN THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL

By ALEXANDER J. ZASLAVSKI The Technion—Israel Institute of Technology, Haifa, Israel

13

Library of Congress Cataloging-in-Publication Data Zaslavski, Alexander J. Turnpike properties in the calculus of variations and optimal control / by Alexander J. Zaslavski. p. cm. — (Nonconvex optimization and its applications ; v. 80) Includes bibliographical references and index. ISBN-13: 978-0-387-28155-1 (alk. paper) ISBN-10: 0-387-28155-X (alk. paper) ISBN-13: 978-0-387-28154-4 (ebook) ISBN-10: 0-387-28154-1 (ebook) 1. Calculus of variations. 2. Mathematical optimization. I. Title. II. Series QA316.Z37 2005 515´.64—dc22 2005050039 AMS Subject Classifications: 49–02

ISBN-10: 0-387-28155-X e-ISBN-10: 0-387-28154-1

ISBN-13: 978-0387-28155-1 e-ISBN-13: 978-0387-28154-4

© 2006 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springeronline.com

SPIN 11405689

Contents

Preface Introduction

ix xiii

1. INFINITE HORIZON VARIATIONAL PROBLEMS 1.1 Preliminaries 1.2 Main results 1.3 Auxiliary results 1.4 Discrete-time control systems 1.5 Proofs of Theorems 1.1-1.3

1 1 3 7 17 20

2. EXTREMALS OF NONAUTONOMOUS PROBLEMS 2.1 Main results 2.2 Preliminary lemmas 2.3 Proofs of Theorems 2.1.1-2.1.4 2.4 Periodic variational problems 2.5 Spaces of smooth integrands 2.6 Examples

33 33 37 54 59 62 69

3. EXTREMALS OF AUTONOMOUS PROBLEMS 3.1 Main results 3.2 Proof of Proposition 3.1.1 3.3 Weakened version of Theorem 3.1.3 3.4 Continuity of the function U f (T1 , T2 , x, y) 3.5 Discrete-time control systems 3.6 Proof of Theorem 3.1.2 3.7 Preliminary lemmas for Theorem 3.1.1

71 71 76 79 83 88 90 94

vi

TURNPIKE PROPERTIES

3.8 3.9 3.10 3.11 3.12

Preliminary lemmas for Theorems 3.1.3 and 3.1.4 Proof of Theorem 3.1.4 Proof of Theorem 3.1.3 Proofs of Theorems 3.1.1 and 3.1.5 Examples

99 106 112 114 114

4. INFINITE HORIZON AUTONOMOUS PROBLEMS 4.1 Main results 4.2 Proofs of Theorems 4.1.1-4.1.3 4.3 Proof of Theorem 4.1.4

115 115 119 150

5. TURNPIKE FOR AUTONOMOUS PROBLEMS 5.1 Main results 5.2 Proof of Theorem 5.1.1 5.3 Proof of Theorem 5.1.2 5.4 Examples

153 153 158 169 172

6. LINEAR PERIODIC CONTROL SYSTEMS 6.1 Main results 6.2 Preliminary results 6.3 Discrete-time control systems 6.4 Proof of Theorem 6.1.1 6.5 Proof of Theorem 6.1.2 6.6 Proof of Theorem 6.1.3 6.7 Proof of Theorem 6.1.4

173 173 176 183 186 188 190 193

7. LINEAR SYSTEMS WITH NONPERIODIC INTEGRANDS 197 7.1 Main results 197 7.2 Preliminary results 201 7.3 Discrete-time control systems 203 7.4 Proof of Theorem 7.1.1 204 7.5 Proof of Theorem 7.1.2 209 7.6 Proofs of Theorems 7.1.3 and 7.1.4 215 8. DISCRETE-TIME CONTROL SYSTEMS 8.1 Convex inﬁnite dimensional control systems 8.2 Preliminary results 8.3 Proofs of Theorems 8.1.1 and 8.1.2

223 223 226 230

vii

Contents

8.4 8.5 8.6

Nonautonomous control systems in metric spaces An auxiliary result Proof of Theorem 8.4.1

236 239 248

9. CONTROL PROBLEMS IN HILBERT SPACES 9.1 Main results 9.2 Preliminary results 9.3 Proof of Theorems 9.1.1-9.1.3 9.4 Proof of Theorems 9.1.4 and 9.1.5 9.5 Systems with distributed and boundary controls

257 257 261 262 273 277

10. A CLASS OF DIFFERENTIAL INCLUSIONS 10.1 Main result 10.2 Preliminary results 10.3 Suﬃcient condition for the turnpike property 10.4 Preliminary lemmas 10.5 Proof of Theorem 10.1.1 10.6 Example

283 283 288 294 298 310 318

11. CONVEX PROCESSES 11.1 Preliminaries 11.2 Asymptotic turnpike property 11.3 Turnpike theorems 11.4 Proofs of Theorems 11.3.1 and 11.3.2 11.5 Stability of the turnpike phenomenon 11.6 Proofs of Theorems 11.5.1, 11.5.2 and 11.5.3

321 321 322 324 325 334 337

12. A DYNAMIC ZERO-SUM GAME 12.1 Preliminaries 12.2 Main results 12.3 Deﬁnitions and notation 12.4 Preliminary results 12.5 The existence of a minimal pair of sequences 12.6 Preliminary lemmas for Theorem 12.2.1 12.7 Preliminary lemmas for Theorem 12.2.2 12.8 Proofs of Theorems 12.2.1 and 12.2.2 Comments

349 349 351 352 353 354 357 366 372 381

viii References

TURNPIKE PROPERTIES

387

Index 395

Preface

This monograph is devoted to recent progress in the turnpike theory. Turnpike properties are well known in mathematical economics. The term was ﬁrst coined by Samuelson who showed that an eﬃcient expanding economy would for most of the time be in the vicinity of a balanced equilibrium path (also called a von Neumann path) [78, 79]. These properties were studied by many authors for optimal trajectories of a Neumann–Gale model determined by a superlinear set-valued mapping. In the monograph we discuss a number of results concerning turnpike properties in the calculus of variations and optimal control which were obtained by the author in the last ten years. These results show that the turnpike properties are a general phenomenon which holds for various classes of variational problems and optimal control problems. Turnpike properties are studied for optimal control problems on ﬁnite time intervals [T1 , T2 ] of the real line. Solutions of such problems (trajectories) always depend on the time interval [T1 , T2 ], an optimality criterion which is usually determined by a cost function, and on data which is some initial conditions. In the turnpike theory we are interested in the structure of solutions of optimal problems. We study the behavior of solutions when an optimality criterion is ﬁxed while T1 , T2 and the data vary. To have turnpike properties means, roughly speaking, that the solutions of a problem are determined mainly by the optimality criterion (a cost function), and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. If a point t does not belong to these regions, then the value of a solution at t is closed to a trajectory (“turnpike”) which is deﬁned on the inﬁnite time interval and depends only on the optimality criterion. This phenomenon has the following interpretation. If one wishes to reach a point A from a point B by a car in an optimal way, then one should enter onto a turnpike, spend most of one’s time on it and then leave the turnpike to reach the required point. The turnpike phenomenon was discovered by Samuelson in a speciﬁc situation. In further numerous studies turnpike properties were established under strong assumptions on an optimality criterion (a cost function). The usual assumptions were that a cost function is time independent and is convex as a function of all its variables. Under these

x

TURNPIKE PROPERTIES

assumptions the “turnpike” is a stationary trajectory (a singleton). The simple form of the “turnpike” with a convex cost function allowed one to discover the turnpike property in this case. Since convexity plays an important role in mathematical economics, turnpike theory has many applications in this area of research. It should be mentioned that there are several interesting results concerning turnpike properties without convexity assumptions. In these results convexity was replaced by other assumptions. The veriﬁcation of these assumptions was rather diﬃcult and they hold for a narrow class of problems. Thus the turnpike phenomenon was considered by experts as an interesting property of some very particular problems arising in mathematical economics for which a “turnpike” was usually a singleton or a half-ray. This situation has changed in the last ten years. In this monograph we discuss results which were obtained during this period and allow us today to think about turnpike properties as a general phenomenon which holds for various classes of variational problems and optimal control problems. To establish these properties we do not need convexity of a cost function and its time independence. It was my great pleasure to receive on October 2000 the following letter from Paul A. Samuelson, the discoverer of the turnpike phenomenon. Dear Professor Zaslavski: I note with interest your long paper “The Turnpike Property ...Functions” in Nonlinear Analysis 42 (2000), 1465-98. It may be of interest to report that this property and name originated just over half a century ago when, as a Guggenheim Fellow on a 194849 sabbatical leave from MIT, I conjectured it in a memo written at the RAND Corporation in Santa Monica, California. In The Collected Scientiﬁc Papers of Paul A. Samuelson, MIT Press, 1966, 1972, 1977, 1986, it is reproduced. R. Dorfman, P.A. Samuelson, R.M. Solow, Linear Programming and Economic Analysis, McGraw-Hill, 1958 gives a preRoy Radner exposition. I believe that somewhere Lionel McKenzie has given a nice survey of the relevant mathematical-economics literature. With admiration, Paul A. Samuelson Our studies are based on the following ideas. A “turnpike” is not necessarily a singleton or a half-ray. It can be an absolutely continuous time-dependent function (trajectory) or a compact subset of Rn . To establish a turnpike property we consider a space of cost functions

PREFACE

xi

equipped with a natural complete metric and show that a turnpike property holds for most elements of this space in the sense of Baire categories. We obtain a turnpike theorem in the following way. We consider an optimality criterion (a cost function f ) and show that for a problem with this criterion there exists an optimal trajectory, say Xf , on an inﬁnite time interval. Then we perturb our cost function by some nonnegative small perturbation which is zero only on Xf . We show that for our new cost function f¯ the trajectory Xf is a turnpike, and that optimal solutions of the problem with a cost function g which is closed to f¯, are also most of the time close to Xf .

Alexander J. Zaslavski June 2005

Introduction

Let us consider the following problem of the calculus of variations: T 0

f (v(t), v (t))dt → min,

(P0 )

v : [0, T ] → Rn is an absolutely continuous function such that v(0) = y, v(T ) = z. Here T is a positive number, y and z are elements of the n-dimensional Euclidean space Rn and an integrand f : Rn × Rn → R1 is a continuous function. We are interested in the structure of solutions of the problem (P0 ) when y, z and T vary and T is suﬃciently large. Assume that the function f is strictly convex and diﬀerentiable and satisﬁes the following growth condition: f (y, z)/(|y| + |z|) → ∞ as |y| + |z| → ∞. Here we denote by | · | the Euclidean norm in Rn and by < ·, · > the scalar product in Rn . In order to analyse the structure of minimizers of the problem (P0 ) we consider the auxiliary minimization problem: f (y, 0) → min, y ∈ Rn .

(P1 )

It follows from the growth condition and the strict convexity of f that the problem (P1 ) has a unique solution which will be denoted by y¯. Clearly, ∂f /∂y(¯ y , 0) = 0. Deﬁne an integrand L : Rn × Rn → R1 by L(y, z) = f (y, z) − f (¯ y , 0)− < ∇f (¯ y , 0), (y, z) − (¯ y , 0) > = f (y, z) − f (¯ y , 0)− < (∂f /∂z)(¯ y , 0), z > .

xiv

TURNPIKE PROPERTIES

Clearly L is also diﬀerentiable and srictly convex and satisﬁes the same growth condition as f : L(y, z)/(|y| + |z|) → ∞ as |y| + |z| → ∞. Since f and L are strictly convex we obtain that L(y, z) ≥ 0 for all (y, z) ∈ Rn × Rn and L(y, z) = 0 if and only if y = y¯, z = 0. Consider the following auxiliary problem of the calculus of variations: T 0

L(v(t), v (t))dt → min,

(P2 )

v : [0, T ] → Rn is an absolutely continuous function such that v(0) = y, v(T ) = z, where T > 0 and y, z ∈ Rn . It is easy to see that for any absolutely continuous function x : [0, T ] → Rn with T > 0, T 0

T

= T

=

0

0

L(x(t), x (t))dt

[f (x(t), x (t)) − f (¯ y , 0)− < (∂f /∂z)(¯ y , 0), x (t) >]dt

f (x(t), x (t))dt + T f (¯ y , 0)− < (∂f /∂z)(¯ y ), x(T ) − x(0) > .

These equations imply that the problems (P0 ) and (P2 ) are equivalent: a function x : [0, T ] → Rn is a solution of the problem (P0 ) if and only if it is a solution of the problem (P2 ). The integrand L : Rn × Rn → R1 has the following property: n n (C) If {(yi , zi )}∞ i=1 ⊂ R × R satisﬁes limi→∞ L(yi , zi ) = 0, then limi→∞ yi = y¯ and limi→∞ zi = 0. Indeed, assume that n n {(yi , zi )}∞ i=1 ⊂ R × R and lim L(yi , zi ) = 0. i→∞

By the growth condition the sequence {(yi , zi )}∞ i=1 is bounded. Let (y, z) ∞ be a limit point of the sequence {(yi , zi )}i=1 . Then, L(y, z) = lim L(yi , zi ) = 0 i→∞

xv

INTRODUCTION

and (y, z) = (¯ y , 0). This implies that (¯ y , 0) = limi→∞ (yi , zi ). ¯ : [0, T ] → Rn be an optimal Let y, z ∈ Rn , T > 2 and a function x ¯ is also an optimal solution of the solution of the problem (P0 ). Then x problem (P2 ). We will show that T 0

L(¯ x(t), x ¯ (t))dt ≤ 2c0 (|y|, |z|)

where c0 (|y|, |z|) is a constant which depends only on |y| and |z|. Deﬁne a function x : [0, T ] → Rn by x(t) = y + t(¯ y − y), t ∈ [0, 1], x(t) = y¯, t ∈ [1, T − 1], x(t) = y¯ + (t − (T − 1))(z − y¯), t ∈ [T − 1, T ]. It follows from the deﬁnition of x ¯ and x that T 0

1

=

0

L(¯ x(t), x ¯ (t))dt ≤

L(x(t), y¯ − y)dt + 1

=

0

T 0

L(x(t), x (t))dt

T −1 1

T

L(¯ y , 0)dt +

L(x(t), y¯ − y)dt +

T T −1

T −1

L(x(t), z − y¯)dt

L(x(t), z − y¯)dt.

It is not diﬃcult to see that the integrals 1 0

L(x(t), y¯ − y)dt and

T T −1

L(x(t), z − y¯)dt

do not exceed a constant c0 (|y|, |z|) which depends only on |y|, |z|. Thus T 0

L(¯ x(t), x ¯ (t))dt ≤ 2c0 (|y|, |z|).

It is very important that in this inequality the constant c0 (|y|, |z|) does not depend on T . We denote by mes(E) the Lebesgue measure of a Lebesgue mesurable set E ⊂ R1 . Now let be a positive number. By the property (C) there is δ > 0 such that if (y, z) ∈ Rn × Rn and L(y, z) ≤ δ, then |y − y¯| + |z| ≤ . Then x(t), x ¯ (t))dt ≤ 2c0 (|y|, |z|), by the choice of δ and the inequality 0T L(¯ y , 0)| > } mes{t ∈ [0, T ] : |(¯ x(t), x ¯ (t)) − (¯

xvi

TURNPIKE PROPERTIES

≤ mes{t ∈ [0, T ] : L(¯ x(t), x ¯ (t)) > δ} ≤δ and

−1

T 0

L(¯ x(t), x ¯ (t))dt ≤ δ −1 2c0 (|y|, |z|)

mes{t ∈ [0, T ] : |¯ x(t) − y¯| > } ≤ δ −1 2c0 (|y|, |z|).

Therefore the optimal solution x ¯ spends most of the time in an neighbor- hood of the point y¯. The Lebesgue measure of the set of all points t, for which x ¯(t) does not belong to this -neighborhood, does not exceed the constant 2δ −1 c0 (|y|, |z|) which depends only on |y|, |z| and and does not depend on T . Following the tradition, the point y¯ is called the turnpike. Moreover we can show that the set {t ∈ [0, T ] : |¯ x(t) − y¯| > } is contained in the union of two intervals [0, τ1 ] ∪ [T − τ2 , T ], where 0 < τ1 , τ2 ≤ 2δ −1 c0 (|y|, |z|). Under the assumptions posed on f , the structure of optimal solutions of the problem (P0 ) is rather simple and the turnpike y¯ is calculated easily. On the other hand the proof is strongly based on the convexity of f and its time independence. The approach used in the proof cannot be employed to extend the turnpike result for essentially larger classes of variational problems. For such extensions we need other approaches and ideas. The question of what happens if the integrand f is nonconvex and nonautonomous seems very interesting. What kind of turnpike and what kind of convergence to the turnpike do we have for general nonconvex nonautonomous integrands? The following example helps to understand the problem. Let f (t, x, u) = (x − cos(t))2 + (u + sin(t))2 , (t, x, u) ∈ R1 × R1 × R1 and consider the family of the variational problems T2 T1

[(v(t) − cos(t))2 + (v (t) + sin(t))2 ]dt → min,

(P3 )

v : [T1 , T2 ] → R1 is an absolutely continuous function such that v(T1 ) = y, v(T2 ) = z, where y, z, T1 , T2 ∈ R1 and T2 > T1 . The integrand f depends on t, for each t ∈ R1 the function f (t, ·, ·) : R2 → R1 is convex, and for each x, u ∈ R1 \ {0} the functon f (·, x, u) : R1 → R1 is nonconvex. Thus the function f : R1 × R1 × R1 → R1 is also nonconvex and depends on t.

xvii

INTRODUCTION

Assume that y, z, T1 , T2 ∈ R1 , T2 > T1 + 2 and v : [T1 , T2 ] → R1 is an optimal solution of the problem (P3 ). Note that the problem (P3 ) has a solution since f is continuous and f (t, x, ·) : R1 → R1 is convex and grows superlinearly at inﬁnity for each (t, x) ∈ [0, ∞) × R1 . Deﬁne v : [T1 , T2 ] → R1 by v(t) = y + (cos(1) − y)(t − T1 ), t ∈ [T1 , T1 + 1], v(t) = cos(t), t ∈ [T1 + 1, T2 − 1], v(t) = cos(T2 − 1) + (t − T2 + 1)(z − cos(T2 )), t ∈ [T2 − 1, T2 ]. It is easy to see that T2 −1 T1 +1

and

T2 T1

=

f (t, v(t), v (t))dt = 0

f (t, v(t), v (t))dt ≤

T1 +1 T1

T2

f (t, v(t), v (t))dt +

T1

f (t, v(t), v (t))dt

T2 T2 −1

f (t, v(t), v (t))dt

≤ 2 sup{|f (t, x, u)| : t, x, u ∈ R1 , |x|, |u| ≤ |y| + |z| + 1}. Thus

T2 T1

f (t, v(t), v (t))dt ≤ c1 (|y|, |z|),

where c1 (|y|, |z|) = 2 sup{|f (t, x, u)| : t, x, u ∈ R1 , |x|, |u| ≤ |y| + |z| + 1}. For any ∈ (0, 1) we have mes{t ∈ [T1 , T2 ] : |v(t) − cos(t)| > } ≤ −2

T2 T1

f (t, v(t), v (t))dt ≤ −2 c1 (|y|, |z|).

Since the constant c1 (|y|, |z|) does not depend on T2 and T1 we conclude that if T2 − T1 is suﬃciently large, then the optimal solution v(t) is equal to cos(t) up to for most t ∈ [T1 , T2 ]. Again, as in the case of convex time independent problems we can show that {t ∈ [T1 , T2 ] : |x(t) − cos(t)| > } ⊂ [T1 , T1 + τ ] ∪ [T2 − τ, T2 ] where τ > 0 is a constant which depends only on , |y| and |z|.

xviii

TURNPIKE PROPERTIES

This example shows that there exist nonconvex time dependent integrands which have the turnpike property with the same type of convergence as in the case of convex autonomous variational problems. The diﬀerence is that the turnpike is not a singleton but an absolutely continuous time dependent function deﬁned on the inﬁnite interval [0, ∞). This leads us to the following deﬁnition of the turnpike property for general integrands. Let us consider the following variational problem: T2 T1

f (t, v(t), v (t))dt → min,

(P )

v : [T1 , T2 ] → Rn is an absolutely continuous function such that v(T1 ) = y, v(T2 ) = z. Here T1 < T2 are real numbers, y and z are elements of the n-dimensional Euclidean space Rn and an integrand f : [0, ∞) × Rn × Rn → R1 is a continuous function. We say that the integrand f has the turnpike property if there exists a locally absolutely continuous function Xf : [0, ∞) → Rn (called the “turnpike”) which depends only on f and satisﬁes the following condition: For each bounded set K ⊂ Rn and each > 0 there exists a constant T (K, ) > 0 such that for each T1 ≥ 0, each T2 ≥ T1 + 2T (K, ), each y, z ∈ K and each optimal solution v : [T1 , T2 ] → Rn of variational problem (P), the inequality |v(t) − Xf (t)| ≤ holds for all t ∈ [T1 + T (K, ), T2 − T (K, )]. The turnpike property is very important for applications. Suppose that the integrand f has the turnpike property, K and are given, and we know a ﬁnite number of “approximate” solutions of the problem (P). Then we know the turnpike Xf , or at least its approximation, and the constant T (K, ) which is an estimate for the time period required to reach the turnpike. This information can be useful if we need to ﬁnd an “approximate” solution of the problem (P) with a new time interval [T1 , T2 ] and the new values y, z ∈ K at the end points T1 and T2 . Namely instead of solving this new problem on the “large” interval [T1 , T2 ] we can ﬁnd an “approximate” solution of problem (P) on the “small” interval [T1 , T1 + T (K, )] with the values y, Xf (T1 + T (K, )) at the end points and an “approximate” solution of problem (P) on the “small” interval [T2 − T (K, ), T2 ] with the values Xf (T2 − T (K, )), z at the end points. Then the concatenation of the ﬁrst solution, the function Xf : [T1 + T (K, ), T2 − T (K, )] and the second solution is an

xix

INTRODUCTION

“approximate” solution of problem (P) on the interval [T1 , T2 ] with the values y, z at the end points. We begin our monograph with a discussion of the problem (P). In Chapter 1 we introduce a space M of continuous integrands f : [0, ∞) × Rn × Rn → R1 . This space is equipped with a natural complete metric. We show that for any initial condition x0 ∈ Rn there exists a locally absolutely continuous function x : [0, ∞) → Rn with x(0) = x0 such that for each T1 ≥ 0 and T2 > T1 the function x : [T1 , T2 ] → Rn is a solution of problem (P) with y = x(T1 ) and z = x(T1 ). We also establish that for every bounded set E ⊂ Rn the C([T1 , T2 ]) norms of approximate solutions x : [T1 , T2 ] → Rn for the problem (P) with y, z ∈ E are bounded by some constant which does not depend on T1 and T2 . In Chapter 2 we establish the turnpike property stated above for a generic integrand f ∈ M. We establish the existence of a set F ⊂ M which is a countable intersection of open everywhere dense sets in M such that for each f ∈ F the turnpike property holds. Moreover we show that the turnpike property holds for approximate solutions of variational problems with a generic integrand f and that the turnpike phenomenon is stable under small pertubations of a generic integrand f . In Chapters 3-5 we study turnpike properties for autonomous problems (P) with integrands f : Rn × Rn → R1 which do not depend on t. Since the turnpike theorems of Chapter 2 are of generic nature and the subset of M which consists of all time independent integrands are nowhere dense, the results of Chapter 2 can not be applied for this subset. Moreover, we cannot expect to obtain the turnpike property stated above for the general autonomous case. Indeed, if an integrand f does not depend on t and has a turnpike, then this turnpike should also be time independent. It means that the turnpike is a stationary trajectory (a singleton). But it is not true when a time independent integrand f is not a convex function. Consider the following example. Let f (x1 , x2 , u1 , u2 ) = (x21 + x22 − 1)2 + (u1 + x2 )2 + (u2 − x1 )2 , (x1 , x2 , u1 , u2 ) ∈ R2 × R2 and consider the family of the variational problems T 0

f (v1 (t), v2 (t), v1 (t), v2 (t))dt → min,

(v1 , v2 ) : [0, T ] → R2 is an absolutely continuous function such that (v1 , v2 )(0) = y, (v1 , v2 )(T ) = z,

(P4 )

xx

TURNPIKE PROPERTIES

where y = (y1 , y2 ), z = (z1 , z2 ) ∈ R2 and T > 0. The integrand f does not depend on t. Since f is continuous and for each x = (x1 , x2 ) ∈ R2 the function f (x, ·) : R2 → R1 is convex and grows superlinearly at inﬁnity, the problem (P4 ) has a solution for each T > 0 and each y, z ∈ R2 . Clearly, if T > 0, y = (cos(0), sin(0)) and z = (cos(T ), sin(T )), then the function 1 (t) = cos(t), x 2 (t) = sin(t), t ∈ [0, T ] x is a solution of the problem (P4 ). Thus, if the integrand f has a turnpike property, then the turnpike is not a singleton. v1 , v¯2 ) : [0, T ] → R2 be a solution Let T > 2, y, z ∈ R2 and let v¯ = (¯ of the problem (P4 ). Deﬁne a function v = (v1 , v2 ) : [0, T ] → Rn by v(t) = y + t((cos(1), sin(1)) − y), t ∈ [0, 1], v(t) = (cos(t), sin(t)), t ∈ [1, T − 1], v(t) = (cos(T − 1), sin(T − 1)) + (t − T + 1)(z − (cos(T − 1), sin(T − 1)), t ∈ [T − 1, T ]. Then

T −1 1

and

T 0

2

f (v(t), v (t))dt = 0 2

2

(¯ v1 (t) + v¯2 (t) − 1) dt ≤ ≤ 1

=

0

T 0

T 0

f (¯ v (t), v¯ (t))dt

f (v(t), v (t))dt

f (v(t), v (t))dt +

T T −1

f (v(t), v (t))dt

≤ sup{f (x1 , x2 , u1 , u2 ) : x1 , x2 , u1 , u2 ∈ R1 and |xi |, |ui | ≤ 2|y| + 2|z| + 2, i = 1, 2}. Thus

T 0

with

(¯ v1 (t)2 + v¯2 (t)2 − 1)2 dt ≤ c2 (|y|, |z|)

c2 (|y|, |z|) = sup{f (x1 , x2 , u1 , u2 ) : x1 , x2 , u1 , u2 ∈ R1 and |xi |, |ui | ≤ 2|y| + 2|z| + 2}.

Here c2 (|y|, |z|) depends only on |y|, |z| and does not depend on T . For any ∈ (0, 1) we have mes{t ∈ [0, T ] : ||(¯ v1 (t), v¯2 (t))| − 1| > }

xxi

INTRODUCTION

≤ mes{t ∈ [0, T ] : |¯ v1 (t)2 + v¯2 (t)2 − 1| > 2 } ≤

−4

T 0

(¯ v1 (t)2 + v¯22 − 1)2 dt

≤ −4 c2 (|y|, |z|). It means that for most t ∈ [0, T ], v¯(t) belongs to the -neighborhood of the set {x ∈ R2 : |x| = 1}. Thus we can say that the integrand f has a weakened version of the turnpike property and the set {|x| = 1} can be considered as the turnpike for f . For a general autonomous nonconvex problem (P) we also have a version of the turnpike property in which a turnpike is a compact subset of Rn . This subset depends only on the integrand f . Consider the following autonomous variational problem: T 0

f (z(t), z (t))dt → min, z(0 = x, z(T ) = y,

(Pa )

z : [0, T ] → Rn is an absolutely continuous function where T > 0, x, y ∈ Rn and f : R2n → R1 is an integrand. We say that a time independent integrand f = f (x, u) ∈ C(R2n ) has the turnpike property if there exists a compact set H(f ) ⊂ Rn such that for each bounded set K ⊂ Rn and each > 0 there exist numbers L1 > L2 > 0 such that for each T ≥ 2L1 , each x, y ∈ K and an optimal solution v : [0, T ] → Rn for the variational problem (Pa ), the relation dist(H(f ), {v(t) : t ∈ [τ, τ + L2 ]}) ≤ holds for each τ ∈ [L1 , T − L1 ]. (Here dist(·, ·) is the Hausdorﬀ metric). We also consider a weak version of this turnpike property for a time independent integrand f (x, u). In this weak version, for an optimal solution of the problem (Pa ) with x, y ∈ Rn and large enough T , the relation dist(H(f ), {v(t) : t ∈ [τ, τ + L2 ]}) ≤ with L2 , which depends on and |x|, |y| and a compact set H(f ) ⊂ Rn depending only on the integrand f , holds for each τ ∈ [0, T ] \ E where E ⊂ [0, T ] is a measurable subset such that the Lebesgue measure of E does not exceed a constant which depends on and on |x|, |y|. These two turnpike properties for autonomous problems (Pa ) are considered in Chapters 3-5. In Chapter 3 we consider the space A of all time independent integrands f ∈ M. We establish the existence of a set F ⊂ A which is a

xxii

TURNPIKE PROPERTIES

countable intersection of open everywhere dense sets in A such that for each f ∈ F the weakened version of the turnpike property holds. The turnpike property for time independent integrands is established in Chapter 5 for a generic element of a subset N of the space A. The space N is a subset of all integrands f ∈ A which satisfy some diﬀerentiability assumptions. In the other chapters of the monograph we establish a number of turnpike results (generic and individual) for various classes of optimal control problems. We study optimal control of linear periodic systems with convex integrands (Chapter 6) and optimal solutions of linear systems with convex nonperiodic integrands (Chapter 7). In Chapter 8 we establish turnpike theorems for discrete-time control systems in Banach spaces and in complete metric spaces. Inﬁnite-dimensional continuoustime optimal control problems in a Hilbert space are studied in Chapter 9. A turnpike theorem for a class of diﬀerential inclusions arising in economic dynamics is proved in Chapter 10 and structure of optimal trajectories of convex processes is studied in Chapter 11. In Chapter 12 we establish a turnpike property for a dynamic discrete-time zero-sum game.

Chapter 1 INFINITE HORIZON VARIATIONAL PROBLEMS

In this chapter we study existence and uniform boundedness of extremals of variational problems with integrands which belong to a complete metric space of functions. We establish that for every bounded set E ⊂ Rn the C([0, T ]) norms of approximate solutions x : [0, T ] → Rn for the minimization problem on an interval [0, T ] with x(0), x(T ) ∈ E are bounded by some constant which does not depend on T . Given an x0 ∈ Rn we study the inﬁnite horizon problem of minimizing the expression 0T f (t, x(t), x (t))dt as T grows to inﬁnity, where x : [0, ∞) → Rn satisﬁes the initial condition x(0) = x0 . We analyse the existence and the properties of approximate solutions for every prescribed initial value x0 .

1.1.

Preliminaries

Variational and optimal control problems deﬁned on inﬁnite intervals are of interest in many areas of mathematics and its applications [10, 11, 16, 32, 62, 63, 88, 89, 95]. These problems arise in engineering [1, 3], in models of economic growth [14, 26, 27, 28, 29, 45, 46, 49-52, 60, 61, 67, 68, 72, 74, 80, 86, 94], in dynamic games theory [15, 17], in inﬁnite discrete models of solid-state physics related to dislocations in one-dimensional crystals [6, 85] and in the theory of thermodynamical equilibrium of materials [20, 44, 53-55, 90-92, 95]. We consider the inﬁnite horizon problem of minimizing the expression T 0

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

2

TURNPIKE PROPERTIES

as T grows to inﬁnity where a function x : [0, ∞) → K is locally absolutely continuous (a.c.) and satisﬁes the initial condition x(0) = x0 , K ⊂ Rn is a closed convex set and f belongs to a complete metric space of functions to be described below. We say that an a.c. function x : [0, ∞) → K is (f )-overtaking optimal if T

lim sup T →∞

0

[f (t, x(t), x (t)) − f (t, y(t), y (t))]dt ≤ 0

for any a.c. function y : [0, ∞) → K satisfying y(0) = x(0). This notion, known as the overtaking optimality criterion, was introduced in the economics literature by Atsumi [4], Gale [33] and von Weizsacker [81] and has been used in control theory [3, 13, 14, 16, 39, 40]. In general, overtaking optimal solutions may fail to exist. Most studies that are concerned with their existence assume convex integrands [13, 40, 72]. Another type of optimality criterion for inﬁnite horizon problems was introduced by Aubry and Le Daeron [6] in their study of the discrete Frenkel–Kontorova model related to dislocations in one-dimensional crystals. More recently this optimality criterion was used in [44, 65, 66, 85]. A similar notion was introduced in Halkin [34] for his proof of the maximum principle. Let I be either [0, ∞) or (−∞, ∞). We say that an a.c. function x : I → K is an (f )-minimal solution if T2 T1

f (t, x(t), x (t))dt ≤

T2 T1

f (t, y(t), y (t))dt ≤ 0

for each T1 ∈ I, T2 > T1 and each a.c. function y : [T1 , T2 ] → K which satisﬁes y(Ti ) = x(Ti ), i = 1, 2. It is easy to see that every (f )-overtaking optimal function is an (f )minimal solution. In this chapter we consider a functional space of integrands M described in Section 1.1. We show that for each f ∈ M and each z ∈ Rn there exists a bounded (f )-minimal solution Z : [0, ∞) → Rn satisfying Z(0) = z such that any other a.c. function Y : [0, ∞) → Rn is not “better” than Z. We also establish that given f ∈ M and a bounded set E ⊂ Rn the C([0, T ]) norms of approximate solutions x : [0, T ] → Rn for the minimization problem on an interval [0, T ] with x(0), x(T ) ∈ E are bounded by some constant which depends only on f and E.

3

Inﬁnite horizon variational problems

1.2.

Main results

Let a > 0 be a constant and ψ : [0, ∞) → [0, ∞) be an increasing function such that ψ(t) → ∞ as t → ∞. Let K ⊂ Rn be a closed convex set. Denote by |·| the Euclidean norm in Rn and denote by M the set of continuous functions f : [0, ∞) × K × Rn → R1 which satisfy the following assumptions: A(i) for each (t, x) ∈ [0, ∞) × K the function f (t, x, ·) : Rn → R1 is convex; A(ii) the function f is bounded on [0, ∞) × E for any bounded set E ⊂ K × Rn ; A(iii) for each (t, x, u) ∈ [0, ∞) × K × Rn , f (t, x, u) ≥ max{ψ(|x|), ψ(|u|)|u|} − a; A(iv) for each M, > 0 there exist Γ, δ > 0 such that |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ max{f (t, x1 , u1 ), f (t, x2 , u2 )} for each t ∈ [0, ∞), each u1 , u2 ∈ Rn and each x1 , x2 ∈ K which satisfy |xi | ≤ M, |ui | ≥ Γ, i = 1, 2,

max{|x1 − x2 |, |u1 − u2 |} ≤ δ;

A(v) for each M, > 0 there exist δ > 0 such that |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ for each t ∈ [0, ∞), each u1 , u2 ∈ Rn and each x1 , x2 ∈ K which satisfy |xi |, |ui | ≤ M, i = 1, 2,

max{|x1 − x2 |, |u1 − u2 |} ≤ δ.

When K = Rn it is an elementary exercise to show that an integrand f = f (t, x, u) ∈ C 1 ([0, ∞) × Rn × Rn ) belongs to M if f satisﬁes Assumptions A(i), A(iii), sup{|f (t, 0, 0)| : t ∈ [0, ∞)} < ∞ and there exists an increasing function ψ0 : [0, ∞) → [0, ∞) such that sup{|∂f /∂x(t, x, u)|, |∂f /∂u(t, x, u)|} ≤ ψ0 (|x|)(1 + ψ(|u|)|u|) for each t ∈ [0, ∞), x, u ∈ Rn . Therefore the space M contains many functions. Example 1. It is not diﬃcult to see that if ψ(t) = t for all t ≥ 0, n = 1, K = R1 , if functions h1 , h2 , h3 ∈ C 1 (R1 ) satisfy h1 (t) ≥ 0, t ∈ [0, ∞), sup{h1 (t) : t ∈ [0, ∞)} < ∞,

4

TURNPIKE PROPERTIES

h2 (x) ≥ |x| + 1, x ∈ R1 and if the function h3 : R1 → R1 is convex and u2 + 1 ≤ h3 (u) ≤ c0 (u2 + 1), |h3 (u)| ≤ c0 (u2 + 1) for all u ∈ R1 , where c0 is a positive constant, then the function f (t, x, u) = h1 (t) + h2 (x)h3 (u), (t, x, u) ∈ [0, ∞) × R1 × R1 belongs to M. In Chapters 1-5 we consider variational problems with integrands belonging to the space M or to its subspaces. The Assumption A(i) and the inequality f (t, x, u) ≥ ψ(|u|)|u| − a in the Assumption A(iii) guarantee the existence of minimizers of the variational problems. These assumptions are common in the literature. We need the inequality f (t, x, u) ≥ ψ(|x|) − a in A(iii) in order to show that for every bounded set E ⊂ Rn the C([0, T ]) norms of approximate solutions x : [0, T ] → Rn for the variational problems on intervals [0, T ] with x(0), x(T ) ∈ E are bounded by some constant which does not depend on T . We need the Assumptions A(ii) and A(v) in order to obtain certain properties of approximate solutions for variational problems on intervals [T1 , T2 ] which depend on T2 − T1 and do not depend of T1 and T2 . Note that if a function f is Frechet diﬀerentiable, then the Assumption A(v) means that the growth of the partial derivatives of f does not exceed the growth of f . We use it in order to establish the continuity of the function U f which is deﬁned below. We equip the set M with the uniformity which is determined by the following base: E(N, , λ) = {(f, g) ∈ M × M : |f (t, x, u) − g(t, x, u)| ≤

(2.1)

for each t ∈ [0, ∞), each u ∈ Rn each x ∈ K satisfying |x|, |u| ≤ N } ∩{(f, g) ∈ M × M : (|f (t, x, u)| + 1)(|g(t, x, u)| + 1)−1 ∈ [λ−1 , λ] for each t ∈ [0, ∞), each u ∈ Rn and each x ∈ K satisfying |x| ≤ N } where N > 0, > 0, λ > 1 [37]. Clearly, the uniform space M is Hausdorﬀ and has a countable base. Therefore M is metrizable. We will prove in Secton 1.3 that the uniform space M is complete. Put I f (T1 , T2 , x) =

T2

T1

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

(2.2)

Inﬁnite horizon variational problems

5

where f ∈ M, 0 ≤ T1 < T2 < ∞ and x : [T1 , T2 ] → K is an a.c. function. For f ∈ M, a, b ∈ K and numbers T1 , T2 satisfying 0 ≤ T1 < T2 , put U f (T1 , T2 , a, b) = inf{I f (T1 , T2 , x) : x : [T1 , T2 ] → K

(2.3)

is an a.c. function satisfying x(T1 ) = a, x(T2 ) = b}, σ f (T1 , T2 , a) = inf{U f (T1 , T2 , a, b) : b ∈ K}.

(2.4)

U f (T1 , T2 , a, b)

< ∞ for each f ∈ M, each It is easy to see that −∞ < a, b ∈ K and each pair of numbers T1 , T2 satisfying 0 ≤ T1 < T2 . Let f ∈ M. We say that an a.c. function x : [0, ∞) → K is an (f )-good function if for any a.c. function y : [0, ∞) → K, inf{I f (0, T, y) − I f (0, T, x) : T ∈ (0, ∞)} > −∞.

(2.5)

In this chapter we study the set of (f )-good functions and prove the following results. Theorem 1.2.1 For each h ∈ M and each z ∈ K there exists an (h)good function Z h : [0, ∞) → K satisfying Z h (0) = z such that: 1. For each f ∈ M, each z ∈ K and each a.c. function y : [0, ∞) → K one of the following properties holds: (i) I f (0, T, y) − I f (0, T, Z f ) → ∞ as T → ∞; (ii) sup{|I f (0, T, y) − I f (0, T, Z f )| : T ∈ (0, ∞)} < ∞, sup{|y(t)| : t ∈ [0, ∞)} < ∞. 2. For each f ∈ M and each number M > inf{|u| : u ∈ K} there exist a neighborhood U of f in M and a number Q > 0 such that sup{|Z g (t)| : t ∈ [0, ∞)} ≤ Q for each g ∈ U and each z ∈ K satisfying |z| ≤ M . 3. For each f ∈ M and each number M > inf{|u| : u ∈ K} there exist a neighborhood U of f in M and a number Q > 0 such that for each g ∈ U , each z ∈ K satisfying |z| ≤ M , each T1 ≥ 0, T2 > T1 and each a.c. function y : [T1 , T2 ] → K satisfying |y(T1 )| ≤ M the following relation holds: I g (T1 , T2 , Z g ) ≤ I g (T1 , T2 , y) + Q. 4. If K = Rn , then for each f ∈ M and each z ∈ Rn the function Z f : [0, ∞) → Rn is an (f )-minimal solution. Corollary 1.2.1 Let f ∈ M, z ∈ K and let y : [0, ∞) → K be an a.c. function. Then y is an (f )-good function if and only if condition (ii) of Assertion 1 of Theorem 1.2.1 holds.

6

TURNPIKE PROPERTIES

Theorem 1.2.2 For each f ∈ M there exist a neighborhood U of f in M and a number M > 0 such that for each g ∈ U and each (g)-good function x : [0, ∞) → K, lim sup |x(t)| < M. t→∞

Our next result shows that for every bounded set E ⊂ K the C([0, T ]) norms of approximate solutions x : [0, T ] → K for the minimization problem on an interval [0, T ] with x(0), x(T ) ∈ E are bounded by some constant which does not depend on T . Theorem 1.2.3 Let f ∈ M and M1 , M2 , c be positive numbers. Then there exist a neighborhood U of f in M and a number S > 0 such that for each g ∈ U , each T1 ∈ [0, ∞) and each T2 ∈ [T1 + c, ∞) the following properties hold: (i) if x, y ∈ K satisfy |x|, |y| ≤ M1 and if an a.c. function v : [T1 , T2 ] → K satisﬁes v(T1 ) = x, v(T2 ) = y, I g (T1 , T2 , v) ≤ U g (T1 , T2 , x, y) + M2 , then |v(t)| ≤ S, t ∈ [T1 , T2 ];

(2.6)

(ii) if x ∈ K satisﬁes |x| ≤ M1 and if an a.c. function v : [T1 , T2 ] → K satisﬁes v(T1 ) = x, I g (T1 , T2 , v) ≤ σ g (T1 , T2 , x) + M2 , then the inequality (2.6) is valid. Theorems 1.2.1-1.2.3 have been proved in [98]. In the sequel we use the following notation: B(x, r) = {y ∈ Rn : |y − x| ≤ r}, x ∈ Rn , r > 0,

(2.7)

B(r) = B(0, r), r > 0. Chapter 1 is organized as follows. In Section 1.3 we study the space M and the dependence of the functionals U f and I f of f . In Section 1.4 we associate with any f ∈ M a related discrete-time control system and study its approximate solutions. Theorems 1.2.1-1.2.3 are proved in Section 1.5.

7

Inﬁnite horizon variational problems

1.3.

Auxiliary results

In this section we study the space M and continuity properties of the functionals I f and U f . The next proposition follows from Assumption A(iv). Proposition 1.3.1 Let f ∈ M. Then for each pair of positive numbers M and there exist Γ, δ > 0 such that the following property holds: If t ∈ [0, ∞) and if u1 , u2 ∈ Rn and x1 , x2 ∈ K satisfy |xi | ≤ M, |ui | ≥ Γ, i = 1, 2, |u1 − u2 |, |x1 − x2 | ≤ δ,

(3.1)

then |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ min{f (t, x1 , u1 ), f (t, x2 , u2 )}. Proof. Let M, > 0. Choose 0 ∈ (0, 8−1 inf{1, }).

(3.2)

It follows from Assumption A(iv) that there exist Γ, δ > 0 such that the following property holds: If t ∈ [0, ∞) and if u1 , u2 ∈ Rn and x1 , x2 ∈ K satisfy (3.1), then |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ 0 sup{f (t, x1 , u1 ), f (t, x2 , u2 )}.

(3.3)

Assume that t ∈ [0, ∞), u1 , u2 ∈ Rn and x1 , x2 ∈ K satisfy (3.1). By the deﬁnition of Γ, δ, (3.2) and (3.3), min{f (t, x1 , u1 ), f (t, x2 , u2 )} ≥ (1 − 0 ) max{f (t, x1 , u1 ), f (t, x2 , u2 )} −1 ≥ (1 − 0 )−1 0 |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≥ |f (t, x1 , u1 ) − f (t, x2 , u2 )|.

Proposition 1.3.1 is proved. Proposition 1.3.2 The uniform space M is complete. Proof. Assume that {fi }∞ i=1 ⊂ M is a Cauchy sequence. Clearly, for each (t, x, u) ∈ [0, ∞) × K × Rn the sequence {fi (t, x, u)}∞ i=1 is a Cauchy sequence. Then there exists a function f : [0, ∞) × K × Rn → R1 such that (3.4) f (t, x, u) = lim fi (t, x, u) i→∞

Rn .

for each (t, x, u) ∈ [0, ∞) × K × In order to prove the proposition it is suﬃcient to show that f satisﬁes Assumption A(iv).

8

TURNPIKE PROPERTIES

Let M, be positive numbers. Choose a number λ > 1 for which λ2 − 1 < 8−1 .

(3.5)

Since {fi }∞ i=1 is a Cauchy sequence there exists an integer j ≥ 1 such that (3.6) (fi , fj ) ∈ E(M, , λ) for any integer i ≥ j. By (3.5) and the properties of ψ there exists a number Γ0 such that Γ0 > 1, ψ(Γ0 ) ≥ 2a, λ2 (1 + 2ψ(Γ0 )−1 )2 − 1 < 8−1 .

(3.7)

Choose 1 > 0 such that 81 [λ(1 + 2ψ(Γ0 )−1 )]2 < .

(3.8)

By Proposition 1.3.1 there exist numbers Γ, δ > 0 such that Γ > Γ0 and that for each t ∈ [0, ∞), each u1 , u2 ∈ Rn and each x1 , x2 ∈ K which satisfy (3.1) the inequality |fj (t, x1 , u1 ) − fj (t, x2 , u2 )| ≤ 1 min{fj (t, x1 , u1 ), fj (t, x2 , u2 )}

(3.9)

is true. Assume that t ∈ [0, ∞), u1 , u2 ∈ Rn , x1 , x2 ∈ K satisfy (3.1). Then the inequality (3.9) follows from the deﬁnition of Γ, δ. (2.1), (3.4), (3.6) and (3.1) imply that (|f (t, xi , ui )| + 1)(|fj (t, xi , ui )| + 1)−1 ∈ [λ−1 , λ], i = 1, 2.

(3.10)

It follows from Assumption A(iii), (3.1), (3.7) and (3.9) that min{f (t, xi , ui ), fj (t, xi , ui )} ≥ 2−1 ψ(Γ0 ), i = 1, 2.

(3.11)

By (3.11) and (3.10), f (t, xi , ui )fj (t, xi , ui )−1 ∈ [(λ(1 + 2ψ(Γ0 )−1 ))−1 , λ(1 + 2ψ(Γ0 )−1 )], i = 1, 2.

(3.12)

We may assume without loss of generality that f (t, x1 , u1 ) ≥ f (t, x2 , u2 ). It follows from (3.12), (3.9), (3.8) and (3.7) that f (t, x1 , u1 ) − f (t, x2 , u2 ) ≤ λ(1 + 2ψ(Γ0 )−1 )fj (t, x1 , u1 )

(3.13)

9

Inﬁnite horizon variational problems

−(λ(1 + 2ψ(Γ0 )−1 ))−1 fj (t, x2 , u2 ) = λ(1 + 2ψ(Γ0 )−1 )[fj (t, x1 , u1 ) − fj (t, x2 , u2 )] +fj (t, x2 , u2 )[λ(1 + 2ψ(Γ0 )−1 ) − (λ(1 + 2ψ(Γ0 )−1 ))−1 ] ≤ λ(1 + 2ψ(Γ0 )−1 )1 fj (t, x2 , u2 ) + fj (t, x2 , u2 )[λ(1 + 2ψ(Γ0 )−1 ) −(λ(1 + 2ψ(Γ0 )−1 ))−1 ] ≤ 1 [λ(1 + 2ψ(Γ0 )−1 )]2 f (t, x2 , u2 ) +f (t, x2 , u2 )[λ2 (1 + 2ψ(Γ0 )−1 )2 − 1] ≤ f (t, x2 , u2 ). Therefore the function f satisﬁes Assumption A(iv). This completes the proof of the proposition. The next auxiliary result will be used in order to establish the continuous dependence of the functional U f (T1 , T2 , y, z) of T1 , T2 , y, z and the continuous dependence of the functional I f (T1 , T2 , x) of f . Proposition 1.3.3 Let M1 > 0 and let 0 < τ0 < τ1 . Then there exists a number M2 > 0 such that the following property holds: If f ∈ M, numbers T1 , T2 satisfy 0 ≤ T1 , T2 ∈ [T1 + τ0 , T1 + τ1 ]

(3.14)

and if an a.c. function x : [T1 , T2 ] → K satisﬁes I f (T1 , T2 , x) ≤ M1 ,

(3.15)

|x(t)| ≤ M2 , t ∈ [T1 , T2 ].

(3.16)

then Proof. By Assumption A(iii) and the properties of the function ψ there exists a number c0 > 0 such that f (t, x, u) ≥ |u|

(3.17)

for each f ∈ M and each (t, x, u) ∈ [0, ∞) × K × Rn satisfying |u| ≥ c0 , and (3.18) f (t, x, u) ≥ 2M1 (min{1, τ0 })−1 for each f ∈ M and each (t, x, u) ∈ [0, ∞) × K × Rn satisfying |x| ≥ c0 . Fix a number (3.19) M2 > 1 + M1 + aτ1 + c0 (1 + τ1 ) (recall a in Assumption A(iii)). Let f ∈ M, T1 , T2 be numbers satisfying (3.14) and let x : [T1 , T2 ] → K be an a.c. function satisfying (3.15). We will show that (3.16) holds.

10

TURNPIKE PROPERTIES

Assume the contrary. Then there exists t0 ∈ [T1 , T2 ] such that |x(t0 )| > M2 .

(3.20)

By the deﬁnition of c0 , (3.18), (3.14) and (3.15) there exists t1 ∈ [T1 , T2 ] satisfying |x(t1 )| ≤ c0 . (3.21) Set E = [inf{t0 , t1 }, sup{t0 , t1 }], E1 = {t ∈ E : |x (t)| ≥ c0 }, E2 = E \ E1 . (3.22) By the deﬁnition of c0 , Assumption A(iii), (3.15), (3.22), (3.14) and (3.17), |x(t1 ) − x(t0 )| ≤

≤ τ1 c0 +

E1

E1

|x (t)|dt +

|x (t)|dt ≤ τ1 c0 +

E1

E2

|x (t)|dt

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

f

≤ τ1 c0 + I (T1 , T2 , x) + aτ1 ≤ τ1 (c0 + a) + M1 . It follows from this inequality, (3.20) and (3.21) that M2 − c0 ≤ τ1 (c0 + a) + M1 . This is contradictory to (3.19). The obtained contradiction proves the proposition. The following propositon establishes an important property which will be used in Chapter 2. Proposition 1.3.4 Let M1 , > 0 and let 0 < τ0 < τ1 . Then there exists a positive number δ such that for each f ∈ M and each pair of numbers T1 , T2 satisfying (3.14) the following property holds: If an a.c. function x : [T1 , T2 ] → K satisﬁes (3.15) and if t1 , t2 ∈ [T1 , T2 ] satisﬁes |t1 − t2 | ≤ δ, then |x(t1 ) − x(t2 )| ≤ . Proof. By Assumption A(iii) and the properties of the function ψ there exists a number c0 > 0 such that for each f ∈ M and each (t, x, u) ∈ [0, ∞) × K × Rn satisfying |u| ≥ c0 the inequality

is true. Choose

f (t, x, u) ≥ 4−1 (M1 + 2 + aτ1 )|u|

(3.23)

δ ∈ (0, 8−1 (c0 + 1)−1 ).

(3.24)

11

Inﬁnite horizon variational problems

Assume that f ∈ M, numbers T1 , T2 satisfy (3.14), an a. c. function x : [T1 , T2 ] → K satisﬁes (3.15) and t1 , t2 ∈ [T1 , T2 ], 0 < |t1 − t2 | ≤ δ.

(3.25)

Set E = [min{t1 , t2 }, max{t1 , t2 }], E1 = {t ∈ E : |x (t)| ≥ c0 }, E2 = E \ E1 . By Assumption A(iii), the choice of c0 , (3.14), (3.25) and (3.23), |x(t2 ) − x(t1 )| ≤

E1

|x (t)|dt +

E2 −1

≤ δc0 + [4(M1 + 2 + aτ1 )]

|x (t)|dt ≤ δc0 +

E1

E1

|x (t)|dt

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

≤ δc0 + [4(M1 + 2 + aτ1 )]−1 (I f (T1 , T2 , x) + aτ1 ). Combined with (3.15), (3.14) and (3.24) this inequality implies that |x(t2 ) − x(t1 )| ≤ δc0 + 4−1 ≤ . This completes the proof of the proposition. We have the following result (see [9]). Proposition 1.3.5 Assume that f ∈ M, M1 > 0, 0 ≤ T1 < T2 , xi : [T1 , T2 ] → K, i = 1, 2, . . . is a sequence of a.c. functions such that I f (T1 , T2 , xi ) ≤ M1 , i = 1, 2, . . . . Then there exists a subsequence {xik }∞ k=1 and an a.c. function x : [T1 , T2 ] → K such that I f (T1 , T2 , x) ≤ M1 , xik → x(t) as k → ∞ uniformly in [T1 , T2 ] and xik → x as k → ∞ weakly in L1 (Rn ; (T1 , T2 )). Corollary 1.3.1 For each f ∈ M, each pair of numbers T1 , T2 satisfying 0 ≤ T1 < T2 and each z1 , z2 ∈ K there exists an a.c. function x : [T1 , T2 ] → K such that x(Ti ) = zi , i = 1, 2, I f (T1 , T2 , x) = U f (T1 , T2 , z1 , z2 ). Corollary 1.3.2 For each f ∈ M, each T1 , T2 satisfying 0 ≤ T1 < T2 and each z ∈ K there exists an a.c. function x : [T1 , T2 ] → K such that x(T1 ) = z, I f (T1 , T2 , x) = σ f (T1 , T2 , z).

12

TURNPIKE PROPERTIES

It is an elementary exercise to prove the following result. Proposition 1.3.6 Let f ∈ M, 0 < c1 < c2 < ∞ and let c3 > 0. Then there exists a neighborhood U of f in M such that the set {U g (T1 , T2 , z1 , z2 ) : g ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + c1 , T1 + c2 ], z1 , z2 ∈ K ∩ B(c3 ), i = 1, 2} is bounded. The next auxiliary result establishes the continuity of the functional (T1 , T2 , y, z) → U f (T1 , T2 , y, z). Proposition 1.3.7 Assume that K = Rn , f ∈ M, 0 < c1 < c2 < ∞ and M, > 0. Then there exists δ > 0 such that the following property holds: If T1 , T2 ≥ 0 satisfy T2 ∈ [T1 + c1 , T1 + c2 ]

(3.26)

and if y1 , y2 , z1 , z2 ∈ Rn satisfy |yi |, |zi | ≤ M, i = 1, 2, then

sup{|y1 − y2 |, |z1 − z2 |} ≤ δ,

|U f (T1 , T2 , y1 , z1 ) − U f (T1 , T2 , y2 , z2 )| ≤ .

(3.27) (3.28)

Proof. By Proposition 1.3.6 there exists a number M0 > sup{|U f (T1 , T2 , y, z)| : T1 ∈ [0, ∞), T2 ∈ [T1 +c1 , T1 +c2 ], (3.29) y, z ∈ B(M )}. It follows from Proposition 1.3.3 that there exists a number M1 > 0 such that the following property holds: If a pair of numbers T1 , T2 ≥ 0 satisﬁes (3.26) and an a.c. function x : [T1 , T2 ] → Rn satisﬁes I f (T1 , T2 , x) ≤ 4M0 + 1, then |x(t)| ≤ M1 , t ∈ [T1 , T2 ].

(3.30)

Choose a number δ1 > 0 such that 4δ1 (2c2 + 2a + 4ac2 + 1 + M0 ) <

(3.31)

Inﬁnite horizon variational problems

13

(see Assumption A(iii)). By Proposition 1.3.1 there exist Γ0 > 2 and δ2 ∈ (0, 8−1 )

(3.32)

such that |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ δ1 inf{f (t, x1 , u1 ), f (t, x2 , u2 )}

(3.33)

for each t ∈ [0, ∞) and each u1 , u2 , x1 , x2 ∈ Rn which satisfy |xi | ≤ M1 + 1, |ui | ≥ Γ0 − 1, i = 1, 2, |u1 − u2 |, |x1 − x2 | ≤ δ2 . (3.34) By Assumption A(iv) there exists δ3 ∈ (0, 4−1 inf{δ1 , δ2 })

(3.35)

|f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ δ1

(3.36)

such that for each t ∈ [0, ∞), each u1 , u2 , x1 , x2 ∈ Rn which satisfy |xi |, |ui | ≤ Γ0 + M1 + 4, i = 1, 2, sup{|x1 − x2 |, |u1 − u2 |} ≤ δ3 . (3.37) Choose a number δ > 0 for which 8(c−1 1 + 1)δ < δ3 .

(3.38)

Assume that numbers T1 , T2 ≥ 0 satisfy (3.26) and y1 , y2 , z1 , z2 ∈ Rn satisfy (3.27). By Corollary 1.3.1 there exists an a.c. function x1 : [T1 , T2 ] → Rn such that x1 (T1 ) = y1 , x1 (T2 ) = z1 , I f (T1 , T2 , x1 ) = U f (T1 , T2 , y1 , z1 ).

(3.39)

Put x2 (t) = x1 (t)+y2 −y1 +(t−T1 )(T2 −T1 )−1 (z2 −z1 −y2 +y1 ), t ∈ [T1 , T2 ]. (3.40) Clearly (3.41) x2 (T1 ) = y2 , x2 (T2 ) = z2 . It follows from (3.26), (3.27), (3.39), (3.29) and the deﬁnition of M1 that |x1 (t)| ≤ M1 , t ∈ [T1 , T2 ].

(3.42)

(3.40), (3.27) and (3.26) imply that |x1 (t) − x2 (t)| ≤ 3δ, |x1 (t) − x2 (t)| ≤ 2c−1 1 δ, t ∈ [T1 , T2 ].

(3.43)

14

TURNPIKE PROPERTIES

Set E1 = {t ∈ [T1 , T2 ] : |x1 (t)| ≥ Γ0 }, E2 = [T1 , T2 ] \ E1 . We have where

|I f (T1 , T2 , x2 ) − I f (T1 , T2 , x1 )| ≤ σ1 + σ2

σj =

Ej

|f (t, x1 (t), x1 (t)) − f (t, x2 (t), x2 (t))|dt, j = 1, 2.

(3.44) (3.45)

(3.46)

We will estimate σ1 , σ2 separately. By (3.42), (3.43), (3.44), (3.38), (3.35), (3.32) and the deﬁnition of δ2 for each t ∈ E1 , |f (t, x1 (t), x1 (t)) − f (t, x2 (t), x2 (t))| ≤ δ1 f (t, x1 (t), x1 (t)) and σ1 ≤ δ 1

E1

f (t, x1 (t), x1 (t))dt.

It follows from this inequality, (3.39), (3.27), (3.29), (3.26) and Assumption A(iii) that σ1 ≤ δ1 (I f (T1 , T2 , x1 ) + a(T2 − T1 )) ≤ δ1 (M0 + ac2 ).

(3.47)

By the deﬁnition of δ3 , (3.42), (3.43), (3.38) and (3.44), |f (t, x1 (t), x1 (t)) − f (t, x2 (t), x2 (t))| ≤ δ1 for each t ∈ E2 and

σ2 ≤ δ1 c2 .

(3.48)

Combining (3.45), (3.47), (3.48) and (3.31) we obtain that |I f (T1 , T2 , x2 ) − I f (T1 , T2 , x1 )| ≤ δ1 (M0 + ac2 + c2 ) ≤ . Together with (3.39) and (3.41) this implies that U f (T1 , T2 , y2 , z2 ) ≤ U f (T1 , T2 , y1 , z1 ) + . This completes the proof of the proposition. The next proposition is an important tool which will be used in Chapters 1-5. It establishes that the integral functional I f (T1 , T2 , x) depends continuously on f . Proposition 1.3.8 Let f ∈ M, 0 < c1 < c2 < ∞, D, > 0. Then there exists a neighborhood V of f in M such that for each g ∈ V , each

15

Inﬁnite horizon variational problems

pair of numbers T1 , T2 ≥ 0 satisfying T2 − T1 ∈ [c1 , c2 ] the following property holds: If an a. c. function x : [T1 , T2 ] → K satisﬁes inf{I f (T1 , T2 , x), I g (T1 , T2 , x)} ≤ D, then

(3.49)

|I f (T1 , T2 , x) − I g (T1 , T2 , x)| ≤ .

Proof. It follows from Proposition 1.3.3 there exists a number S > 0 such that (3.50) |x(t)| ≤ S, t ∈ [T1 , T2 ] for each g ∈ M, each T1 , T2 ≥ 0 satisfying T2 − T1 ∈ [c1 , c2 ] and each a.c. function x : [T1 , T2 ] → K which satisﬁes I g (T1 , T2 , x) ≤ D + 1. Choose δ ∈ (0, 1), N > S and Γ > 1 such that δ(c2 + 1) ≤ 4−1 , ψ(N )N > 4a, (Γ − 1)(c2 + D + ac2 + 1) ≤ 4−1 (3.51) and put V = {g ∈ M : (f, g) ∈ E(N, δ, Γ)} (see (2.1)). Assume that g ∈ V , T1 , T2 ≥ 0, T2 − T1 ∈ [c1 , c2 ]

(3.52)

and x : [T1 , T2 ] → K is an a.c. function satisfying (3.49). By the choice of S the inequality (3.50) is true. Put E1 = {t ∈ [T1 , T2 ] : |x (t)| ≤ N }, E2 = [T1 , T2 ] \ E1 . It follows from (3.50) and the choice of V and N that |f (t, x(t), x (t)) − g(t, x(t), x (t))| ≤ δ, t ∈ E1 .

(3.53)

Deﬁne h(t) = inf{f (t, x(t), x (t)), g(t, x(t), x (t))}, t ∈ [T1 , T2 ].

(3.54)

It follows from (3.50), (3.51), Assumption A(iii) and the deﬁnition of V , N that for t ∈ E2 (f (t, x(t), x (t)) + 1)(g(t, x(t), x (t)) + 1)−1 ∈ [Γ−1 , Γ],

(3.55)

|f (t, x(t), x (t)) − g(t, x(t), x (t))| ≤ (Γ − 1)(h(t) + 1). By (3.53), (3.52), (3.55), (3.49), (3.54), Assumption A(iii) and (3.51), f

g

|I (T1 , T2 , x) − I (T1 , T2 , x)| ≤

E1

|f (t, x(t), x (t)) − g(t, x(t), x (t))|dt

16

TURNPIKE PROPERTIES

+

E2

|f (t, x(t), x (t)) − g(t, x(t), x (t))|dt ≤ δc2 + (Γ − 1)

E2

(h(t) + 1)dt

≤ δc2 + (Γ − 1)c2 + (Γ − 1)(D + ac2 ) ≤ . The proposition is proved. The next result establishes that the functional U f (T1 , T2 , y, z) depends continuously on f . It is also an important tool which will be used in Chapters 1-5. Proposition 1.3.9 Let f ∈ M, 0 < c1 < c2 < ∞, c3 , > 0. Then there exists a neighborhood V of f in M such that |U f (T1 , T2 , y, z) − U g (T1 , T2 , y, z)| ≤ for each g ∈ V , each T1 , T2 ≥ 0 satisfying T2 − T1 ∈ [c1 , c2 ] and each y, z ∈ K ∩ B(c3 ). Proof. By Proposition 1.3.6 there exist a neighborhood V1 of f in M and a positive number D0 such that |U g (T1 , T2 , z1 , z2 )| + 1 < D0 for each g ∈ V1 , each T1 ∈ [0, ∞), T2 ∈ [T1 + c1 , T1 + c2 ] and each z1 , z2 ∈ K ∩ B(c3 ), i = 1, 2. It follows from Proposition 1.3.8 that there is a neighborhood V of f in M such that V ⊂ V1 and that |I f (T1 , T2 , x) − I g (T1 , T2 , x)| ≤ inf{1, } for each g ∈ V , each T1 , T2 ≥ 0 satisfying T2 − T1 ∈ [c1 , c2 ] and each a.c. function x : [T1 , T2 ] → K which satisfy min{I f (T1 , T2 , x), I g (T1 , T2 , x)} ≤ D0 + 2. The validity of the proposition now follows from the equality U g (T1 , T2 , y, z) = inf{I g (T1 , T2 , x) : x : [T1 , T2 ] → K is an a.c. function satisfying x(T1 ) = y, x(T2 ) = z, I g (T1 , T2 , x) ≤ D0 + 1} which holds for g ∈ V , T1 ≥ 0, T2 ∈ [T1 + c1 , T1 + c2 ] and y, z ∈ K satisfying |y|, |z| ≤ c3 .

Inﬁnite horizon variational problems

1.4.

17

Discrete-time control systems

In this section we associate with f ∈ M a related discrete-time control system. We establish a boundedness of approximate solutions of this system (see Proposition 1.4.2). This result plays a crucial role in the proof of Theorem 1.2.3. Let f ∈ M, z¯ ∈ K and let 0 < c1 < c2 < ∞. It follows from Proposition 1.3.6 that there exist a positive number M0 and a neighborhood U0 of f in M such that |U g (T1 , T2 , y, z)| ≤ M0 for each g ∈ U0 , each T1 ∈ [0, ∞), T2 ∈ [T1 + c1 , T1 + c2 ]

(4.1)

and each y, z ∈ K ∩ B(2|¯ z | + 1). Proposition 1.3.3 implies that there is a number M1 > 0 such that 2M0 + 2 ≤ U g (T1 , T2 , y, z) for each g ∈ M, each T1 ∈ [0, ∞), T2 ∈ [T1 + c1 , T1 + c2 ],

(4.2)

and each y, z ∈ K satisfying |y| + |z| ≥ M1 . Proposition 1.4.1 Let a number M1 > 0 satisfy (4.2) and let M2 > 0. Then there are an integer N > 2 and a neighborhood U of f in M such that for each g ∈ U , each ∆ ∈ [0, ∞), each T ∈ [c1 , c2 ] and each pair of integers q1 , q2 satisfying 0 ≤ q1 < q2 , q2 −q1 ≥ N the following assertions hold: 2 ⊂ K satisﬁes 1. If {zi }qi=q 1 {i ∈ {q1 , . . . , q2 } : |zi | ≤ M1 } = {q1 , q2 } and if yi = zi , i = q1 , q2 , yi = z¯, i = q1 + 1, . . . , q2 − 1, then q 2 −1

[U g (∆ + iT, ∆ + (i + 1)T, zi , zi+1 )

i=q1

−U g (∆ + iT, ∆ + (i + 1)T, yi , yi+1 )] ≥ M2 ; 2. If

2 {zi }qi=q 1

(4.3)

⊂ K satisﬁes {i ∈ {q1 , . . . , q2 } : |zi | ≤ M1 } = {q1 }

and if yq1 = zq1 , yi = z¯, i = q1 + 1, . . . , q2 , then the inequality (4.3) is valid.

18

TURNPIKE PROPERTIES

Proof. It follows from Proposition 1.3.6 that there exist a positive number M3 and a neighborhood U of f in M such that U ⊂ U0 , |U g (T1 , T2 , y, z)| ≤ M3 for each g ∈ U, each T1 ∈ [0, ∞), T2 ∈ [T1 + c1 , T1 + c2 ] and each y, z ∈ K ∩ B(2|¯ z | + 1 + 2M1 ). Fix an integer N ≥ M2 + 4M3 + 4. The validity of the proposition now follows from the deﬁnition of U , M3 , N , (4.1) and (4.2). Proposition 1.4.2 Assume that a positive number M1 satisﬁes (4.2) and M3 > 0. Then there exist a neighborhood V of f in M and a number M4 > M1 such that for each g ∈ V , each ∆ ∈ [0, ∞), each T ∈ [c1 , c2 ] and each pair of integers q1 , q2 satisfying 0 ≤ q1 < q2 the following assertions hold: 2 ⊂ K satisﬁes 1. If a sequence {zi }qi=q 1 |zq1 |, |zq2 | ≤ M1 , max{|zi | : i = q1 , . . . , q2 } > M4 ,

(4.4)

2 then there is a sequence {yi }qi=q ⊂ K such that yqj = zqj , j = 1, 2 and 1

q 2 −1

[U g (∆ + iT, ∆ + (i + 1)T, zi , zi+1 )

i=q1

−U g (∆ + iT, ∆ + (i + 1)T, yi , yi+1 )] ≥ M3 .

(4.5)

2 ⊂ K satisﬁes 2. If a sequence {zi }qi=q 1

|zq1 | ≤ M1 , max{|zi | : i = q1 , . . . , q2 } > M4 ,

(4.6)

2 then there is a sequence {yi }qi=q ⊂ K such that yq1 = zq1 and the in1 equality (4.5) is true.

Proof. There exist a neighborhood U ⊂ U0 of f in M and an integer N > 2 such that Proposition 1.4.1 holds with M2 = 4(M3 + 1). It follows from Proposition 1.3.6 that there exist a positive number r1 and a neighborhood V of f in M such that V ⊂ U, |U g (T1 , T2 , y, z)| + 1 < r1 for each g ∈ V, each T1 ∈ [0, ∞), (4.7) z | + 1 + M1 ). T2 ∈ [T1 + c1 , T1 + c2 ] and each y, z ∈ K ∩ B(|¯ By Proposition 1.3.3 there exists M4 > M1 such that inf{U g (T1 , T2 , y, z) : g ∈ M, T1 ∈ [0, ∞), T2 ∈ [T1 + c1 , T1 + c2 ], (4.8)

Inﬁnite horizon variational problems

19

y, z ∈ K, |y| + |z| ≥ M4 } > 3r1 N + 4 + 4M3 + 3ac2 N (recall a in Assumption A(iii)). 2 Let g ∈ V , ∆ ∈ [0, ∞), T ∈ [c1 , c2 ], 0 ≤ q1 < q2 , {zi }qi=q ⊂ K. 1 We prove Assertion 1. Assume that (4.4) holds. Then there is j ∈ {q1 , . . . , q2 } such that |zj | > M4 . Set i1 = max{i ∈ {q1 , . . . , j} : |zi | ≤ M1 }, i2 = min{i ∈ {j, . . . , q2 } : |zi | ≤ M1 }. If i2 − i1 ≥ N , then by the deﬁnition of V , U , N and Proposition 1.4.1 2 ⊂ K which satisﬁes (4.5) and yqi = zqi , there exists a sequence {yi }qi=q 1 i = 1, 2. Now assume that i2 − i1 < N . Put yi = zi , i ∈ {q1 , . . . , i1 }∪{i2 , . . . , q2 }, yi = z¯, i = i1 +1, . . . , i2 −1. (4.9) It follows from (4.9), (4.7), Assumption A(iii) and the deﬁnition of i1 , i2 , j that q 2 −1

[U g (∆ + iT, ∆ + (i + 1)T, zi , zi+1 ) − U g (∆ + iT, ∆ + (i + 1)T, yi , yi+1 )]

i=q1

(4.10) =

i 2 −1

[U g (∆+iT, ∆+(i+1)T, zi , zi+1 )−U g (∆+iT, ∆+(i+1)T, yi , yi+1 )]

i=i1

≥ U g (∆ + (j − 1)T, ∆ + jT, zj−1 , zj ) − a(i2 − i1 − 1)c2 − (i2 − i1 )r1 . By this relation and the deﬁnition of j, M4 (see (4.8)) q 2 −1

[U g (∆ + iT, ∆ + (i + 1)T, zi , zi+1 )

(4.11)

i=q1

−U g (∆ + iT, ∆ + (i + 1)T, yi , yi+1 )] ≥ 4M3 + 4. This completes the proof of Assertion 1. We prove Assertion 2. Assume that (4.6) holds. Then there is j ∈ {q1 , . . . , q2 } such that |zj | > M4 . Set i1 = sup{i ∈ {q1 , . . . , j} : |zi | ≤ M1 }. There are two cases: 1) |zi | > M1 , i = j, . . . , q2 ; 2) inf{|zi | : i = j, . . . , q2 } ≤ M1 . Consider the ﬁrst case. We set yi = zi , i = q1 , . . . , i1 , yi = z¯, i = i1 + 1, . . . , q2 .

20

TURNPIKE PROPERTIES

If q2 − i1 ≥ N , then (4.5) follows from the deﬁnition of V , U , N and Proposition 1.4.1. If q2 − i1 < N , then (4.5) follows from the deﬁnition 2 , i1 , j, M4 , (4.7) (see (4.10), (4.11) with i2 = q2 ). of {yi }qi=q 1 Consider the second case. Set i2 = inf{i ∈ {j, . . . , q2 } : |zi | ≤ M1 }. If i2 − i1 ≥ N , then by the deﬁnition of V, U, N and Proposition 1.4.1 2 ⊂ K which satisﬁes (4.5) and yqi = zqi , there exists a sequence {yi }qi=q 1 2 ⊂ K by (4.9). Then i = 1, 2. If i2 −i1 < N we deﬁne a sequence {yi }qi=q 1 (4.10) and (4.11) follows from (4.9), the deﬁnition of i1 , i2 , j, M4 , (4.7). Assertion 2 is proved. This completes the proof of the proposition.

1.5.

Proofs of Theorems 1.1-1.3

Construction of a neighborhood U . Let f ∈ M, z¯ ∈ K, M > 2|¯ z |. It follows from Proposition 1.3.6 that there exist a positive number M0 and a neighborhood U0 of f in M such |U g (T1 , T2 , y, z)| ≤ M0 for each g ∈ U0 , each T1 ∈ [0, ∞),

(5.1)

z | + 1). T2 ∈ [T1 + 4−1 , T1 + 4] and each y, z ∈ K ∩ B(2|¯ It follows from Proposition 1.3.3 that there exists a number M1 > M for which 2M0 + 1 < inf{U g (T1 , T2 , y, z) : g ∈ M, T1 ∈ [0, ∞), T2 ∈ [T1 + 4−1 , T1 + 4], y, z ∈ K, |y| + |z| ≥ M1 }.

(5.2)

By (5.1), (5.2) there exists a neighborhood U1 of f in M and a number M2 such that U1 ⊂ U0 , M2 > M1 and Proposition 1.4.2 holds with M3 = 1,

(5.3)

c1 = 4−1 , c2 = 4, V = U1 , M4 = M2 . Proposition 1.3.6 implies that there exist a positive number Q0 and a neighborhood U2 of f in M such that U2 ⊂ U1 , |U g (T1 , T2 , y, z)| + 1 ≤ Q0 for each g ∈ U2 , each T1 ∈ [0, ∞), (5.4) T2 ∈ [T1 + 4−1 , T1 + 4] and each y, z ∈ K ∩ B(M2 + 1). By Proposition 1.3.3 there exists a number Q1 > Q0 + M2 + 1 such that the following property holds:

(5.5)

Inﬁnite horizon variational problems

21

If g ∈ M, T1 , T2 satisfy 0 ≤ T1 < T2 , T2 − T1 ∈ [4−1 , 4] and if an a.c. function x : [T1 , T2 ] → K satisﬁes I g (T1 , T2 , x) ≤ 2Q0 + 2, then (5.6) |x(t)| ≤ Q1 , t ∈ [T1 , T2 ]. It follows from Proposition 1.3.6 that there exist a positive number Q2 and a neighborhood U of f in M such that U ⊂ U2 , Q2 > Q1 , |U g (T1 , T2 , y, z)| + 1 < Q2 for each g ∈ U,

(5.7)

each T1 ∈ [0, ∞), T2 ∈ [T1 + 4−1 , T1 + 4] and each y, z ∈ K ∩ B(2Q1 + 4). We may assume without loss of generality that there exists a positive number Q3 such that |g(t, y, u)| + 1 < Q3 for each g ∈ U, each t ∈ [0, ∞)

(5.8)

and each y ∈ K ∩ B(2M2 + 2), u ∈ B(2M2 + 2). Construction of a function Z g : [0, ∞) → K. Let g ∈ U , z ∈ K, |z| ≤ M . By Corollary 1.3.2 for any integer q ≥ 1 there exists an a. c. function Zqg : [0, q] → K such that Zqg (0) = z, I g (0, q, Zqg ) = σ g (0, q, z).

(5.9)

It follows from Proposition 1.4.2 and the deﬁnition of Zqg , U1 , M2 that |Zqg (i)| ≤ M2 , i = 0, . . . , q, q = 1, 2, . . . .

(5.10)

There exists a subsequence {Zggj }∞ j=1 such that for any integer i ≥ 0 there exists (5.11) zig = lim Zqgj (i). j→∞

By Corollary 1.3.1 there exists an a.c. function Z g : [0, ∞) → K such that for each integer i ≥ 0, g ). Z g (i) = zig , I g (i, i + 1, Z g ) = U g (i, i + 1, zig , zi+1

(5.12)

It follows from (5.9), (5.10) and (5.4) that I g (i, i + 1, Zqg ) < Q0 , i = 0, . . . , q − 1, q = 1, 2, . . . .

(5.13)

22

TURNPIKE PROPERTIES

(5.10), (5.11), (5.12) and (5.4) imply that I g (i, i + 1, Z g ) < Q0 , i = 0, 1, . . . .

(5.14)

By (5.13), (5.14) and the deﬁnition of Q1 (see (5.5), (5.6)) |Zqg (t)| ≤ Q1 , t ∈ [0, q], q = 1, 2, . . . ,

|Z g (t)| ≤ Q1 , t ∈ [0, ∞). (5.15)

Therefore for each g ∈ U and each z ∈ K satisfying |z| ≤ M we deﬁne a.c. functions Zqg : [0, q] → K, q = 1, 2, . . . and Z g : [0, ∞) → K satisfying (5.9)-(5.15). The next auxiliary result shows that the sequence {ziq }∞ i=0 is (g)-good for each g ∈ U . Lemma 1.5.1 Let g ∈ U , z ∈ K, |z| ≤ M and let a pair of integers 2 q1 , q2 satisfy 0 ≤ q1 < q2 . Then if a sequence {yi }qi=q ⊂ K satisﬁes 1 |yq1 | ≤ M1 , then q 2 −1

g [U g (i, i + 1, zig , zi+1 ) − U g (i, i + 1, yi , yi+1 )] ≤ 4 + 4Q2 .

(5.16)

i=q1 2 Proof. Assume that a sequence {yi }qi=q ⊂ K satisﬁes |yq1 | ≤ M1 . We 1 will show that (5.16) holds. Let us assume the converse. Then

q 2 −1

g [U g (i, i + 1, zig , zi+1 ) − U g (i, i + 1, yi , yi+1 )] > 4 + 4Q2 .

(5.17)

i=q1

By Corollaries 1.3.1 and 1.3.2 we may assume without loss of generality 2 ⊂ K satisﬁes y¯q1 = yq1 , then that if a sequence {¯ yi }qi=q 1 q 2 −1

[U g (i, i + 1, yi , yi+1 ) − U g (i, i + 1, y¯i , y¯i+1 )] ≤ 0.

i=q1

(5.3) and (5.5) imply that |yi | ≤ M2 < Q1 , i = q1 , . . . , q2 .

(5.18)

By Proposition 1.3.5, (5.9), (5.11) and (5.13) for any integer i ≥ 0, g U g (i, i + 1, zig , zi+1 ) ≤ lim inf U g (i, i + 1, Zqgj (i), Zqgj (i + 1)). j→∞

Therefore there exists an integer q > q2 + 1 such that q2 i=q1

g [U g (i, i + 1, zig , zi+1 ) − U g (i, i + 1, Zqg (i), Zqg (i + 1))] ≤ 1.

(5.19)

23

Inﬁnite horizon variational problems

We deﬁne a sequence {hi }qi=0 ⊂ K as follows: hi = Zqg (i), i ∈ {0, . . . , q1 } ∪ {q2 + 1, . . . , q}, hi = yi , i = q1 + 1, . . . , q2 . (5.20) It follows from (5.20), (5.9), Corollary 1.3.1, (5.19) and (5.17) that 0≥

q−1

[U g (i, i + 1, Zqg (i), Zqg (i + 1)) − U g (i, i + 1, hi , hi+1 )]

i=0

=

q2

[U g (i, i + 1, Zqg (i), Zqg (i + 1)) − U g (i, i + 1, hi , hi+1 )]

i=q1

=

q2

g [U g (i, i + 1, Zqg (i), Zqg (i + 1)) − U g (i, i + 1, zig , zi+1 )]

i=q1

+

q2

g

U (i, i +

g ) 1, zig , zi+1

−

q 2 −1

U g (i, i + 1, yi , yi+1 )

i=q1

i=q1 g

+U (q1 , q1 + 1, yq1 , yq1 +1 ) g

−U (q1 , q1 + 1, hq1 , hq1 +1 ) − U g (q2 , q2 + 1, hq2 , hq2 +1 ) ≥ 3 + 4Q2 +U g (q2 , q2 + 1, zqg2 , zqg2 +1 ) + U g (q1 , q1 + 1, yq1 , yq1 +1 ) −U g (q1 , q1 + 1, hq1 , hq1 +1 ) − U g (q2 , q2 + 1, hq2 , hq2 +1 ). Combined with (5.20), (5.18), (5.10), (5.11), (5.5) and (5.7) this relation implies that 0 ≥ 3 + 4Q + U g (q2 , q2 + 1, zqg2 , zqg2 +1 ) + U g (q1 , q1 + 1, yq1 , yq1 +1 ) −U g (q1 , q1 + 1, Zqg (q1 ), yq1 +1 ) −U g (q2 , q2 + 1, yq2 , Zqg (q2 + 1)) ≥ 3 + 4Q2 − 4Q2 . The contradiction we have reached proves the lemma. Lemma 1.5.1 implies the following result. Lemma 1.5.2 Let g ∈ U , z ∈ K, |z| ≤ M , an integer q ≥ 0 and let T ∈ (q, ∞). Then I g (q, T, Z g ) ≤ I g (q, T, x) + 4 + 4Q2 + Q0 + 2a

(5.21)

for each a.c. function x : [q, T ] → K such that |x(q)| ≤ M1 (recall a in Assumption A(iii)).

24

TURNPIKE PROPERTIES

Proof. Assume that an a.c. function x : [q, T ] → K satisﬁes |x(q)| ≤ M1 . There exists an integer q1 ≥ q such that q1 < T ≤ q1 + 1. It follows from Lemma 1.5.1 and (5.12) that I g (q, q1 , Z g ) ≤ I g (q, q1 , x) + 4 + 4Q2 .

(5.22)

By Assumption A(iii) and (5.14) I g (q1 , T, x) ≥ −a,

I g (q1 , T, Z g ) ≤ Q0 + a.

(5.23)

(5.22) and (5.23) imply (5.21). The lemma is proved. The next lemma, which follows from Lemma 1.5.2, establishes that the function Z g is (g)-good for each g ∈ U and each z ∈ K satisfying |z| ≤ M . Lemma 1.5.3 Let g ∈ U , z ∈ K, |z| ≤ M and 0 ≤ T1 < T2 . Then I g (T1 , T2 , Z g ) ≤ I g (T1 , T2 , x) + 4 + 4Q2 + Q0 + Q3 + 3a

(5.24)

for each a.c. function x : [T1 , T2 ] → K such that |x(T1 )| ≤ M1 . Proof. Assume that an a.c. function x : [T1 , T2 ] → K satisﬁes |x(T1 )| ≤ M1 . There exists an integer q ≥ 0 such that q ≤ T1 < q + 1. Set x1 (t) = x(T1 ), t ∈ [q, T1 ], x1 (t) = x(t), t ∈ [T1 , T2 ].

(5.25)

By Lemma 1.5.2, I g (q, T2 , Z g ) ≤ I g (q, T2 , x1 ) + 4 + 4Q2 + Q0 + 2a.

(5.26)

By Assumption A(iii) and (5.26), I g (T1 , T2 , Z g ) = I g (q, T2 , Z g ) − I g (q, T1 , Z g ) ≤ I g (q, T2 , Z g ) + a (5.27) ≤ I g (q, T2 , x1 ) + 4 + 4Q2 + Q0 + 3a. It follows from (5.25) and (5.8) that |I g (q, T1 , x1 )| ≤ Q3 . (5.24) now follows from this relation and (5.27), (5.25). The lemma is proved. The following auxiliary result shows that a sequence {yi }∞ i=0 ⊂ K is not good if lim supi→∞ |yi | > M2 . It plays an important role in the proof of Theorem 1.2.2. Lemma 1.5.4 Let g ∈ U , z ∈ K, |z| ≤ M , {yi }∞ i=0 ⊂ K, lim sup |yi | > M2 . i→∞

(5.28)

25

Inﬁnite horizon variational problems

Then N −1

g [U g (i, i + 1, yi , yi+1 ) − U g (i, i + 1, zig , zi+1 )] → ∞ as N → ∞. (5.29)

i=0

Proof. There are two cases: a) lim inf |yi | > 2−1 M1 ; b) lim inf |yi | ≤ 2−1 M1 . i→∞

i→∞

Consider the case a). Set hi = z¯ for i = 0, 1, . . .. It follows from (5.1), (5.2) that U g (i, i + 1, yi , yi+1 ) − U g (i, i + 1, hi , hi+1 ) ≥ M0 + 1 for all large i. (5.29) now follows from this relation and Lemma 1.5.1. Consider the case b). By (5.28) there exists a subsequence {yik }∞ k=1 such that 0 < i1 , |yik | < M1 , sup{|yj | : j = ik , . . . , ik+1 } > M2 , k = 1, 2, . . . . (5.30) It follows from (5.3), (5.30) and Proposition 1.4.2 that for any integer ik+1 k ≥ 1 there exists a sequence {hj }j=i ⊂ K such that hj = yj , j ∈ k {ik , ik+1 }, ik+1 −1

[U g (j, j + 1, yj , yj+1 ) − U g (j, j + 1, hj , hj+1 )] ≥ 1.

(5.31)

j=ik

Fix an integer q ≥ 4. By (5.30), Lemma 1.5.1 and (5.31) for an integer N > iq N −1

g [U g (j, j + 1, zjg , zj+1 ) − U g (j, j + 1, yj , yj+1 )] ≤ 4 + 4Q2 ,

j=iq iq −1

g [U g (j, j + 1, zjg , zj+1 ) − U g (j, j + 1, hj , hj+1 )] ≤ 4 + 4Q2 ,

j=i1 N −1

g [U g (j, j + 1, zjg , zj+1 ) − U g (j, j + 1, yj , yj+1 )]

j=0

=

i 1 −1

g [U g (j, j + 1, zjg , zj+1 ) − U g (j, j + 1, yj , yj+1 )]

j=0

26

TURNPIKE PROPERTIES iq −1

+

g [U g (j, j + 1, zjg , zj+1 ) − U g (j, j + 1, hj , hj+1 )]

j=i1 iq −1

+

[U g (j, j + 1, hj , hj+1 ) − U g (j, j + 1, yj , yj+1 )]

j=i1

+

N −1

g [U g (j, j + 1, zjg , zj+1 ) − U g (j, j + 1, yj , yj+1 )]

j=iq

≤

i 1 −1

g ) − U g (j, j + 1, yj , yj+1 )] + 2(4 + 4Q2 ) − (q − 1). [U g (j, j + 1, zjg , zj+1

j=0

This completes the proof of the lemma. The next lemma implies Theorem 1.2.2. Its proof is based on Lemma 1.5.4. Lemma 1.5.5 Assume that g ∈ U , z ∈ K, |z| ≤ M and y : [0, ∞) → K is an a.c. function which satisﬁes lim sup |y(t)| > Q1 .

(5.32)

I g (0, T, y) − I g (0, T, Z g ) → ∞ as T → ∞.

(5.33)

t→∞

Then

Proof. There are two cases: a) lim sup |y(i)| > M2 ; b) lim sup |y(i)| ≤ M2 i→∞

i→∞

where i is an integer. Consider the case a). It follows from Lemma 1.5.4, (5.12) that I g (0, q, y) − I g (0, q, Z g ) → ∞ as an integer q → ∞.

(5.34)

Let T > 0. There exists an integer q(T ) ≥ 0 such that q(T ) < T ≤ q(T ) + 1.

(5.35)

By Assumption A(iii) and (5.14), I g (q(T ), T, y) ≥ −a, I g (q(T ), T, Z g ) = I g (q(T ), q(T ) + 1, Z g ) −I g (T, q(T ) + 1, Z g ) ≤ Q0 + a. Together with (5.34) these relations imply that I g (0, T, y) − I g (0, T, Z g ) ≥ I g (0, q(T ), y)

(5.36)

27

Inﬁnite horizon variational problems

−I g (0, q(T ), Z g ) − Q0 − 2a → ∞ as T → ∞. Consider the case b). There exists an integer i0 ≥ 2 such that |y(i)| ≤ M2 + 2−1 for all integers i ≥ i0 .

(5.37)

By (5.37), (5.32), (5.4) and the deﬁnition of Q1 (see (5.5)), N

[I g (i, i + 1, y) − U g (i, i + 1, y(i), y(i + 1))] → ∞ as N → ∞. (5.38)

i=0

Deﬁne a sequence {di }∞ i=i0 ⊂ K as di0 = z, di = y(i) for all integers i > i0 . By Lemma 1.5.1 and the deﬁnition of {di }∞ i=i0 for any integer N ≥ i0 + 1 N

g [U g (i, i + 1, y(i), y(i + 1)) − U g (i, i + 1, zig , zi+1 )]

i=i0 +1 N

=

g [U g (i, i + 1, di , di+1 ) − U g (i, i + 1, zig , zi+1 )]

i=i0

+U (i0 , i0 + 1, zig0 , zig0 +1 ) − U g (i0 , i0 + 1, z, y(i0 + 1)) g

≥ −4 − 4Q2 + U g (i0 , i0 + 1, zig0 , zig0 +1 ) − U g (i0 , i0 + 1, z, y(i0 + 1)). Together with (5.28), (5.12) this implies that N

[I g (i, i + 1, y) − I g (i, i + 1, Z g )] → +∞ as N → ∞.

(5.39)

i=0

Let T > 0. There exists an integer q(T ) ≥ 0 satisfying (5.35). Clearly (5.36) holds. (5.33) now follows from (5.36) and (5.39). The lemma is proved. The following lemma implies Assertion 1 of Theorem 1.2.1. Lemma 1.5.6 Let g ∈ U , z ∈ K, |z| ≤ M and let y : [0, ∞) → K be an a.c. function. Then one of the relations below holds: (i) I g (0, T, y) − I g (0, T, Z g ) → ∞ as T → ∞; (ii) sup{|I g (0, T, y) − I g (0, T, Z g )| : T ∈ (0, ∞)} < ∞. Proof. By Lemma 1.5.5 we may assume that lim supt→∞ |y(t)| ≤ Q1 . There exists an integer i0 > 0 such that |y(t)| ≤ Q1 + 2−1 , t ∈ [i0 , ∞).

(5.40)

28

TURNPIKE PROPERTIES

Fix an integer i > i0 . By Corollary 1.3.1 there exists an a.c. function y¯ : [i − 1, ∞) → K such that y¯(i − 1) = z, y¯(t) = y(t), t ∈ [i, ∞), I g (i − 1, i, y¯) = U g (i − 1, i, z, y(i)). (5.41) (5.7), (5.41), (5.40), (5.5) imply that |U g (i−1, i, z, y(i))| ≤ Q2 . It follows from this relation, (5.41), Lemma 1.5.2 and Assumption A(iii) that for each T > i, I g (i, T, y) − I g (i, T, Z g ) = I g (i − 1, T, y¯) − I g (i − 1, T, Z g )

(5.42)

−I g (i − 1, i, y¯) + I g (i − 1, i, Z g ) ≥ −4 − 4Q2 − Q0 − 2a −I g (i − 1, i, y¯) + I g (i − 1, i, Z g ) ≥ −4 − 5Q2 − Q0 − 3a. (5.42) holds for each integer i > i0 and each T > i. Let S > i0 + 1, T > S + 1. There exists an integer i > i0 + 1 such that i − 1 ≤ S < i. Clearly (5.42) holds. By Assumption A(iii) and (5.14), I g (S, i, y) ≥ −a, I g (S, i, Z g ) = I g (i − 1, i, Z g ) − I g (i − 1, S, Z g ) ≤ Q0 + a. Together with (5.42) this implies that I g (S, T, y) − I g (S, T, Z g ) = I g (i, T, y) − I g (i, T, Z g ) + I g (S, i, y) − I g (S, i, Z g )

(5.43)

≥ −4 − 5Q2 − 2Q0 − 5a. We established (5.43) for each S > i0 + 1 and each T > S + 1. Assume that (ii) does not hold. It follows from (5.14), Assumption A(iii) and (5.43) which holds for each S > i0 + 1, T > S + 1 that inf{I g (0, T, y) − I g (0, T, Z g ) : T ∈ (0, ∞)} > −∞. Therefore sup{I g (0, T, y)−I g (0, T, Z g ) : T ∈ (0, ∞)} = ∞. By Assumption A(iii) and (5.14) sup{I g (0, i, y) − I g (0, i, Z g ) : i = 1, 2, . . .} = ∞. Together with (5.43) which holds for each S > i0 + 1, T > S + 1 this implies (i). The lemma is proved. The next auxiliary result implies Assertion 4 of Theorem 1.2.1. Lemma 1.5.7 Let K = Rn , g ∈ U and let z ∈ K satisfy |z| ≤ M . Then the function Z g is a (g)-minimal solution. Proof. Let us assume the converse. Then there exist T1 ≥ 0 and T2 > T1 such that I g (T1 , T2 , Z g ) > U g (T1 , T2 , Z g (T1 ), Z g (T2 )).

29

Inﬁnite horizon variational problems

Choose a number ∈ (0, 8−1 [I g (T1 , T2 , Z g ) − U g (T1 , T2 , Z g (T1 ), Z g (T2 ))]

(5.44)

and an integer q0 > T2 + 5. By Corollary 1.3.1 there exists an a.c. function y : [T1 , T2 ] → K such that y(Ti ) = Z g (Ti ), i = 1, 2,

I g (T1 , T2 , y) = U g (T1 , T2 , Z g (T1 ), Z g (T2 )). (5.45) It follows from (5.10), (5.11), (5.12) and Proposition 1.3.7 that there exists an integer k > 2q0 + 4 for which |U g (i, i + 1, Z g (i), Z g (i + 1)) − U g (i, i + 1, Zkg (i), Zkg (i + 1))| ≤ (5.46) (2q0 + 1)−1 , i = 0, . . . , 2q0 + 1, |U g (q0 , q0 + 1, Z g (q0 ), Zkg (q0 + 1)) g

−U (q0 , q0 +

1, Zkg (q0 ), Zkg (q0

(5.47) −1

+ 1))| ≤ (2q0 + 1)

.

By Corollary 1.3.1 and (5.45) there exists an a.c. function x : [0, k] → K such that x(t) = Z g (t), t ∈ [0, T1 ] ∪ [T2 , q0 ], x(t) = y(t), t ∈ [T1 , T2 ],

(5.48)

x(t) = Zkg (t), t ∈ [q0 + 1, k], I g (q0 , q0 + 1, x) = U g (q0 , q0 + 1, x(q0 ), x(q0 + 1)). It follows from (5.48), (5.9) that I g (0, k, x) ≥ I g (0, k, Zkg ).

(5.49)

By (5.48), (5.9), (5.12), (5.46), (5.47) and (5.44), I g (0, k, x) − I g (0, k, Zkg ) = I g (0, q0 + 1, x) − I g (0, q0 + 1, Zkg ) = (I g (0, q0 , x) − I g (0, q0 , Z g )) + (I g (0, q0 , Z g ) − I g (0, q0 , Zkg )) +I g (q0 , q0 + 1, x) − I g (q0 , q0 + 1, Zkg ) ≤ I g (T1 , T2 , y) − I g (T1 , T2 , Z g ) +

q 0 −1 i=0

[U g (i, i + 1, Z g (i), Z g (i + 1)) − U g (i, i + 1, Zkg (i), Zkg (i + 1))]

+U (q0 , q0 + 1, Z g (q0 ), Zkg (q0 + 1)) − U g (q0 , q0 + 1, Zkg (q0 ), Zkg (q0 + 1)) g

≤ I g (T1 , T2 , y) − I g (T1 , T2 , Z g ) + . It follows from this relation, (5.44), (5.45) that I g (0, k, x) − I g (0, k, Zkg ) ≤ I g (T1 , T2 , y) − I g (T1 , T2 , Z g ) + < −.

30

TURNPIKE PROPERTIES

This is contradictory to (5.49). The obtained contradiction proves the lemma. Proof of Theorem 1.2.1. At the begining of Section 1.5 for each f ∈ M and each M > 2|¯ z | we constructed a neighborhood U of f in M and for each g ∈ U and each z ∈ K satisfying |z| ≤ M we deﬁned a.c. functions Z g : [0, ∞) → K, Zqg : [0, q] → K, q = 1, 2, . . . satisfying (5.9)-(5.15). Clearly an a.c. function Z f : [0, ∞) → K was deﬁned for every f ∈ M and every z ∈ K. By Lemmas 1.5.5, 1.5.6 for each f ∈ M and each z ∈ K the function Z f is (f )-good and Assertion 1 of Theorem 1.2.1 holds. Assertion 2 of Theorem 1.2.1 follows from (5.15) which holds for every g ∈ U (U is a neighborhood of f in M) and each z ∈ K satisfying |z| ≤ M . Assertion 3 of Theorem 1.2.1 follows from Lemma 1.5.3. Lemma 1.5.7 implies Assertion 4 of Theorem 1.2.1. Theorem 1.2.1 is proved. Theorem 1.2.2 follows from Lemma 1.5.5. Proof of Theorem 1.2.3 Fix z¯ ∈ K. It follows from Proposition 1.3.6 that there exists a positive number M0 and a neighborhood U0 of f in M such that |U g (T1 , T2 , y, z)| ≤ M0 for each g ∈ U0 , each T1 ∈ [0, ∞), T2 ∈ [T1 + c, T1 + 2c + 2],

(5.50)

and each y, z ∈ K ∩ B(2|¯ z | + 1). By Proposition 1.3.3 we may assume without loss of generality that inf{U g (T1 , T2 , y, z) : g ∈ M, T1 ∈ [0, ∞), T2 ∈ [T1 + c, T1 + 2c + 2], (5.51) y, z ∈ K, |y| + |z| ≥ M1 } > 2M0 + 1. There exist a positive number S1 and a neighborhood U1 of f in M such that U1 ⊂ U0 , S1 > M1 and Proposition 1.4.2 holds with

(5.52)

M3 = M2 + 2, M4 = S1 , V = U1 , c1 = c, c2 = 2c + 2. It follows from Proposition 1.3.6 that there exist a positive number M3 and a neighborhood U of f in M such that U ⊂ U1 , |U g (T1 , T2 , y, z)| + 1 < M3 for each g ∈ U, each T1 ∈ [0, ∞), (5.53)

Inﬁnite horizon variational problems

31

T2 ∈ [T1 + c, T1 + 2c + 2] and y, z ∈ K ∩ B(S1 ). It follows from Proposition 1.3.3 that there exists S > S1 + 1 such that the following property holds: If g ∈ M, T1 ∈ [0, ∞), T2 ∈ [T1 + c, T1 + 2c + 2] and if an a.c. function v : [T1 , T2 ] → K satisﬁes I g (T1 , T2 , v) ≤ 2M3 + 2M2 + 2, then |v(t)| ≤ S, t ∈ [T1 , T2 ]. Assume that g ∈ U , T1 ∈ [0, ∞), T2 ≥ c + T1 . We will show that property (i) holds. Let x, y ∈ K, |x|, |y| ≤ M1 and let v : [T1 , T2 ] → K be an a.c. function which satisﬁes v(T1 ) = x, v(T2 ) = y, I g (T1 , T2 , v) ≤ U g (T1 , T2 , x, y) + M2 .

(5.54)

There is a natural number p such that pc ≤ T2 − T1 < (p + 1)c. Set T = p−1 (T2 − T1 ). Clearly T ∈ [c, 2c]. By (5.54) and Corollary 1.3.1, p−1

[U g (T1 + iT, T1 + (i + 1)T, v(T1 + iT ), v(T1 + (i + 1)T ))

i=0

−U g (T1 + iT, T1 + (i + 1)T, yi , yi+1 )] ≤ M2 for each sequence {yi }pi=0 ⊂ K satisfying y0 = v(T1 ), yp = v(T2 ). It follows from this, (5.52), (5.54) and Proposition 1.4.2 that |v(T1 + iT )| ≤ S1 , i = 0, . . . , p. By this relation and (5.54), (5.53) for i = 0, . . . , p − 1, I g (T1 + iT, T1 + (i + 1)T, v) ≤ U g (T1 + iT, T1 + (i + 1)T, v(T1 + iT ), v(T1 + (i + 1)T )) + M2 < M3 + M2 . It follows from this relation and the deﬁnition of S that |v(t)| ≤ S, t ∈ [T1 , T2 ]. Therefore property (i) holds. Analogously to this we can show that property (ii) holds. The theorem is proved.

Chapter 2 EXTREMALS OF NONAUTONOMOUS PROBLEMS

In this chapter we show that the turnpike property is a general phenomenon which holds for a large class of nonautonomous variational problems with nonconvex integrands. We consider the complete metric space of integrands M introduced in Section 1.1 and establish the existence of a set F ⊂ M which is a countable intersection of open everywhere dense sets in M such that for each f ∈ F and each z ∈ Rn the following properties hold: (i) there exists an (f )-overtaking optimal function Z f : [0, ∞) → Rn satisfying Z f (0) = z; (ii) the integrand f has the turnpike property with the trajectory {Z f (t) : t ∈ [0, ∞)} being the turnpike. Moreover we show that the turnpike property holds for approximate solutions of variational problems with a generic integrand f and that the turnpike phenomenon is stable under small pertubations of a generic integrand f .

2.1.

Main results

Let a > 0 be a constant and let ψ : [0, ∞) → [0, ∞) be an increasing function such that ψ(t) → +∞ as t → ∞. Denote by | · | the Euclidean norm in Rn . We consider the space of integrands M introduced in Section 1.1. This space consists of all continuous functions f : [0, ∞) × Rn × Rn → R1 which satisfy the following assumptions:

34

TURNPIKE PROPERTIES

A(i) for each (t, x) ∈ [0, ∞) × Rn the function f (t, x, ·) : Rn → R1 is convex; A(ii) the function f is bounded on [0, ∞) × E for any bounded set E ⊂ Rn × Rn ; A(iii) f (t, x, u) ≥ max{ψ(|x|), ψ(|u|)|u|} − a for each (t, x, u) ∈ [0, ∞) × Rn × Rn ; A(iv) for each pair of positive numbers M, there exist Γ, δ > 0 such that if t ∈ [0, ∞) and if u1 , u2 , x1 , x2 ∈ Rn satisfy |xi | ≤ M, |ui | ≥ Γ, i = 1, 2,

max{|x1 − x2 |, |u1 − u2 |} ≤ δ,

then |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ max{f (t, x1 , u1 ), f (t, x2 , u2 )}; A(v) for each pair of positive numbers M, there is a positive number δ such that if t ∈ [0, ∞) and if u1 , u2 , x1 , x2 ∈ Rn satisfy |xi |, |ui | ≤ M, i = 1, 2,

max{|x1 − x2 |, |u1 − u2 |} ≤ δ,

then |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ . We equip the set M with two topologies where one is weaker than the other. We refer to them as the weak and the strong topologies, respectively. For the set M we consider the uniformity determined by the following base: Es () = {(f, g) ∈ M × M : |f (t, x, u) − g(t, x, u)| ≤ for each t ∈ [0, ∞) and each x, u ∈ Rn }, where > 0. It is not diﬃcult to see that the uniform space M with this uniformity is metrizable and complete. This uniformity generates in M the strong topology. We also equip the set M with the uniformity which is determined by the following base: E(N, , λ) = {(f, g) ∈ M × M : |f (t, x, u) − g(t, x, u)| ≤ for each t ∈ [0, ∞) and each x, u ∈ Rn satisfying |x|, |u| ≤ N, (|f (t, x, u)| + 1)(|g(t, x, u)| + 1)−1 ∈ [λ−1 , λ] for each t ∈ [0, ∞) and each x, u ∈ Rn satisfying |x| ≤ N },

35

Extremals of nonautonomous problems

where N > 0, > 0, λ > 1. This uniformity which was introduced in Section 1.2, generates in M the weak topology. By Proposition 1.3.2 the space M with this uniformity is complete. We consider functionals of the form I f (T1 , T2 , x) =

T2 T1

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

(1.1)

where f ∈ M, 0 ≤ T1 < T2 < +∞ and x : [T1 , T2 ] → Rn is an a.c. function. For each f ∈ M, each pair of vectors y, z ∈ Rn , each T1 ≥ 0 and each T2 > T1 we set U f (T1 , T2 , y, z) = inf{I f (T1 , T2 , x) : x : [T1 , T2 ] → Rn

(1.2)

is an a.c. function satisfying x(T1 ) = y, x(T2 ) = z}, σ f (T1 , T2 , y) = inf{U f (T1 , T2 , y, u) : u ∈ Rn }.

(1.3)

It is not diﬃcult to see that U f (T1 , T2 , y, z) is ﬁnite for each f ∈ M, each y, z ∈ Rn and all numbers T1 , T2 satisfying 0 ≤ T1 < T2 . Recall the deﬁnition of an overtaking optimal function given in Section 1.1 and the deﬁnition of a good function introduced in Section 1.2. Let f ∈ M. An a.c. function x : [0, ∞) → Rn is called (f )-overtaking optimal if for any a.c. function y : [0, ∞) → Rn satisfying y(0) = x(0), T

lim sup T →∞

0

[f (t, x(t), x (t)) − f (t, y(t), y (t))]dt ≤ 0.

Let f ∈ M. We say that an a.c. function x : [0, ∞) → Rn is an (f )-good function if for any a.c. function y : [0, ∞) → Rn , the function T → I f (0, T, y) − I f (0, T, x), T ∈ (0, ∞) is bounded from below. In this chapter we establish the existence of a set F ⊂ M which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets of M such that the following theorems are valid. Theorem 2.1.1 1. For each g ∈ F and each pair of (g)-good functions vi : [0, ∞) → Rn , i = 1, 2, |v2 (t) − v1 (t)| → 0 as t → ∞. 2. For each g ∈ F and each y ∈ Rn there exists a (g)-overtaking optimal function Y : [0, ∞) → Rn satisfying Y (0) = y.

36

TURNPIKE PROPERTIES

3. Let g ∈ F, > 0 and Y : [0, ∞) → Rn be a (g)-overtaking optimal function. Then there exists a neighborhood U of g in M with the weak topology such that the following property holds: If h ∈ U and if v : [0, ∞) → Rn is an (h)-good function, then |v(t) − Y (t)| ≤ for all large t. Theorem 2.1.2 Let g ∈ F, M, > 0 and let Y : [0, ∞) → Rn be a (g)overtaking optimal function. Then there exists a neighborhood U of g in M with the weak topology and a number τ > 0 such that for each h ∈ U and each (h)-overtaking optimal function v : [0, ∞) → Rn satisfying |v(0)| ≤ M , |v(t) − Y (t)| ≤ for all t ∈ [τ, ∞). Theorems 2.1.1 and 2.1.2 establish the existence of (g)-overtaking optimal functions and describe the asymptotic behavior of (g)-good functions for g ∈ F. Theorem 2.1.3 Let g ∈ F, S1 , S2 , > 0 and let Y : [0, ∞) → Rn be a (g)-overtaking optimal function. Then there exists a neighborhood U of g in M with the weak topology, a number L > 0 and an integer Q ≥ 1 such that if h ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + LQ, ∞) and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes one of the following relations below: (a) |v(Ti )| ≤ S1 , i = 1, 2, (b) |v(T1 )| ≤ S1 ,

I h (T1 , T2 , v) ≤ U h (T1 , T2 , v(T1 ), v(T2 )) + S2 ; I h (T1 , T2 , v) ≤ σ h (T1 , T2 , v(T1 )) + S2 ,

then the following property holds: There exist sequences of numbers {di }qi=1 , {bi }qi=1 ⊂ [T1 , T2 ] such that q ≤ Q, bi < di ≤ bi + L, i = 1, . . . , q, |v(t) − Y (t)| ≤ for each t ∈ [T1 , T2 ] \ ∪qi=1 [bi , di ]. Theorem 2.1.4 Let g ∈ F, S, > 0 and let Y : [0, ∞) → Rn be a (g)-overtaking optimal function. Then there exist a neighborhood U of g in M with the weak topology and numbers δ, L > 0 such that for each h ∈ U, each pair of numbers T1 ∈ [0, ∞), T2 ∈ [T1 +2L, ∞) and each a.c. function v : [T1 , T2 ] → Rn which satisﬁes one of the following relations below: (a) |v(Ti )| ≤ S, i = 1, 2, (b) |v(T1 )| ≤ S,

I h (T1 , T2 , v) ≤ U h (T1 , T2 , v(T1 ), v(T2 )) + δ; I h (T1 , T2 , v) ≤ σ h (T1 , T2 , v(T1 )) + δ

Extremals of nonautonomous problems

37

the inequality |v(t) − Y (t)| ≤ is valid for all t ∈ [T1 + L, T2 − L]. Theorem 2.1.4 establishes the turnpike property for any g ∈ F. The results of this chapter have been established in [103]. In the sequel we use the notation B(x, r) = {y ∈ Rn : |y − x| ≤ r}, x ∈ Rn , r > 0,

(1.4)

B(r) = B(0, r), r > 0. Chapter 2 is organized as follows. In Section 2.2 for a given f ∈ M and a given neighborhood of f in M with the strong topology we construct an integrand f ∗ which belongs to this neighborhood and establishes the turnpike property for f ∗ . We also study the structure of approximate solutions of variational problems with integrands belonging to a small neighborhood of f ∗ in the weak topology. Theorems 2.1.1-2.1.4 are proved in Section 2.3. In Section 2.4 we discuss analogs of Theorems 2.1.1-2.1.4 for a class of periodic variational problems. In Section 2.5 we show that Theorems 2.1.1-2.1.4 also hold for certain subspaces of M which consist of smooth integrands. In Section 2.6 we consider an example of an integrand which has the turnpike property and an example of an integrand which does not have the turnpike property.

2.2.

Preliminary lemmas

Fix f ∈ M and z∗ ∈ Rn . Let > 0, M > |z∗ | and let an a.c. function : [0, ∞) → Rn be as guaranteed by Theorem 1.2.1. We have that Z∗f is an (f )-good function, Z∗f (0) = z∗ and for each T1 ≥ 0, T2 > T1 , Z∗f

U f (T1 , T2 , Z∗f (T1 ), Z∗f (T2 )) = I f (T1 , T2 , Z∗f ).

(2.1)

First we deﬁne functions frM for r > 0 such that frM → f as r → 0+ in the strong topology and such that each frM has the turnpike property. Fix a continuous bounded function φM : [0, ∞) × Rn → [0, ∞) which satisﬁes the following assumptions: B(i) {(t, x) ∈ [0, ∞) × Rn : φM (t, x) = 0} = {(t, Z∗f (t)) : t ∈ [0, ∞)} ∪ {(t, x) ∈ [0, ∞) × Rn : |x| ≥ M + 2}; B(ii) for any δ > 0 there is a positive number γ such that if t ∈ [0, ∞) and if x1 , x2 ∈ Rn satisfy |x1 − x2 | ≤ γ, then |φM (t, x1 ) − φM (t, x2 )| ≤ δ;

38

TURNPIKE PROPERTIES

B(iii) for any positive number δ there is γ > 0 such that if t ∈ [0, ∞) and if x ∈ Rn satisﬁes |x − Z∗f (t)| ≥ δ and |x| ≤ M + 1, then φM (t, x) ≥ γ. Remark 2.2.1 Consider a continuous function θ : R1 → [0, 1] for which θ(t) = 1, t ∈ (−∞, M + 1], θ(t) = 0, t ∈ [M + 2, ∞), θ(t) > 0, t ∈ (M + 1, M + 2). Let q be a natural number. Deﬁne a bounded continuous function φM : [0, ∞) × Rn → R1 by φM (t, x) = |x − Z∗f (t)|q θ(|x|),

t ∈ [0, ∞), x ∈ Rn .

(2.2).

It is easy to verify that the function φM satisﬁes assumption (B). Deﬁne a function fM : [0, ∞) × Rn × Rn → R1 by fM (t, x, u) = f (t, x, u) + φM (t, x), t ∈ [0, ∞), x, u ∈ Rn .

(2.3)

It is easy to verify that fM ∈ M and to prove the following result. Lemma 2.2.1 Let M > |z∗ | and V be a neighborhood of f in M with the strong topology. Then there exists a number r0 > 0 such that frM ∈ V for every number r ∈ (0, r0 ). Fix a natural number p. It follows from Theorem 1.2.1 and Theorem 1.2.2 that there exist a number M f > 0 and a neighborhood W f of f in M with the weak topology such that

and

|z∗ |, sup{|Z∗f (t)| : t ∈ [0, ∞)} < M f

(2.4)

lim sup |x(t)| < M f

(2.5)

t→∞

for each g ∈ W f and each (g)-good function x : [0, ∞) → Rn . There exist a positive number M0 (f, p) and an open neighborhood W0 (f, p) of f in M with the weak topology such that W0 (f, p) ⊂ W f , M0 (f, p) > 2M f + 2p + 2

(2.6)

and Theorem 1.2.3 holds with M1 , M2 = 2M f + 2p + 2, c = 4−1 , S = M0 (f, p), U = W0 (f, p). (2.7)

39

Extremals of nonautonomous problems

There exists a neighborhood W (f, p) of f in M with the weak topology and a number M (f, p) such that W (f, p) ⊂ W0 (f, p), M (f, p) > 2M0 (f, p) + 2

(2.8)

and Theorem 1.2.3 holds with M1 , M2 = 2M0 (f, p) + 2, c = 4−1 , S = M (f, p), U = W (f, p).

(2.9)

It follows from Lemma 2.2.1 that there is a positive number r(f, p) such that frM (f,p)+1 ∈ W (f, p) for each r ∈ (0, r(f, p)). (2.10) Fix r ∈ (0, r(f, p)) and set f ∗ = frM (f,p)+1 .

(2.11)

We study the structure of approximate solutions of variational problems with integrands belonging to a small neighborhood of f ∗ in the strong topology. We show that the integrand f ∗ has the turnpike property and the function Z∗f is its turnpike. The next lemma establishes that each approximate solution deﬁned on an interval [T1 , T2 ] is close enough to the turnpike Z∗f at a certain point of [T1 , T2 ] if the integrand is close enough to f ∗ and T2 − T1 is large enough. Lemma 2.2.2 Let 0 ∈ (0, 1). Then there exist a neighborhood U of f ∗ in M with the weak topology and an integer N ≥ 8 such that if g ∈ U, T ≥ 0 and if an a.c. function v : [T, T + N ] → Rn satisﬁes max{|v(T )|, |v(T + N )|} ≤ 2M0 (f, p) + 2,

(2.12)

I g (T, T + N, v) ≤ U g (T, T + N, v(T ), v(T + N )) + 2M0 (f, p) + 2, then there is an integer i0 ∈ [0, N − 6] such that |v(t) − Z∗f (t)| ≤ 0 , t ∈ [i0 + T, i0 + T + 6].

(2.13)

Proof. It follows from Proposition 1.3.6 that there exist a positive number S0 and an open neighborhood U0 of f ∗ in M with the weak topology such that U0 ⊂ W (f, p), |U g (T1 , T2 , y1 , y2 )| + 1 < S0 for each g ∈ U 0 , each T1 ∈ [0, ∞), T2 ∈ [T1 + 4−1 , T1 + 8] and each y1 , y2 ∈ B(M (f, p) + 1), i = 1, 2.

(2.14)

40

TURNPIKE PROPERTIES

By Theorem 1.2.1 there is a positive number S1 such that the inequality I f (T1 , T2 , Z∗f ) ≤ I f (T1 , T2 , v) + S1 holds for each T1 ≥ 0, T2 > T1 and each a.c. function v : [T1 , T2 ] → Rn which satisﬁes |v(T1 )| ≤ M (f, p) + 1. (2.4), Proposition 1.3.6 and Assertion 4 of Theorem 1.2.1 imply that there exists S2 > sup{|I f (T1 , T2 , Z∗f )| : T1 ∈ [0, ∞), T2 ∈ [T1 + 4−1 , T1 + 8]} + 2M (f, p).

(2.15)

By Proposition 1.3.4 there exists δ ∈ (0, 8−1 ) such that for each g ∈ M, each T1 , T2 ∈ [0, ∞) satisfying 4−1 ≤ T2 − T1 ≤ 8 and each a.c. function v : [T1 , T2 ] → Rn satisfying I g (T1 , T2 , v) ≤ 2S0 + 2S1 + 2S2 + 2,

(2.16)

the following property holds: If t1 , t2 ∈ [T1 , T2 ] and if |t1 − t2 | ≤ δ, then |v(t1 ) − v(t2 )| ≤ 16−1 0 .

(2.17)

There exists a number 1 ∈ (0, 4−1 0 ) such that Assumption B(iii) holds with M = M (f, p) + 1, δ = 4−1 0 , γ = 1 . Fix a natural number N > 48 for which 4−1 (6−1 N − 8)δ1 r > 2M (f, p) + 2S0 + 6a + 4 + S1

(2.18)

(recall a in Assumption A(iii)). By Proposition 1.3.6 there exist a number S3 > 0 and a neighborhood U1 of f ∗ in M with the weak topology such that U1 ⊂ U0 , |U g (T1 , T2 , y1 , y2 )| + 1 < S3 for each g ∈ U1 each T1 ∈ [0, ∞),

(2.19)

T2 ∈ [T1 + 4−1 , T1 + N + 4] and each y1 , y2 ∈ B(M (f, p) + 2), i = 1, 2. By Propositions 1.3.8 and 1.3.9 there is an open neighborhood U of f ∗ in M with the weak topology such that U ⊂ U1 and that for each g ∈ U and each T1 ∈ [0, ∞), T2 ∈ [T1 + 4−1 , T1 + N + 4] the following properties hold: a) if an a.c. function v : [T1 , T2 ] → Rn satisﬁes ∗

min{I f (T1 , T2 , v), I g (T1 , T2 , v)} ≤ S3 + 2M (f, p) + 2, ∗

then |I f (T1 , T2 , v) − I g (T1 , T2 , v)| ≤ 4−1 ;

41

Extremals of nonautonomous problems

b)

∗

|U f (T1 , T2 , y1 , y2 ) − U g (T1 , T2 , y1 , y2 )| ≤ 4−1

for each y1 , y2 ∈ B(M (f, p) + 2), i = 1, 2. Assume that g ∈ U, T ∈ [0, ∞) and v : [T, T + N ] → Rn is an a.c. function which satisﬁes (2.12). We show that (2.13) holds with an integer i0 ∈ [0, N − 6]. Let us assume the converse. Then for any integer i ∈ [0, N − 6], sup{|v(t) − Z∗f (t)| : t ∈ [i + T, i + T + 6]} > 0 .

(2.20)

By the inequalities (2.12), Theorem 1.2.3 and the choice of W (f, p), M (f, p) (see (2.8) and (2.9)), |v(t)| ≤ M (f, p), t ∈ [T, T + N ].

(2.21)

(2.12), (2.19) and (2.21) imply that I g (T, T + N, v) ≤ U g (T, T + N, v(T ), v(T + N )) + 2M0 (f, p) + 2 ≤ 2M0 (f, p) + S3 + 2.

(2.22)

It follows from property (b) and the inequality (2.12) that ∗

|U f (T, T + N, v(T ), v(T + N )) − U g (T, T + N, v(T ), v(T + N ))| ≤ 4−1 . (2.23) Combined with the inequality (2.22) the property (a) implies that ∗

|I f (T, T + N, v) − I g (T, T + N, v)| ≤ 4−1 .

(2.24)

By (2.12), (2.23) and (2.24), ∗

∗

I f (T, T +N, v) ≤ U f (T, T +N, v(T ), v(T +N ))+2M0 (f, p)+3. (2.25) There exists an integer j1 such that j1 − 2 < T ≤ j1 − 1.

(2.26)

Fix an integer i ∈ [0, N − 6]. By (2.20) there exists a number ti such that (2.27) ti ∈ [i + T, i + T + 6], |v(ti ) − Z∗f (ti )| > 0 . The inequality (2.15) implies that |I f (T + i, T + i + 6, Z∗f )| ≤ S2 . It follows from (2.12), (2.21) and (2.14) that I g (T + i, T + i + 6, v)

(2.28)

42

TURNPIKE PROPERTIES

≤ U g (T + i, T + i + 6, v(T + i), v(T + i + 6)) + 2M0 (f, p) + 2 (2.29) ≤ 2M0 (f, p) + S0 + 2. By (2.28), (2.29), (2.15), (2.27) and the deﬁnition of δ (see (2.16), (2.17)), for each (2.30) t ∈ [i + T, i + T + 6] ∩ [ti − δ, ti + δ] the inequalitites |v(ti )−v(t)| ≤ 16−1 0 , |Z∗f (ti )−Z∗f (t)| ≤ 16−1 0 , |v(t)−Z∗f (t)| ≥ 3·4−1 0 are true. By these inequalities, the deﬁnition of 1 , (2.21) and assumption B(iii), (2.31) φM (f,p)+1 (t, v(t)) ≥ 1 for each integer i ∈ [0, N − 6] and each number t sastisfying (2.30). (2.11), (2.3) and (2.21) imply that ∗

I f (T, T + N, v) = I f (T, T + N, v) + r

T +N T

φM (f,p)+1 (t, v(t))dt.

By this equality and (2.31) which is true for each integer i ∈ [0, N − 6] and each t satisfying (2.30), ∗

I f (T, T + N, v) ≥ I f (T, T + N, v) + r(N 6−1 − 2)δ1 .

(2.32)

By Corollary 1.3.1 there exists an a.c. function w : [T, T + N ] → Rn for which w(T ) = v(T ), w(T + N ) = v(T + N ), w(t) = Z∗f (t), t ∈ [j1 , j1 + N − 3], (2.33) f∗ f∗ f∗ I (T, j1 , w) = U (T, j1 , w(T ), w(j1 )), I (j1 + N − 3, T + N, w) = U f∗ (j1 + N − 3, T + N, w(j1 + N − 3), w(T + N )). Combined with the inequality (2.25) the relations (2.33) imply that ∗

∗

I f (T, T + N, v) ≤ I f (T, T + N, w) + 2M0 (f, p) + 3.

(2.34)

By (2.33), (2.21), (2.4) and (2.14), ∗

∗

I f (T, T + N, w) ≤ I f (j1 , j1 + N − 3, Z∗f ) + 2S0 .

(2.35)

It follows from Assumption A(iii), the deﬁnition of S1 and the inequalities (2.26) and (2.21) that I f (T, T + N, v) ≥ I f (j1 , j1 + N − 3, v) − 6a

43

Extremals of nonautonomous problems

≥ −6a + I f (j1 , j1 + N − 3, Z∗f ) − S1 .

(2.36)

Combined with (2.32) and (2.35) the inequality (2.36) implies that ∗

I f (T, T + N, v) ≥ r(N 6−1 − 2)δ1 − 6a + I f (j1 , j1 + N − 3, Z∗f ) − S1 ∗

≥ I f (T, T + N, w) + r(N 6−1 − 2)δ1 − 6a − S1 − 2S0 . Together with (2.34) this inequality implies that 2M0 (f, p) + 3 ≥ r(N 6−1 − 2)δ1 − 6a − S1 − 2S0 . This is contradictory to (2.18). proves the lemma.

The contradiction we have reached

The following auxiliary result shows that an approximate solution of a variational problem deﬁned on an interval [T1 , T2 ] is close to the turnpike Z∗f at any point of [T1 , T2 ] if it is close enough to the turnpike at the points T1 and T2 . Lemma 2.2.3 For each ∈ (0, 1) there is δ ∈ (0, ) such that the following property holds: If T1 ∈ [0, ∞), T2 ∈ [T1 + 1, ∞) and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes |v(Ti ) − Z∗f (Ti )| ≤ δ, i = 1, 2, ∗

then

∗

I f (T1 , T2 , v) ≤ U f (T1 , T2 , v(T1 ), v(T2 )) + δ,

(2.37)

|v(t) − Z∗f (t)| ≤ , t ∈ [T1 , T2 ].

(2.38)

Proof. Let ∈ (0, 1). It follows from Proposition 1.3.6 that there exists a positive number S0 such that ∗

|U f (T1 , T2 , y1 , y2 )|+6 < S0 for each T1 ∈ [0, ∞), T2 ∈ [T1 +4−1 , T1 +10] (2.39) and each y1 , y2 ∈ B(M (f, p) + 1), i = 1, 2. It follows from (2.4), Proposition 1.3.6 and Assertion 4 of Theorem 1.2.1 that there is a positive number S1 such that |I f (T1 , T2 , Z∗f )| + 1 < S1 for each T1 ∈ [0, ∞), T2 ∈ [T1 + 4−1 , T1 + 10]}. (2.40) By Proposition 1.3.4 there is δ0 ∈ (0, 8−1 ) such that for each g ∈ M, each pair of numbers T1 , T2 ∈ [0, ∞) satisfying 4−1 ≤ T2 − T1 ≤ 10 and each a.c. function v : [T1 , T2 ] → Rn which satisﬁes I g (T1 , T2 , v) ≤ 2S0 + 2S1 + 2,

(2.41)

44

TURNPIKE PROPERTIES

the following property holds: |v(t1 ) − v(t2 )| ≤ 16−1

(2.42)

for each t1 , t2 ∈ [T1 , T2 ] satisfying |t1 − t2 | ≤ δ0 . There exists a number 1 ∈ (0, 4−1 ) such that Assumption B(iii) holds with M = M (f, p) + 1, δ = 4−1 , γ = 1 . By Proposition 1.3.7 there exists a number δ ∈ (0, inf{δ0 , 8−1 1 , 8−1 1 δ0 r})

(2.43)

such that if T1 ≥ 0, T2 ∈ [T1 + 4−1 , T1 + 10] and if yi , xi ∈ B(M (f, p) + 2), i = 1, 2, max{|y1 − y2 |, |x1 − x2 |} ≤ δ, (2.44) then

|U f (T1 , T2 , y1 , x1 ) − U f (T1 , T2 , y2 , x2 )|, |U

f∗

(T1 , T2 , y1 , x1 ) − U

f∗

(2.45)

(T1 , T2 , y2 , x2 )| ≤ 2−7 1 δ0 r.

Assume that T1 ∈ [0, ∞), T2 ∈ [T1 + 1, ∞) and an a.c. function v : [T1 , T2 ] → Rn satisﬁes (2.37). We show that the inequality (2.38) is valid. Let us assume the converse. Then there exists a number t1 for which t1 ∈ [T1 , T2 ], |v(t1 ) − Z∗f (t1 )| > .

(2.46)

(2.37), (2.7), Theorem 1.2.3 and (2.4) imply that |v(t)| ≤ M0 (f, p), t ∈ [T1 , T2 ].

(2.47)

It is not diﬃcult to see that there exist d1 , d2 ∈ [T1 , T2 ] such that d2 − d1 = 1, t1 ∈ [d1 , d2 ].

(2.48)

It follows from the inequalities (2.37), (2.47) and (2.39) that ∗

∗

I f (d1 , d2 , v) ≤ U f (d1 , d2 , v(d1 ), v(d2 )) + δ ≤ S0 + 1.

(2.49)

The inequality (2.40) implies that I f (d1 , d2 , Z∗f ) < S1 . By the choice of δ0 (see (2.41), (2.42)), (2.50) and (2.49) |v(t) − v(t1 )| ≤ 16−1 , |Z∗f (t) − Z∗f (t1 )| ≤ 16−1 for each t ∈ [d1 , d2 ] ∩ [t1 − δ0 , t1 + δ0 ].

(2.50)

45

Extremals of nonautonomous problems

Combined with (2.46) this fact implies that |v(t) − Z∗f (t)| ≥ 3 · 4−1 , t ∈ [d1 , d2 ] ∩ [t1 − δ0 , t1 + δ0 ]. It follows from this inequality, assumption B(iii), the deﬁnition of 1 and the inequality (2.47) that φM (f,p)+1 (t, v(t)) ≥ 1 for each t ∈ [d1 , d2 ] ∩ [t1 − δ0 , t1 + δ0 ].

(2.51)

We prove that the inequality |U g (T1 , T2 , v(T1 ), v(T2 )) − U g (T1 , T2 , Z∗f (T1 ), Z∗f (T2 ))| ≤ δ + 32−1 1 δ0 r (2.52) is true with g = f, f ∗ . Corollary 1.3.1 implies that for g = f, f ∗ there exist a.c. functions vig : [T1 , T2 ] → Rn , i = 1, 2 such that v1g (Ti ) = Z∗f (Ti ), i = 1, 2, v1g (t) = v(t), t ∈ [T1 + 2−1 , T2 − 2−1 ], (2.53) I g (S, S + 2−1 , v1g ) = U g (S, S + 2−1 , v1g (S), v1g (S + 2−1 )), S = T1 , T2 − 2−1 , v2g (Ti ) = v(Ti ), i = 1, 2, v2g (t) = Z∗f (t), t ∈ [T1 + 2−1 , T2 − 2−1 ], I g (S, S + 2−1 , v2g ) = U g (S, S + 2−1 , v2g (S), v2g (S + 2−1 )), S = T1 , T2 − 2−1 . By the deﬁnition of δ (see (2.43)) and the inequalities (2.53), (2.47), (2.4) and (2.37), |U g (S, S + 2−1 , v2g (S), v2g (S + 2−1 )) − U g (S, S + 2−1 , Z∗f (S), Z∗f (S + 2−1 ))| (2.54) ≤ 2−7 1 δ0 r, g = f, f ∗ , S = T1 , T2 − 2−1 , |U g (S, S + 2−1 , v1g (S), v1g (S + 2−1 )) − U g (S, S + 2−1 , v(S), v(S + 2−1 ))| (2.55) ≤ 2−7 1 δ0 r, g = f, f ∗ , S = T1 , T2 − 2−1 . It follows from (2.3), (2.53), the inequalities (2.54) and (2.55) and Assertion 4 of Theorem 1.2.1 that for g = f, f ∗ , U g (T1 , T2 , v(T1 ), v(T2 )) − U g (T1 , T2 , Z∗f (T1 ), Z∗f (T2 )) ≤ I g (T1 , T2 , v2g ) − I g (T1 , T2 , Z∗f ) = I g (T1 , T1 + 2−1 , v2g ) − I g (T1 , T1 + 2−1 , Z∗f ) +I g (T2 − 2−1 , T2 , v2g ) − I g (T2 − 2−1 , T2 , Z∗f ) = U g (T1 , T1 + 2−1 , v2g (T1 ), v2g (T1 + 2−1 )) −U g (T1 , T1 + 2−1 , Z∗f (T1 ), Z∗f (T1 + 2−1 ))

(2.56)

46

TURNPIKE PROPERTIES

+U g (T2 − 2−1 , T2 , v2g (T2 − 2−1 ), v2g (T2 )) −U g (T2 − 2−1 , T2 , Z∗f (T2 − 2−1 ), Z∗f (T2 )) ≤ 2−6 1 δ0 r. (2.55), (2.37) and (2.53) imply that ∗

∗

U f (T1 , T2 , v(T1 ), v(T2 )) − U f (T1 , T2 , Z∗f (T1 ), Z∗f (T2 )) ∗

∗

∗

(2.57)

∗

≥ I f (T1 , T2 , v) − δ − I f (T1 , T2 , v1f ) = −δ + I f (T1 , T1 + 2−1 , v) ∗

∗

∗

∗

∗

−I f (T1 , T1 + 2−1 , v1f ) + I f (T2 − 2−1 , T2 , v) − I f (T2 − 2−1 , T2 , v1f ) ∗

≥ −δ + U f (T1 , T1 + 2−1 , v(T1 ), v(T1 + 2−1 )) ∗

∗

∗

−U f (T1 , T1 + 2−1 , v1f (T1 ), v1f (T1 + 2−1 )) ∗

+U f (T2 − 2−1 , T2 , v(T2 − 2−1 ), v(T2 )) ∗

∗

∗

−U f (T2 − 2−1 , T2 , v1f (T2 − 2−1 ), v1f (T2 )) ≥ −δ − 2−6 1 δ0 r. It follows from Corollary 1.3.1 that there are a.c. functions vi : [T1 , T2 ] → Rn , i = 3, 4 such that I f (T1 , T2 , v3 ) = U f (T1 , T2 , v(T1 ), v(T2 )), (2.58) f −1 −1 v4 (Ti ) = Z∗ (Ti ), i = 1, 2, v4 (t) = v3 (t), t ∈ [T1 + 2 , T2 − 2 ],

v3 (Ti ) = v(Ti ), i = 1, 2,

I f (S, S + 2−1 , v4 ) = U f (S, S + 2−1 , v4 (S), v4 (S + 2−1 )), S = T1 , T2 − 2−1 . By Theorem 1.2.3, (2.9), (2.58) and (2.47), |v3 (t)| ≤ M (f, p), t ∈ [T1 , T2 ].

(2.59)

By the deﬁnition of δ (see (2.45) and (2.44)), (2.58), (2.37), (2.4) and (2.59), |U f (S, S + 2−1 , v3 (S), v3 (S + 2−1 )) − U f (S, S + 2−1 , v4 (S), v4 (S + 2−1 ))| (2.60) ≤ 2−7 1 δ0 r, S = T1 , T2 − 2−1 . Combined with (2.58) the inequality (2.60) implies that U f (T1 , T2 , v(T1 ), v(T2 )) − U f (T1 , T2 , Z∗f (T1 ), Z∗f (T2 )) ≥ I f (T1 , T2 , v3 ) − I f (T1 , T2 , v4 ) = U f (T1 , T1 + 2−1 , v3 (T1 ), v3 (T1 + 2−1 )) +U f (T2 − 2−1 , T2 , v3 (T2 − 2−1 ), v3 (T2 ))

Extremals of nonautonomous problems

47

−U f (T1 , T1 + 2−1 , v4 (T1 ), v4 (T1 + 2−1 )) −U f (T2 − 2−1 , T2 , v4 (T2 − 2−1 ), v4 (T2 )) ≥ −2−6 1 δ0 r. By these relations, (2.57) and (2.56) which holds for g = f, f ∗ , the inequality (2.52) is valid with g = f, f ∗ . Combined with Assertion 4 of Theorem 1.2.1, (2.3) and (2.37) this implies that U f (T1 , T2 , v(T1 ), v(T2 )) ≥ U f (T1 , T2 , Z∗f (T1 ), Z∗f (T2 )) − δ − 32−1 1 δ0 r (2.61) = I f (T1 , T2 , Z∗f ) − δ − 32−1 1 δ0 r ∗

≥ U f (T1 , T2 , Z∗f (T1 ), Z∗f (T2 )) − δ − 32−1 1 δ0 r ∗

≥ U f (T1 , T2 , v(T1 ), v(T2 )) − 2(δ + 32−1 1 δ0 r) ∗

≥ I f (T1 , T2 , v) − δ − 2(δ + 32−1 1 δ0 r). It follows from (2.3), (2.11), (2.47) and (2.51) that ∗

I f (T1 , T2 , v) ≥ I f (T1 , T2 , v) + 1 δ0 r. This inequality and (2.61) imply that U f (T1 , T2 , v(T1 ), v(T2 )) ≥ I f (T1 , T2 , v) + 1 δ0 r − δ − 2(δ + 32−1 1 δ0 r) ≥ U f (T1 , T2 , v(T1 ), v(T2 )) − 3δ + 15 · 16−1 1 δ0 r. This is contradictory to (2.43). proves the lemma.

The contradiction we have reached

Now we will prove an auxiliary result which generalizes Lemma 2.2.3. This result shows that the convergence property established in Lemma 2.2.3 for the integrand f ∗ is also valid for all integrands from a small neighborhood of f ∗ . Lemma 2.2.4 For each ∈ (0, 1) there exist an open neighborhood U of f ∗ in M with the weak topology and δ ∈ (0, ) such that the following property holds: If g ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + 1, ∞) and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes |v(Ti ) − Z∗f (Ti )| ≤ δ, i = 1, 2,

then

I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + δ,

(2.62)

|v(t) − Z∗f (t)| ≤ , t ∈ [T1 , T2 ].

(2.63)

48

TURNPIKE PROPERTIES

Proof. Let ∈ (0, 1). It follows from Lemma 2.2.3 that there exists δ ∈ (0, ) such that if T1 ∈ [0, ∞), T2 ∈ [T1 +1, ∞) and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes |v(Ti ) − Z∗f (Ti )| ≤ 8δ, i = 1, 2, ∗

∗

I f (T1 , T2 , v) ≤ U f (T1 , T2 , v(T1 ), v(T2 )) + 8δ,

(2.64)

|v(t) − Z∗f (t)| ≤ , t ∈ [T1 , T2 ].

(2.65)

then

Lemma 2.2.2 implies that there are an integer N ≥ 8 and an open neighborhood U0 of f ∗ in M with the weak topology such that U0 ⊂ W (f, p) and for each g ∈ U0 , each T ≥ 0 and each a.c. function v : [T, T + N ] → Rn the following property holds: If (2.66) sup{|v(T )|, |v(T + N )|} ≤ 2M0 (f, p) + 2, I g (T, T + N, v) ≤ U g (T, T + N, v(T ), v(T + N )) + 2M0 (f, p) + 2, then there exists an integer i0 ∈ [0, N − 6] such that |v(t) − Z∗f (t)| ≤ δ, t ∈ [i0 + T, i0 + T + 6].

(2.67)

By Proposition 1.3.6 there exist an open neighborhood U1 of f ∗ in M with the weak topology and a positive number S such that U1 ⊂ U0 , |U g (T1 , T2 , y1 , y2 )| + 1 < S for each g ∈ U1 , each T1 ∈ [0, ∞), (2.68) −1 T2 ∈ [T1 + 2 , T1 + 8N + 8] and each y1 , y2 ∈ B(M (f, p) + 2), i = 1, 2. It follows from Propositions 1.3.8 and 1.3.9 that there exist an open neighborhood U of f ∗ in M with the weak topology such that U ⊂ U1 and that for each g ∈ U, each T1 ∈ [0, ∞) and each T2 ∈ [T1 +4−1 , T1 +8N +8] the following properties hold: a) If an a.c. function v : [T1 , T2 ] → Rn satisﬁes ∗

min{I f (T1 , T2 , v), I g (T1 , T2 , v)} ≤ 2S + 2M (f, p) + 4, then

∗

|I f (T1 , T2 , v) − I g (T1 , T2 , v)| ≤ 4−1 δ;

(2.69) (2.70)

b) If y1 , y2 ∈ B(M (f, p) + 2), i = 1, 2, then ∗

|U f (T1 , T2 , y1 , y2 ) − U g (T1 , T2 , y1 , y2 )| ≤ 4−1 δ.

(2.71)

49

Extremals of nonautonomous problems

Let g ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + 1, ∞) and let an a.c. function v : [T1 , T2 ] → Rn satisfy (2.62). We show that the inequality (2.63) is true. We have two cases: (i) T2 − T1 ≤ 6N ; (ii) T2 − T1 > 6N . Consider the case (i). (2.4), (2.62) and (2.68) imply the inequality (2.69). The inequality (2.70) follows from (2.69) and property a). By property b) and (2.62), ∗

|U f (T1 , T2 , v(T1 ), v(T2 )) − U g (T1 , T2 , v(T1 ), v(T2 ))| ≤ 4−1 δ. Combined with (2.62) and (2.70) this inequality implies that ∗

I f (T1 , T2 , v) ≤ I g (T1 , T2 , v) +4−1 δ ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + δ + 4−1 δ ∗

≤ U f (T1 , T2 , v(T1 ), v(T2 )) + 3 · 2−1 δ. It follows from this inequality, the inequality (2.62) and the deﬁnition of δ (see (2.64), (2.65)) that the inequality (2.63) is true. Consider the case (ii). By Theorem 1.2.3, the deﬁnitions of M0 (f, p) and W0 (f, p) (see (2.7)) and the inequality (2.62), |v(t)| ≤ M0 (f, p) + 1, t ∈ [T1 , T2 ].

(2.72)

It follows from the choice of U0 and N (see (2.66) and (2.67)) and the inequalities (2.72) and (2.62) that the following property holds: For each τ ∈ [T1 , T2 − N ] there exists an integer iτ ∈ [0, N − 6] such that (2.73) |v(t) − Z∗f (t)| ≤ δ, t ∈ [iτ + τ, iτ + τ + 6]. It is easy to see that there exists a ﬁnite sequence {ti }qi=0 ⊂ [T1 , T2 ] such that t0 = T1 , tq = T2 , ti+1 − ti ∈ [6, N ], i = 0, . . . , q − 2, tq − tq−1 ∈ [N, 2N ], (2.74) f (2.75) |v(ti ) − Z∗ (ti )| ≤ δ, i = 0, . . . , q. Fix an integer i ∈ {0, . . . , q − 1}. (2.62), (2.74), (2.75) and (2.68) imply that I g (ti , ti+1 , v) ≤ U g (ti , ti+1 , v(ti ), v(ti+1 )) + δ ≤ δ + S.

(2.76)

It follows from (2.76), (2.74), (2.72) and the properties a), b) that ∗

|I f (ti , ii+1 , v) − I g (ti , ti+1 , v)| ≤ 4−1 δ, ∗

|U f (ti , ti+1 , v(ti ), v(ti+1 )) − U g (ti , ti+1 , v(ti ), v(ti+1 ))| ≤ 4−1 δ,

50

TURNPIKE PROPERTIES ∗

I f (ti , ii+1 , v) ≤ I g (ti , ti+1 , v) + 4−1 δ ∗

≤ U g (ti , ti+1 , v(ti ), v(ti+1 )) + δ + δ/4 ≤ U f (ti , ti+1 , v(ti ), v(ti+1 )) + 3δ/2. (2.77) Combined with the choice of δ (see (2.64), (2.65)) and (2.75) the inequality (2.77) implies that |v(t) − Z∗f (t)| ≤ for all t ∈ [ti , ti+1 ]. This completes the proof of the lemma. We need the next lemma in order to establish the convergence property of Theorem 2.1.3. This lemma follows from Lemmas 2.2.4 and 2.2.2. Lemma 2.2.5 For each ∈ (0, 1) there exist an open neighborhood U of f ∗ in M with the weak topology, a number l∗ ≥ 8 and an integer q∗ ≥ 4 such that for each g ∈ U, each T1 ∈ [0, ∞), T2 ∈ [T1 + l∗ q∗ , ∞) and each a.c. function v : [T1 , T2 ] → Rn the following property holds: If |v(t)| ≤ 2M0 (f, p) + 2, t ∈ [T1 , T2 ] and

I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + p,

(2.78)

then there exist sequences of numbers {di }qi=1 , {d¯i }qi=1 such that q ≤ q∗ , di < d¯i ≤ di + l∗ , i = 1, . . . , q,

(2.79)

|v(t) − Z∗f (t)| ≤ , t ∈ [T1 , T2 ] \ ∪qi=1 [di , d¯i ].

(2.80)

Proof. Let ∈ (0, 1). It follows from Lemma 2.2.4 that there exist δ ∈ (0, ) and a neighborhood U0 of f ∗ in M with the weak topology such that the inequality (2.63) is true for each g ∈ U0 , each T1 ∈ [0, ∞), T2 ∈ [T1 + 1, ∞) and each a.c. function v : [T1 , T2 ] → Rn which satisﬁes (2.62). By Lemma 2.2.2 there exist a neighborhood U of f ∗ in M with the weak topology and an integer N ≥ 8 such that U ⊂ U0 and if g ∈ U, T ≥ 0 and if an a.c. function v : [T, T + N ] → Rn satisﬁes (2.66) then there exists an integer i0 ∈ [0, N − 6] such that (2.67) is true. Fix an integer (2.81) q∗ > 4 + δ −1 p and a number l∗ ≥ 2N. Assume that g ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + l∗ q∗ , ∞) and v : [T1 , T2 ] → Rn is an a.c. function satisfying (2.78). It follows from the deﬁnition of U and (2.78) that for each τ ∈ [T1 , T2 −N ] there is an integer iτ ∈ [0, N −6] such that |v(t) − Z∗f (t)| ≤ δ for all t ∈ [iτ + τ, iτ + τ + 6].

51

Extremals of nonautonomous problems

This implies that there exists a sequence of numbers t0 , . . . , tG such that t0 = T1 , tG = T2 , ti+1 − ti ∈ [3, N ], i = 0, . . . , G − 1, |v(ti ) − Z∗f (ti )| ≤ δ, i = 1, . . . , G − 1.

(2.82) (2.83)

Set C = {i ∈ {1, . . . , G − 2} : I g (ti , ti+1 , v) > U g (ti , ti+1 , v(ti ), v(ti+1 )) + δ} (2.84) and denote by Card(C) the cardinality of C. By (2.78), p ≥ I g (T1 , T2 , v) − U g (T1 , T2 , v(T1 ), v(T2 )) ≥

[I g (ti , ti+1 , v) (2.85)

i∈C

−U g (ti , ti+1 , v(ti ), v(ti+1 ))] ≥ δ Card(C),

Card(C) ≤ δ −1 p.

Let i ∈ {1, . . . , G − 2} \ C. It follows from the deﬁnition of δ, U0 and (2.84), (2.83), (2.82) that |v(t)−Z∗f (t)| ≤ for all t ∈ [ti , ti+1 ]. Therefore |v(t) − Z∗f (t)| ≤ , t ∈ [ti , ti+1 ], i ∈ {1, . . . , G − 2} \ C. It is easy to see that [T1 , T2 ] \ ∪{[ti , ti+1 ] : i ∈ {1, . . . , G − 2} \ C} ⊂ ∪{[ti , ti+1 ] : i ∈ C ∪ {0, G − 1}} and by (2.82), (2.81), (2.85), ti+1 − ti ≤ N ≤ l∗ , i = 0, . . . , G − 1, Card(C ∪ {0, G − 1}) ≤ q∗ . This completes the proof of the lemma. Lemma 2.2.6 Let ∈ (0, 1). Then there exist a neighborhood U of f ∗ in M with the weak topology, a number l∗ ≥ 8, an integer q∗ ≥ 4 such that for each g ∈ U, each T1 ∈ [0, ∞), T2 ∈ [T1 + l∗ q∗ , ∞) and each a.c. function v : [T1 , T2 ] → Rn which satisﬁes one of the following relations below: (i)|v(Ti )| ≤ 2M f + 2p + 2, i = 1, 2, I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + p; (ii)|v(T1 )| ≤ 2M f + 2p + 2, I g (T1 , T2 , v) ≤ σ g (T1 , T2 , v(T1 )) + p, there are sequences of numbers {di }qi=1 , {d¯i }qi=1 for which (2.79) and (2.80) hold.

52

TURNPIKE PROPERTIES

Lemma 2.2.6, which is an extension of Lemma 2.2.5, now follows from Lemma 2.2.5, Theorem 1.2.3 and the deﬁnition of W0 (f, p), M0 (f, p) (see (2.6), (2.7)). The next auxiliary result which follows from Lemmas 2.2.4 and 2.2.2 will be used in order to establish the convergence property of Theorems 2.1.1 and 2.1.2. Lemma 2.2.7 Let ∈ (0, 1). Then there exist a neighborhood U of f ∗ in M with the weak topology such that, for each g ∈ U and each (g)-good function v : [0, ∞) → Rn , the inequality |v(t) − Z∗f (t)| ≤ is valid for all suﬃciently large t. Proof. It follows from Lemma 2.2.4 that there exist a number δ ∈ (0, ) and a neighborhood U0 of f ∗ in M with the weak topology such that the inequality (2.63) is true for each g ∈ U0 , each T1 ∈ [0, ∞), T2 ∈ [T1 + 1, ∞) and each a.c. function v : [T1 , T2 ] → Rn which satisﬁes (2.62). It follows from Lemma 2.2.2 that there exist a neighborhood U of f ∗ in M with the weak topology and an integer N ≥ 8 such that U ⊂ U0 ∩ W (f, p) and that the following property holds: If g ∈ U, T ≥ 0 and if an a.c. function v : [T, T + N ] → Rn satisﬁes (2.12), then there is an integer i0 ∈ [0, N − 6] such that |v(t) − Z∗f (t)| ≤ δ, t ∈ [i0 + T, i0 + T + 6].

(2.86)

Assume that g ∈ U and v : [0, ∞) → Rn is a (g)-good function. By the deﬁnition of W f (see (2.5)), |v(t)| ≤ M f for all large t.

(2.87)

Since v is a (g)-good function there exists T0 > 0 such that I g (t1 , t2 , v) ≤ U g (t1 , t2 , v(t1 ), v(t2 )) + δ

(2.88)

for each t1 ≥ T0 , t2 > t1 . We may assume that |v(t)| ≤ M f for all t ∈ [T0 , ∞). Let T ≥ T0 . It follows from the deﬁnition of U, N and (2.88) which holds with t1 = T , t2 = T + N that there exists an integer iT ∈ [0, N − 6] such that |v(t) − Z∗f (t)| ≤ δ, t ∈ [iT + T, iT + T + 6]. Therefore there exists a sequence {Ti }∞ i=1 ⊂ (T0 , ∞) such that Ti+1 − Ti ∈ [6, N ], i = 0, 1, . . . , |v(Ti ) − Z∗f (Ti )| ≤ δ, i = 1, 2, . . . .

Extremals of nonautonomous problems

53

It follows from these relations, (2.88) which holds with t1 = Ti , t2 = Ti+1 and the deﬁnition of U0 , δ that |v(t) − Z∗f (t)| ≤ , t ∈ [Ti , Ti+1 ], i = 1, 2, . . . . The lemma is proved. The next lemma plays a crucial role in the proof of Theorem 2.1.4. Its proof is based on Lemmas 2.2.4 and 2.2.2. Lemma 2.2.8 Let ∈ (0, 1). Then there exist a neighborhood U of f ∗ in M with the weak topology, δ ∈ (0, ) and ∆ > 1 such that the following property holds: For each g ∈ U, each T1 ∈ [0, ∞), T2 ∈ [T1 + 2∆, ∞) and each a.c. function v : [T1 , T2 ] → Rn which satisﬁes |v(t)| ≤ M0 (f, p), t ∈ [T1 , T2 ], I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + δ,

(2.89)

there exist τ1 ∈ [T1 , T1 + ∆] and τ2 ∈ [T2 − ∆, T2 ] such that |v(t) − Z∗f (t)| ≤ , t ∈ [τ1 , τ2 ].

(2.90)

Moreover if |v(T1 ) − Z∗f (T1 )| ≤ δ, then τ1 = T1 . Proof. By Lemma 2.2.4 there exist δ ∈ (0, ) and a neighborhood U0 of f ∗ in M with the weak topology such that for each g ∈ U0 , each T1 ∈ [0, ∞), T2 ∈ [T1 + 1, ∞) and each a.c. function v : [T1 , T2 ] → Rn which satisﬁes (2.62), relation (2.63) holds. By Lemma 2.2.2 there exist a neighborhood U of f ∗ in M with the weak topology and an integer N ≥ 8 such that U ⊂ U0 and for each g ∈ U, each T ≥ 0 and each a.c. function v : [T, T + N ] → Rn which satisﬁes (2.12), there is an integer i0 ∈ [0, N − 6] for which (2.86) holds. Set ∆ = 2N. (2.91) Assume that g ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + 2∆, ∞) and v : [T1 , T2 ] → Rn is an a.c. function satisfying (2.89). It follows from the deﬁnition of U, N and (2.89) that for each T ∈ [T1 , T2 − N ] there is an integer i0 ∈ [0, N − 6] for which (2.86) holds. Therefore there exists a sequence of numbers {ti }G i=0 ⊂ [T1 , T2 ] such that t0 = T1 , ti+1 − ti ∈ [6, N ], i = 0, . . . , G − 1, T2 − tG ≤ N, |v(ti ) − Z∗f (ti )| ≤ δ, i = 1, . . . , G.

(2.92) (2.93)

54

TURNPIKE PROPERTIES

It follows from the deﬁnition of δ, U0 , (2.89), (2.93), (2.92) that |v(t) − Z∗f (t)| ≤ , t ∈ [ti , ti+1 ]

(2.94)

for all i = 1, . . . , G − 1 and if |v(T1 ) − Z∗f (T1 )| ≤ δ, then (2.94) holds for i = 0, . . . , G − 1. This completes the proof of the lemma. The following lemma is an extension of Lemma 2.2.8. Lemma 2.2.9 Let ∈ (0, 1). Then there exist a neighborhood U of f ∗ in M with the weak topology, δ ∈ (0, ), ∆ > 1 such that for each g ∈ U, each T1 ∈ [0, ∞), T2 ∈ [T1 + 2∆, ∞) and each a.c. function v : [T1 , T2 ] → Rn the following properties hold: (i) If |v(Ti )| ≤ 2M f + 2 + 2p, i = 1, 2 and

I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + δ,

then (2.90) holds with τ1 ∈ [T1 , T1 + ∆] and τ2 ∈ [T2 − ∆, T2 ]. Moreover if |v(T1 ) − Z∗f (T1 )| ≤ δ, then τ1 = T1 . (ii) If |v(T1 )| ≤ 2M f + 2 + 2p, I g (T1 , T2 , v) ≤ σ g (T1 , T2 , v(T1 )) + δ, then (2.90) holds with τ1 ∈ [T1 , T1 + ∆] and τ2 ∈ [T2 − ∆, T2 ]. Moreover if |v(T1 ) − Z∗f (T1 )| ≤ δ, then τ1 = T1 . Lemma 2.2.9 now follows from Lemma 2.2.8, Theorem 1.2.3 and the deﬁnition of W0 (f, p), M0 (f, p) (see (2.6), (2.7)). Lemma 2.2.10 Let ∈ (0, 1). Then there exist a neighborhood U of f ∗ in M with the weak topology, ∆ > 1 such that for each g ∈ U and each (g)-overtaking optimal function v : [0, ∞) → Rn satisfying |v(0)| ≤ 2M f + 2 + 2p the relation |v(t) − Z∗f (t)| ≤ holds for all t ∈ [∆, ∞). Lemma 2.2.10 follows from the deﬁnition of W f , M f (see (2.5)) and Lemma 2.2.9.

2.3.

Proofs of Theorems 2.1.1-2.1.4

Construction of the set F. Fix z∗ ∈ Rn and an integer p ≥ 1. For each f ∈ M we deﬁne a function Z∗f : [0, ∞) → Rn , numbers M f , M0 (f, p), M (f, p), a function φM (f,p)+1 , a number r(f, p) > 0 and neighborhoods W f , W0 (f, p), W (f, p) of f in M with the weak topology as in Section 2.2 (see (2.3)-(2.10)). Set Ep = {frM (f,p)+1 : f ∈ M, r ∈ (0, r(f, p))}.

(3.1)

Extremals of nonautonomous problems

55

Clearly for each f ∈ M and each r ∈ (0, r(f, p)) Lemmas 2.2.2-2.2.10 M (f,p)+1 hold with f ∗ = fr . By Lemma 2.2.1 Ep is everywhere dense in M with the strong topology. For each f ∈ M, each r ∈ (0, r(f, p)) and each integer k ≥ 1 there M (f,p)+1 exist an open neighborhood V (f, p, r, k) of fr in M with the weak topology, an integer q(f, p, r, k) ≥ 4, numbers δ(f, p, r, k) ∈ (0, (4k)−1 ), l(f, p, r, k) ≥ 8, ∆(f, p, r, k) > 1 such that: (i) Lemma 2.2.6 holds with f ∗ = frM (f,p)+1 , = (4k)−1 , U = V (f, p, r, k), q∗ = q(f, p, r, k), l∗ = l(f, p, r, k); (ii) Lemma 2.2.7 holds with f ∗ = frM (f,p)+1 , = (4k)−1 , U = V (f, p, r, k); (iii) Lemmas 2.2.9 and 2.2.10 hold with f ∗ = frM (f,p)+1 , U = V (f, p, r, k), = (4k)−1 , δ = δ(f, p, r, k), ∆ = ∆(f, p, r, k). We deﬁne Fp = ∩∞ k=1 ∪ {V (f, p, r, k) : f ∈ M, r ∈ (0, r(f, p))},

(3.2)

F = ∩∞ p=1 Fp .

(3.3)

Clearly F is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets of M. Proof of Theorem 2.1.1. We prove Assertion 1. Assume that g ∈ F and vi : [0, ∞) → Rn , i = 1, 2 are (g)-good functions. Let > 0. Fix an integer k ≥ 4−1 . There exist f ∈ M and r ∈ (0, r(f, 1)) such that g ∈ V (f, 1, r, k). By condition (ii) and Lemma 2.2.7, |v2 (t) − v1 (t)| ≤ (2k)−1 < for all large t. Since is an arbitrary positive number we conclude that |v2 (t) − v1 (t)| → 0 as t → ∞. Assertion 1 is proved.

56

TURNPIKE PROPERTIES

We prove Assertion 2. Let g ∈ F and let y ∈ Rn . By Theorem 1.2.1 there exists a (g)-good function Y : [0, ∞) → Rn such that Y (0) = y and (3.4) I g (0, T, Y ) = U g (0, T, Y (0), Y (T )) for each T ≥ 0. We show that Y is a (g)-overtaking optimal function. Let us assume the converse. Then there exists a number > 0 and an a.c. function Z : [0, ∞) → Rn such that Z(0) = y and lim sup[I g (0, T, Y ) − I g (0, T, Z)] > . T →∞

(3.5)

There exists a sequence of positive numbers {Ti }∞ i=1 such that Ti → +∞ as i → ∞, I g (0, Ti , Y ) − I g (0, Ti , Z) > , i = 1, 2, . . . .

(3.6)

By Theorem 1.2.1, Z is a bounded (g)-good function. Therefore Y (t) − Z(t) → 0 as t → ∞.

(3.7)

The functions Y and Z are (f )-good and bounded. Therefore we can choose a number S > sup{|Z(t)|, Y (t)| : t ∈ [0, ∞)}.

(3.8)

It follows from Proposition 1.3.7 that there exists δ > 0 such that the following property holds: If T1 ∈ [0, ∞), T2 ∈ [T1 + 4−1 , T1 + 4] and if y1 , y2 , z1 , z2 ∈ Rn satisfy |yi |, |zi | ≤ S, i = 1, 2, |y1 − y2 |, |z1 − z2 | ≤ δ, then

|U g (T1 , T2 , y1 , z1 ) − U g (T1 , T2 , y2 , z2 | ≤ 8−1 .

(3.9)

Since |Y (t) − Z(t)| → 0 as t → ∞ there exists τ > 0 such that |Z(t) − Y (t)| ≤ 2−1 δ, t ∈ [τ, ∞).

(3.10)

Fix a natural number j such that Tj > τ . There exists an a.c. function X : [0, ∞) → Rn such that X(t) = Z(t), t ∈ [0, Tj ], X(t) = Y (t), t ∈ [Tj + 1, ∞),

(3.11)

I g (Tj , Tj + 1, X) = U g (Tj , Tj + 1, X(Tj ), X(Tj + 1)). It follows from (3.11), (3.4), (3.8), (3.10) and the deﬁnition of δ that X(0) = Y (0), X(Tj + 1) = Y (Tj + 1),

57

Extremals of nonautonomous problems

|I g (Tj , Tj + 1, X) − I g (Tj , Tj + 1, Y )| = |U g (Tj , Tj + 1, X(Tj ), X(Tj + 1)) −U g (Tj , Tj + 1, Y (Tj ), Y (Tj + 1))| ≤ 8−1 . Together with (3.11) and (3.6) these relations imply that I g (0, Tj + 1, Y ) − U g (0, Tj + 1, Y (0), Y (Tj + 1)) ≥ I g (0, Tj + 1, Y ) − I g (0, Tj + 1, X) = I g (0, Tj , Y )−I g (0, Tj , Z)+I g (Tj , Tj +1, Y )−I g (Tj , Tj +1, X) ≥ −8−1 . This is contradictory to (3.4). The obtained contradiction proves Assertion 2. We prove Assertion 3. Let g ∈ F, > 0 and Y : [0, ∞) → Rn be a (g)-overtaking optimal function. Fix an integer k ≥ 4−1 . There exist f ∈ M, r ∈ (0, r(f, 1)) such that g ∈ V (f, 1, r, k). Assume that h ∈ V (f, 1, r, k) and v : [0, ∞) → Rn is an (h)-good function. It follows from condition (ii) and Lemma 2.2.7 that |v(t) − Y (t)| < (2k)−1 for all large t. Assertion 3 is proved. Proof of Theorem 2.1.2. Let g ∈ F, M, > 0 and let Y : [0, ∞) → Rn be a (g)-overtaking function. Fix integers p > 2M + 2|Y (0)| + 2, k > 4−1 .

(3.12)

There exists f ∈ M and r ∈ (0, r(f, p)) such that g ∈ V (f, p, r, k). By condition (ii) and Lemma 2.2.7 there exists a number τ0 > 0 such that |Y (t) − Z∗f (t)| ≤ (4k)−1 , t ∈ [τ0 , ∞).

(3.13)

τ = τ0 + ∆(f, p, r, k) + 1.

(3.14)

Set Rn

Assume that h ∈ V (f, p, r, k) and v : [0, ∞) → is an (h)-overtaking optimal function such that |v(0)| ≤ M . By condition (iii), Lemma 2.2.10 and (3.12), |v(t) − Z∗f (t)| ≤ (4k)−1 , t ∈ [∆(f, p, r, k), ∞). Together with (3.13), (3.14) and (3.12) this implies that |v(t) − Y (t)| ≤ for all t ∈ [τ, ∞). The theorem is proved. Proof of Theorem 2.1.3. Let g ∈ F, S1 , S2 , > 0 and Y : [0, ∞) → Rn be a (g)-overtaking optimal function. By Theorem 1.2.1, Y is a (g)-good function and sup{|Y (t)| : t ∈ [0, ∞)} < ∞. Fix integers p > 2S1 + 2S2 + 1 + sup{|Y (t)| : t ∈ [0, ∞)}, k > 8−1 .

(3.15)

58

TURNPIKE PROPERTIES

There exist f ∈ M and r ∈ (0, r(f, p)) such that g ∈ V (f, p, r, k). By condition (ii) and Lemma 2.2.7 there exists a number τ0 > 0 such that |Y (t) − Z∗f (t)| ≤ (4k)−1 , t ∈ [τ0 , ∞).

(3.16)

Set U = V (f, p, r, k), L = τ0 + l(f, p, r, k), Q = q(f, p, r, k) + 1.

(3.17)

Assume that h ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + LQ, ∞) and an a.c. function v : [T1 , T2 ] → Rn satisﬁes one of the conditions (a), (b) of the theorem. By condition (i) and Lemma 2.2.6 there are numbers {di }qi=1 , {d¯i }qi=1 such that q ≤ q(f, p, r, k), di < d¯i ≤ di + l(f, p, r, k), i = 1, . . . , q, |v(t) − Z∗f (t)| ≤ (4k)−1 , t ∈ [T1 , T2 ] \ ∪qi=1 [di , d¯i ]. By these relations and (3.15), (3.16), |v(t) − Y (t)| ≤ , t ∈ [T1 , T2 ] \ ([0, τ0 ] ∪qi=1 [di , d¯i ]). This completes the proof of the theorem. Proof of Theorem 2.1.4. Let g ∈ F, S, > 0 and let Y : [0, ∞) → Rn be a (g)-overtaking optimal function. By Theorem 1.2.1, Y is a (g)-good function and sup{|Y (t)| : t ∈ [0, ∞)} < ∞. Fix integers p > 2S + 1 + sup{|Y (t)| : t ∈ [0, ∞)}, k > 8−1 .

(3.18)

There exist f ∈ M and r ∈ (0, r(f, p)) such that g ∈ V (f, p, r, k). By condition (ii) and Lemma 2.2.7 there exists a number τ0 such that (3.16) holds. Set U = V (f, p, r, k), L = τ0 + ∆(f, p, r, k), δ = δ(f, p, r, k).

(3.19)

Assume that h ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + 2L, ∞) and an a.c. function v : [T1 , T2 ] → Rn satisﬁes one of the conditions (a), (b) of the theorem. It follows from condition (iii) and Lemma 2.2.9 that |v(t) − Z∗f (t)| ≤ (4k)−1 , t ∈ [T1 + ∆(f, p, r, k), T2 − ∆(f, p, r, k)]. By this relation, (3.16), (3.18) and (3.19), |v(t) − Y (t)| ≤ for all t ∈ [T1 + L, T2 − L]. The theorem is proved.

59

Extremals of nonautonomous problems

2.4.

Periodic variational problems

Let a > 0, Z = {0, ±1, ±2, . . .} and let ψ : [0, ∞) → [0, ∞) be an increasing function such that ψ(t) → +∞ as t → ∞. Denote by Mp the set of continuous functions f : [0, ∞) × Rn × Rn → R1 which satisfy the following assumptions: A (i) f (t, x + q, u) = f (t, x, u) for each t ∈ [0, ∞), x, u ∈ Rn , q ∈ Zn ; A (ii) for each (t, x) ∈ [0, ∞) × Rn the function f (t, x, ·) : Rn → R1 is convex; A (iii) the function f is bounded on [0, ∞) × Rn × E for any bounded set E ⊂ Rn ; A (iv) f (t, x, u) ≥ ψ(|u|)|u| − a for each (t, x, u) ∈ [0, ∞) × Rn × Rn ; A (v) for each > 0 there exist positive numbers Γ, δ such that if t ∈ [0, ∞) and if u1 , u2 , x1 , x2 ∈ Rn are such that |ui | ≥ Γ, i = 1, 2,

max{|x1 − x1 |, |u1 − u2 |} ≤ δ,

then |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ max{f (t, x1 , u1 ), f (t, x2 , u2 )}; A (vi) for each M, > 0 there exists a positive number δ such that if t ∈ [0, ∞) and if u1 , u2 , x1 , x2 ∈ Rn satisfy |ui | ≤ M, i = 1, 2,

max{|x1 − x2 |, |u1 − u2 |} ≤ δ,

then |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ . We equip the set Mp with the uniformity which is determined by the following base: E(N, , λ) = {(f, g) ∈ Mp × Mp : |f (t, x, u) − g(t, x, u)| ≤ for each t ∈ [0, ∞) and each x, u ∈ Rn satisfying |u| ≤ N, (|f (t, x, u)| + 1)(|g(t, x, u)| + 1)−1 ∈ [λ−1 , λ] for each t ∈ [0, ∞) and each x, u ∈ Rn }, where N > 0, > 0, λ > 1. We can show that the uniform space Mp is metrizable and complete. Consider functionals of the form f

I (T1 , T2 , x) =

T2 T1

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

60

TURNPIKE PROPERTIES

where f ∈ Mp , 0 ≤ T1 < T2 < +∞ and x : [T1 , T2 ] → Rn is an a.c. function. For f ∈ Mp , y, z ∈ Rn and numbers T1 , T2 satisfying 0 ≤ T1 < T2 we deﬁne U f (T1 , T2 , y, z) and σ f (T1 , T2 , y) by (1.2) and (1.3) and set ˜ f (T1 , T2 , y, z) = inf{U f (T1 , T2 , y, z + m) : m ∈ Zn }. U It is easy to see that −∞ < U f (T1 , T2 , y, z) < +∞ for each f ∈ Mp , each y, z ∈ Rn and each pair of numbers T1 , T2 satisfying 0 ≤ T1 < T2 . For f ∈ Mp we use the notions of an (f )-good function and an (f )overtaking optimal function. The methods used in the proofs of Theorems 1.2.1-1.2.3 and Theorems 2.1.1-2.1.4 are applicable to the space Mp . The following results are valid. Theorem 2.4.1 For each h ∈ Mp and each z ∈ Rn there exists an (h)-good function Z h : [0, ∞) → Rn satisfying Z h (0) = z such that: 1. ˜ f (T1 , T2 , Z f (T1 ), Z f (T2 )) = I f (T1 , T2 , Z f ) U for each f ∈ M, each z ∈ Rn and each T1 ≥ 0, T2 > T1 . 2. For each f ∈ Mp , each z ∈ Rn and each a.c. function y : [0, ∞) → n R either I f (0, T, y) − I f (0, T, Z f ) → +∞ as T → ∞ or

sup{|I f (0, T, y) − I f (0, T, Z f )| : T ∈ (0, ∞)} < ∞

(4.1)

and if (4.1) is valid, then sup{|y(t1 ) − y(t2 )| : t1 ∈ [0, ∞), t2 ∈ [t1 , t1 + 1]} < ∞. 3. For each f ∈ Mp there exist a neighborhood U of f in Mp and a number Q > 0 such that sup{|Z g (t1 ) − Z g (t2 )| : t1 ∈ [0, ∞), t2 ∈ [t1 , t1 + 1]} ≤ Q for each g ∈ U and each z ∈ Rn . 4. For each f ∈ Mp there exist a neighborhood U of f in Mp and a number Q > 0 such that I g (T1 , T2 , Z g ) ≤ I g (T1 , T2 , y) + Q for g ∈ U, each z ∈ Rn , each T1 ≥ 0, T2 > T1 and each a.c. function y : [T1 , T2 ] → Rn .

Extremals of nonautonomous problems

61

Theorem 2.4.2 For each f ∈ Mp there exist a neighborhood U of f in Mp and M > 0 such that if g ∈ U and if x : [0, ∞) → Rn is a (g)-good function, then lim sup(sup{|x(t1 ) − x(t2 )| : T →∞

t1 ∈ [T, ∞), t2 ∈ [t1 , t1 + 1]}) < M. We can show that there exists a set F ⊂ Mp which is a countable intersection of open everywhere dense sets in Mp such that the following theorems are valid. Theorem 2.4.3 1. For each g ∈ F and each pair of (g)-good functions vi : [0, ∞) → Rn , i = 1, 2, |v2 (t) − v1 (t) − m| → 0 as t → ∞ with m ∈ Zn . 2. For each g ∈ F and each y ∈ Rn there exists a (g)-overtaking optimal function Y : [0, ∞) → Rn such that Y (0) = y. 3. Let g ∈ F, > 0 and Y : [0, ∞) → Rn be a (g)-overtaking optimal function. Then there exists a neighborhood U of g in Mp such that the following property holds: If h ∈ U and if v : [0, ∞) → Rn is an (h)-good function, then there is m ∈ Zn such that |v(t) − Y (t) − m| ≤ for all large t. Theorem 2.4.4 Let g ∈ F, > 0 and let Y : [0, ∞) → Rn be a (g)overtaking optimal function. Then there exists a neighborhood U of g in Mp and a number τ > 0 such that the following property holds: If h ∈ U and if v : [0, ∞) → Rn is an (h)-overtaking optimal function, then there exists m ∈ Zn such that |v(t) − Y (t) − m| ≤ , t ∈ [τ, ∞). Theorem 2.4.5 Let g ∈ F, S, > 0 and let Y : [0, ∞) → Rn be a (g)overtaking optimal function. Then there exists a neighborhood U of g in Mp , a number L > 0 and an integer Q ≥ 1 such that for each h ∈ U and each pair of numbers T1 ∈ [0, ∞), T2 ∈ [T1 + LQ, ∞) the following property holds: If an a.c. function v : [T1 , T2 ] → Rn satisﬁes ˜ h (T1 , T2 , v(T1 ), v(T2 )) + S, I h (T1 , T2 , v) ≤ U

62

TURNPIKE PROPERTIES

then there exist sequences of numbers {di }qi=1 , {bi }qi=1 ⊂ [T1 , T2 ] such that q ≤ Q, bi < di ≤ bi + L, i = 1, . . . , q and that for each interval J ⊂ [T1 , T2 ] \ ∪qi=1 [bi , di ] there is m ∈ Zn for which |v(t) − Y (t) − m| ≤ , t ∈ J . Theorem 2.4.6 Let g ∈ F, > 0 and let Y : [0, ∞) → Rn be a (g)overtaking optimal function. Then there exists a neighborhood U of g in Mp and numbers δ, L > 0 such that for each h ∈ U, each pair of numbers T1 ∈ [0, ∞), T2 ∈ [T1 + 2L, ∞) and each a.c. function v : [T1 , T2 ] → Rn ˜ h (T1 , T2 , v(T1 ), v(T2 )) + δ the following which satisﬁes I h (T1 , T2 , v) ≤ U property holds: There exists m ∈ Zn such that |v(t) − Y (t) − m| ≤ , t ∈ [T1 + L, T2 − L].

2.5.

Spaces of smooth integrands

Consider the complete metric space M deﬁned in Section 2.1. For any function g : R1 × Rn × Rn → R1 denote by L(g) the restriction of g to [0, ∞) × R2n and for an integer k ≥ 1 denote by C(k, M) the space of all integrands f = f (t, x, u) ∈ C k (R2n+1 ) such that L(f ) ∈ M. Let k ≥ 1 be an integer. For p = (p1 , . . . , p2n+1 ) ∈ {0, . . . , k}2n+1 and f ∈ C k (R2n+1 ) we set |p| =

2n+1 i=1

p

2n+1 pi , Dp f = ∂ |p| f /∂y1p1 . . . ∂y2n+1 .

For the set C(k, M) we consider the uniformity which is determined by the following base: E(N, , λ) = {(f, g) ∈ C(k, M) × C(k, M) : |Dp f (t, x, u) − Dp g(t, x, u)| ≤

63

Extremals of nonautonomous problems

for each (t, x, u) ∈ R2n+1 satisfying |t|, |x|, |u| ≤ N and each p ∈ {0, . . . k}2n+1 such that |p| ≤ k, |f (t, x, u) − g(t, x, u)| ≤ for each t ∈ [0, ∞) and each x, u ∈ Rn for which |x|, |u| ≤ N, (|f (t, x, u)| + 1)(|g(t, x, u)| + 1)−1 ∈ [λ−1 , λ] for each t ∈ [0, ∞) and each x, u ∈ Rn such that |x| ≤ N }, where N > 0, > 0, λ > 1. Clearly the uniform space C(k, M) is Hausdorﬀ and has a countable base. Therefore C(k, M) is metrizable. It is easy to verify that the uniform space C(k, M) is complete and the operator L : C(k, M) → M is continuous. Denote by Mk the space of all functions f ∈ C(k, M) which satisfy the following conditions: f ∈ C k (R2n+1 ), ∂f /∂ui ∈ C k (R2n+1 ) for i = 1, . . . , n;

(5.1)

the matrix (∂ 2 f /∂ui ∂uj )(t, x, u), i, j = 1, . . . , n is positive deﬁnite (5.2) for all (t, x, u) ∈ R2n+1 ; there exist a number c0 > 1 and monotone increasing functions φi : [0, ∞) → [0, ∞), i = 0, 1, 2 such that φ0 (t)t−1 → +∞ as t → +∞, f (t, x, u) ≥ φ0 (c0 |u|) − φ1 (|x|), t ∈ R1 , x, u ∈ Rn ; sup{|∂f /∂xi (t, x, u)|, |∂f /∂ui (t, x, u)|} ≤ φ2 (|x|)(1 + φ0 (|u|)),

(5.3) (5.4)

t ∈ R1 , x, u ∈ Rn , i = 1, . . . , n. ¯ k the closure of Mk in C(k, M). We consider the topoDenote by M ¯ k ⊂ C(k, M) with the relative topology. In order logical subspace M to show that the conclusions of Theorems 2.1.1-2.1.4 also hold for a ¯ k we need the following result. Gδ -subset of the space M Proposition 2.5.1 Let k ≥ 1 be an integer, a number c0 > 1, φi : [0, ∞) → [0, ∞), i = 0, 1, 2 be monotone increasing functions and let an integrand f : R2n+1 → R1 satisfy (5.1)-(5.4). Assume that T1 ∈ [0, ∞), T2 > T1 and an a.c. function w : [T1 , T2 ] → Rn satisﬁes I f (T1 , T2 , w) = U f (T1 , T2 , w(T1 ), w(T2 )) < ∞.

(5.5)

64

TURNPIKE PROPERTIES

Then w ∈ C k+1 ([T1 , T2 ]; Rn ) and (d/dt)(∂f /∂ui (t, w(t), w (t)) = (∂f /∂xi (t, w(t), w (t))

(5.6)

for each i ∈ {1, . . . , n} and each t ∈ [T1 , T2 ]. Denote by < x, y > the scalar product of x, y ∈ Rn . In the proof of Proposition 2.5.1 we need the following simple result. Lemma 2.5.1 Let n ≥ 1 be an integer, −∞ < T1 < T2 < +∞ and let f ∈ L1 ([T1 , T2 ]; Rn ) have the following property: T2 T1

< f (t), g(t) > dt = 0

for every function g ∈ L∞ ([T1 , T2 ]; Rn ) such that TT12 g(t)dt = 0. Then there exists d ∈ Rn such that f (t) = d for almost all t ∈ [T1 , T2 ]. Proof of Proposition 2.5.1. Put x = w(T1 ), y = w(T2 ). In proving Proposition 2.5.1 we follow [64, Theorem 1.10.1]. For t ∈ [T1 , T2 ] we set B(t) = (t, w(t), w (t)). Analogously to the proof of Theorem 1.10.1 of [64] we can show that if an a.c. function h : [T1 , T2 ] → Rn satisﬁes h(T1 ) = h(T2 ) = 0, h ∈ L∞ ([T1 , T2 ]; Rn ), (5.7) then n

∂f /∂xi (B(t))hi (t) +

i=1

∂f /∂ui (B(t))hi (t) ∈ L1 (T1 , T2 ),

(5.8)

i=1

T2 n T1

n

[

∂f /∂xi (B(t))hi (t) +

i=1

n

∂f /∂ui (B(t))hi (t)]dt = 0.

(5.9)

i=1

It follows from (5.3)-(5.5) that the function t → |∂f /∂xi (B(t))| + |∂f /∂ui (B(t))|, t ∈ [T1 , T2 ] belongs to the space L1 (T1 , T2 ) for i = 1, . . . , n. Consider a function g ∈ L∞ (T1 , T2 ); Rn ) such that T2 T1

g(t)dt = 0

(5.10)

65

Extremals of nonautonomous problems

and put

t

h(t) = t

Ei (t) =

T1

T1

g(τ )dτ, t ∈ [T1 , T2 ],

(∂f /∂xi )(B(τ ))dτ, t ∈ [T1 , T2 ], i = 1, . . . , n.

Clearly h satisﬁes (5.7). Thus (5.9) and (5.8) are true. The Fubini theorem implies that for i = 1, . . . , n, T2 T1

(∂f /∂xi )(B(t))hi (t)dt =

T2 T1

gi (τ )(Ei (T2 ) − Ei (τ ))dτ.

Combined with (5.9) this equality implies that T2 n T1

[

∂f /∂ui (B(t)) + Ei (T2 ) − Ei (t)]gi (t)dt = 0.

i=1

We have shown that this equality holds for every g ∈ L∞ ([T1 , T2 ]; Rn ) satisfying (5.10). Therefore Lemma 2.5.1 implies that there is d = (d1 , . . . , dn ) ∈ Rn such that ∂f /∂ui (B(t)) + Ei (T2 ) − Ei (t) = di

(5.11)

for each i ∈ {1, . . . , n} and almost all t ∈ [T1 , T2 ]. Deﬁne a mapping G : R1 × Rn × Rn → R1 × Rn × Rn by G(t, x, u) = (t, x, (∂f /∂ui (t, x, u))ni=1 ). Assume that (ti , xi , ui ) ∈ R2n+1 , i = 1, 2 and G(t1 , x1 , u1 ) = G(t2 , x2 , u2 ). Clearly t1 = t2 , x1 = x2 . We show that u1 = u2 . For λ ∈ [0, 1] we denote by A(λ) the matrix (∂ 2 f /∂ui ∂uj )(t1 , x1 , u1 + λ(u2 − u1 )), i, j = 1, . . . , n. For i = 1, . . . , n we have 0 = ∂f /∂ui (t1 , x1 , u2 ) − ∂f /∂ui (t1 , x1 , u1 ) 1

= 1

=

0

0

(d/dλ)(∂f /∂ui (t1 , x1 , u1 + λ(u2 − u1 ))dλ

< (∂ 2 f /∂ui ∂uj )(t1 , x1 , u1 + λ(u2 − u1 ))nj=1 , u2 − u1 > dλ.

66

TURNPIKE PROPERTIES

This implies that

1

1 0

0

A(λ)(u2 − u1 )dλ = 0,

< A(λ)(u2 − u1 ), u2 − u1 > dλ = 0.

By the deﬁnition of A(λ) and (5.2) u2 = u1 . Therefore the mapping G is injective. By the inverse function theorem and the conditions of the proposition, G(R2n+1 ) is an open subset of R2n+1 , there exists G−1 : G(R2n+1 ) → R2n+1 ∈ C 1 and

We show that

(G−1 ) (y) = [G · G−1 (y)]−1 , y ∈ G(R2n+1 ).

(5.12)

G(R2n+1 ) = R2n+1 .

(5.13)

Let (t, x, U ), (ti , xi , ui ) ∈ R2n+1 , i = 1, 2, . . ., G(ti , xi , ui ) → (t, x, U ) as i → ∞.

(5.14)

It is suﬃcient to prove that (t, x, U ) ∈ G(R2n+1 ). Clearly (ti , xi ) → (t, x) as i → ∞.

(5.15)

We show that the sequence {ui }∞ i=1 is bounded. Let us assume the converse. By (5.14) and (5.15) there exists a number M > 0 such that |G(ti , xi , ui )|, ti , |xi | ≤ M as i = 1, 2, . . . .

(5.16)

There exist M0 > 0 such that |f (τ, y, 0)|, |(∂f /∂uj )(τ, y, 0)| ≤ M0 , j = 1, . . . , n, |τ | ≤ M, |y| ≤ M. (5.17) We may assume that (5.18) |ui | → ∞ as i → ∞. It follows from (5.2) and (5.16)-(5.18) that for any integer i ≥ 1 f (ti , xi , 0) ≥ f (ti , xi , ui )− < (∂f /∂uj (ti , xi , ui ))nj=1 , ui >,

(5.19)

f (ti , xi , ui ) ≤ M0 + M |ui |, lim sup f (ti , xi , ui )/|ui | ≤ M. i→∞

On the other hand by (5.16), (5.18) and (5.3) for any integer i ≥ 1 f (ti , xi , ui ) ≥ φ0 (c0 |ui |) − φ1 (M ),

67

Extremals of nonautonomous problems

lim sup f (ti , xi , ui )/|ui | ≥ lim sup φ0 (c0 |ui |)/|ui | = +∞. i→∞

i→∞

This is contradictory to (5.19). The obtained contradiction proves the n boundedness of {ui }∞ i=1 . We may assume that ui → u ∈ R as i → ∞. Together with (5.14), (5.15) this implies that (t, x, U ) = lim G(ti , xi , ui ) = G(t, x, u) ∈ G(R2n+1 ). i→∞

Therefore (5.13) holds. By (5.11) for almost all t ∈ [T1 , T2 ], (t, w(t), w (t)) = G−1 (t, w(t), (di − Ei (T2 ) − Ei (t))ni=1 ).

(5.20)

It is now easy to see that the last relation holds for all t ∈ [T1 , T2 ] and w ∈ C 2 ([T1 T2 ]; Rn ). (5.9) implies that for each h ∈ C 1 ([T1 , T2 ]; Rn ) satisfying (5.7), n T2 i=1 T1

hi (t)[∂f /∂xi (B(t)) − (d/dt)(∂f /∂ui (B(t)))]dt = 0.

This implies (5.6) for each t ∈ [T1 , T2 ] and each i ∈ {1, . . . , n}. By (5.12) G−1 ∈ C k . Together with (5.20) this implies that w ∈ C k+1 ([T1 , T2 ]; Rn ). This completes the proof of the proposition. Theorem 2.5.1 Let k ≥ 1 be an integer. Then there exists a Gδ -set F ⊂ M which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) sets in M and for which the ¯ k which is conclusions of Theorems 2.1.1.-2.1.4 hold and a set Fk ⊂ M ¯ k such that a countable intersection of open everywhere dense sets in M L(Fk ) ⊂ F. Proof. Fix z∗ ∈ Rn . For each f ∈ M let Z∗f : [0, ∞) → Rn be as guaranteed by Theorem 1.2.1. For each M > 0 there exists a function ψ M ∈ C ∞ (R1 ) such that ψ M (t) = 1, t ∈ [−M − 1, M + 1], ψ M (t) = 0, |t| ≥ M + 2, ψ M (t) ∈ (0, 1), t ∈ (−M − 2, −M − 1) ∪ (M + 1, M + 2). For f ∈ M and M > |z∗ | we deﬁne φM : [0, ∞) × Rn → R1 as follows: φM (t, x) = |x − Z∗f (t)|2 ψ M (|x|), t ∈ [0, ∞), x ∈ Rn . By Remark 2.2.1 the function φM satisﬁes Assumption B.

68

TURNPIKE PROPERTIES

For each f ∈ M and each integer p ≥ 1 we deﬁne numbers M f , M0 (f, p), M (f, p), r(f, p) > 0 and neighborhoods W f , W0 (f, p), W (f, p) of f in M as in Section 2.2 (see (2.3)-(2.10)). Consider the set Ep ⊂ M deﬁned by (3.1), (2.3). For each f ∈ M, each pair of integers p, q ≥ 1 and each r ∈ (0, r(f, p)) we deﬁne an open neighborhood V (f, p, r, q) of M (f,p)+1 fr in M as in Section 2.3 and deﬁne a set F by (5.2), (5.3). In Section 2.5, Theorems 2.2.1-2.2.4 were established for the set F. Let g ∈ Mk , p ≥ 1 be an integer and r ∈ (0, r(L(g), p)). By Theorem L(g) 1.2.1 and Proposition 2.5.1, Z∗ ∈ C k+1 . There exists Y g : R1 → Rn ∈ k+1 such that C L(g) Y g (t) = Z∗ (t) for t ∈ [0, ∞), (dk+1 Y g /dtk+1 )(t) = (dk+1 Y g /dtk+1 )(0) for t ∈ (−∞, 0). We deﬁne a function grp : R2n+1 → R1 as follows: grp (t, x, u) = g(t, x, u) + r|x − Y g (t)|2 ψ S (|x|), t ∈ R1 , x, u ∈ Rn where S = M (L(g), p) + 1. It is easy to verify that grp ∈ Mk , grp → g as r → 0 in C(k, M), L(grp ) ∈ Ep for g ∈ Mk , p ≥ 1 and r ∈ (0, r(L(g), p)). For each integer p ≥ 1 we set Gp = {grp : g ∈ Mk , r ∈ (0, r(L(g), p))}. For each g ∈ Mk , each pair of integers p, q ≥ 1 and each r ∈ (0, r(L(g), p)) we set

¯ k. U (g, p, r, q) = L−1 (V (L(g), p, r, q)) ∩ M

¯ k . We deﬁne Evidently U (g, p, r, q) is an open neighborhood of grp in M Fkp = ∩∞ q=1 ∪ {U (g, p, r, q) : g ∈ Mk , r ∈ (0, r(L(g), p))}, Fk = ∩∞ p=1 Fkp . Clearly Fk is a countable intersection of open everywhere dense sets in ¯ k and L(Fk ) ⊂ F. This completes the proof of the theorem. M

Extremals of nonautonomous problems

2.6.

69

Examples

Let n ≥ 1. Fix a positive constant a and set ψ(t) = t, t ∈ [0, ∞). Consider a complete metric space M of integrands f : [0, ∞) × R1 × R1 → R1 deﬁned in Section 2.1 and a Gδ -subset F ⊂ M constructed in Section 2.3. Deﬁne by F0 the set of all integrands g ∈ M for which the conclusions of Theorems 2.1.1-2.1.4 are valid. Clearly the set F is everywhere dense in M and F ⊂ F0 . Example 6.1. Consider an integrand f (t, x, u) = x2 + u2 , t, x, u ∈ R1 . It is easy to see that f ∈ M. Applying the methods used in the proofs of Theorems 2.1.1-2.1.4 we can show that f ∈ F0 . Example 6.2. Fix a number q > 0 and consider an integrand g(t, x, u) = qx2 (x − 1)2 + u2 , t, x, u ∈ R1 . It is easy to see that g ∈ M if a is large enough. Clearly the function v1 (t) = 0, v2 (t) = 1, t ∈ [0, ∞) are (g)-overtaking optimal. Assertion 1 of Theorem 2.1.1 implies that g ∈ F0 . It is easy to verify that f, g ∈ Mk for any integer k ≥ 1.

Chapter 3 EXTREMALS OF AUTONOMOUS PROBLEMS

In this chapter we establish the turnpike property for autonomous variational problems with nonconvex integrands. For this class of integrands the “turnpike” is a compact subset of Rn . We consider the complete metric space of integrands M introduced in Section 2.1 and the subspace A ⊂ M of all integrands f ∈ M which do not depend on t. We establish the existence of a set F ⊂ A which is a countable intersection of open everywhere dense sets in A such that each f ∈ F has the turnpike property. Moreover we show that the turnpike property holds for approximate solutions of variational problems with a generic integrand f and that the turnpike phenomenon is stable under small perturbations of a generic integrand f .

3.1.

Main results

Let a > 0 be a constant and let ψ : [0, ∞) → [0, ∞) be an increasing function such that ψ(t) → +∞ as t → ∞. Denote by | · | the Euclidean norm in Rn and denote by A the set of continuous functions f : Rn × Rn → R1 which satisfy the following assumptions: A(i) for each x ∈ Rn the function f (x, ·) : Rn → R1 is convex; A(ii) f (x, u) ≥ max{ψ(|x|), ψ(|u|)|u|} − a for each (x, u) ∈ Rn × Rn ; A(iii) for each M, > 0 there exist Γ, δ > 0 such that if u1 , u2 , x1 , x2 ∈ n R satisfy |xi | ≤ M, |ui | ≥ Γ, i = 1, 2,

max{|x1 − x2 |, |u1 − u2 |} ≤ δ,

72

TURNPIKE PROPERTIES

then |f (x1 , u1 ) − f (x2 , u2 )| ≤ max{f (x1 , u1 ), f (x2 , u2 )}. Clearly, A is the set of all integrands f ∈ M which do not depend on t.

It is easy to show that an integrand f = f (x, u) ∈ C 1 (Rn × Rn ) belongs to A if f satisﬁes Assumptions A(i), A(ii) and also there exists an increasing function ψ0 : [0, ∞) → [0, ∞) such that sup{|∂f /∂x(x, u)|, |∂f /∂u(x, u)|} ≤ ψ0 (|x|)(1 + ψ(|u|)|u|)

for each x, u ∈ Rn . We consider the topological subspace A ⊂ M with the relative weak and strong topologies introduced in Section 2.1. Note that A is the closed subset of M with the weak topology. The strong topology is induced by the uniformity which is determined by the following base: Es () = {(f, g) ∈ A × A : |f (x, u) − g(x, u)| ≤ for each x, u ∈ Rn }, where > 0. It is easy to see that the space A with this uniformity is metrizable and complete. The weak topology is induced by the uniformity which is determined by the following base: E(N, , λ) = {(f, g) ∈ A × A : |f (x, u) − g(x, u)| ≤ for each x, u ∈ Rn satisfying |x|, |u| ≤ N, (|f (x, u)| + 1)(|g(x, u)| + 1)−1 ∈ [λ−1 , λ] for each x, u ∈ Rn satisfying |x| ≤ N }, where N > 0, > 0, λ > 1. Clearly, the space A with this uniformity is metrizable and complete (see Proposition 1.3.2). We consider functionals of the form f

I (T1 , T2 , x) =

T2 T1

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

(1.1)

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

(1.2)

73

Extremals of autonomous problems

is an a.c. function satisfying x(T1 ) = y, x(T2 ) = z}, σ f (T1 , T2 , y) = inf{U f (T1 , T2 , y, u) : u ∈ Rn }.

(1.3)

It is easy to see that −∞ < U f (T1 , T2 , y, z) < +∞ for each f ∈ A, each y, z ∈ Rn and all numbers T1 , T2 satisfying 0 ≤ T1 < T2 . Recall the deﬁnition of a good function given in Section 1.2 and the deﬁnition of an overtaking optimal function introduced in Section 1.1. Let f ∈ A. We say that an a.c. function x : [0, ∞) → Rn is an (f )-good function if for any a.c function y : [0, ∞) → Rn the function T → I f (0, T, y) − I f (0, T, x), T ∈ (0, ∞) is bounded from below. Let f ∈ A. We say that an a.c. function x : [0, ∞) → Rn is (f )overtaking optimal if T

lim sup T →∞

0

[f (x(t), x (t)) − f (y(t), y (t))]dt ≤ 0

for any a.c. function y : [0, ∞) → Rn satisfying y(0) = x(0). In this paper we employ the following weakened version of this criterion [12, 42, 88, 89]. Let f ∈ A. We say that an a.c. function x : [0, ∞) → Rn is (f )-weakly optimal if T

lim inf T →∞

0

[f (x(t), x (t)) − f (y(t), y (t))]dt ≤ 0

for any a.c. function y : [0, ∞) → Rn satisfying y(0) = x(0). Let f ∈ A. For any a.c. function x : [0, ∞) → Rn we set J(x) = lim inf T −1 I f (0, T, x). T →∞

(1.4)

Of special interest is the minimal long-run average cost growth rate µ(f ) = inf{J(x) : x : [0, ∞) → Rn is an a.c. function}.

(1.5)

Clearly −∞ < µ(f ) < +∞ and for every (f )-good function x : [0, ∞) → Rn , µ(f ) = J(x). (1.6) In Section 3.2 we will establish the following result. Proposition 3.1.1 For any a.c. function x : [0, ∞) → Rn either I f (0, T, x) − T µ(f ) → ∞ as T → ∞

74 or

TURNPIKE PROPERTIES

sup{|I f (0, T, x) − T µ(f )| : T ∈ (0, ∞)} < ∞.

(1.7)

Moreover (1.7) holds if and only if x is an (f )-good function. We denote d(x, B) = inf{|x − y| : y ∈ B} for x ∈ Rn , B ⊂ Rn . Denote by dist(A, B) the Hausdorﬀ metric for two sets A ⊂ Rn , B ⊂ Rn and denote by Card(A) the cardinality of a set A. For every bounded a.c. function x : [0, ∞) → Rn deﬁne Ω(x) = {y ∈ Rn : there exists a sequence {ti }∞ i=0 ⊂ (0, ∞) for which ti → ∞, x(ti ) → y as i → ∞}.

(1.8)

We say that an integrand f ∈ A has the asymptotic turnpike property, or brieﬂy (ATP), if Ω(v1 ) = Ω(v2 ) for each pair of (f )-good functions vi : [0, ∞) → Rn , i = 1, 2. Let f ∈ A have the asymptotic turnpike property. Put H(F ) = Ω(v)

(1.9)

where v : [0, ∞) → Rn is an (f )-good function. Clearly, H(f ) does not depend on v. By Theorem 1.2.1, H(f) is a compact subset of Rn . We say that H(f ) is the turnpike of f . In this chapter we prove the following results. Theorem 3.1.1 There exists a set F ⊂ A which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets of A such that each f ∈ F has the asymptotic turnpike property. Theorem 3.1.1 describes the limit behavior of (f )-good functions for a generic f ∈ A. The following result establishes the existence of an (f )-weakly optimal function for each f ∈ A which has (ATP) and for each initial state x ∈ Rn . Theorem 3.1.2 Assume that f ∈ A has the asymptotic turnpike property. Then for each x ∈ Rn there exists an (f )-weakly optimal function X : [0, ∞) → Rn satisfying X(0) = x. It follows from Theorems 3.1.1 and 3.1.2 that for a generic f ∈ A and every x ∈ Rn there exists an (f )-weakly optimal function X : [0, ∞) → Rn satisfying X(0) = x. Theorem 3.1.3 Assume that f ∈ A has (ATP) and H(f ) is the turnpike of f . Let be a positive number. Then there exist an integer L ≥ 1

Extremals of autonomous problems

75

and a neighborhood U of f in A with the weak topology such that the following property holds: If g ∈ U and if v : [0, ∞) → Rn is a (g)-good function, then dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ for all large T. Theorem 3.1.4 Assume that f ∈ A has (ATP) and H(f ) ⊂ Rn is the turnpike of f . Let M0 , M1 , > 0. Then there exists a neighborhood U of f in A with the weak topology, numbers l, S > 0 and integers L,∗ Q ≥ 1 such that for each g ∈ U, each pair of numbers T1 ∈ [0, ∞), T2 ∈ [T1 + L + lQ∗ , ∞) and each a.c. function v : [T1 , T2 ] → Rn which satisﬁes |v(Ti )| ≤ M1 , i = 1, 2,

I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + M0

the following properties hold: |v(t)| ≤ S for all t ∈ [T1 , T2 ]; Q there exist sequences of numbers {bi }Q i=1 , {ci }i=1 ⊂ [T1 , T2 ] such that

Q ≤ Q∗ , 0 ≤ ci − bi ≤ l, i = 1, . . . , Q, dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ for each T ∈ [T1 , T2 − L] \ ∪Q i=1 [bi , ci ]. Theorem 3.1.4 shows that if an integrand f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn , then for any ﬁnite horizon problem the turnpike property holds with the set H(f ) being the attractor. Let k ≥ 1 be an integer. Denote by Ak the space of all integrands f ∈ A ∩ C k (R2n ). For p = (p1 , . . . , p2n ) ∈ {0, . . . , k}2n and f ∈ C k (R2n ) we set |p| =

2n i=1

p2n pi , Dp f = ∂ |p| f /∂y1p1 . . . ∂y2n .

For the set Ak we consider the uniformity which is determined by the following base: E(N, , λ) = {(f, g) ∈ Ak × Ak : |Dp f (x, u) − Dp g(x, u)| ≤ for each (x, u) ∈ R2n satisfying |x|, |u| ≤ N and each p ∈ {0, . . . , k}2n such that |p| ≤ k, |f (x, u) − g(x, u)| ≤ for each x, u ∈ Rn satisfying |x|, |u| ≤ N,

76

TURNPIKE PROPERTIES

(|f (x, u)| + 1)(|g(x, u)| + 1)−1 ∈ [λ−1 , λ] for each x, u ∈ Rn such that |x| ≤ N }, where N > 0, > 0, λ > 1. Clearly the uniform space Ak is Hausdorﬀ and has a countable base. Therefore Ak is metrizable [37]. It is easy to verify that the uniform space Ak is complete. We establish the following result. Theorem 3.1.5 Let k ≥ 1 be an integer. Then there exists a set Fk ⊂ Ak which is a countable intersection of open everywhere dense subsets of Ak such that each f ∈ Fk has the asymptotic turnpike property. The results of this chapter have been established in [93]. Note that the turnpike result of this chapter (Theorem 3.1.4) is weaker than the turnpike result of Chapter 2 (Theorem 2.1.4). In Chapter 2 for nonautonomous integrands we obtained that an approximate solution v on an interval [T1 , T2 ] is close to the turnpike Z f for most points t ∈ [T1 , T2 ]. In this chapter for autonomous integrands we obtain only that the image v([T, T + L]) of the interval [T, T + L] ⊂ [T1 , T2 ] is close to the turnpike H(f ) for most points T ∈ [T1 , T2 ]. (Here L > 0 is a constant.) Moreover, for the nonautonomous case the set of all bad points is contained in the union of two intervals [T1 , T1 + L0 ] and [T2 − L0 , T2 ] where L0 is a positive constant. In the autonomous case considered in this chapter the set of all bad points of the interval [T1 , T2 ] is contained in the union of a ﬁnite number of closed intervals and we can only say that the length of each of these intervals and their number are bounded by a constant which do not depend on [T1 , T2 ] and v(T1 ), v(T2 ). Chapter 3 is organized as follows. Proposition 3.1.1 is proved in Section 3.2. In Section 3.3 we will establish a weakened version of Theorem 3.1.3. The continuity of the function U f (T1 , T2 , x, y) is studied in Section 3.4. Section 3.5 contains some useful results on discrete-time control systems while Theorem 3.1.2 is proved in Section 3.6. Section 3.7 contains auxiliary results for Theorem 3.1.1 while Section 3.8 contains auxiliary results for Theorems 3.1.3 and 3.1.4. Theorem 3.1.4 is proved in Section 3.9 and Theorem 3.1.3 is proved in Section 3.10. Section 3.11 contains the proofs of Theorems 3.1.1 and 3.1.5. Certain examples are given in Section 3.12.

3.2.

Proof of Proposition 3.1.1

In the sequel we associate with any f ∈ A a related discrete-time control system. We need the following result established in [39] (see also [16]) for such discrete-time control systems.

77

Extremals of autonomous problems

Proposition 3.2.1 Let K ⊂ Rn be a compact set, v : K × K → R1 be a continuous function and deﬁne

µ(v) = inf lim inf N

−1

N −1

N →∞

v(zi , zi+1 ) :

{zi }∞ i=0

⊂K ,

i=0

π v (x)

= inf lim inf N −1 N →∞

N −1

[v(zi , zi+1 ) − µ(v)] : {zi }∞ i=0 ⊂ K, z0 = x ,

i=0 v

θ (x, y) = v(x, y) − µ(v) + π v (y) − π v (x) for x, y ∈ K. Then π v : K → R1 , θv : K × K → R1 are continuous functions, θv is nonnegative and E(x) = {y ∈ K : θv (x, y) = 0} is nonempty for every x ∈ K. Proof of Proposition 3.1.1. Let f ∈ A, z ∈ Rn and let an a.c. function : [0, ∞) → Rn be as guaranteed in Theorem 1.2.1. It follows from Theorem 1.2.1, (1.4), (1.6) and assumption A(ii) that Zf

µ(f ) = lim inf N −1 I f (0, N, Z f ) N →∞

where N is an integer. Combined with Theorem 1.2.1 this equality implies that µ(f ) = lim inf N −1 N →∞

N −1

U f (i, i + 1, Z f (i), Z f (i + 1)).

(2.1)

i=0

It follows from Theorems 1.2.1 and 1.2.2 that there is a number Q > sup{|Z f (t)| : t ∈ [0, ∞)}

(2.2)

lim sup |x(t)| < Q

(2.3)

such that t→∞

for each (f )-good function x : [0, ∞) → Rn . Put BQ = {x ∈ Rn : |x| ≤ Q}, v(x, y) = U f (0, 1, x, y) (x, y ∈ BQ ).

(2.4)

It is easy to see that for each x, y ∈ Rn and each integer i ≥ 0, U f (i, i + 1, x, y) = U f (0, 1, x, y).

(2.5)

78

TURNPIKE PROPERTIES

Proposition 1.3.7 implies that the function v : BQ × BQ → R1 is continuous. We show that N −1

sup |

f

f

[v(Z (i), Z (i + 1)) − µ(f )]| : N = 1, 2, . . .

< ∞.

(2.6)

i=0

Fix an integer N ≥ 2. By Proposition 3.2.1 there exists a sequence {xj }N j=0 ⊂ BQ such that x0 = z, xN = Z f (N ), θv (xi , xi+1 ) = 0, (0 ≤ i < N − 1).

(2.7)

Proposition 3.2.1 implies that N −1

[v(xi , xi+1 ) − µ(v)] = π v (z) − π v (xN −1 ) + v(xN −1 , xN ) − µ(v). (2.8)

i=0

By (2.2) and Proposition 3.2.1, N −1

[v(Z f (i), Z f (i + 1)) − µ(v)] ≥ π v (z) − π v (Z f (N )).

(2.9)

i=0

On the other hand it follows from Assertion 4 of Theorem 1.2.1, (2.4), (2.5), (2.7), and Corollary 1.3.1 that N −1

v(Z f (i), Z f (i+1)) =

i=0

N −1

U f (i, i+1, Z f (i), Z f (i+1)) = I f (0, N, Z f )

i=0

(2.10) ≤

N −1

U f (i, i + 1, xi , xi+1 ) =

i=0

N −1

v(xi , xi+1 ).

i=0

Combining (2.8)-(2.10) we obtain that for any integer N ≥ 2, |

N −1

[v(Z f (i), Z f (i + 1)) − µ(v)]| ≤ 2 sup{|π v (y)| : y ∈ BQ }

(2.11)

i=0

+2 sup{|v(x, y)| : x, y ∈ BQ }. By (2.1), (2.4), (2.5) and (2.11), µ(f ) = µ(v).

(2.12)

Combined with (2.11) this equality implies (2.6). We show that sup{|I f (0, T, Z f ) − T µ(f )| : T ∈ (0, ∞)} < ∞.

(2.13)

Extremals of autonomous problems

79

Let T > 0. There exists an integer N ≥ 0 such that N ≤ T < N + 1. By Assumption A(ii), I f (T, N + 1, Z f ) ≥ −a.

(2.14)

On the other hand it follows from (2.2), (2.4), (2.5), Theorem 1.2.1 and Assumption A(ii) that I f (T, N + 1, Z f ) = I f (N, N + 1, Z f ) − I f (N, T, Z f ) ≤ v(Z f (N ), Z f (N + 1)) + a. By these relations, Theorem 1.2.1, (2.4), (2.5) and (2.14) |I f (0, T, Z f ) − T µ(f )| ≤ |I f (0, N + 1, Z f ) − (N + 1)µ(f )| +|I f (T, N + 1, Z f ) − (N + 1 − T )µ(f )| ≤|

N

[v(Z f (i), Z f (i+1))−µ(f )]|+|µ(f )|+a+sup{|v(x, y)| : x, y ∈ BQ }.

i=0

Together with (2.6) this implies (2.13). Proposition 3.1.1 now follows from (2.13) and Theorem 1.2.1.

3.3.

Weakened version of Theorem 3.1.3

The proof of Theorem 3.1.3 is diﬃcult and it is based on a number of auxiliary results. Now we are ready to prove its weakened version which establishes the convergence property of Theorem 3.1.3 when g = f . We begin this section with the following useful property of good functions. Proposition 3.3.1 Let g ∈ A and let y : [0, ∞) → Rn be a (g)-good function. Then for each > 0 there exists T0 > 0 such that the following property holds: If T ≥ T0 and T¯ > T , then I g (T, T¯, y) ≤ U g (T, T¯, y(T ), y(T¯)) + . Proof. Let us assume the converse. Then there exist > 0 and ¯ ∞ sequences {Ti }∞ i=1 , {Ti }i=1 ⊂ (0, ∞) such that for each natural number i, Ti < T¯i < Ti+1 , I g (Ti , T¯i , y) > U g (Ti , T¯i , y(Ti ), y(T¯i )) + .

80

TURNPIKE PROPERTIES

Put T¯0 = 0. It follows from Corollary 1.3.1 that there is an a.c. function x : [0, ∞) → Rn such that for each natural number i, x(t) = y(t), t ∈ [T¯i , Ti+1 ], I g (Ti , T¯i , x) = U g (Ti , T¯i , x(Ti ), x(T¯i )). It is not diﬃcult to verify that I g (0, T¯i , y) − I g (0, T¯i , x) → ∞ as i → ∞. Therefore y is not a (g)-good function. reached proves the proposition.

The contradiction we have

We will use the next auxiliary result in the proof of the weakened version of Theorem 3.1.3 in order to show that a certain function is good. Proposition 3.3.2 Let f ∈ A and let x : [0, ∞) → Rn be an a.c. function such that sup{|x(t)| : t ∈ [0, ∞)} < ∞ and

sup{I f (0, i, x) − U f (0, i, x(0), x(i)), i = 1, 2, . . .} < ∞.

Then x is an (f )-good function. Proof. Choose positive numbers S0 and S1 such that |x(t)| ≤ S0 for all t ∈ [0, ∞) and

I f (0, i, x) ≤ U f (0, i, x(0), x(i)) + S1 , i = 1, 2, . . . .

Fix z ∈ Rn satisfying |z| ≤ S0 and let an a.c. function Z f : [0, ∞) → Rn be as guaranteed in Theorem 1.2.1. It follows from Theorems 1.2.1 and 1.2.2 that there exists a number Q > sup{|Z f (t)| : t ∈ [0, ∞)} + S0 such that for each (f )-good function y : [0, ∞) → Rn , lim sup |y(t)| < Q. t→∞

Put

BQ = {y ∈ Rn : |y| ≤ Q},

81

Extremals of autonomous problems

v(y1 , y2 ) = U f (0, 1, y1 , y2 ),

y1 , y2 ∈ BQ .

By Proposition 1.3.7 the function v : BQ × BQ → R1 is continuous. It was shown in Section 3.2 that µ(f ) = µ(v). Fix an integer N ≥ 2. By Proposition 3.2.1 there exists a sequence {yi }N i=0 ⊂ BQ such that y0 = x(0), θv (yi , yi+1 ) = 0 (0 ≤ i < N − 1) , yN = x(N ). It follows from these relations, the conditions of the proposition, Proposition 3.2.1 and Corollary 1.3.1 that N −1

[v(yi , yi+1 ) − µ(v)] = π v (x(0)) − π v (yN −1 ) + v(yN −1 , x(N )) − µ(v),

i=0

I f (0, N, x) − N µ(f ) ≤ U f (0, N, x(0), x(N )) + S1 − N µ(v) ≤

N −1

[v(yi , yi+1 ) − µ(v)] + S1

i=0

≤ S1 + 2 sup{|π v (h)| : h ∈ BQ } + 2 sup{|v(h1 , h2 )| : h1 , h2 ∈ BQ }. It follows from these relations and Proposition 3.1.1 that x is an (f )-good function. The proposition is proved. The following result is a weakened version of Theorem 3.1.3. Theorem 3.3.1 Assume that f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Then for each > 0 there exists an integer L ≥ 1 such that the following property holds: If v : [0, ∞) → Rn is an (f )-good function, then dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ for all large T. Proof. Let be a positive number. Assume that the assertion of the theorem does not hold. Then for every integer N ≥ 1 there exists an (f )-good function xN : [0, ∞) → Rn such that lim sup dist(H(f ), {xN (t) : t ∈ [T, T + N ]}) ≥ . T →∞

(3.1)

It follows from Theorem 1.2.2 that there exists M0 > 0 such that lim sup |x(t)| < M0 t→∞

(3.2)

82

TURNPIKE PROPERTIES

for each (f )-good function x : [0, ∞) → Rn . By the deﬁnition of M0 , Propositions 3.1.1 and 3.3.1 we may assume that (3.3) |xN (t)| < M0 , (t ∈ [0, ∞), N = 1, 2, . . .) and that for each integer N ≥ 1 and each pair of numbers T1 ≥ 0, T2 > T1 , I f (T1 , T2 , xN ) ≤ U f (T1 , T2 , xN (T1 ), xN (T2 )) + N −1 .

(3.4)

It is not diﬃcult to see that if x : [0, ∞) → Rn is an (f )-good function and γ is the positive number, then d(x(t), H(f )) ≤ γ for all large t. Therefore we may assume that d(xN (t), H(f )) ≤ 4−1 (t ∈ [0, ∞), N = 1, 2, . . .).

(3.5)

Let N ≥ 4 be an integer. By (3.1) there exists TN ∈ [0, ∞) such that dist(H(f ), {xN (t) : t ∈ [TN , TN + N ]}) ≥ 7 · 8−1 .

(3.6)

By (3.5) and (3.6) there exists hN ∈ H(f ) for which d(hN , {xN (t) : t ∈ [TN , TN + N ]}) ≤ 2−1 .

(3.7)

Fix an integer jN ≥ 0 for which jN ≤ TN < jN + 1 and put vN (t) = xN (t + 1 + jN )

(t ∈ [0, N − 2]).

(3.8)

It follows from the inequalities (3.3), (3.4) and (3.7) that |vN (t)| < M0

(t ∈ [0, N − 2]),

I f (T1 , T2 , vN ) ≤ U f (T1 , T2 , vN (T1 ), vN (T2 )) + N −1

(3.9) (3.10)

for each T1 ∈ [0, N − 2), T2 ∈ (T1 , N − 2] and that d(hN , {vN (t) : t ∈ [0, N − 2]}) ≥ 2−1 .

(3.11)

Fix an integer j ≥ 1. By (3.9), (3.10) and Proposition 1.3.7, sup{I f (0, j, vN ) : N is an integer, N > 4 + j} ≤ sup{U f (0, j, vN (0), vN (j)) : N is an integer, N ≥ 4 + j} + 1 < ∞. By this inequality and Proposition 1.3.5 there exist an a.c. function v∗ : [0, ∞) → Rn and a subsequence {vNi }∞ i=1 such that for any integer j ≥ 1, (3.12) vNk (t) → v∗ (t) as k → ∞ uniformly in [0, j],

83

Extremals of autonomous problems vN → v∗ as k → ∞ weakly in L1 (Rn ; (0, j)) k

and

I f (0, j, v∗ ) ≤ lim inf I f (0, j, vNk ). k→∞

(3.13)

(3.9) and (3.12) imply that |v∗ (t)| ≤ M0 for all t ∈ [0, ∞). It follows from Proposition 1.3.7, (3.12), (3.13) and (3.10) that for any integer j ≥ 1, I f (0, j, v∗ ) = U f (0, j, v∗ (0), v∗ (j)). Combined with Proposition 3.3.2 this equality implies that v∗ is an (f )good function. We may assume without loss of generality that there exists h∗ = lim hNk ∈ H(f ). k→∞

(3.14)

Since v∗ is an (f )-good function we have Ω(v∗ ) = H(f ). Therefore there exists a sequence {ti }∞ i=1 ⊂ (0, ∞) such that ti → ∞ and v∗ (ti ) → h∗ as i → ∞. On the other hand it follows from (3.11), (3.12), (3.14) that d(h∗ , {v∗ (t) : t ∈ [0, ∞)}) ≥ 2−1 . The contradiction we have reached proves the theorem.

3.4.

Continuity of the function U f (T1 , T2 , x, y)

Theorem 3.4.1 Assume that f ∈ A. Then the mapping (T1 , T2 , x, y) → U f (T1 , T2 , x, y) is continuous for T1 ∈ [0, ∞), T2 ∈ (T1 , ∞), x, y ∈ Rn . Proof. Lower semicontinuity. Let x, y ∈ Rn , T1 ∈ [0, ∞), T2 ∈ ∞ n ∞ ∞ (T1 , ∞), {xi }∞ i=1 , {yi }i=1 ⊂ R , {T1i }i=1 , {T2i }i=1 ⊂ [0, ∞), T2i > T1i (i = 1, 2, . . .), xi → x, yi → y, T1i → T1 , T2i → T2 as i → ∞.

(4.1)

By Corollary 1.3.1 for each integer i ≥ 1 there exists an a.c. function zi : [T1i , T2i ] → Rn such that zi (T1i ) = xi , zi (T2i ) = yi , I f (T1i , T2i , zi ) = U f (T1i , T2i , xi , yi ).

(4.2)

84

TURNPIKE PROPERTIES

By Proposition 1.3.6 and (4.1), sup{|U f (T1i , T2i , xi , yi )| : i = 1, 2, . . .} < ∞.

(4.3)

We may assume that there exists limi→∞ U f (T1i , T2i , xi , yi ). Fix δ ∈ (0, 1) and set (4.4) T1 (δ) = sup{0, T1 − δ}. There exists a natural number N (δ) such that T1 + δ ≥ T1i ≥ T1 (δ), T2 − δ ≤ T2i ≤ T2 + δ for all integers i ≥ N (δ). (4.5) Set γ = sup{|f (h, 0)| : h ∈ Rn , |h| ≤ sup{|xi | + |yi | : i = 1, 2, . . .}}. (4.6) For every integer i ≥ N (δ) we deﬁne an a.c. function zδi : [T1 (δ), T2 + δ] → Rn as follows: zδ1 (t) = zi (T1i ), t ∈ [T1 (δ), T1i ], zδi (t) = zi (t), t ∈ [T1i , T2i ], zδi (t) = zi (T2i ), t ∈ [T2i , T2 + δ].

(4.7)

It follows from the relations (4.2), (4.3), (4.5) and (4.7) that the sequence {I f (T1 (δ), T2 +δ, zδi )}∞ i=N (δ) is bounded. By Proposition 1.3.5 there exist an a.c. function zδ : [T1 (δ), T2 + δ] → Rn and a subsequence {zδik }∞ k=1 (i1 ≥ N (δ)) such that zδik (t) → zδ (t) as k → ∞ uniformly in [T1 (δ), T2 + δ], zδi → zδ as k → ∞ weakly in L1 (Rn ; (T1 , T2 )), k

I f (T1 (δ), T2 + δ, zδ ) ≤ lim inf I f (T1 (δ), T2 + δ, zδik ). k→∞

(4.8)

By (4.6)-(4.8) and (4.2) for any integer k ≥ 1, I f (T1 (δ), T2 + δ, zδik ) = I f (T1 (δ), T1ik , zδik ) + I f (T1ik , T2ik , zδik ) +I f (T2ik , T2 + δ, zδik ) ≤ (T1ik − T1 (δ))γ + U f (T1ik , T2ik , xik , yik ) +(T2 + δ − T2ik )γ ≤ U f (T1ik , T2ik , xik , yik ) + 4γδ. By (4.8) and (4.9), I f (T1 (δ), T2 + δ, zδ ) ≤ lim U f (T1i , T2i , xi , yi ) + 4γδ. i→∞

By this relation, (4.4) and Assumption A(ii), I f (T1 , T2 , zδ ) ≤ lim U f (T1i , T2i , xi , yi ) + 4γδ + 2aδ. i→∞

(4.9)

85

Extremals of autonomous problems

To complete the proof of the lower semicontinuity it is suﬃcient to show that (4.10) Zδ (T1 ) = x, zδ (T2 ) = y. By (4.8), zδik (T1 ) → zδ (T1 ), zδik (T2 ) → zδ (T2 ) as k → ∞.

(4.11)

It follows from (4.1), (4.3), (4.9) and Proposition 1.3.4 that zδik (T1 ) − zδik (T1ik ) → 0, zδik (T2 ) − zδik (T2ik ) → 0 as k → ∞. Together with (4.11), (4.7), (4.2) and (4.1) this relation implies (4.10). The lower semicontinuity is proved. Upper semicontinuity. Let x, y ∈ Rn , T1 ∈ [0, ∞), T2 ∈ (T1 , ∞), ∞ n {xi }∞ i=1 , {yi }i=1 ⊂ R , ∞ {T1i }∞ i=1 , {T2i }i=1 ⊂ [0, ∞),

T2i > T1i (i = 1, 2, . . .), xi → x, yi → y, T1i → T1 , T2i → T2 as i → ∞.

(4.12)

By Corollary 1.3.1 there exists an a.c. function z : [T1 , T2 ] → Rn such that z(T1 ) = x, x(T2 ) = y, I f (T1 , T2 , z) = U f (T1 , T2 , x, y).

(4.13)

Fix δ ∈ (0, 1) and deﬁne γ, T1 (δ) by (4.6) and (4.4). There is an integer N (δ) ≥ 1 satisfying (4.5). Deﬁne an a.c. function zδ : [T1 (δ), T2 + δ] → Rn as follows: zδ (t) = x (t ∈ [T1 (δ), T1 ]), zδ (t) = z(t) (t ∈ [T1 , T2 ]), zδ (t) = y (t ∈ [T2 , T2 + δ]).

(4.14)

For an integer i ≥ N (δ) we set bi = (T2i − T1i )−1 [yi + zδ (T1i ) − xi − zδ (T2i )], ai = xi − zδ (T1i ) − bi T1i , zδi (t) = zδ (t) + ai + bi t (t ∈ [T1 (δ), T2 + δ]).

(4.15)

Clearly zδi (T1i ) = xi , zδi (T2i ) = yi , (i is an integer, i ≥ N (δ)), ai → 0, bi → 0 as i → ∞.

(4.16) (4.17)

86

TURNPIKE PROPERTIES

We will show that

I f (T1 (δ), T2 + δ, zδi ) → I f (T1 (δ), T2 + δ, zδ ) as i → ∞.

(4.18)

It is easy to see that |I f (T1 (δ), T2 + δ, zδ )| < ∞. Set S0 = sup{|zδ (t)| : t ∈ [T1 (δ), T2 + δ]}.

(4.19)

Fix > 0. There exists a number ∆ ∈ (0, 1) such that ∆(|I f (T1 (δ), T2 + δ, zδ )| + a(T2 − T1 + 2δ)) < 8−1

(4.20)

(recall a in Assumption A(ii)). By Proposition 1.3.1 there exist Γ0 > 0, δ0 ∈ (0, 1) such that the following property holds: If u1 , u2 , h1 , h2 ∈ Rn satisfy |hi | ≤ S0 + 4, |ui | ≥ Γ0 (i = 1, 2), |u1 − u2 |, |h1 − h2 | ≤ δ0 ,

(4.21)

then |f (h1 , u1 ) − f (h2 , u2 )| ≤ ∆ min{f (h1 , u1 ), f (h2 , u2 )}.

(4.22)

Since the function f is continuous there exists δ1 ∈ (0, δ0 ) such that the following property holds: If h1 , h2 , u1 , u2 ∈ Rn satisfy |hi |, |ui | ≤ Γ0 + S0 + 6, i = 1, 2, max{|h1 − h2 |, |u1 − u2 |} ≤ δ1 , (4.23) then

|f (h1 , u1 ) − f (h2 , u2 )| ≤ [8(T2 − T1 + 2)]−1 .

(4.24)

It follows from (4.17) that there is an integer N1 > N (δ) such that |bi | ≤ 2−1 δ1 , |ai + bi t| ≤ 2−1 δ1 (t ∈ [T1 (δ), T2 + δ])

(4.25)

for any integer i ≥ N1 . Assume that an integer i ≥ N1 and estimate I f (T1 (δ), T2 + δ, zδi ) − I f (T1 (δ), T2 + δ, zδ ). Put E1 = {t ∈ [T1 (δ), T2 + δ] : |zδ (t)| ≥ Γ0 + 1}, E2 = [T1 (δ), T2 + δ] \ E1 . (4.26) Clearly |I f (T1 (δ), T2 + δ, zδi ) − I f (T1 (δ), T2 + δ, zδ )| = σ1 + σ2

(4.27)

87

Extremals of autonomous problems

with

σi =

Ei

|f (zδ (t), zδ (t)) − f (zδi (t), zδi (t))|dt, i = 1, 2.

(4.28)

By (4.26), (4.19), (4.25), (4.15) and the deﬁnition of Γ0 , δ0 (see (4.21), (4.22)) for each t ∈ E1 , (t))| ≤ ∆f (zδ (t), zδ (t)). |f (zδ (t), zδ (t)) − f (zδi (t), zδi

Combined with (4.20) and Assumption A(ii) this inequality implies that σ1 ≤ ∆

E1

f (zδ (t), zδ (t))dt

≤ ∆(I f (T1 (δ), T2 + δ, zδ ) + a(T2 − T1 + 2δ)) ≤ 8−1 .

(4.29)

It follows from (4.26), (4.19), (4.25), (4.15) and the deﬁnition of δ1 (see (4.23), (4.24)) that |f (zδ (t), zδ (t)) − f (zδi (t), zδi (t))| ≤ [8(T2 − T1 + 2)]−1

for each t ∈ E2 and

σ2 ≤ 8−1 .

(4.30)

By (4.27), (4.29) and (4.30) for each integer i ≥ N1 , |I f (T1 (δ), T2 + δ, zδ ) − I f (T1 (δ), T2 + δ, zδi )| ≤ . Therefore we have proved (4.18). By (4.16), (4.4), (4.5), Assumption A(ii) and (4.18) for any integer i ≥ N (δ), U f (T1i , T2i , xi , yi ) ≤ I f (T1i , T2i , zδi ) ≤ I f (T1 (δ).T2 + δ, zδi ) + 4δa → I f (T1 (δ), T2 + δ, zδ ) + 4aδ as i → ∞. Together with (4.14), (4.4), (4.6) and (4.13) this implies that lim sup U f (T1i , T2i , xi , yi ) ≤ I f (T1 (δ), T2 + δ, zδ ) + 4aδ i→∞

≤ I f (T1 , T2 , z) + 4aδ + 2δγ ≤ 4aδ + 2δγ + U f (T1 , T2 , x, y). This completes the proof of the upper semicontinuity. The theorem is proved.

88

TURNPIKE PROPERTIES

3.5.

Discrete-time control systems

In the sequel we associate with any f ∈ A a related discrete-time control system. In this section we establish some useful properties of such systems. Consider a continuous function v : Rn × Rn → R1 satisfying v(x, y) → ∞ as |x| + |y| → ∞. We have the following result [39]. Proposition 3.5.1 Given a compact set C ⊂ Rn there is a ball B ⊂ Rn n such that for every sequence {zk }∞ k=0 ⊂ R not included in B with z0 ∈ C ∞ there exists a sequence {sk }k=1 ⊂ B with s0 = z0 such that N

v(sk , sk+1 ) <

k=0

N

v(zk , zk+1 ) for all large N.

k=0

Let x ∈ Rn . Deﬁne

µ(v) = inf lim inf N −1 N →∞

Clearly µ(v) ∈

R1

N −1

n v(zk , zk+1 ) : {zk }∞ k=0 ⊂ R , z0 = x .

k=0

and is independent of x. For x, y ∈

v

π (x) = inf lim inf N →∞

(5.1)

Rn

we set

N −1

[v(zk , zk+1 ) − µ(v)] :

{zk }∞ k=0

n

⊂ R , z0 = x ,

k=0

(5.2) (5.3)

θv (x, y) = v(x, y) − µ(v) + π v (y) − π v (x). For a compact set C ⊂ Rn we deﬁne v C : C × C → R1 by v C (x, y) = v(x, y), x, y ∈ C. Propositions 3.2.1 and 3.5.1 imply the following result.

Proposition 3.5.2 1. Given a compact set C ⊂ Rn there is a ball B ⊃ C such that B

B

µ(v B ) = µ(v), π v (x) = π v (x), θv (x, y) = θv (x, y) B

π v (x) ≥ π v (x) (x ∈ B),

(x, y ∈ C).

B

θv (x, y) ≥ θv (x, y) (x ∈ C, y ∈ B); B

for every x ∈ C there exists y ∈ B satisfying θv (x, y) = θv (x, y) = 0. 2. π v , θv are continuous functions, θv is nonnegative.

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Extremals of autonomous problems

Proposition 3.5.3 π v (x) → ∞ as |x| → ∞. Proof. There is a number c0 > 0 such that inf{v(z, y) : z, y ∈ Rn , |z| + |y| ≥ c0 } ≥ 4|µ(v)| + 4. Let

n x ∈ Rn , {zk }∞ k=0 ⊂ R , z0 = x,

lim inf N →∞

N

[v(zk , zk+1 ) − µ(v)] ≤ π v (x) + 1.

k=0

It is easy to see that the set E := {k ∈ {1, 2, . . .} : |zk | ≤ c0 } is nonempty. Denote by m the minimal element of the set E. We have π v (x) + 1 ≥ lim inf N →∞

≥

N

[v(zk , zk+1 ) − µ(v)]

k=0

m−1

[v(zk , zk+1 ) − µ(v)]

k=0

+π v (zm ) ≥ v(x, z1 ) − µ(v) + inf{π v (y) : y ∈ Rn , |y| ≤ c0 } → ∞ as |x| → ∞. This completes the proof of the proposition. n Proposition 3.5.4 Let {xk }∞ k=1 ⊂ R . Then either N

[v(xk , xk+1 ) − µ(v)] → ∞ as N → ∞

k=0

or the sequences {xk }∞ k=1 and {

N

k=0

[v(xk , xk+1 ) − µ(v)]}∞ N =0

are bounded. Proof. Assume that the sequence {

N

k=0

[v(xk , xk+1 ) − µ(v)]}∞ N =1

90

TURNPIKE PROPERTIES

does not tend to +∞. Then for every integer k ≥ 1, ∞ > lim inf N →∞

N

[v(xq , xq+1 ) − µ(v)] ≥

q=0

[v(xq , xq+1 ) − µ(v)] + π v (xk )

q=0

≥ π v (x0 ) + ∞

k−1

k−1

θv (xq , xq+1 ).

q=0

Hence q=0 θv (xq , xq+1 ) < ∞. The validity of the proposition follows from this inequality and Proposition 3.5.3.

3.6.

Proof of Theorem 3.1.2

In this section we associate with any f ∈ A a related discrete-time control system. There is a simple correspondence between solutions of variational problems with the integrand f and solutions of this control system. Let f ∈ A. It is easy to see that U f (t, t + T, x, y) = U f (0, T, x, y) for each x, y ∈ Rn , T, t ∈ (0, ∞). For every T > 0 deﬁne a function vTf : Rn × Rn → R1 as follows: vTf (x, y) = U f (0, T, x, y) (x, y ∈ Rn ).

(6.1)

Theorem 3.4.1 and Proposition 1.3.3 imply that for any T > 0 the function vTf is continuous and vTf (x, y) → ∞ as |x| + |y| → ∞. We deﬁne µfT = T −1 µ(u), πTf = π u , θTf = θu

(6.2)

where u = vTf (see (5.1-5.3)). We consider a discrete-time control system studied in Section 3.5 with the cost function vTf : Rn × Rn → R1 , where T > 0. Theorem 3.1.2 will be proved in the following way. We ﬁx T > 0 and obtain an optimal n solution {xi }∞ i=0 ⊂ R of the discrete-time control system. Then the function v : [0, ∞) → Rn which satisﬁes v(iT ) = xi , I f (iT, (i + 1)T, v) = U f (0, 1, xi , xi+1 ), i = 0, 1, . . . will be an (f )-weakly optimal function. We begin with the following auxiliary result which establishes an important relation between the inﬁnite horizon variational problem with the integrand f and the inﬁnite horizon discrete-time control system with the cost function vTf .

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Extremals of autonomous problems

Proposition 3.6.1 µfτ = µ(f ) for every τ > 0 (recall µ(f ) in (1.5)). Proof. Let τ > 0. Consider any (f )-good function x : [0, ∞) → Rn . By Proposition 3.1.1 the function I f (0, T, x) − T µ(f ), T ∈ (0, ∞) is bounded. Then relations (5.1) and (6.2) imply that µfτ

≤τ

−1

lim inf N

−1

N −1

N →∞

≤ τ −1 lim inf N −1 N →∞

≤τ

N −1

vτf (x(iτ ), x((i + 1)τ ))

i=0

U f (iτ, (i + 1)τ, x(iτ ), x((i + 1)τ ))

i=0 −1

lim inf N −1 I f (0, N τ, x) ≤ µ(f ). N →∞

n By Proposition 3.5.2 there exists a sequence {yi }∞ i=0 ⊂ R such that f θτ (yi , yi+1 ) = 0, i = 0, 1, . . .. Propositions 3.5.3 and 3.5.4 imply that

N −1

sup |

[vτf (yi , yi+1 )

−

τ µfτ ]|

: N = 0, 1, . . .

< ∞.

(6.3)

i=0

By Corollary 1.3.1 there exists an a.c. function x : [0, ∞) → Rn such that x(iτ ) = yi , I f (iτ, (i + 1)τ, x) = U f (iτ, (i + 1)τ, yi , yi+1 ) = vτf (yi , yi+1 )

(i = 0, 1, . . .).

It follows from these relations, (1.4), (1.5) and (6.3) that µ(f ) ≤ lim inf T −1 I f (0, T, x) ≤ lim inf (τ N )−1 T →∞

N →∞

N −1

vτf (yi , yi+1 ) = µfτ .

i=0

Combined with (6.3) this inequality completes the proof of the proposition. Let m(ds) be the Lebesgue measure. The following result was established in [44]. Lemma 3.6.1 Let {dk }∞ k=1 be an increasing sequence of positive numbers such that dk → ∞ as k → ∞. Consider numbers T > 0 with the property that for every p ≥ 1, inf {dk − iT : k ≥ p, i ≥ 0, dk ≥ iT } = 0. k,i

(6.4)

Then there is a set D ⊂ [0, ∞) with m([0, ∞) \ D) = 0 such that every T ∈ D satisﬁes (6.4) for every integer p ≥ 1.

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By a simple modiﬁcation of the proof of Proposition 4.4 in [44] we can establish the following result which is a continuous-time version of Proposition 3.2.1. Theorem 3.6.1 There exist a continuous function π f : Rn → R1 , a continuous nonnegative function (T, x, y) → θ¯Tf (x, y) ∈ R1 deﬁned for T > 0, x, y ∈ Rn , and a set D ⊂ [0, ∞) with m([0, ∞) \ D) = 0 such that π f (x) = πTf (x) for every x ∈ Rn and every T ∈ D; π f (x) ≤ πTf (x) for every x ∈ Rn and every T > 0; U f (0, T, x, y) = T µ(f ) + π f (x) − π f (y) + θ¯Tf (x, y) for each x, y ∈ Rn and each T > 0; for every T > 0 and every x ∈ Rn there is y ∈ Rn satisfying θ¯Tf (x, y) = 0. For each f ∈ A, each pair of numbers T1 ≥ 0, T2 > T1 and each a.c. function v : [T1 , T2 ] → Rn put σ f (T1 , T2 , v) = I f (T1 , T2 , v) − (T2 − T1 )µ(f ) + π f (v(T2 )) − π f (v(T1 )), (6.5) f f Φ (T1 , T2 , v) = I (T1 , T2 , v) − (T2 − T1 )µ(f ). Theorem 3.6.2 For every x ∈ Rn , π f (x) = inf{lim inf [I f (0, T, v) − µ(f )T ] : v : [0, ∞) T →∞

→ Rn is an a.c. function satisfying v(0) = x}. Proof. Let x ∈ Rn and v : [0, ∞) → Rn be an a.c. function such that v(0) = x. By Theorem 3.6.1 and Proposition 3.6.1 we need to show that π f (x) ≤ lim inf Φf (0, T, v). T →∞

By Proposition 3.1.1 we may assume that v is an (f )-good function and that the function T → Φf (0, T, v), T ∈ (0, ∞) is bounded. Theorem 1.2.2 implies that the function v : [0, ∞) → Rn is bounded. There exists a sequence of positive numbers {dk }∞ k=1 such that dk+1 − dk ≥ 1 (k = 1, 2, . . .) and lim Φf (0, dk , v) = lim inf Φf (0, T, v).

k→∞

T →∞

(6.6)

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Extremals of autonomous problems

By Lemma 3.6.1 there is a set D ⊂ (0, ∞) with m([0, ∞) \ D) = 0 such that every T ∈ D satisﬁes (6.5) for every integer p ≥ 1. Let T ∈ D. It follows from the deﬁnition of D and (6.5) that there exists a sequence of positive integers Mj → ∞ as j → ∞ (j = 1, 2, . . .) and a subsequence {dkj }∞ j=1 such that dkj −1 < Mj T ≤ dkj (j = 1, 2, . . .), dkj − Mj T → 0 as j → ∞.

(6.7)

It follows from Assumption A(ii) and (6.7) that for any natural number j Φf (0, T Mj , v) − Φf (0, dkj , v) ≤ −I f (T Mj , dkj , v) +|µ(f )|(dkj − T Mj ) ≤ (dkj − T Mj )(|µ(f )| + a) → 0 as j → ∞. It follows from this relation, Proposition 3.6.1 and Theorem 3.6.1 that π f (x) ≤ πTf (x) ≤ lim inf N →∞

N

[U f (iT, (i + 1)T, v(iT ), v((i + 1)T )) − T µ(f )]

i=0

≤ lim inf Φf (0, T Mj , v) ≤ lim inf Φf (0, dkj , v). j→∞

j→∞

The theorem now follows from this relation and (6.6). Theorems 3.6.1 and 3.6.2 are important tools in our study of autonomous variational problems. It is not diﬃcult to see that a minimization problem with the functional I f (T1 , T2 , v) is equivalent to the corresponding minimization problem with the functional σ f (T1 , T2 , v). In the sequel we prefer to work with the functional σ f (T1 , T2 , v) which is always nonnegative. The next theorem establishes the existence of an a.c. function v : [0, ∞) → Rn such that σ f (0, T, v) = 0 for all T > 0. We will show (see Theorem 3.6.4) that if the integrand f has the asymptotic turnpike property, then any such function is (f )-weakly optimal. Theorem 3.6.3 For every x ∈ Rn there exists an (f )-good function v : [0, ∞) → Rn such that v(0) = x and for each T1 ∈ [0, ∞) and each T2 ∈ (T1 , ∞), σ f (T1 , T2 , v) = 0. Proof. Let x ∈ Rn . Choose a number T > 0 such that πTf (y) = π f (y) for all y ∈ Rn . It follows from Proposition 3.5.2 that there exists a n sequence {xi }∞ i=0 ⊂ R such that x0 = x, θTf (xi , xi+1 ) = 0 (i = 0, 1, . . .).

(6.8)

Propositions 3.5.3 and 3.5.4 imply that the sequence {xi }∞ i=0 is bounded. By Corollary 1.3.1 there is an a.c. function v : [0, ∞) → Rn such that

94

TURNPIKE PROPERTIES

for every integer i ≥ 0, v(iT ) = xi , I f (iT, (i + 1)T, v) = U f (iT, (i + 1)T, v(iT ), v((i + 1)T )). (6.9) is bounded it follows from (6.9) and ProposiSince the sequence {xi }∞ i=0 tions 1.3.6 and 1.3.3 that the function v : [0, ∞) → Rn is also bounded. It follows from Propositions 3.1.1 and 3.6.1, (6.8) and (6.9) that v is an (f )-good function. By Theorem 3.6.1, σ f (τ1 , τ2 , v) ≥ 0 for each τ1 ≥ 0, τ2 > τ1 .

(6.10)

It follows from (6.8) and (6.9) and the deﬁnition of T that for any integer i ≥ 0, σ f (iT, (i + 1)T, v) = 0. (6.11) The theorem now follows from (6.10) and the boundedness of v. Theorem 3.6.4 Assume that the integrand f has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Let v : [0, ∞) → Rn be an a.c. function such that σ f (T1 , T2 , v) = 0 for each T1 ≥ 0, T2 > T1 . Then v is an (f )-weakly optimal function. Moreover, there exists a sequence of numbers tj → ∞ as j → ∞ such that lim sup[I f (0, tj , v) − I f (0, tj , w)] ≤ 0 j→∞

for each a.c. function w : [0, ∞) → Rn satisfying v(0) = w(0). Proof. There exists x∗ ∈ H(f ) such that π f (x∗ ) ≥ π f (z) for all z ∈ H(f ). It follows from Theorem 3.6.1, Propositions 3.1.1 and 3.5.3 that v is an (f )-good function. To prove the theorem it is suﬃcient to note that there exists a sequence of numbers {tj }∞ j=1 such that tj → ∞, v(tj ) → x∗ as j → ∞. Theorem 3.1.2 now follows from Theorems 3.6.3 and 3.6.4.

3.7.

Preliminary lemmas for Theorem 3.1.1

In this section we show that the set of all integrands f ∈ A which have the asymptotic turnpike property is an everywhere dense subset of A with the strong topology. The following result was established in [5, Chapter 2, section 3]. Proposition 3.7.1 Let Ω be a closed subset of Rq . Then there exists a bounded nonnegative function φ ∈ C ∞ (Rq ) such that Ω = {x ∈ Rq :

95

Extremals of autonomous problems

φ(x) = 0} and for each sequence of nonnegative integers p1 , p2 , . . . , pq the p function ∂ |p| φ/∂xp11 . . . ∂xq q : Rq → R1 is bounded where |p| = qi=1 pi . Let f ∈ A. For any r ∈ (0, 1) we construct fr ∈ A which has the asymptotic turnpike property such that fr → f as r → 0+ in the strong topology. By Theorem 1.2.2 there exists a number M > 0 such that lim sup |v(t)| < M t→∞

(7.1)

for each (f )-good function v : [0, ∞) → Rn . Deﬁne D(f ) = {Ω(v) : v is an (f )-good function}.

(7.2)

Lemma 3.7.1 There exists H ∗ ∈ D(f ) such that for every D ∈ D(f ) \ {H ∗ } D \ H ∗ = ∅. Proof. Let D1 , D2 ∈ D(f ). We will say that D1 ≤ D2 if and only if D1 ⊂ D2 . We will show that there exists a minimal element of the ordered set D(f ). Consider a nonempty set E ⊂ D(f ) such that for each D1 , D2 ∈ E one of the relations below holds: D1 ≤ D2 ; D2 ≤ D1 . By Zorn’s lemma ˜ ∈ D(f ) such that D ˜ ⊂ D for any it is suﬃcient to show that there is D D ∈ E. Set ¯ = ∩D∈E D. D ¯ = ∅. We will show that there exists an (f )-good function Clearly D ¯ v : [0, ∞) → Rn such that Ω(v) ⊂ D. For every integer p ≥ 1 there exists Dp ∈ E such that ¯ Dp ) ≤ p−1 dist(D,

(7.3)

and there exists an (f )-good function vp : [0, ∞) → Rn such that Ω(vp ) = Dp .

(7.4)

By Proposition 3.3.1 we may assume without loss of generality that I f (T1 , T2 , vp ) ≤ U f (T1 , T2 , vp (T1 ), vp (T2 )) + 1

(7.5)

for each p ≥ 1 and each T1 ∈ [0, ∞), T2 ∈ (T1 , ∞) and d(vp (t), Dp ) ≤ p−1

(t ∈ [0, ∞), p = 1, 2, . . .).

(7.6)

96

TURNPIKE PROPERTIES

It follows from (7.5), (7.6), Theorem 3.4.1 and Proposition 1.3.5 that n there exist a subsequence {vpj }∞ j=1 and an a.c. function u : [0, ∞) → R such that for every integer N ≥ 1, vpj (t) → u(t) as j → ∞ uniformly in [0, N ] and

(7.7)

I f (0, N, u) ≤ U f (0, N, u(0), u(N )) + 1. It follows from (7.3) and (7.6) that ¯ for all t ∈ [0, ∞). u(t) ∈ D

(7.8)

By Proposition 3.3.2, (7.7) and (7.8), u is an (f )-good function. This completes the proof of the lemma. Lemma 3.7.2 Let u : [0, ∞) → Rn be an (f )-good function. Then there exists an (f )-good function v : [0, ∞) → Ω(u) such that for each T1 ∈ [0, ∞), T2 ∈ (T1 , ∞), I f (T1 , T2 , v) = U f (T1 , T2 , v(T1 ), v(T2 )). Proof. We may assume that d(u(t), Ω(u)) ≤ 1 for all t ∈ [0, ∞).

(7.9)

By Proposition 3.3.1 we may assume without loss of generality that I f (T1 , T2 , u) ≤ U f (T1 , T2 , u(T1 ), u(T2 )) + 1

(7.10)

for each T1 ∈ [0, ∞), T2 ∈ (T2 , ∞). For every integer p ≥ 1 we set (t ∈ [0, ∞)).

vp (t) = u(t + p)

(7.11)

By (7.9), (7.10), Theorem 3.4.1 and Proposition 1.3.5, there exist a n subsequence {vpj }∞ j=1 and an a.c. function v : [0, ∞) → R such that for every integer N ≥ 1, vpj (t) → v(t) as j → ∞ uniformly in [0, N ] and I f (0, N, v) ≤ lim inf I f (0, N, vpj ). j→∞

(7.12)

Together with (7.11) this implies that v(t) ∈ Ω(u)

(t ∈ [0, ∞)).

(7.13)

It follows from (7.11), (7.12), Proposition 3.3.1 and Theorem 3.4.1 that I f (0, N, v) = U f (0, N, v(0), v(N )) for any integer N ≥ 1.

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Extremals of autonomous problems

Together with (7.13) and Proposition 3.3.2 this implies that v is an (f )good function. The lemma is proved. By Lemma 3.7.1 there exists H ∗ ∈ D(f ) such that D \ H ∗ = ∅ for every D ∈ D(f ) \ {H ∗ }. By Lemma 3.7.2 there exists an (f )-good function wf : [0, ∞) → H ∗ such that for each number T > 0, I f (0, T, wf ) = U f (0, T, wf (0), wf (T )).

(7.14)

Ω(wf ) = H ∗ .

(7.15)

Clearly

Lemma 3.7.3 Suppose that φ : Rn → [0, ∞) is a continuous bounded function such that H ∗ ⊂ {x ∈ Rn : φ(x) = 0} ⊂ H ∗ ∪ {x ∈ Rn : |x| ≥ M + 3}.

(7.16)

For r ∈ (0, 1] we set fr (x, u) = f (x, u) + rφ(x)

(x, u ∈ Rn ).

(7.17)

Then fr ∈ A for all r ∈ (0, 1] and the following property holds: For any neighborhood U of f in A with the strong topology there exists a number r0 ∈ (0, 1) such that fr ∈ U for every r ∈ (0, r0 ). The next auxiliary result shows that for any r ∈ (0, 1] the integrand fr has the asymptotic turnpike property. Lemma 3.7.4 Suppose that φ : Rn → [0, ∞) is a continuous bounded function which satisﬁes (7.16), r ∈ (0, 1] and a function fr : Rn × Rn → R1 is deﬁned by (7.17). Then Ω(v) = H ∗ for each (fr )-good function v : [0, ∞) → Rn . Proof. Let us assume the converse. Then there exists an (fr )-good function v : [0, ∞) → Rn such that Ω(v) = H ∗ .

(7.18)

It is easy to see that µ(fr ) = µ(f ) and wf is an (fr )-good function. Then Proposition 3.1.1 implies that v is an (f )-good function and sup{|Φfr (0, T, v)|} < ∞, sup{|Φf (0, T, v)|} < ∞.

T >0

T >0

(7.19)

98

TURNPIKE PROPERTIES

It follows from (7.18) and the deﬁnition of H ∗ that there exists a sequence {Ti }∞ i=1 ⊂ (0, ∞) such that Ti → ∞, v(Ti ) → z ∈ Ω(v) \ H ∗ as i → ∞.

(7.20)

By the deﬁnition of M (see (7.1)) and Proposition 3.3.1 we may assume that |v(t)| < M (t ∈ [0, ∞)), I f (T1 , T2 , v) ≤ U f (T1 , T2 , v(T1 ), v(T2 )) + 1

(7.21)

for each T1 ∈ [0, ∞), T2 ∈ (T1 , ∞). We may assume without loss of generality that Ti+1 − Ti ≥ 10, d(v(Ti ), H ∗ ) ≥ γ, i = 1, 2, . . .

(7.22)

with a constant γ > 0. It follows from Propositions 1.3.4 and 1.3.6 and (7.21) that there exists δ ∈ (0, 1) such that the following property holds: If t1 , t2 ∈ [0, ∞) satisfy |t1 − t2 | ≤ δ , then |v(t1 ) − v(t2 )| ≤ 8−1 γ. By the deﬁnition of δ, (7.21) and (7.22), d(v(t), H ∗ ) ≥ 2−1 γ for every integer i ≥ 1 and every t ∈ [Ti , Ti + δ]. Combined with (7.21) and (7.16) this implies that there exists a constant γ1 > 0 such that φ(v(t)) ≥ γ1 for every integer i ≥ 1 and every t ∈ [Ti , Ti + δ]. By this property, (7.19) and (7.17), Φfr (0, TN , v) ≥ Φf (0, TN , v) + rγ1 δ(N − 1) → ∞ as N → ∞. This is contradictory to (7.19). The obtained conradiction proves the lemma. Lemmas 3.7.3 and 3.7.4 imply the following result. Lemma 3.7.5 There exists an everywhere dense subset E of the space A with the strong topology such that each f ∈ E has the asymptotic turnpike property. Propositions 3.7.1 and Lemmas 3.7.3 and 3.7.4 imply the following result. Lemma 3.7.6 Let k ≥ 1 be an integer. Then there exists an everywhere dense subset Ek ⊂ Ak such that each f ∈ Ek has the asymptotic turnpike property.

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Extremals of autonomous problems

3.8.

Preliminary lemmas for Theorems 3.1.3 and 3.1.4

Assume that f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Then Ω(v) = H(f ) for every (f )-good function v : [0, ∞) → Rn . We begin with the following auxiliary result which shows that an approximate solution of a variational problem with the integrand f deﬁned on a ﬁnite interval is close to the turnpike H(f ) at some points if the length of the interval is large enough. Lemma 3.8.1 Let 0 ∈ (0, 1), K0 , M0 > 0 and let l be a positive integer such that for each (f )-good function x : [0, ∞) → Rn , dist(H(f ), {x(t) : t ∈ [T, T + l]}) ≤ 8−1 0

(8.1)

for all large T (the existence of l follows from Theorem 3.3.1). Then there exists an integer N ≥ 10 such that the following property holds: If an a.c. function x : [0, N l] → Rn satisﬁes |x(0)|, |x(N l)| ≤ K0 and I f (0, N l, x) ≤ U f (0, N l, x(0), x(N l)) + M0 , then there exists an integer i0 ∈ [0, N − 8] such that dist(H(f ), {x(t) : t ∈ [T, T + l]}) ≤ 0 for all T ∈ [i0 l, (i0 + 7)l]. Proof. Assume that the lemma is wrong. Then for every integer N ≥ 10 there exists an a.c. function xN : [0, N ] → Rn satisfying |xN (0)|, |xN (N l)| ≤ K0 , I f (0, N l, xN ) ≤ U f (0, N l, xN (0), xN (N l))+M0 (8.2) such that the following property holds: If an integer i ∈ [0, N − 8], then there is T (i) ∈ [il, (i + 7)l] for which dist(H(f ), {x(t) : t ∈ [T (i), T (i) + l]}) > 0 .

(8.3)

By Theorem 1.2.3, sup{|xN (t)| : t ∈ [0, N l], N is an integer, N ≥ 10} < ∞.

(8.4)

It follows from (8.2), (8.4) and Theorem 3.4.1 that for any integer j ≥ 1, sup{I f (0, jl, xN ) : N is an integer, N ≥ max{j, 10}} < ∞.

(8.5)

By (8.5) and Proposition 1.3.5 there exist a subsequence {xNi }∞ i=1 and an a.c. function x : [0, ∞) → Rn such that for each integer q ≥ 1, xNi (t) → x(t) as i → ∞ uniformly in [0, ql],

100

TURNPIKE PROPERTIES

xNi → x as i → ∞ weakly 1

n

f

(8.6) f

in L (R ; (0, ql)), I (0, ql, x) ≤ lim inf I (0, ql, xNi ). i→∞

(8.2), (8.6) and Theorem 3.4.1 imply that I f (0, ql, x) ≤ U f (0, ql, x(0), x(ql)) + M0 for any integer q ≥ 1.

(8.7)

It follows from Proposition 3.3.2, (8.7), (8.4) and (8.6) that x : [0, ∞) → Rn is an (f )-good function. By the deﬁnition of l there exists an integer Q ≥ 1 such that (8.1) holds for all T ∈ [Ql, ∞). By (8.6) there exists an integer p ≥ 1 such that Np ≥ 2Q + 20, |xNp (t) − x(t)| ≤ 8−1 0 , t ∈ [0, (2Q + 20)l].

(8.8)

It follows from (8.1) which holds for every T ∈ [Ql, ∞) and (8.8) that dist(H(f ), {xNp (t) : t ∈ [T, T + l]}) ≤ 4−1 0 for each T ∈ [Ql, Ql + 10l]. This is contradictory to the deﬁnition of xNp (see (8.3)). The obtained contradiction proves the lemma. The following auxiliary result is an extension of Lemma 3.8.1. It shows that the property established in Lemma 3.8.1 for the variational problems with the integrand f also holds for variational problems with integrands which belong to a small neighborhood of f . Lemma 3.8.2 Let 0 ∈ (0, 1), K0 , M0 > 0 and let l be a positive integer such that each (f )-good function x : [0, ∞) → Rn satisﬁes (8.1) for all large T . Then there exist an integer N ≥ 10 and a neighborhood U of f in A with the weak topology such that the following property holds: If g ∈ U, S ∈ [0, ∞) and if an a.c. function x : [S, S + N l] → Rn satisﬁes |x(S)|, |x(S + N l)| ≤ K0 , I g (S, S + N l, x) ≤ U g (S, S + N l, x(S), x(S + N l)) + M0 ,

(8.9)

then there exists an integer i0 ∈ [0, N − 8] such that dist(H(f ), {x(t) : t ∈ [T, T + l]}) ≤ 0

(8.10)

for all T ∈ [S + i0 l, S + (i0 + 7)l]. Proof. It follows from Lemma 3.8.1 that there exists an integer N ≥ 10 such that the following property holds: If an a.c. function x : [0, N l] → Rn satisﬁes |x(0)|, |x(N l)| ≤ K0 , I f (0, N l, x) ≤ U f (0, N l, x(0), x(N l)) + M0 + 8, (8.11)

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then there exists an integer i0 ∈ [0, N − 8] such that (8.10) holds for all T ∈ [i0 l, (i0 + 7)l]. Put S˜ = sup{|U f (0, N l, y, z)| : y, z ∈ Rn , |y|, |z| ≤ 2K0 + 1}.

(8.12)

By Theorem 3.4.1, S˜ is ﬁnite. It follows from Proposition 1.3.9 that there exists a neighborhood U1 of f in A with the weak topology such that the following property holds: If g ∈ U1 , S ∈ [0, ∞) and if y, z ∈ Rn satisfy |y|, |z| ≤ 2K0 + 1, then |U f (S, S + N l, y, z) − U g (S, S + N l, y, z)| ≤ 8−1 .

(8.13)

It follows from Proposition 1.3.8 that there is a neighborhood U of f in A with the weak topology such that U ⊂ U1 and that the following property holds: If g ∈ U, S ∈ [0, ∞) and if an a.c. function x : [S, S + N l] → Rn satisﬁes ˜ min{I f (S, S + N l, x), I g (S, S + N l, x)} ≤ 4 + M0 + S, then

|I f (S, S + N l, x) − I g (S, S + N l, x)| ≤ 8−1 .

(8.14) (8.15)

Assume that g ∈ U, S ∈ [0, ∞) and an a.c. function x : [S, S + N l] → satisﬁes (8.9). It follows from (8.9) and the choice of U1 that

Rn

|U f (S, S + N l, x(S), x(S + N l)) − U g (S, S + N l, x(S), x(S + N l))| ≤ 8−1 . (8.16) By (8.16), (8.12) and (8.9), I g (S, S + N l, x) ≤ U f (S, S + N l, x(S), x(S + N l)) + M0 + 8−1 ≤ S˜ + M0 + 8−1 . (8.17) By the choice of U and the inequality (8.17), |I g (S, S + N l, x) − I f (S, S + N l, x)| ≤ 8−1 .

(8.18)

(8.17) and (8.18) imply that I f (S, S + N l, x) ≤ U f (S, S + N l, x(S), x(S + N l)) + M0 + 4−1 . By the deﬁnition of N there exists an integer i0 ∈ [0, N − 8] such that (8.10) holds for all T ∈ [S + i0 l, S + (i0 + 7)l]. The lemma is proved. In general for given x1 , x2 ∈ Rn we cannot guarantee the existence of T > 0 and v : [0, T ] → Rn such that v(0) = x1 , v(T ) = x2 and σ f (0, T, v)

102

TURNPIKE PROPERTIES

is small. The following lemma shows that such T and v exist if x1 and x2 are close to the turnpike H(f ). Lemma 3.8.3 For each > 0 there exists δ > 0 such that the following property holds: If x1 , x2 ∈ Rn satisfy d(xi , H(f )) ≤ δ, i = 1, 2, then there exists an a.c. function v : [0, T ] → Rn such that T ≥ 1, v(0) = x1 , v(T ) = x2 ,

(8.19)

σ f (0, T, v) ≤ .

(8.20)

Proof. Let > 0. It follows from Theorems 3.4.1 and 3.6.1 that there exists δ0 ∈ (0, min{1, }) for which the following property holds: If y1 , y2 , z1 , z2 ∈ Rn satisfy

then

d(yi , H(f )) ≤ 1, d(zi , H(f )) ≤ 1, i = 1, 2,

(8.21)

|yi − zi | ≤ δ0 , i = 1, 2,

(8.22)

|U f (0, 1, y1 , y2 ) − U f (0, 1, z1 , z2 )| ≤ 8−1 , |π f (yi ) − π f (zi )| ≤ 8−1 , i = 1, 2.

(8.23)

It follows from Theorem 3.3.1 that there exists a natural number G such that for each (f )-good function v : [0, ∞) → Rn the inequality dist(H(f ), {v(t) : t ∈ [T, T + G]}) ≤ 4−1 δ0

(8.24)

is valid for all large T . Theorem 3.6.3 implies that there exists an (f )good function w : [0, ∞) → Rn such that w(0) = 0 and that for each T1 ≥ 0, T2 > T1 , (8.25) σ f (T1 , T2 , w) = 0. By the choice of G there exists T0 > 0 such that for any T ≥ T0 ,

Put

dist(H(f ), {w(t) : t ∈ [T, T + G]}) ≤ 4−1 δ0 .

(8.26)

δ = 8−1 δ0 .

(8.27)

Let x1 , x2 ∈ Rn , d(xi , H(f )) ≤ δ, i = 1, 2. There exist h1 , h2 ∈ H(f ) such that (8.28) |xi − hi | ≤ δ, i = 1, 2. It follows from (8.26), which holds for any T ≥ T0 , that there exist T1 ∈ [T0 , T0 + G] and T2 ∈ [T0 + 10G, T0 + 11G] such that |w(Ti ) − hi | ≤ 4−1 δ0 , i = 1, 2.

(8.29)

103

Extremals of autonomous problems

There exists an a.c. function v : [0, T2 − T1 ] → Rn such that v(0) = x1 , v(t) = w(T + T1 )

(t ∈ [1, T2 − T1 − 1]), v(T2 − T1 ) = x2 ,

I f (0, 1, v) = U f (0, 1, v(0), v(1)), I f (T2 − T1 − 1, T2 − T1 , v) = U f (T2 − T1 − 1, T2 − T1 , v(T2 − T1 − 1), v(T2 − T1 )).

(8.30)

By the deﬁnition of δ0 (see (8.21)-(8.23)), (8.30), (8.28), (8.27) and (8.29), |U f (0, 1, v(0), v(1)) − U f (T1 , T1 + 1, w(T1 ), w(T1 + 1))| ≤ 8−1 , |U f (T2 − T1 − 1, T2 − T1 , v(T2 − T1 − 1), v(T2 − T1 )) −U f (T2 − 1, T2 , w(T2 − 1), w(T2 ))| ≤ 8−1 .

(8.31)

(8.30) implies that σ f (0, T2 −T1 , v) = σ f (0, 1, v)+σ f (1, T2 −T1 , v)+σ f (T2 −T1 −1, T2 −T1 , v). (8.32) It follows from (8.30) and (8.25) that σ f (1, T2 − T1 − 1, v) = 0.

(8.33)

By the deﬁnition of δ0 (see (8.21)-(8.23)), (8.28), (8.27), (8.29) and (8.31), |π f (w(Ti )) − π f (xi )| ≤ 8−1 , i = 1, 2. Combined with (8.30), (8.31) and (8.25) this inequality implies that |σ f (0, 1, v)| ≤ |U f (0, 1, v(0), v(1)) − U f (T1 , T1 + 1, w(T1 ), w(T1 + 1))| +|π f (w(T1 )) − π f (x1 )| + |U f (T1 , T1 + 1, w(T1 ), w(T1 + 1)) − π f (w(T1 )) +π f (w(T1 + 1)) − µ(f )| ≤ 4−1 , |σ f (T2 − T1 − 1, T2 − T1 , v)| ≤ |U f (T2 − T1 − 1, T2 − T1 , v(T2 − T1 − 1), v(T2 − T1 )) −U f (T2 − 1, T2 , w(T2 − 1), w(T2 ))| + |π f (w(T2 )) − π f (v(T2 − T1 ))| +|U f (T2 − 1, T2 , w(T2 − 1), w(T2 )) − π f (w(T2 − 1)) +π f (w(T2 )) − µ(f )| ≤ 4−1 . It follows from these inequalities, (8.32) and (8.33) that 0 ≤ σ f (0, T2 − T1 , v) ≤ .

104

TURNPIKE PROPERTIES

This completes the proof of the lemma. The next lemma is an important tool in the proof of Theorem 3.1.4. It establishes the turnpike property for approximate solutions v : [0, T ] → Rn such that v(0) and v(T ) are close to the turnpike and σ f (0, T, v) is small enough. Lemma 3.8.4 Let ∈ (0, 1) and let L be a positive integer such that for each (f )-good function v : [0, ∞) → Rn the inequality dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤

(8.34)

is valid for all large T (the existence of L follows from Theorem 3.3.1). Then there exists δ > 0 such that the following property holds: If T ∈ [L, ∞) and if an a.c. function v : [0, T ] → Rn satisﬁes d(v(0), H(f )) ≤ δ, d(v(T ), H(f )) ≤ δ,

(8.35)

I f (0, T, v) − T µ(f ) − π f (v(0)) + π f (v(T )) ≤ δ,

(8.36)

then for every S ∈ [0, T − L], dist(H(f ), {v(t) : t ∈ [S, S + L]}) ≤ .

(8.37)

Proof. It follows from Lemma 3.8.3 that for every integer k ≥ 1 there exists (8.38) δk ∈ (0, 2−k ) such that the following property holds: If x1 , x2 ∈ Rn satisfy d(xi , H(f )) ≤ δk , i = 1, 2,

(8.39)

then there is an a.c. function v : [0, T ] → Rn such that T ≥ 1, v(0) = x1 , v(T ) = x2 , σ f (0, T, v) ≤ 2−k .

(8.40)

Assume that the lemma is wrong. Then for every integer k ≥ 1 there exist Tk ≥ L, an a.c. function vk : [0, Tk ] → Rn and Sk ∈ [0, Tk − L] such that (8.41) d(vk (s), H(f )) ≤ δk , s = 0, Tk , σ f (0, Tk , vk ) ≤ δk ,

(8.42)

dist(H(f ), {vk (t) : t ∈ [Sk , Sk + L]}) > .

(8.43)

105

Extremals of autonomous problems

By the choice of {δk }∞ t=1 (see (8.38)-(8.40)) and (8.41) for each natural number k there exists an a.c. function uk : [0, τk ] → Rn such that τk ≥ 1, uk (0) = vk (Tk ), uk (τk ) = vk+1 (0),

(8.44)

σ f (0, τk , uk ) ≤ 2−k . By (8.44) there exists an a.c. function v : [0, ∞) → Rn such that v(t) = v1 (t) (t ∈ [0, T1 ]), v(t) = u1 (t − T1 ) (t ∈ [T1 , T1 + τ1 ]),

v(t) = vk t −

k−1

(Ti + τi ) , t ∈

i=1

v(t) = uk t − t∈

k−1

k−1

k−1

i=1

i=1

(Ti + τi ),

k−1

(Ti + τi ) + Tk ,

i=1

(Ti + τi ) + Tk ,

(Ti + τi ) − Tk ,

i=1 k

(Ti + τi ) , k = 2, 3, . . . .

(8.45)

i=1

Put T0 , τ0 = 0. It follows from (8.38), (8.42), (8.44) and (8.45) that for any natural number k ≥ 2,

σ

f

0,

k

(Ti + τi ), v

=

⎛

f

k

+

⎛

i=0 q

i=0

i=0

(Ti + τi ) + Tq ,

f

(Ti + τi ) + Tq , v ⎠

i=0

q−1

⎝σ f ⎝

⎞

q−1

(Ti + τi ),

⎛

q=1 k

q−1

σ ⎝

q=1

i=0

=

k

f

[σ (0, Tq , vq ) + σ (0, τq , uq )] ≤

q=1

⎞⎞

(Ti + τi ), v ⎠⎠

k

(δq + 2−q ) ≤ 4.

q=1

It follows from this relation, Proposition 3.1.1, (8.45), (8.44) and (8.41) that v is an (f )-good function. By the deﬁnition of L, (8.34) holds for all large T . On the other hand by (8.45) and (8.43) for every integer k ≥ 2,

dist H(f ), v(t) : t ∈

k−1

(Ti + τi ) + Sk ,

i=1

k−1

(Ti + τi ) + Sk + L

i=1

> . The obtained contradiction proves the lemma.

106

3.9.

TURNPIKE PROPERTIES

Proof of Theorem 3.1.4

Assume that f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Let M0 , M1 , > 0. We may assume without loss of generality that M1 > 4 + sup{|y| : y ∈ H(f )}, < 1.

(9.1)

It follows from Theorem 3.3.1 that there is a natural number L such that if v : [0, ∞) → Rn is an (f )-good function, then for all large T , dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 4−1 .

(9.2)

It follows from Theorem 1.2.3 that there exist a number S > M0 + M1 and a neighborhood U0 of f in A with the weak topology such that the following property holds: If g ∈ U0 , T1 ∈ [0, ∞), T2 ∈ [T1 + 1, ∞) and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes |v(Ti )| ≤ M1 , i = 1, 2, I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + M0 , (9.3) then (9.4) |v(t)| ≤ S t ∈ [T1 , T2 ]. It follows from Lemma 3.8.4 that there exists a number δ0 ∈ (0, 8−1 )

(9.5)

such that if T ∈ [L, ∞) and if an a.c. function v : [0, T ] → Rn satisﬁes d(v(τ ), H(f )) ≤ δ0 , τ = 0, T,

(9.6)

σ f (0, T, v) ≤ δ0 , then for every τ ∈ [0, T − L], dist(H(f ), {v(t) : t ∈ [τ, τ + L]}) ≤ 4−1 .

(9.7)

Choose an integer Q∗ > 20 + 2(M0 + 4 + 2 sup{|Uf (0, 1, x, y)| : x, y ∈ Rn , |x|, |y| ≤ S} (9.8) +2 sup{|π f (x)| : x ∈ Rn , |x| ≤ S} + 2|µ(f )|)δ0−1 . By Theorems 3.4.1 and 3.6.1 there exists δ1 ∈ (0, 8−1 δ0 )

(9.9)

107

Extremals of autonomous problems

such that the following property holds: If y1 , y2 , z1 , z2 ∈ Rn satisfy d(yi , H(f )) ≤ 4, d(zi , H(f )) ≤ 4, |yi − zi | ≤ 8δ1 , i = 1, 2, then and

|U f (0, 1, y1 , y2 ) − U f (0, 1, z1 , z2 )| ≤ (16Q∗ )−1 δ0

(9.10) (9.11)

|π f (yi ) − π f (zi )| ≤ (16Q∗ )−1 δ0 , i = 1, 2.

It follows from Theorem 3.3.1 that there is a natural number L1 such that if v : [0, ∞) → Rn is an (f )-good function, then for all large T dist(H(f ), {v(t) : t ∈ [T, T + L1 ]}) ≤ 8−1 δ1 .

(9.12)

We may assume without loss of generality that L1 ≥ 8L.

(9.13)

It follows from Lemma 3.8.2 that there exist a natural number N ≥ 10 and a neighborhood U1 of f in A with the weak topology such that U1 ⊂ U0

(9.14)

and that the following property holds: If g ∈ U1 , T ∈ [0, ∞), and if an a.c. function x : [T, T + N L1 ] → Rn satisﬁes |x(T )|, |x(T + N L1 )| ≤ S, I g (T, T +N L1 , x) ≤ U g (T, T +N L1 , x(T ), x(T +N L1 ))+M0 +4, (9.15) then there exists an integer i0 ∈ [0, N − 8] such that for all τ ∈ [T + i0 L1 , T + (i0 + 7)L1 ], dist(H(f ), {x(t) : t ∈ [τ, τ + L1 ]}) ≤ δ1 .

(9.16)

Put D0 = sup{|π f (x)| : x ∈ Rn , d(x, H(f )) ≤ 4}, l = 50N L1 .

(9.17)

It follows from Propositions 1.3.8 and 1.3.9 that there exists a neighborhood U of f in A with the weak topology such that U ⊂ U1 and that the following properties hold:

(9.18)

108

TURNPIKE PROPERTIES

(i) If g ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + 1, T1 + 100N (L1 + l)] and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes min{I f (T1 , T2 , v), I g (T1 , T2 , v)} ≤ 4D0 + 8 + 200N L1 |µ(f )|,

(9.19)

+2 sup{|U f (0, 1, x, y)| : x, y ∈ Rn , |x|, |y| ≤ S}, then

|I f (T1 , T2 , v) − I g (T1 , T2 , v)| ≤ (64Q∗ )−1 δ1 ;

(9.20)

(ii) if g ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + 1, T1 + 100N (L1 + l)] and if y, z ∈ Rn satisfy |y|, |z| ≤ 2S + 8, then |U f (T1 , T2 , y, z) − U g (T1 , T2 , y, z)| ≤ (64Q∗ )−1 δ1 .

(9.21)

Let g ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 +L+lQ∗ , ∞) and let v : [T1 , T2 ] → Rn be an a.c. function such that |v(Ti )| ≤ M1 , i = 1, 2, I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + M0 . (9.22) By the deﬁnition of S and U0 , (9.4) holds. Put E = {h ∈ R1 : T1 + 10N L1 ≤ h ≤ T2 − 10N L1 and dist(H(f ), {v(t) : t ∈ [h, h + L]}) > }.

(9.23)

If E = ∅, then Theorem 3.1.4 is valid. Hence we may assume that E = ∅. We set ˜ = inf{h : h ∈ E} h ˜ + 4−1 . It follows from the deﬁnition and choose h1 ∈ E satisfying h1 ≤ h of U1 , N (see (9.14)-(9.16)) that there are integers i1 , i2 ∈ {0, . . . , N − 8} such that (9.24) dist(H(f ), {v(t) : t ∈ [T, T + L1 ]}) ≤ δ1 for any number T ∈ [h1 − (2N − i1 )L1 , h1 − (2N − i1 − 7)L1 ] ∪ [h1 + (N + i2 )L1 , h1 + (N + i2 + 7)L1 ]. Set b1 = h1 − 2N L1 + i1 L1 , c1 = h1 + N Li + i2 L1 . By induction we deﬁne a sequence of intervals [bq , cq ] (q ≥ 1) such that: (B) N L1 ≤ cq − bq ≤ 4N L1 , bq ≥ cq−1 if q ≥ 2, [bq , cq − N L1 ] ∩ E = ∅, E \ ∪qj=1 [bj , cj ] ⊂ (cq , T2 ]; (C) for h ∈ {bq , cq }, (9.24) holds for every T ∈ [h, h + 7L1 ]. Clearly for q = 1 properties (B) and (C) hold.

(9.25)

Extremals of autonomous problems

109

Let k be a natural number. Assume that a sequence of intervals {[bq , cq ]}kq=1 was deﬁned and properties (B), (C) hold for q = 1, . . . , k. If E \ ∪kq=1 [bq , cq ] = ∅, then the construction of the sequence is completed and [bk , ck ] is its last element. Let us assume the converse. Choose h2 ∈ E \ ∪kq=1 [bq , cq ] such that h2 ≤ 4−1 + inf{h : h ∈ E \ ∪kq=1 [bq , cq ]}. It follows from the deﬁnition of U1 , N that there are integers j1 , j2 ∈ [0, N − 8] such that (9.24) holds for every T ∈ [h2 − (2N − j1 )L1 , h2 − (2N − j1 − 7)L1 ] ∪[h2 + (N + j2 )L1 , h2 + (N + j2 + 7)L1 ]. Set ck+1 = h2 + N L1 + j2 L1 , bk+1 = max{ck , h2 − 2N L1 + j1 L1 }. It is easy to verify that properties (B), (C) hold with q = k + 1. Evidently the construction of the sequence will be completed in a ﬁnite number of steps. Let [bQ , cQ ] be the last element of the sequence. Clearly (9.26) E ⊂ ∪Q q=1 [bq , cq ]. It follows from Theorem 3.6.3 that there exists an (f )-good function w : [0, ∞) → Rn such that w(0) = 0 and σ f (T1 , T2 , w) = 0

(9.27)

for each T1 ∈ [0, ∞), T2 ∈ (T1 , ∞). By the choice of L1 (see (9.12)) there exists a positive number τ0 > 0 such that dist(H(f ), {w(t) : t ∈ [T, T + L1 ]}) ≤ 8−1 δ1

(9.28)

for each T ≥ τ0 . Let i ∈ {1, . . . , Q}. By properties (B) and (C), (9.13), (9.9), (9.24), the choice of δ0 (see (9.5)-(9.7)) and the deﬁnition of E, σ f (bi , ci , v) > δ0 .

(9.29)

By properties (B) and (C), (9.4), (9.7), (9.29) and the deﬁnition of U (see (9.19) and (9.20)), σ g (bi , ci , v) > δ0 − 64−1 δ0 .

(9.30)

110

TURNPIKE PROPERTIES

It follows from (9.24), (9.28) and property (C) that there exist t(i, 1) ∈ [τ0 , ∞) and t(i, 2) ∈ [t(i, 1) + 2L1 , t(i, 1) + 3L1 ] such that |w(t(i, 1)) − v(bi )|, |w(t(i, 2)) − v(ci )| ≤ 2δ1 .

(9.31)

t(i, 3) = t(i, 2) − t(i, 1).

(9.32)

Put By Corollary 1.3.1 there exists an a.c. function vi : [t(i, 1), t(i, 2)] → Rn such that vi (t(i, 1)) = v(bi ), vi (t) = w(t) (t ∈ [t(i, 1) + 1, t(i, 2) − 1]), vi (t(i, 2)) = v(ci ), I f (t(i, 1), t(i, 1) + 1, vi ) = U f (t(i, 1), t(i, 1) + 1, vi (t(i, 1)), vi (t(i, 1) + 1)), I f (t(i, 2) − 1, t(i, 2), vi ) = U f (t(i, 2) − 1, t(i, 2), vi (t(i, 2) − 1), vi (t(i, 2))). (9.33) By (9.33), (9.31), (9.27), (9.28) and the choice of δ1 (see (9.9)-(9.11)), σ f (t(i, 1), t(i, 2), vi ) = σ f (t(i, 1), t(i, 1) + 1, vi ) + σ f (t(i, 2) − 1, t(i, 2), vi ) ≤ |U f (t(i, 1), t(i, 1) + 1, vi (t(i, 1)), vi (t(i, 1) + 1)) −U f (t(i, 1), t(i, 1) + 1, w(t(i, 1)), w(t(i, 1) + 1))| +|π f (w(t(i, 1))) − π f (vi (t, 1))| +|U f (t(i, 2) − 1, t(i, 2), vi (t(i, 2) − 1), vi (t(i, 2))) −U f (t(i, 2) − 1, t(i, 2), w(t(i, 2) − 1), w(t(i, 2)))| +|π f (w(t(i, 2))) − π f (vi (t(i, 2)))| ≤ (4Q∗ )−1 δ0 .

(9.34)

Property (C), (9.34), (9.33) and (9.17) imply that I f (t(i, 1), t(i, 2), vi ) ≤ 3L1 |µ(f )| + 2D0 + 1.

(9.35)

By induction we deﬁne sequences of numbers {¯bi }Q ci }Q i=1 , {¯ i=1 as follows: ¯b1 = b1 , c¯i = ¯bi + t(i, 3) (i = 1, . . . , Q),

(9.36)

¯bi+1 = bi+1 + c¯i − ci (i is an integer, i ≤ Q − 1). Deﬁne an a.c. function u : [T1 , c¯Q ] → Rn by u(t) = v(t) (t ∈ [T1 , b1 ]), u(t) = vi (t − ¯bi + t(i, 1)) (t ∈ [¯bi , c¯i ], i = 1, . . . , Q), u(t) = v(t − c¯i + ci ) (t ∈ [¯ ci , ¯bi+1 ], i ≤ Q − 1).

(9.37)

111

Extremals of autonomous problems

(9.23) and property (B) imply that cQ ≤ T2 − 6N L1 , c¯Q = cQ −

Q

(ci − bi − t(i, 3))

i=1

∈ [cQ − 4QN L1 , cQ − (N − 3)QL1 ].

(9.38)

By (9.34), (9.35), (9.36) and the deﬁnitions of u and U (see (9.19)-(9.21)) for i = 1, . . . , Q, σ g (¯bi , c¯i , u) ≤ σ f (t(i, 1), t(i, 2), vi ) + (64Q∗ )−1 δ1 ≤ (64Q∗ )−1 δ1 + (4Q∗ )−1 δ0 .

(9.39)

There exists an a.c. function u0 : [T1 , T2 ] → Rn such that cQ +t(Q, 2)) (t ∈ [¯ cQ +1, cQ −1]), u0 (t) = u(t) (t ∈ [T1 , c¯Q ]), u0 (t) = w(t−¯ u0 (t) = v(t) (t ∈ [cQ , T2 ]), cQ , cQ −1}). (9.40) I f (h, h+1, u0 ) = U f (h, h+1, u0 (h), u0 (h+1)) (h ∈ {¯ It follows from (9.40), (9.37), (9.33), (9.28), (9.4) and (9.1) that cQ , c¯Q + 1, cQ − 1, cQ }). |u0 (t)| ≤ S (t ∈ {¯

(9.41)

By this relation, the deﬁnition of U (see (9.19)-(9.21) and (9.40)) I g (h, h + 1, u0 ) ≤ sup{|U f (0, 1, x, y)| : x, y ∈ Rn , |x|, |y| ≤ S} +(64Q∗ )−1 δ1 (h ∈ {¯ cQ , cQ − 1}).

(9.42)

By (9.17), (9.27), (9.28), (9.40) and the choice of U (see (9.19) and (9.20)) cQ + 1, ∞) and if t2 ∈ (−∞, cQ − 1] satisfy 1 ≤ t2 − t1 ≤ 100N L1 , if t1 ∈ [¯ then (9.43) |σ g (t1 , t2 , u0 )| ≤ (64Q∗ )−1 δ1 . (9.40), (9.37) and (9.42) imply that I g (T1 , T2 , u0 ) − I g (T1 , T2 , v) = I g (b1 , cQ , u0 ) − I g (b1 , cQ , v) = I g (cQ − 1, cQ , u0 ) cQ + 1, cQ − 1, u0 ) + I g (¯ cQ , c¯Q + 1, u0 ) + +I g (¯

Q i=1

[I g (¯bi , c¯i , u) − I g (bi , ci , v)]

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TURNPIKE PROPERTIES

≤

Q

[I g (¯bi , c¯i , u) − I g (bi , ci , v)] + I g (¯ cQ + 1, cQ − 1, u0 )

i=1

+2 sup{|U f (0, 1, x, y)| : x, y ∈ Rn , |x|, |y| ≤ S} + (32Q∗ )−1 δ1 . It follows from this relation, (9.39), (9.33), (9.37), (9.36) and (9.30) that I g (T1 , T2 , u0 ) − I g (T1 , T2 , v) ≤ I g (¯ cQ + 1, cQ − 1, u0 ) +2 sup{|U f (0, 1, x, y)| : x, y ∈ Rn , |x|, |y| ≤ S} + (32Q∗ )−1 δ1 +

Q

[δ1 (64Q∗ )−1 +(4Q∗ )−1 δ0 +µ(f )(t(i, 3)−ci +bi )−δ0 +64−1 δ0 ]. (9.44)

i=1

Put

ti = c¯Q + 1 + iQ−1 (cQ − c¯Q − 2), i = 0, . . . , Q.

By (9.27), (9.38), (9.40), (9.41) and (9.43), I g (¯ cQ + 1, cQ − 1, u0 ) ≤

Q−1

[I g (ti , ti+1 , u0 ) − π f (u0 (ti )) + π f (u0 (ti+1 ))]

i=0

+2 sup{|π f (x)| : x ∈ Rn , |x| ≤ S} ≤ 2 sup{|π f (x)| : x ∈ Rn , |x| ≤ S} +µ(f )(cQ − c¯Q − 2) + Qδ1 (64Q∗ )−1 . By this relation, (9.44), (9.22), (9.40), (9.37) and (9.38), −M0 ≤ I g (T1 , T2 , u0 ) − I g (T1 , T2 , v) ≤ 2 sup{|π f (x)| : x ∈ Rn , |x| ≤ S} +2 sup{|U f (0, 1, x, y)| : x, y ∈ Rn , |x|, |y| ≤ S} + (32Q∗ )−1 δ1 +Q[δ1 (64Q∗ )−1 + δ0 (4Q∗ )−1 + δ0 (64)−1 − δ0 ] + Qδ1 (64Q∗ )−1 + 2|µ(f )|. It follows from this relation, (9.8) and (9.9) that Q + 20 ≤ Q∗ . This completes the proof of the theorem.

3.10.

Proof of Theorem 3.1.3

Assume that f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . It follows from Theorem 1.2.2 that there exist a neighborhood U1 of f in A with the weak topology and a number M1 > 0 such that (10.1) lim sup |v(t)| < M1 t→∞

Extremals of autonomous problems

113

for each g ∈ U1 and each (g)-good function v : [0, ∞) → Rn . It follows from Theorem 3.1.4 that there exist a number l > 0, integers L, Q∗ ≥ 1 and a neighborhood U of f in A with the weak topology such that U ⊂ U1 and the following property holds: If g ∈ U, T1 ∈ [0, ∞), T2 ∈ [T1 + L, +lQ∗ , ∞) and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes |v(Ti )| ≤ M1 , i = 1, 2, I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + 1, (10.2) Q Q then there exist sequences of numbers {bi }i=1 , {ci }i=1 ⊂ [T1 , T2 ] such that (10.3) Q ≤ Q∗ , 0 ≤ ci − bi ≤ l (i = 1, . . . , Q), and for every T ∈ [T1 , T2 − L] \ ∪Q i=1 [bi , ci ], dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ .

(10.4)

Let g ∈ U and v : [0, ∞) → Rn be a (g)-good function. It is suﬃcient to show that (10.4) holds for all large T . Let us assume the converse. Then there exists a sequence {ti }∞ i=1 ⊂ (0, ∞) such that ti+1 − ti ≥ 4l + 4, dist(H(f ), {v(t) : t ∈ [ti , ti + L]}) > , i = 1, 2, . . . . (10.5) By the deﬁnition of U1 , M1 (see (10.1)) and Proposition 3.3.1 we may assume that (10.6) |v(t)| < M1 (t ∈ [0, ∞)), I g (T1 , T2 , v) ≤ U g (T1 , T2 , v(T1 ), v(T2 )) + 1

(10.7)

for each T1 ∈ [0, ∞), T2 ∈ (T1 , ∞). Fix an integer τ > (L + l)Q∗ + t2Q∗ +2 + 4L + 4.

(10.8)

It follows from (10.6), (10.7), (10.8) and the deﬁnition of U, l, L, Q∗ that Q there exist sequences of numbers {bi }Q i=1 , {ci }i=1 ⊂ [0, τ ] such that (10.3) holds and (10.4) is valid for every T ∈ [0, τ − L] \ ∪Q i=1 [bi , ci ]. Together with (10.5) and (10.8) this implies that for every i ∈ {1, . . . , 2Q∗ + 2} there exists p(i) ∈ {1, . . . , Q} for which ti ∈ [bp(i) , cp(i) ]. By (10.3) there exist i, j ∈ {1, . . . , 2Q∗ + 2} such that i < j, p(i) = p(j), tj − ti ≤ cp(i) − bp(i) ≤ l. This is contradictory to (10.5). The obtained contradiction proves the theorem.

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3.11.

Proofs of Theorems 3.1.1 and 3.1.5

Proof of Theorem 3.1.1. By Lemma 3.7.5 there exists an everywhere dense subset E of A with the strong topology such that each f ∈ E has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Let f ∈ E and p ≥ 1 be an integer. It follows from Theorem 3.1.3 that there exist an integer L(f, p) ≥ 1 and an open neighborhood U(f, p) of f in A with the weak topology such that if g ∈ U(f, p) and if v : [0, ∞) → Rn is a (g)-good function, then dist(H(f ), {v(t) : t ∈ [T, T + L(f, p)]}) ≤ p−1 for all large T. Put

F = ∩∞ p=1 ∪ {U(f, p) : f ∈ E}.

Let g ∈ F and vi : [0, ∞) → Rn (i = 1, 2) be (g)-good functions. Let ∈ (0, 1). There exist an integer p ≥ 1 and f ∈ E such that p > 8−1 , g ∈ U(f, p). It follows from the deﬁnition of U(f, p) and L(f, p) that dist(H(f ), Ω(vi )) ≤ p−1 , i = 1, 2, dist(Ω(v1 ), Ω(v2 )) ≤ 2p−1 < . This completes the proof of the theorem. By a simple modiﬁcation of the above proof (basically by replacing Lemma 3.7.5 by Lemma 3.7.6) we can easily establish Theorem 3.1.5.

3.12.

Examples

Fix a positive constant a and set ψ(t) = t, t ∈ [0, ∞). Consider a complete metric space A of integrands f : Rn × Rn → R1 deﬁned in Section 3.1. Example 1. Consider an integrand f (x, u) = |x|2 + |u|2 , x, u ∈ Rn . It is easy to see that f ∈ A. Applying the methods used in the proofs of Theorems 3.1.1-3.1.4 we can show that Ω(v) = {0} for every (f )-good function v : [0, ∞) → Rn . Example 2. Fix a number q > 0 and consider an integrand g(x, u) = q|x|2 |x − e|2 + |u|2 , x, u ∈ Rn where e = (1, 1, . . . , 1) ∈ Rn . It is easy to see that g ∈ A if a is large enough. Clearly the function v1 (t) = 0, v2 (t) = e, t ∈ [0, ∞) are (g)-good and g does not have the asymptotic turnpike property (see Theorem 3.1.1).

Chapter 4 INFINITE HORIZON AUTONOMOUS PROBLEMS

In this chapter we study the structure of weakly optimal solutions of inﬁnite-horizon autonomous variational problems with vector-valued functions and with integrands which belong to the space A introduced in Section 3.1. The results of this chapter have been established in [97].

4.1.

Main results

Denote by | · | the Euclidean norm in Rn . Let a > 0 be a constant and let ψ : [0, ∞) → [0, ∞) be an increasing function such that ψ(t) → +∞ as t → ∞. We consider the space A introduced in Section 3.1 which consists of all continuous functions f : Rn × Rn → R1 which satisfy the following assumptions: A(i) for each x ∈ Rn the function f (x, ·) : Rn → R1 is convex; A(ii) f (x, u) ≥ max{ψ(|x|), ψ(|u|)|u|} − a for each (x, u) ∈ Rn × Rn ; A(iii) for each M, > 0 there exist Γ, δ > 0 such that if u1 , u2 , x1 , x2 ∈ Rn satisfy |xi | ≤ M, |ui | ≥ Γ, i = 1, 2,

max{|x1 − x2 |, |u1 − u2 |} ≤ δ,

then |f (x1 , u1 ) − f (x2 , u2 )| ≤ max{f (x1 , u1 ), f (x2 , u2 )}. It is easy to show that an integrand f = f (x, u) ∈ C 1 (Rn × Rn ) belongs to A if f satisﬁes Assumptions A(i), A(ii), and also there exists an increasing function ψ0 : [0, ∞) → [0, ∞) such that for each x, u ∈ Rn , sup{|∂f /∂x(x, u)|, |∂f /∂u(x, u)|} ≤ ψ0 (|x|)(1 + ψ(|u|)|u|).

116

TURNPIKE PROPERTIES

For the set A we consider the uniformity introduced in Section 3.1. This uniformity is determined by the following base: E(N, , λ) = {(f, g) ∈ A × A : |f (x, u) − g(x, u)| ≤ for each x, u ∈ Rn satisfying |x|, |u| ≤ N, (|f (x, u)| + 1)(|g(x, u)| + 1)−1 ∈ [λ−1 , λ] for each x, u ∈ Rn satisfying |x| ≤ N }, where N > 0, > 0, λ > 1 [37]. It is known that the uniform space A is metrizable and complete (see Section 3.1). This uniformity induces a topology in A which is called the weak topology. We consider functionals of the form f

I (T1 , T2 , x) =

T2 T1

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

(1.1)

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

(1.2)

is an a.c. function satisfying x(T1 ) = y, x(T2 ) = z}. It is easy to see that −∞ < U f (T1 , T2 , y, z) < +∞ for each f ∈ A, each y, z ∈ Rn and all numbers T1 , T2 satisfying 0 ≤ T1 < T2 . Let f ∈ A. For any a.c. function x : [0, ∞) → Rn we set J(x) = lim inf T −1 I f (0, T, x). T →∞

(1.3)

Of special interest is the minimal long-run average cost growth rate µ(f ) = inf{J(x) : x : [0, ∞) → Rn is an a.c. function}.

(1.4)

Clearly −∞ < µ(f ) < +∞. Recall the deﬁnition of a good function given in Section 1.2 and the deﬁnition of a weakly optimal function introduced in Section 3.1. An a.c. function x : [0, ∞) → Rn is called an (f )-good function if the function φfx : T → I f (0, T, x) − µ(f )T , T ∈ (0, ∞) is bounded. An a.c. function x : [0, ∞) → Rn is called (f )-weakly optimal if for any a.c. function y : [0, ∞) → Rn satisfying y(0) = x(0), T

lim inf T →∞

0

[f (x(t), x (t)) − f (y(t), y (t))]dt ≤ 0.

Inﬁnite horizon autonomous problems

117

Proposition 3.1.1 and Theorem 1.2.2 imply the following result. Proposition 4.1.1 For any a.c. function x : [0, ∞) → Rn either I f (0, T, x) − T µ(f ) → ∞ as T → ∞ or

sup{|I f (0, T, x) − T µ(f )| : T ∈ (0, ∞)} < ∞.

Moreover any (f )-good function x : [0, ∞) → Rn is bounded. We denote d(x, B) = inf{|x−y| : y ∈ B} for x ∈ Rn , B ⊂ Rn . Denote by dist(A, B) the Hausdorﬀ metric for two sets A ⊂ Rn , B ⊂ Rn . For every bounded a.c. function x : [0, ∞) → Rn deﬁne Ω(x) = {y ∈ Rn : there exists a sequence {ti }∞ i=0 ⊂ (0, ∞) for which ti → ∞, x(ti ) → y as i → ∞}.

(1.5)

We say that an integrand f ∈ A has the asymptotic turnpike property if Ω(v1 ) = Ω(v2 ) for each pair of (g)-good functions vi : [0, ∞) → Rn , i = 1, 2. Let f ∈ A have the asymptotic turnpike property. Put H(F ) = Ω(v) where v : [0, ∞) → Rn is an (f )-good function. Clearly, H(f ) does not depend on v. By Theorem 1.2.1, H(f) is a compact subset of Rn . We say that H(f ) is the turnpike of f . Consider any f ∈ A. In analyzing the inﬁnite-horizon variational problem with the integrand f we study the function U f (T1 , T2 , y, z) (T2 > T1 ≥ 0, y, z ∈ Rn ) deﬁned by (1.2). By Theorems 3.6.1 and 3.6.2, U f (0, T, x, y) = T µ(f )+π f (x)−π f (y)+ θ¯Tf (x, y), x, y ∈ Rn , T ∈ (0, ∞), (1.6) f f n 1 ¯ where π : R → R is a continuous function and (T, x, y) → θT (x, y) ∈ R1 is a continuous function deﬁned for T > 0, x, y ∈ Rn , π f (x) = inf{lim inf [I f (0, T, v) − µ(f )T ] : v : [0, ∞) T →∞

→ Rn is an a.c. function satisfying v(0) = x}, x ∈ Rn ,

(1.7)

and for every T > 0 and every x ∈ Rn there is y ∈ Rn satisfying θ¯Tf (x, y) = 0.

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TURNPIKE PROPERTIES

Assume that the integrand f has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . (By Theorem 3.1.1 a generic f ∈ A has the asymptotic turnpike property.) By Theorems 3.6.3 and 3.6.4 for every x ∈ Rn there exists an (f )-good function v : [0, ∞) → Rn such that v(0) = x and the relation I f (T1 , T2 , v) = (T2 − T1 )µ(f ) + π f (v(T1 )) − π f (v(T2 ))

(1.8)

holds for each T1 ∈ [0, ∞), T2 ∈ (T1 , ∞), and moreover, each a.c. function v : [0, ∞) → Rn such that (1.8) holds for each T1 ∈ [0, ∞), T2 ∈ (T1 , ∞) is an (f )-weakly optimal function. For each function f ∈ A denote by A(f ) the set of all a.c. functions v : [0, ∞) → Rn which satisfy (1.8) for each T1 ∈ [0, ∞), T2 ∈ (T1 , ∞). If f ∈ A has the turnpike property, then by Theorem 3.6.4 any function v ∈ A(f ) is an (f )-weakly optimal function. We establish the following results. Theorem 4.1.1 Assume that f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Then f is a continuity point of the mapping g → (µ(g), π g ) ∈ R1 × C(Rn ), g ∈ A, where C(Rn ) is the space of all continuous functions φ : Rn → R1 with the topology of the uniform convergence on bounded subsets and A is equipped with the weak topology. Theorems 4.1.2 and 4.1.3 establish turnpike properties of good functions which satisfy (1.8) for a generic f ∈ A. Theorem 4.1.2 Assume that f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Let be a positive number. Then there exist numbers L, δ > 0 and a neighborhood U of f in A with the weak topology such that the following property holds: If g ∈ U and if v ∈ A(g) satisﬁes d(v(0), H(f )) ≤ δ, then dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤

(1.9)

for each T ≥ 0. Theorem 4.1.2 shows that if f has the asymptotic turnpike property, g belongs to a small neighborhood of f , v ∈ A(g) and v(0) is close to the turnpike H(f ), then v is close to the turnpike H(f ) at any point t ≥ 0. The following result shows that the convergence property established in Theorem 4.1.2 holds for all T ≥ Q where Q is a positive constant, if we do not assume that v(0) is close to the turnpike H(f ).

Inﬁnite horizon autonomous problems

119

Theorem 4.1.3 Assume that f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Let and K be positive numbers. Then there exist numbers L, Q > 0 and a neighborhood U of f in A with the weak topology such that if g ∈ U and if v ∈ A(g) satisﬁes |v(0)| ≤ K, then the inequality (1.9) is valid for all T ∈ [Q, ∞). For each f ∈ A denote by B(f ) the set of all a.c. functions v : → Rn such that (1.8) holds for each T1 ∈ R1 , T2 ∈ (T1 , ∞), and lim inf t→−∞ |v(t)| < ∞. R1

Theorem 4.1.4 Assume that f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Then the following properties hold: (1) for each h ∈ H(f ) there exists v ∈ B(f ) satisfying v(0) = h; (2) for each v ∈ B(f ) the relation v(t) ∈ H(f ) holds for all t ∈ R1 ; (3) for each > 0 there exists L > 0 such that (1.9) holds for each v ∈ B(f ) and each T ∈ R1 . Chapter 4 is organized as follows. Theorems 4.1.1-4.1.3 are proved in Section 4.2. Section 4.3 contains the proof of Theorem 4.1.4.

4.2.

Proofs of Theorems 4.1.1-4.1.3

We preface the proofs of Theorems 4.1.1-4.1.3 by a number of auxiliary results. Theorem 3.6.1, Proposition 3.5.3 and equation (6.2) in Section 3.6 imply the following result. Proposition 4.2.1 Let f ∈ A. Then π f (x) → ∞ as |x| → ∞. Assume that f ∈ A has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . For each g ∈ A, each r1 ≥ 0, r2 > r1 and each a.c. function v : [r1 , r2 ] → Rn set σ g (r1 , r2 , v) = I g (r1 , r2 , v) − π f (v(r1 )) + π f (v(r2 )), σ ˜ g (r1 , r2 , v) = I g (r1 , r2 , v) − π g (v(r1 )) + π g (v(r2 )), Φg (r1 , r2 , v) = I g (r1 , r2 , v) − U g (r1 , r2 , v(r1 ), v(r2 )).

(2.1)

Lemma 3.8.2 and Theorem 1.2.3 imply the following result which will be used in the proof of Theorem 3.1.3.

120

TURNPIKE PROPERTIES

Lemma 4.2.1 Let 0 ∈ (0, 1), K0 , M0 > 0 and let l be a positive integer such that each (f )-good function x : [0, ∞) → Rn satisﬁes dist(H(f ), {x(t) : t ∈ [T, T + l]}) ≤ 8−1 0 for all large T . Then there exist an integer N ≥ 10 and a neighborhood U of f in A with the weak topology and a number M1 > 0 such that the following property holds: If g ∈ U, T1 ≥ 0, T2 ≥ T1 + N l and if an a.c. function x : [T1 , T2 ] → Rn satisﬁes |x(Ti )| ≤ K0 , i = 1, 2, Φg (T1 , T2 , x) ≤ M0 ,

(2.2)

then |x(t)| ≤ M1 for all t ∈ [T1 , T2 ]; for each S ∈ [T1 , T2 − N l] there exists an integer i0 ∈ [0, N − 8] such that for all T ∈ [S + i0 l, S + (i0 + 7)l], dist(H(f ), {x(t) : t ∈ [T, T + l]}) ≤ 0 . Put Df = sup{|h| : h ∈ H(f )}.

(2.3)

For each g ∈ A denote by A(g) the set of all a.c. functions v : [0, ∞) → Rn such that (2.4) σ ˜ g (T1 , T2 , v) = (T2 − T1 )µ(g) for each T1 ∈ [0, ∞), T2 ∈ (T1 , ∞). For K, τ > 0 and g ∈ A put l(g, K, τ ) = inf{U g (0, τ, x, y) − π f (x) + π f (y) : x, y ∈ Rn , |x|, |y| ≤ K}. (2.5) It follows from the representation formula (see (1.6), (1.7)) and Theorem 3.6.3 that (2.6) l(f, K, τ ) = µ(f )τ, τ > 0, K > Df . Equations (2.5), (2.6) and Proposition 1.3.9 imply the following result. Lemma 4.2.2 Let K > Df , 0 < τ1 < τ2 , δ > 0. Then there exists a neighborhood U of f in A with the weak topology such that for each g ∈ U and each τ ∈ [τ1 , τ2 ], |l(g, K, τ ) − τ µ(f )| ≤ δ. Lemma 4.2.3 For each h ∈ H(f ) there exists an (f )-good function v : [0, ∞) → H(f ) such that v(0) = h and v ∈ A(f ).

Inﬁnite horizon autonomous problems

121

Proof. Let h ∈ H(f ). By Proposition 4.2.1 there exists an (f )-good function u ∈ A(f ). We may assume that d(u(t), H(f )) ≤ 1 for all t ∈ [0, ∞).

(2.7)

There exists a sequence of positive numbers {Tp }∞ p=0 such that Tp+1 ≥ Tp + 1, p = 0, 1, . . . , u(Tp ) → h as p → ∞.

(2.8)

For every integer p ≥ 1 we set vp (t) = u(t + Tp ), t ∈ [0, ∞).

(2.9)

By Proposition 1.3.5, (2.9) and (2.7) there exist a subsequence {vpj }∞ j=1 and an a.c. function v : [0, ∞) → Rn such that for every integer N ≥ 1, vpj (t) → v(t) as j → ∞ uniformly in [0, N ],

(2.10)

I f (0, N, v) ≤ lim inf I f (0, N, vpj ).

(2.11)

j→∞

Since f has the asymptotic turnpike property with the turnpike H(f ) we have Ω(v) = H(f ). It follows from (2.9) and (2.10) that v(t) ∈ H(f ) for all t ∈ [0, ∞). By (2.8)-(2.10), v(0) = h. Since u ∈ A(f ) it follows from (2.9)-(2.11) that for each integer N ≥ 1, I f (0, N, v) ≤ lim inf I f (Tpj , Tpj + N, u) j→∞

≤ lim inf [N µ(f ) + π f (u(Tpj )) − π f (u(Tpj + N ))] j→∞

= N µ(f ) + π f (v(0)) − π f (v(N )). Together with the representation formula (see (1.6), (1.7)) this implies that v ∈ A(f ). The lemma is proved. In the next two lemmas we study the existence of a function x : [0, T ] → Rn such that σ g (0, T, x) ≤ l(g, K, T ) + where is a given small positive number, g belongs to a small neighborhood of f in A with the weak topology and x(0), x(T ) belong to the turnpike H(f ). Lemma 4.2.4 For each ∈ (0, 1) and each K > Df + 1 there exists a neighborhood U of f in A with the weak topology such that the following property holds: If g ∈ U and T ≥ 1, then there exists an a.c. function x : [0, T ] → Rn for which x(0), x(T ) ∈ H(f ) and σ g (0, T, x) ≤ l(g, K, T ) + .

(2.12)

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Proof. Let ∈ (0, 1) and K > Df + 1. By Lemma 4.2.3 there exists an (f )-good function v0 : [0, ∞) → H(f ) such that v0 ∈ A(f ).

(2.13)

By Theorem 1.2.3 there exist a number M1 > 0 and a neighborhood U1 of f in A with the weak topology such that the following property holds: If g ∈ U1 , T1 ∈ [0, ∞), T2 ∈ [T1 + 8−1 , ∞) and if an a.c. function x : [T1 , T2 ] → Rn satisﬁes |x(Ti )| ≤ K, i = 1, 2, Φg (T1 , T2 , x) ≤ 1,

(2.14)

|x(t)| ≤ M1 , t ∈ [T1 , T2 ].

(2.15)

then It follows from Theorem 3.4.1 that there exists a number δ ∈ (0, 8−1 )

(2.16)

such that the following property holds: If y1 , y2 , z1 , z2 ∈ Rn satisfy |yi |, |zi | ≤ M1 + 2K + 1, i = 1, 2, |yi − zi | ≤ 4δ, i = 1, 2, then

(2.17)

|U f (0, 1, y1 , y2 ) − U f (0, 1, z1 , z2 )| ≤ 2−6 , |π f (yi ) − π f (zi )| ≤ 2−6 , i = 1, 2.

(2.18)

By Theorem 3.3.1 there exists an integer L ≥ 1 such that if v : [0, ∞) → Rn is an (f )-good function, then for all large T , dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 8−1 δ.

(2.19)

It follows from the choice of L and Lemma 4.2.1 that there exist a neighborhood U2 of f in A with the weak topology and an integer N1 ≥ 10 such that the following property holds: If g ∈ U2 , T1 ≥ 0, T2 ≥ T1 + N1 L and if an a.c. function x : [T1 , T2 ] → n R satisﬁes (2.14), then for each S ∈ [T1 , T2 − N1 L] there exists an integer i0 ∈ [0, N1 − 8] such that for all T ∈ [S + i0 L, S + (i0 + 7)L] dist(H(f ), {x(t) : t ∈ [T, T + L]}) ≤ δ.

(2.20)

Since f has the asymptotic turnpike property we have Ω(v0 ) = H(f ). Then it follows from (2.13) that there exist integers N2 ≥ 4N1 L + 4, N3 ≥ 8

(2.21)

Inﬁnite horizon autonomous problems

such that

123

dist(H(f ), {v0 (t) : t ∈ [0, N2 L]}) ≤ 8−1 δ,

dist(H(f ), {v0 (t) : t ∈ [8(N2 +1)L, 8(N2 +1)L+N3 L]}) ≤ 8−1 δ. (2.22) Fix an integer N0 ≥ 8L(N1 + N2 + N3 + 4).

(2.23)

It follows from Proposition 1.3.9 that there exists a neighborhood U3 of f in A with the weak topology such that |U f (0, 1, y, z) − U g (0, 1, y, z)| ≤ 2−6

(2.24)

for each g ∈ U3 and each z, y ∈ Rn which satisfy |y|, |z| ≤ 4M1 + 4K + 4. It follows from Proposition 1.3.8 that there exists a neighborhood U4 of f in A with the weak topology such that the following property holds: If g ∈ U4 , T1 ≥ 0, T2 ∈ [T1 + 1, T1 + N0 + 1] and if and a.c. function x : [T1 , T2 ] → Rn satisﬁes min{I f (T1 , T2 , x), I g (T1 , T2 , x)}

then

≤ N0 (|µ(f )| + 1) + 2 sup{|π f (h)| : h ∈ H(f )},

(2.25)

|I f (T1 , T2 , x) − I g (T1 , T2 , x)| ≤ 2−6 .

(2.26)

It follows from Lemma 4.2.2 that there exists a neighborhood U5 of f in A with the weak topology such that |l(g, K, τ ) − µ(f )τ | ≤ 2−6

(2.27)

for each g ∈ U5 and each τ ∈ [1, 8(N0 + 1)]. Put U = ∩5i=1 Ui .

(2.28)

Let g ∈ U and a number T ≥ 1. There are two cases: (i) T ≤ N0 ; (ii) T > N0 . Consider the case (i). Put x(t) = v0 (t), t ∈ [0, T ].

(2.29)

By the deﬁnition of A(f ) (see (2.4)) and (2.13), I f (0, T, x) ≤ N0 |µ(f )| + 2 sup{|π f (h)| : h ∈ H(f )}. It follows from this inequality and the choice of U4 that |I f (0, T, x) − I g (0, T, x)| ≤ 2−6 .

(2.30)

124

TURNPIKE PROPERTIES

It follows from the choice of U5 that |l(g, K, T ) − µ(f )T | ≤ 2−6 .

(2.31)

Combining (2.30), (2.29) and (2.13) we obtain that σ g (0, T, x) ≤ σ f (0, T, v0 ) + 2−6 ≤ µ(f )T + 2−6 ≤ l(g, K, T ) + 2−5 . Therefore in the case (i) the assertion of the lemma is valid. Consider the case (ii). Then T > N0 .

(2.32)

There exists an a.c. function y : [0, T ] → Rn such that |y(0)|, |y(T )| ≤ K, σ g (0, T, y) ≤ l(g, K, T ) + 16−1 δ. We have

(2.33)

Φg (0, T, y) ≤ 16−1 δ.

It follows from this inequality, (2.23), (2.32), (2.33), and the choice of N1 , U2 (see (2.20)) that there exist integers i1 , i2 ∈ [0, N1 − 8] such that dist(H(f ), {y(t) : t ∈ [S, S + L]}) ≤ δ

(2.34)

for each S ∈ [i1 L, (i1 + 7)L] ∪ [T − 2N1 L + i2 L, T − 2N1 L + (i2 + 7)L]. (2.35) By (2.34), (2.35) and (2.22) there exist t1 ∈ [8(N2 + 1)L, 8(N2 + 1)L + N3 L], t2 ∈ [0, N2 L]

(2.36)

such that |y(i1 L+1)−v0 (t1 )| ≤ δ+4−1 δ, |y(T −2N1 L+i2 L+1)−v0 (t2 )| ≤ δ+4−1 δ. (2.37) By Corollary 1.3.1, (2.13), (2.36), (2.21), (2.32), and (2.23) there exists an a.c. function x : [0, T ] → Rn such that x(t) = v0 (t + t1 − ii L − 1), t ∈ [0, i1 L + 1), x(t) = y(t), t ∈ [i1 L + 2, T − 2N1 L + i2 L], x(t) = v0 (t+t2 −(T −2N1 L+i2 L+1)), t ∈ [T −2N1 L+i2 L+1, T ], (2.38) I g (τ, τ + 1, x) = U g (0, 1, x(τ ), x(τ + 1)), τ = i1 L + 1, T − 2N1 L + i2 L. (2.39)

125

Inﬁnite horizon autonomous problems

Put τ0 = 0, τ1 = i1 L + 1, τ2 = i1 L + 2, τ3 = T − 2N1 L + i2 L, τ4 = T − 2N1 L + i2 L + 1, τ5 = T.

(2.40)

(2.39) and (2.40) imply that σ g (0, T, x) − σ g (0, T, y) =

1

[σ g (τi , τi+1 , x) − σ g (τi , τi+1 , y)]

i=0

+

4

[σ g (τi , τi+1 , x) − σ g (τi , τi+1 , y)].

(2.41)

i=3

Analogously to the case (i) we can show that σ g (t1 − i1 L − 1, t1 , v0 ) ≤ l(g, K, i1 L + 1) + 2−5 , σ g (t2 , t2 + 2N1 L + i2 L + 1, v0 ) ≤ l(g, K, 2N1 L + i2 L + 1) + 2−5 . By these inequalities and (2.38)-(2.40), σ g (τi , τi+1 , x) ≤ l(g, K, τi+1 − τi ) + 2−5 , i = 0, 4.

(2.42)

It follows from (2.13), (2.16), (2.33), (2.37) and (2.40) that σ g (τi , τi+1 , y) ≥ l(g, K, τi+1 − τi ), i = 0, 4.

(2.43)

(2.38)-(2.40) imply that for i = 1, 3, σ g (τi , τi+1 , x) − σ g (τi , τi+1 , y) ≤ U g (0, 1, x(τi ), x(τi+1 )) − π f (x(τi )) + π f (x(τi+1 )) −[U g (0, 1, y(τi ), y(τi+1 )) − π f (y(τi )) + π f (y(τi+1 ))].

(2.44)

By the choice of U1 and M1 (see (2.14), (2.15)) and (2.33), |y(t)| ≤ M1 , t ∈ [0, T ].

(2.45)

It follows from (2.13), (2.37), (2.38), (2.40), and the choice of δ (see (2.16)-(2.18)) that for i = 1, 3, |π f (x(τi+1 )) − π f (x(τi )) − (π f (y(τi+1 )) − π f (y(τi )))| ≤ 2−6 .

(2.46)

By (2.13), (2.37), (2.38), (2.40), (2.45), the choice of U3 (see (2.24)) and the choice of δ (see (2.16)-(2.18)) for i = 1, 3 |U g (0, 1, x(τi ), x(τi+1 )) − U g (0, 1, y(τi ), y(τi+1 ))|

126

TURNPIKE PROPERTIES

≤ 2−5 + |U f (0, 1, x(τi ), x(τi+1 )) − U f (0, 1, y(τi ), y(τi+1 ))| ≤ 2−4 . Combined with (2.44) and (2.46) this inequality implies that for i = 1, 3, σ g (τi , τi+1 , x) − σ g (τi , τi+1 , y) ≤ 2−3 . This inequality and (2.41)-(2.43) imply that σ g (0, T, x) − σ g (0, T, y) ≤ 2−1 . Combined with (2.16), (2.33) and (2.39) this inequality implies (2.12). This completes the proof of the lemma. The following auxiliary result generalizes Lemma 4.2.4. It shows the existence of x : [0, T ] → Rn satisfying (2.48) and x(0) = h for any h ∈ H(f ) while Lemma 4.2.4 establishes the existence of such x only for a certain h ∈ H(f ). Lemma 4.2.5 For each ∈ (0, 1) and each K > Df + 1 there exists a neighborhood U of f in A with the weak topology such that the following property holds: For each g ∈ U, each h ∈ H(f ) and each number T ≥ 1 there exists an a.c. function x : [0, T ] → Rn such that x(0) = h, x(T ) ∈ H(f ),

(2.47)

σ g (0, T, x) ≤ l(g, M1 , T ) + .

(2.48)

Proof. Let ∈ (0, 1) and K > Df + 1. It follows from Theorem 1.2.3 that there exist a number M1 > 2K + 1 and a neighborhood U1 of f in A with the weak topology such that the following property holds: If g ∈ U1 , T1 ∈ [0, ∞), T2 ∈ [T1 + 8−1 , ∞), and if an a.c. function x : [T1 , T2 ] → Rn satisﬁes |x(Ti )| ≤ K, i = 1, 2, Φg (T1 , T2 , x) ≤ 1,

(2.49)

|x(t)| ≤ M1 , t ∈ [T1 , T2 ].

(2.50)

then Lemma 4.2.4 implies that there exists a neighborhood U2 of f in A with the weak topology such that for each g ∈ U2 and each number T ≥ 1 there exists an a.c. function x : [0, T ] → Rn such that x(0), x(T ) ∈ H(f ), σ g (0, T, x) ≤ l(g, M1 , T ) + 2−6 .

(2.51)

It follows from Theorem 3.4.1 that there exists a number δ ∈ (0, 8−1 )

(2.52)

127

Inﬁnite horizon autonomous problems

such that if y1 , y2 , z1 , z2 ∈ Rn satisfy |yi |, |zi | ≤ 2M1 + 4, i = 1, 2, |yi − zi | ≤ 4δ, i = 1, 2, then

(2.53)

|U f (0, 1, y1 , y2 ) − U f (0, 1, z1 , z2 )| ≤ 2−6 , |π f (yi ) − π f (zi )| ≤ 2−6 , i = 1, 2.

(2.54)

It follows from Proposition 1.3.9 that there exists a neighborhood U3 of f in A with the weak topology such that |U f (0, 1, y, z) − U g (0, 1, y, z)| ≤ 2−6

(2.55)

for each g ∈ U3 and each y, z ∈ Rn which satisfy |y|, |z| ≤ 2M1 + 4. By Lemma 4.2.3 there exists an (f )-good function v0 : [0, ∞) → H(f ) such that v0 ∈ A(f ).

(2.56)

Since the integrand f has the asymptotic turnpike property with the turnpike H(f ) we have Ω(v0 ) = H(f ). Then it follows from (2.56) that there exist integers N1 , N2 ≥ 8 for which dist(H(f ), {v0 (t) : t ∈ [4, N1 + 4]}) ≤ 8−1 δ, dist(H(f ), {v0 (t) : t ∈ [2N1 + 16, 2N1 + 16 + N2 ]}) ≤ 8−1 δ.

(2.57)

Fix an integer N0 ≥ 8(N1 + N2 + 20).

(2.58)

It follows from Proposition 1.3.8 that there exists a neighborhood U4 of f in A with the weak topology such that the following property holds: If g ∈ U4 , T1 ≥ 0, T2 ∈ [T1 + 1, T1 + N0 ] and if an a.c. function x : [T1 , T2 ] → Rn satisﬁes min{I f (T1 , T2 , x), I g (T1 , T2 , x)}

then

≤ N0 |µ(f )| + 2 sup{|π f (z)| : z ∈ H(f )},

(2.59)

|I f (T1 , T2 , x) − I g (T1 , T2 , x)| ≤ 2−6 .

(2.60)

By Lemma 4.2.2 there exists a neighborhood U5 of f in A with the weak topology such that |l(g, M1 , T ) − µ(f )T | ≤ 2−6

(2.61)

for each g ∈ U5 and each T ∈ [1, 2N0 + 8]. Put U = ∩5i=1 Ui .

(2.62)

128

TURNPIKE PROPERTIES

Let g ∈ U, h ∈ H(f ), and a number T ≥ 1. There are two cases: (i) T ≤ N0 ; (ii) T > N0 . Consider the case (i). By Lemma 4.2.3 there exists an (f )-good function x : [0, ∞) → H(f ) such that x ∈ A(f ), x(0) = h. Analogously to the case (i) in the proof of Lemma 4.2.4 we can show that the relation (2.48) holds. Therefore in the case (i) the assertion of the lemma is valid. Consider the case (ii). Then T > N0 .

(2.63)

By (2.58), (2.63) and the choice of U2 (see (2.51)) there exists an a.c. function u : [0, T − 4(N1 + N2 + 5)] → Rn such that u(0), u(T − 4(N1 + N2 + 5)) ∈ H(f )),

(2.64)

σ g (0, T −4(N1 +N2 +5), u) ≤ l(g, M1 , T −4(N1 +N2 +5))+2−6 . (2.65) (2.57) and (2.64) imply that there exist t1 , t3 ∈ [4, N1 + 4], t2 ∈ [2N1 + 16, 2N1 + 16 + N2 ] for which

(2.66)

|h − v0 (t1 )| ≤ 4−1 δ, |u(0) − v0 (t2 )| ≤ 4−1 δ, |u(T − 4(N1 + N2 + 5)) − v0 (t3 )| ≤ 4−1 δ.

(2.67)

By Corollary 1.3.1, (2.58), (2.63) and (2.66) there exists an a.c. function x : [0, T ] → Rn such that x(0) = h, x(t) = v0 (t + t1 ), t ∈ [1, t2 − t1 − 1], x(t) = u(t − t2 + t1 ), t ∈ [t2 − t1 , T − 4(N1 + N2 + 5) + t2 − t1 ], x(t) = v0 (t + t3 − (T − 4(N1 + N2 + 5) + t2 − t1 )), t ∈ [T − 4(N1 + N2 + 5) + t2 − t1 + 1, T ],

(2.68)

Φg (τ, τ +1, x) = 0, τ = 0, t2 −t1 −1, T −4(N1 +N2 +5)+t2 −t1 . (2.69) Clearly (2.47) holds. We show that (2.48) holds. Put τ0 = 0, τ1 = 1, τ2 = t2 − t1 − 1, τ3 = t2 − t1 , τ4 = T − 4(N1 + N2 + 5) + t2 − t1 , τ5 = T − 4(N1 + N2 + 5) + t2 − t1 + 1, τ6 = T.

(2.70)

129

Inﬁnite horizon autonomous problems

There exists an a.c. function y : [0, T ] → Rn such that |y(0)|, |y(T )| ≤ K, σ g (0, T, y) ≤ l(g, K, T ) + 2−6 .

(2.71)

By the choice of U1 and M1 (see (2.49), (2.50)) and (2.71), |y(t)| ≤ M1 , t ∈ [0, T ].

(2.72)

It follows from (2.69) and (2.70) that σ g (0, T, x) − σ g (0, T, y) =

5

[σ g (τi , τi+1 , x) − σ g (τi , τi+1 , y)].

(2.73)

i=0

(2.72) and (2.69) imply that σ g (τi , τi+1 , y) ≥ l(g, M1 , τi+1 − τi ), i = 0, . . . , 5.

(2.74)

By (2.65), (2.68), (2.69) and (2.70), σ g (τ3 , τ4 , x) ≤ l(g, M1 , τ4 − τ3 ) + 2−6 .

(2.75)

Analogously to (2.42) (see the proof of Lemma 4.2.4) we can show that σ g (τi , τi+1 , x) ≤ l(g, M1 , τi+1 − τi ) + 2−5 , i = 1, 5.

(2.76)

It follows from (2.56), (2.64), (2.68)-(2.70) and the choice of U3 (see (2.55)) that for i = 0, 2.4, σ g (τi , τi+1 , x) = U g (0, 1, x(τi ), x(τi+1 )) − π f (x(τi )) + π f (x(τi+1 )) ≤ U f (0, 1, x(τi ), x(τi+1 )) − π f (x(τi )) + π f (x(τi+1 )) + 2−6 .

(2.77)

Put γ0 = t1 , γ2 = t2 − 1, γ4 = t3 .

(2.78)

Equations (2.78), (2.70), (2.68) and (2.67) imply that for i = 0, 2, 4, |x(τi ) − v0 (γi )|, |x(τi + 1) − v0 (γi + 1)| ≤ 4−1 δ.

(2.79)

By (2.56), (2.70), (2.77), (2.79) and the choice of δ (see (2.52)-(2.54)) for i = 0, 2, 4, σ g (τi , τi+1 , x) ≤ U f (0, 1, x(τi ), x(τi +1))−π f (x(τi ))+π f (x(τi +1))+2−6 ≤ U f (0, 1, v0 (γi ), v0 (γi + 1)) − π f (v0 (γi )) + π f (v0 (γi + 1)) +2−4 = µ(f ) + 2−4 .

130

TURNPIKE PROPERTIES

It follows from this relation and the choice of U5 that for i = 0, 2, 4, σ g (τi , τi+1 , x) ≤ l(g, M1 , τi+1 − τi ) + 2−4 + 2−6 . Combining this relation and (2.73)-(2.76) we obtain that σ g (0, T, x) − σ g (0, T, y) ≤ 2−6 + 2−4 + 3(2−6 + 2−4 ) ≤ 2−1 . Combined with (2.69) and (2.71) this implies (2.48). This completes the proof of the lemma. The next lemma establishes the turnpike property for a function v : [0, T ] → Rn such that v(0), v(T ) are close to the turnpike H(f ) and σ g (0, T, v) ≤ l(g, K, T ) + δ, where δ is small, T is large and g belongs to a small neighborhood of f in A. Lemma 4.2.6 Let ∈ (0, 1), K > Df + 1 and L be a positive integer such that for each (f )-good function v : [0, ∞) → Rn , dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 8−1

(2.80)

for all large T (the existence of L follows from Theorem 3.3.1). Then there exists a neighborhood U of f in A with the weak topology and δ ∈ (0, 1) such that the following property holds: If g ∈ U, T ∈ [L, ∞) and if an a.c. function v : [0, T ] → Rn satisﬁes d(v(0), H(f )) ≤ δ, d(v(T ), H(f )) ≤ δ,

(2.81)

σ g (0, T, v) ≤ l(g, K, T ) + δ,

(2.82)

then for every S ∈ [0, T − L], dist(H(f ), {v(t) : t ∈ [S, S + L]}) ≤ .

(2.83)

Proof. It follows from Lemma 3.8.4 that there exists δ1 ∈ (0, )

(2.84)

such that the following property holds: If T ∈ [L, ∞) and if an a.c. function v : [0, T ] → Rn satisﬁes d(v(0), H(f )) ≤ δ1 , d(v(T ), H(f )) ≤ δ1 , σ f (0, T, v) − T µ(f ) ≤ δ1 , (2.85) then the inequality (2.83) holds for every S ∈ [0, T − L]. Fix a number (2.86) δ ∈ (0, 8−1 δ1 ).

Inﬁnite horizon autonomous problems

131

It follows from Theorem 3.3.1 that there exists an integer L1 ≥ 1 such that for each (f )-good function v : [0, ∞) → Rn , dist(H(f ), {v(t) : t ∈ [T, T + L1 ]}) ≤ 8−1 δ

(2.87)

for all large T . We may assume that L1 ≥ 10L + 24.

(2.88)

It follows from Lemma 4.2.1 that there exist a neighborhood U1 of f in A with the weak topology and an integer N1 ≥ 10 such that the following property holds: If g ∈ U1 , T1 ≥ 0, T2 ≥ T1 +N1 L1 and if an a.c. function x : [T1 , T2 ] → Rn satisﬁes (2.89) |x(Ti )| ≤ K, i = 1, 2, Φg (T1 , T2 , x) ≤ 1, then for each S ∈ [T1 , T2 − N1 L1 ] there exists an integer i0 ∈ [0, N1 − 8] such that (2.90) dist(H(f ), {x(t) : t ∈ [T, T + L1 ]}) ≤ δ for all T ∈ [S + i0 L1 , S + (i0 + 7)L1 ]. It follows from Lemma 4.2.5 that there exists a neighborhood U2 of f in A with the weak topology such that if g ∈ U2 , T ≥ 1 and if h ∈ H(f ), then there exists an a.c. function x : [0, T ] → Rn such that x(0) = h, x(T ) ∈ H(f ), σ g (0, T, x) ≤ l(g, K, T ) + 8−1 δ.

(2.91)

N0 ≥ 100L1 N1 .

(2.92)

Choose an integer By Lemma 4.2.2 there exists a neighborhood U3 of f in A with the weak topology such that |l(g, K, τ ) − µ(f )τ | ≤ 4−1 δ

(2.93)

for each g ∈ U3 and each τ ∈ [1, N0 ]. It follows from Proposition 1.3.8 that there exists a neighborhood U4 of f in A with the weak topology such that the following property holds: If g ∈ U4 , T1 , T2 ≥ 0 satisfy T2 − T1 ∈ [1, 2N0 + 1] and if an a.c. function x : [T1 , T2 ] → Rn satisﬁes min{I f (T1 , T2 , x), I g (T1 , T2 , x)} ≤ 2N0 |µ(f )| + 4 + 2 sup{|π f (h)| : h ∈ Rn , |h| ≤ K + 2},

(2.94)

132

TURNPIKE PROPERTIES

then Put

|I f (T1 , T2 , x) − I g (T1 , T2 , x)| ≤ δ.

(2.95)

U = ∩4i=1 Ui .

(2.96)

Assume that g ∈ U, T ≥ L and an a.c. function v : [0, T ] → Rn satisﬁes (2.81) and (2.82). There are two cases: (i) T ≤ N0 ; (ii) T > N0 . Consider case (i). By the choice of U3 (see (2.93)) and (2.82), σ g (0, T, v) ≤ |µ(f )|T + 2δ.

(2.97)

It follows from the choice of U4 (see (2.94), (2.95)) and (2.81) that |I f (0, T, v) − I g (0, T, v)| ≤ δ. Combined with (2.86), (2.81) and (2.97) this inequality implies (2.85). It follows from (2.85) and the deﬁnition of δ1 (see (2.84)) that (2.83) holds for all S ∈ [0, T − L]. Therefore in case (i) the assertion of the lemma is valid. Consider case (ii). Then (2.98) T > N0 . It follows from (2.81), (2.82), (2.98), (2.91) and the deﬁnition of N1 and U1 (see (2.89), (2.90)) that there exists a sequence of numbers {Ti }qi=0 such that T0 = 0, Tq = T, Ti+1 −Ti ∈ [2L1 , 2(2N1 −6)L1 ], i = 0, . . . , q −1, (2.99) d(v(Ti ), H(f )) ≤ δ, i = 0, . . . , q.

(2.100)

For each a.c. function y : [0, T ] → Rn and each r ∈ [0, T ] we set σ g (r, r, y) = 0.

(2.101)

l(g, K, 0) = 0.

(2.102)

We set Let integers j, p ∈ [0, q], j < p. We estimate σ g (Tj , Tp , v) − l(g, K, Tp − Tj ). It follows from (2.99)-(2.102) and the choice of U2 (see (2.91)) that there exists an a.c. function y : [0, T ] → Rn such that y(Ti ) ∈ H(f ), i = 0, j, p, q, σ g (0, Tj , y) ≤ l(g, K, Tj ) + 8−1 δ, σ g (Tj , Tp , y) ≤ l(g, K, Tp − Tj ) + 8−1 δ,

133

Inﬁnite horizon autonomous problems

σ g (Tp , Tq , y) ≤ l(g, K, Tq − Tp ) + 8−1 δ. These relations, (2.82) and (2.99)-(2.102) imply that δ ≥ σ g (0, T, v) − σ g (0, T, y) = [σ g (0, Tj , v) − σ g (0, Tj , y)] + [σ g (Tj , Tp , v) − σ g (Tj , Tp , y)] +[σ g (Tp , Tq , v) − σ g (Tp , Tq , y)] ≥ σ g (Tj , Tp , v) − σ g (Tj , Tp , y) − 4−1 δ, σ g (Tj , Tp , v) ≤ δ + 4−1 δ + l(g, K, Tp − Tj ) + 8−1 δ.

(2.103)

We have shown that (2.103) holds for each pair of integers j, p ∈ [0, q] satisfying j < p. Let s ∈ [0, T − L]. By (2.99) and (2.88) there exist integers j, p ∈ [0, q] such that j < p, S ∈ [Tj , Tp − L], Tp − Tj ∈ [2L1 , 8N1 L1 ].

(2.104)

Evidently (2.103) holds. It follows from (2.92), (2.100), (2.103), (2.104) and the choice of U3 (see (2.93)) that σ g (Tj , Tp , v) ≤ µ(f )(Tp − Tj ) + 2δ. It follows from this inequality, the choice of U4 (see (2.94), (2.95)), (2.92), (2.99) and (2.104) that |I f (Tj , Tp , v) − I g (Tj , Tp , v)| ≤ δ, σ f (Tj , Tp , v) ≤ µ(f )(Tp − Tj ) + 3δ. By these inequalities, (2.99), (2.88), (2.86) and the choice of δ1 (see (2.84)), (2.83) holds. This completes the proof of the lemma. The proof of the next result is based on Lemmas 4.2.5 and 4.2.6. Lemma 4.2.7 Let ∈ (0, 1), K > Df + 4 and L be a positive integer such that for each (f )-good function v : [0, ∞) → Rn , dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 8−1 for all large T (the existence of L follows from Theorem 3.3.1). Then there exists a neighborhood U of f in A with the weak topology such that for each g ∈ U and each h ∈ H(f ) there exists an a.c. function x : [0, ∞) → Rn such that x(0) = h; for all S ∈ [0, ∞), dist(H(f ), {x(t) : t ∈ [S, S + L]}) ≤ ;

(2.105)

134

TURNPIKE PROPERTIES

for each T ≥ 1

σ g (0, T, v) ≤ l(g, K, T ) + ;

(2.106)

σ g (t1 , t2 , x) ≤ l(g, K, t2 − t1 ) +

(2.107)

for each t1 ≥ 1, t2 ≥ t1 + 1. Proof. It follows from Lemma 4.2.6 that there exist a neighborhood U1 of f in A with the weak topology and δ1 ∈ (0, 8−1 )

(2.108)

such that the following property holds: If g ∈ U1 , T ∈ [L, ∞), and if an a.c. function v : [0, T ] → Rn satisﬁes d(v(0), H(f )) ≤ δ1 , d(v(T ), H(f )) ≤ δ1 ,

(2.109)

σ g (0, T, v) ≤ l(g, K, T ) + δ1 , then the inequality (2.105) holds for every S ∈ [0, T − L]. By Lemma 4.2.5 there exists a neighborhood U2 of f in A with the weak topology such that for each g ∈ U2 , each h ∈ H(f ) and each number T ≥ 1 there exists an a.c. function v : [0, T ] → Rn such that v(0) = h, v(T ) ∈ H(f ), σ g (0, T, v) ≤ l(g, K, T ) + 8−1 δ1 .

(2.110)

U = U1 ∩ U2 .

(2.111)

Put Assume that g ∈ U and h ∈ H(f ). By the deﬁnition of U2 (see (2.110)) for each N ≥ 1 there exists an a.c. function xN : [0, N ] → Rn such that (2.110) holds with T = N , v = xN . It follows from the deﬁnition of U1 and δ1 (see (2.109)) that for each integer N ≥ L and each number S ∈ [0, N − L], dist(H(f ), {xN (t) : t ∈ [S, S + L]}) ≤ .

(2.112)

Let N ≥ L be an integer. Assume that an integer q ∈ {2, 3, 4}, {ti }qi=0 ⊂ [0, N ], t0 = 0, tq = N , ti+1 − ti ≥ 1, i = 0, . . . , q − 1. Equation (2.112), which holds for each S ∈ [0, N − L], implies that σ g (ti , ti+1 , xN ) ≥ l(g, K, ti+1 − ti ), i = 0, . . . , q − 1.

(2.113)

By the deﬁnition of U2 (see (2.110)) there exists an a.c. function y : [0, N ] → Rn for which y(ti ) ∈ H(f ), i = 0, . . . , q, σ g (ti , ti+1 , y)

135

Inﬁnite horizon autonomous problems

≤ l(g, K, ti+1 − ti ) + 8−1 δ1 , i = 0, . . . , q − 1. Together with (2.110) which holds with T = N , v = xN and (2.113) this implies that for each j ∈ [0, q − 1], −1

8

g

g

δ1 ≥ σ (0, N, xN ) − σ (0, N, y) =

q−1

[σ g (ti , ti+1 , xN ) − σ g (ti , ti+1 , y)]

i=0

≥ σ g (tj , tj+1 , xN ) − l(g, K, tj+1 − tj ) − 8−1 δ1 q. This implies that for each integer N ≥ L+3 and each τ1 ∈ {0}∪[1, N −2], τ2 ∈ [τ1 + 1, N − 1], σ g (τ1 , τ2 , xN ) ≤ l(g, K, τ2 − τ1 ) + 3 · 4−1 δ1 .

(2.114)

By (2.114), (2.112), which holds for each N ≥ L and each number S ∈ [0, N − L], and by Proposition 1.3.5 there exist a subsequence {xNp }∞ p=1 and an a.c. function x : [0, ∞) → Rn such that for each integer N ≥ 1,

and

xN p (t) → x(t) as p → ∞ uniformly in [0, N ]

(2.115)

I g (T1 , T2 , x) ≤ lim inf I g (T1 , T2 , xNp )

(2.116)

p→∞

for each T1 ∈ [1, ∞) ∪ {0 ], T2 ≥ T1 + 1. Equation (2.110), which holds with T = N , v = xN and (2.115) imply that x(0) = h. Equations (2.112) and (2.115) imply (2.105) for all S ∈ [0, ∞). Equations (2.114)-(2.116) and (2.108) imply (2.106) for each T ≥ 1 and (2.107) for each t1 ≥ 1, t2 ≥ t1 + 1. This completes the proof of the lemma. Lemma 4.2.8 sup{π f (h) : h ∈ H(f )} = 0. Proof. There exists h0 ∈ H(f ) for which π f (h0 ) ≥ π f (h), h ∈ H(f ).

(2.117)

Let v : [0, ∞) → Rn be an a.c. function and let v(0) = h0 .

(2.118)

We will show that lim inf T →∞ [I f (0, T, v) − T µ(f )] ≥ 0. By Proposition 4.1.1 we may assume that v is an (f )-good function. Then Ω(v) = H(f ). It follows from this relation, the representation formula (see (1.6)), (2.117), and (2.118) that lim inf [I f (0, T, v) − T µ(f )] ≥ lim inf [π f (v(0)) − π f (v(T ))] ≥ 0. T →∞

T →∞

136

TURNPIKE PROPERTIES

This implies that π f (h0 ) ≥ 0. By Theorem 3.6.3 there exists an (f )good function u ∈ A(f ) satisfying u(0) = h0 . It is easy to see that Ω(v) = H(f ), lim inf [I f (0, T, u) − T µ(f )] = lim inf [π f (u(0)) − π f (u(T ))] = 0. T →∞

T →∞

This completes the proof of the lemma. The next lemma establishes the continuity of the function f → µ(f ), f ∈ A. Lemma 4.2.9 Let ∈ (0, 1). Then there exists a neighborhood U of f in A with the weak topology such that |µ(f ) − µ(g)| ≤ for each g ∈ U. Proof. By Theorem 1.2.2 there exist a neighborhood U1 of f in A with the weak topology and a number M0 > 0 such that for each g ∈ U1 and each (g)-good function x : [0, ∞) → Rn , lim sup |x(t)| < M0 . t→∞

(2.119)

Set M1 = sup{|U f (0, 1, x, y)| : x, y ∈ Rn , |x|, |y| ≤ 2M0 + 2}.

(2.120)

It follows from Proposition 1.3.8 that there exists a neighborhood U2 of f in A with the weak topology such that the following property holds: If g ∈ U2 , T ≥ 0, and if an a.c. function y : [T, T + 1] → Rn satisﬁes min{I f (T, T + 1, y), I g (T, T + 1, y)} ≤ 2M1 + 4, then

|I f (T, T + 1, y) − I g (T, T + 1, y)| ≤ /8.

(2.121)

It follows from Proposition 1.3.9 that there exists a neighborhood U3 of f in A with the weak topology such that the following property holds: If g ∈ U3 and if x, y ∈ Rn satisfy |y|, |x| ≤ 2M0 + 2, then

Put

|U f (0, 1, x, y) − U g (0, 1, x, y)| ≤ /16.

(2.122)

U = ∩3i=1 Ui .

(2.123)

Rn

be a (g1 )-good function. We

Let g1 , g2 ∈ U and let x : [0, ∞) → have

sup{|I g1 (0, T, x) − T µ(g1 )| : T ∈ (0, ∞)} < ∞.

(2.124)

137

Inﬁnite horizon autonomous problems

By the deﬁnition of U1 , M0 (see (2.119)) we may assume that |x(t)| ≤ M0 , t ∈ [0, ∞).

(2.125)

By Proposition 4.1.1 we may assume without loss of generality that Φg1 (T, T + 1, x) ≤ 4−1 for all t ∈ [0, ∞). By this inequality, (2.120), (2.125) and the choice of U3 (see (2.122)), I g1 (T, T + 1, x) ≤ M1 + 2−1 for all T ∈ [0, ∞). It follows from this inequality and the choice of U2 that for each T ∈ [0, ∞), |I f (T, T + 1, x) − I g1 (T, T + 1, x)| ≤ 8−1 , I f (T, T + 1, x) ≤ M1 + 1, |I f (T, T + 1, x) − I g2 (T, T + 1, x)| ≤ 8−1 , |I g1 (T, T + 1, x) − I g2 (T, T + 1, x)| ≤ 4−1 .

(2.126)

Equations (2.124) and (2.126) imply that sup{|I g2 (0, N, x) − 4−1 N − N µ(g)| : N = 1, 2, . . .} < ∞. Together with Proposition 4.1.1 this implies that µ(g2 ) ≤ µ(g1 ) + 4−1 . This completes the proof of the lemma. Lemma 4.2.10 Let K > Df + 1. Then there exists a neighborhood U of f in A with the weak topology such that l(g, K, τ ) ≤ τ µ(g) for each g ∈ U and each τ > 0. Proof. It follows from Theorem 3.1.3 that there exists a neighborhood U of f in A with the weak topology such that if g ∈ U and if x : [0, ∞) → Rn is an (g)-good function, then lim sup |x(t)| < K. t→∞

Let g ∈ U, τ > 0 and let x : [0, ∞) → Rn be a (g)-good function. We may assume that |x(t)| ≤ K, t ∈ [0, ∞). (2.127) By (2.127) for each integer N ≥ 1, σ g (0, N τ, x) =

N −1

[σ g (iτ, (i + 1)τ, x)] ≥ N l(g, K, τ ),

i=0

138

TURNPIKE PROPERTIES

I g (0, N τ, x) ≥ N l(g, K, τ ) + 2 sup{|π f (z)| : z ∈ Rn , |z| ≤ K}. Since x is a (g)-good function we have sup{N l(g, K, τ ) − N τ µ(g) : N = 1, 2, . . .} < ∞. This completes the proof of the lemma. There exists h∗ ∈ H(f ) such that π f (h∗ ) ≥ π f (h), h ∈ H(f ).

(2.128)

The next auxiliary result is an extension of Lemma 4.2.7. It shows the existence of a good function which in addition to the properties established in Lemma 4.2.7 has some other important properties. The proof is based on a number of results established in the book including Lemma 4.2.7. Lemma 4.2.11 For each ∈ (0, 1) and each K > Df + 4 there exist an integer L ≥ 1 and a neighborhood U of f in A with the weak topology such that if g ∈ U and if h ∈ H(f ), then there exists a (g)-good function v : [0, ∞) → Rn such that v(0) = h; (2.129) dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤

(2.130)

for all T ∈ [0, ∞); σ g (T1 , T2 , v) ≤ l(g, K, T2 − T1 ) +

(2.131)

for each T1 ∈ {0} ∪ [1, ∞), T2 ≥ T1 + 1; |σ g (0, T, v) − T µ(g)| ≤

(2.132)

| lim inf [I g (0, T, v) − T µ(g)] − π f (h)| ≤ 2.

(2.133)

for each T ∈ [1, ∞); T →∞

Proof. Let ∈ (0, 1) and K > Df + 4. It follows from (2.128) and Lemma 4.2.3 that there exists an (f )-good function v0 : [0, ∞) → H(f ) such that (2.134) v0 ∈ A(f ), v0 (0) = h∗ . It follows from Theorem 3.4.1 that there exists a positive number δ < 2−8 such that the following property holds:

(2.135)

Inﬁnite horizon autonomous problems

139

If y1 , y2 , z1 , z2 ∈ Rn satisfy |yi |, |zi | ≤ 2K + 8, i = 1, 2, |yi − zi | ≤ 4δ, i = 1, 2, then

|U f (0, 1, y1 , y2 ) − U f (0, 1, z1 , z2 )| ≤ 2−8 , |π f (yi ) − π f (zi )| ≤ 2−8 , i = 1, 2.

(2.136)

It follows from Proposition 1.3.9 that there exists a neighborhood U1 of f in A with the weak topology such that the following property holds: If g ∈ U1 and if y, z ∈ Rn satisfy |y|, |z| ≤ 2K + 8, then |U f (0, 1, y, z) − U g (0, 1, y, z)| ≤ 2−4 δ.

(2.137)

Since f has the asymptotic turnpike property with the turnpike H(f ) we have Ω(v0 ) = H(f ). Therefore there exist integers N1 , N2 ≥ 10 for which dist(H(f ), {v0 (t) : t ∈ [0, N1 ]}) ≤ 2−4 δ, dist(H(f ), {v0 (t) : t ∈ [4N1 , 4N1 + N2 ]}) ≤ 2−4 δ.

(2.138)

By Theorem 3.3.1 there exists an integer L ≥ 1 such that for each (f )good function v : [0, ∞) → Rn , dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 2−4 δ

(2.139)

for all large T . It follows from Lemma 4.2.7 that there exists a neighborhood U2 of f in A with the weak topology such that if g ∈ U2 and if h ∈ H(f ), then there exists an a.c. function v : [0, ∞) → Rn satisfying (2.129) and such that for each T ∈ [0, ∞), dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ δ,

(2.140)

σ g (T1 , T2 , v) ≤ l(g, K, T2 − T1 ) + δ

(2.141)

for each T1 ∈ {0} ∪ [1, ∞), T2 ≥ T1 + 1. Lemma 4.2.10 implies that there exists a neighborhood U3 of f in A with the weak topology such that l(g, K, T ) ≤ T µ(g)

(2.142)

for each g ∈ U3 and each T > 0. It follows from Lemma 4.2.9 that there exists a neighborhood U4 of f in A with the weak topology such that |µ(f ) − µ(g)| ≤ 2−8 δ(8N1 + 8N2 )−1

(2.143)

140

TURNPIKE PROPERTIES

for each g ∈ U4 . By Proposition 1.3.8 there exists a neighborhood U5 of f in A with the weak topology such that the following property holds: If g ∈ U5 , T1 ≥ 0, T2 ∈ [T1 +1, T1 +8(N1 +N2 )], and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes min{I f (T1 , T2 , v), I g (T1 , T2 , v)} ≤ 8(N1 +N2 )|µ(f )|+4+2 sup{|π f (z)| : z ∈ Rn , |z| ≤ 2K +2}, (2.144) then Put

|I f (T1 , T2 , v) − I g (T1 , T2 , v)| ≤ 2−8 δ.

(2.145)

U = ∩5i=1 Ui .

(2.146)

Let g ∈ U and h ∈ H(f ). It follows from the deﬁnition of U2 that there exists an a.c. function v : [0, ∞) → Rn such that (2.129) holds, relation (2.140) holds for each T ∈ [0, ∞), and relation (2.141) holds for each T1 ∈ {0} ∪ [1, ∞), T2 ≥ T1 + 1. By (2.141), which holds for T1 = 0 and each T2 ≥ 1, and the deﬁnition of U3 (see (2.142)) v is a (g)-good function. Fix a number T ≥ 1. We will establish (2.132). It follows from (2.140) and (2.138) that there exist t1 ∈ [0, N1 ], t2 ∈ [4N1 , 4N1 + N2 ]

(2.147)

for which |v(T ) − v0 (t1 )| ≤ δ + 8−1 δ, |h − v0 (t2 )| ≤ δ + 8−1 δ.

(2.148)

By Corollary 1.3.1 there exists an a.c. function x : [0, T +t2 −t1 ] → Rn such that x(t) = v(t), t ∈ [0, T ], x(t) = v0 (t + t1 − T ), t ∈ [T + 1, T + t2 − t1 − 1], x(T + t2 − t1 ) = h, I g (S, S + 1, x) = U g (0, 1, x(S), x(S + 1)), S = T, T + t2 − t1 − 1.

(2.149)

Equations (2.149) and (2.129) imply that I g (0, T + t2 − t1 , x) ≥ µ(g)(T + t2 − t1 ).

(2.150)

By (2.149), σ g (T, T + t2 − t1 , x) = U g (0, 1, x(T ), x(T + 1)) − π f (x(T ))

141

Inﬁnite horizon autonomous problems

+π f (x(T + 1)) + I g (t1 + 1, t2 − 1, v0 ) −π f (v0 (t1 + 1)) + π f (v0 (t2 − 1)) +U g (0, 1, x(T + t2 − t1 − 1), x(T + t2 − t1 )) −π f (x(T + t2 − t1 − 1)) + π f (x(T + t2 − t1 )).

(2.151)

Analogously to the proof of the case (i) in Lemma 4.2.4 (see (2.30) and (2.31)) we can show by using (2.134), (2.147), and the deﬁnition of U5 (see (2.145)) that σ g (t1 + 1, t2 − 1, v0 ) ≤ 2−8 δ + µ(f )(t2 − t1 − 2).

(2.152)

S1 = T, S2 = T + t2 − t1 − 1, r1 = t1 , r2 = t2 − 1.

(2.153)

Set

It follows from this relation, (2.149), the deﬁnition of U1 , δ (see (2.135)-(2.137)), (2.140), which holds for each T ≥ 0 and (2.134) that for i = 1, 2, U g (0, 1, x(Si ), x(Si + 1)) − π f (x(Si )) + π f (x(Si + 1)) ≤ U f (0, 1, x(Si ), x(Si + 1)) − π f (x(Si )) + π f (x(Si + 1)) + 2−4 δ ≤ U f (0, 1, v0 (ri ), v0 (ri + 1)) − π f (v0 (ri )) + π f (v0 (ri + 1)) + 2−6 ≤ µ(f ) + 2−6 . It follows from this relation, (2.147), (2.151)-(2.153) and the choice of U4 (see (2.143)) that σ g (T, T + t2 − t1 , x) ≤ 2µ(f ) + 2−5 + µ(f )(t2 − t1 − 2) + 2−8 ≤ 2−5 + µ(g)(t2 − t1 ) + 2−7 .

(2.154)

By (2.129), (2.149), (2.150) and (2.154), σ g (0, T, v) ≥ µ(g)T − 2−4 . This relation, (2.141), which holds with T1 = 0, T2 = T and (2.142) imply (2.132). Therefore we have shown that (2.132) holds for each T ∈ [1, ∞). Together with (2.129), this implies that ≥ | lim inf [I g (0, T, v) − T µ(g)] − lim inf [π f (h) − π f (v(T ))]|. (2.155) T →∞

T →∞

By (2.140), which holds for each T ∈ [0, ∞), and the deﬁnition of δ (see (2.135), (2.136)) | lim inf [π f (h) − π f (v(T ))] − [π f (h) − sup{π f (z) : z ∈ H(f )}]| ≤ 2−8 . T →∞

142

TURNPIKE PROPERTIES

Equation (2.133) now follows from this relation, (2.155), and Lemma 4.2.8. The lemma is proved. The following lemma is an auxiliary result for Theorem 4.1.1. Lemma 4.2.12 For each ∈ (0, 1) there exist δ ∈ (0, ) and a neighborhood U of f in A with the weak topology such that the followiing property holds: If g ∈ U, h ∈ H(f ) and if y ∈ Rn satisﬁes |y − h| ≤ δ, then |π g (y) − π f (y)| ≤ . Proof. Let ∈ (0, 1). Fix K > Df + 4.

(2.156)

It follows from Theorem 3.4.1 that there exists a positive number δ < 8−1

(2.157)

such that the following property holds: If x1 , x2 , y1 , y2 ∈ Rn satisfy |xi |, |yi | ≤ K + 2, i = 1, 2, |xi − yi | ≤ 8δ, i = 1, 2, then

(2.158)

|U f (0, 1, x1 , x2 ) − U f (0, 1, y1 , y2 )| ≤ 2−8 , |π f (xi ) − π f (yi )| ≤ 2−8 , i = 1, 2.

(2.159)

It follows from Proposition 1.3.9 that there exists a neighborhood U1 of f in A with the weak topology such that the following property holds: If g ∈ U1 and if y1 , y2 ∈ Rn satisfy |yi | ≤ 2K + 2, i = 1, 2, then |U f (0, 1, y1 , y2 ) − U g (0, 1, y1 , y2 )| ≤ 2−8 .

(2.160)

It follows from Lemma 4.2.11 that there exist an integer L ≥ 1 and a neighborhood U2 of f in A with the weak topology such that if g ∈ U2 and if h ∈ H(f ), then there exists a (g)-good function v : [0, ∞) → Rn such that v(0) = h; dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 2−6 δ

(2.161)

for all T ∈ [0, ∞); σ g (T1 , T2 , v) ≤ l(g, K, T2 − T1 )) + 2−6 δ

(2.162)

for each T1 ∈ {0} ∪ [1, ∞), T2 ≥ T1 + 1; |σ g (0, T, v) − T µ(g)| ≤ 2−6 δ

(2.163)

143

Inﬁnite horizon autonomous problems

for each T ∈ [1, ∞); | lim inf [I g (0, T, v) − T µ(g)] − π f (h)| ≤ 2−6 δ. T →∞

(2.164)

Lemmas 4.2.10 and 4.2.9 imply that there exists a neighborhood U3 of f in A with the weak topology such that |µ(g) − µ(f )| ≤ 2−6 , l(g, K, T ) ≤ T µ(g)

(2.165)

for each g ∈ U3 and each T ∈ (0, ∞). It follows from Theorem 3.1.3 that there exist an integer L1 ≥ 1 and a neighborhood U4 of f in A with the weak topology such that the following property holds: If g ∈ U4 and if v : [0, ∞) → Rn is a (g)-good function, then dist(H(f ), {v(t) : t ∈ [T, T + L1 ]}) ≤ 2−8 δ for all large T . Put

(2.166)

U = ∩4i=1 Ui .

(2.167)

g ∈ U, h ∈ H(f ), y ∈ Rn , |y − h| ≤ δ.

(2.168)

Assume that

By the deﬁnition of U2 , L and (2.168) there exists a (g)-good function v : [0, ∞) → Rn such that v(0) = h, (2.161) holds for each T ≥ 0, (2.162) holds for each T1 ∈ {0} ∪ [1, ∞) and each T2 ≥ T1 + 1, and (2.163) holds for each T ∈ [1, ∞). Together with (2.165) this implies that 0 ≤ T µ(g) − l(g, K, T ) ≤ 2−5 δ for all T ∈ [1, ∞).

(2.169)

Consider any (g)-good function u : [0, ∞) → Rn for which u(0) = y. By the deﬁnition of U4 , L1 , (2.168) holds with v = u for all large T . Together with (2.169) and (2.168), this implies that lim inf [I g (0, T, u) − T µ(g)] ≥ lim inf [I g (0, T, u) − l(g, K, T )] − 2−5 δ T →∞

T →∞

≥ lim sup[π f (y) − π f (u(T ))] − 2−5 δ. T →∞

(2.170)

It follows from (2.166), which holds with v = u for all large T ; (2.170); (2.168); the deﬁnition of δ (see (2.157)-(2.159)); and Lemma 3.8.2 that lim inf [π f (y) − π f (u(T ))] ≥ π f (h) − 2−8 − lim sup π f (u(T )) T →∞

T →∞

≥ π f (h) − 2−8 − sup{π f (z) : z ∈ H(f )} − 2−8

144

TURNPIKE PROPERTIES

≥ π f (h) − 2−7 . Together with (2.170) and (2.157) this implies that lim inf [I g (0, T, u) − T µ(g)] ≥ π f (h) − . T →∞

Therefore

π g (y) ≥ π f (h) − .

(2.171)

We will show that π g (y) ≤ π f (h) + . By Lemma 4.2.3 there exists an (f )-good function v0 : [0, ∞) → H(f ) such that v0 ∈ A(f ), v0 = h.

(2.172)

By the deﬁnition of U2 and L and (2.172) there exists a (g)-good function v1 : [0, ∞) → Rn such that v1 (0) = v0 (1).

(2.173)

(2.161) holds with v = v1 for each T ∈ [0, ∞); (2.162) holds with v = v1 for each T1 ∈ {0} ∪ [1, ∞), T2 ≥ T1 + 1; (2.163) holds with v = v1 for each T ∈ [1, ∞); and (2.164) holds with v = v1 , h = v0 (1). By Corollary 1.3.1 there exists an a.c. function w : [0, ∞) → Rn such that w(0) = y, w(t) = v1 (t−1), t ∈ [1, ∞), I g (0, 1, w) = U g (0, 1, w(0), w(1)). (2.174) Equations (2.173), (2.174) and (2.164), which holds with v = v1 , h = v0 (1), imply that π g (y) ≤ lim inf [I g (0, T, w) − T µ(g)] = U g (0, 1, y, v0 (1)) − µ(g) T →∞

+ lim inf [I g (0, T, v1 ) − T µ(g)] T →∞

g

= U (0, 1, y, v0 (1)) − µ(g) + π f (v0 (1)) + 2−6 δ.

(2.175)

It follows from (2.168), (2.156), (2.172), the deﬁnition of U1 (see (2.160)), and the deﬁnition of δ (see (2.157)-(2.159)) that U g (0, 1, y, v0 (1)) ≤ U f (0, 1, y, v0 (1)) +2−8 ≤ 2−8 + U f (0, 1, v0 (0), v0 (1)) + 2−8 = 2−7 + π f (h) − π f (v0 (1)) + µ(f ). Together with (2.175) and (2.165) this implies that π g (y) ≤ 2−7 +2−5 + π f (h) ≤ π f (h) + . This completes the proof of the lemma.

Inﬁnite horizon autonomous problems

145

There exists hf ∈ H(f ) such that π f (hf ) ≥ π f (h), h ∈ H(f ).

(2.176)

The next lemma establishes a useful formula for calculation of π g (x) where x ∈ Rn and g belongs to a small neighborhood of f . Lemma 4.2.13 Let ∈ (0, 1), K > Df + 4. Then there exist a neighborhood U of f in A with the weak topology and integers Q1 ≥ 8, Q2 ≥ 8+Q1 such that for each g ∈ U and each x ∈ Rn satisfying |x| ≤ K, π g (x) = inf{lim inf [I g (0, T, v) − T µ(g)] : T →∞

v : [0, ∞) → Rn is an a.c. function, v(0) = x, inf{|v(t) − hf | : t ∈ [Q1 , Q2 ]} ≤ }.

(2.177)

Proof. It follows from Theorem 3.1.3 that there exist an integer L ≥ 1 and a neighborhood U1 of f in A with the weak topology such that the following property holds: If g ∈ U1 and if v : [0, ∞) → Rn is a (g)-good function, then for all large T , (2.178) dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 16−1 . It follows from Lemma 4.2.1 and the choice of L that there exist an integer N ≥ 10 and a neighborhood U2 of f in A with the weak topology such that the following property holds: If g ∈ U2 , T1 ≥ 0, T2 ≥ T1 + N L, and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes |v(Ti )| ≤ K, i = 1, 2, Φg (T1 , T2 , v) ≤ 4, (2.179) then for each S ∈ [T1 , T2 − N L] there exists an integer i0 ∈ [0, N − 8] such that for all T ∈ [S + i0 L, S + (i0 + 7)L], dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 2−1 .

(2.180)

U = U1 ∩ U2 , Q1 = N L, Q2 = 2N L.

(2.181)

Put Assume that

g ∈ U, x ∈ Rn , |x| ≤ K.

Denote by E the set of all (g)-good functions v : [0, ∞) → Rn for which v(0) = x, lim inf [I g (0, T, v) − T µ(g)] ≤ π g (x) + 1. T →∞

(2.182)

146

TURNPIKE PROPERTIES

It is easy to see that π g (x) = inf{lim inf [I g (0, T, v) − T µ(g)] : v ∈ E}. T →∞

(2.183)

Consider any v ∈ E. By the deﬁnition of U1 , L |v(t)| ≤ K for all large t. By (2.182)

(2.184)

Φg (0, T, v) ≤ 2 for all T ∈ [1, ∞).

It follows from this relation, (2.184), (2.181), and the deﬁnition of N, U2 (see (2.179), (2.180)) that inf{|v(t) − hf | : t ∈ [Q1 , Q2 ]} ≤ . This completes the proof of the lemma. Proof of Theorem 4.1.1. By Lemma 4.2.9, f is a continuity point of the mapping g → µ(g), g ∈ A. We will show that f is a continuity point of the mapping g → π g , g ∈ A. Assume that ∈ (0, 1), K > Df + 4. Lemma 4.2.12 implies that there exist a neighborhood U1 of f in A with the weak topology and a positive number (2.185) δ < 16−1 such that

|π g (y) − π f (h)| ≤ 16−1

(2.186)

for each g ∈ U1 , each h ∈ H(f ), and each y ∈ Rn satisfying |y − h| ≤ δ. It follows from Lemma 4.2.13 that there exist a neighborhood U2 of f in A with the weak topology and integers Q1 ≥ 8, Q2 ≥ 8 + Q1 such that if g ∈ U2 and if x ∈ Rn satisﬁes |x| ≤ K, then relation (2.177) is valid with = 8−1 δ. It follows from Lemma 4.2.9 and Proposition 1.3.9 that there exists a neighborhood U3 of f in A with the weak topology such that the following property holds: If g ∈ U3 , τ ∈ [1, 2Q], and if x, y ∈ Rn satisfy |x|, |y| ≤ 2K + 2, then

Put

|µ(g) − µ(f )| ≤ (16(Q1 + Q2 ))−1 ,

(2.187)

|U f (0, τ, x, y) − U g (0, τ, x, y)| ≤ 16−1 .

(2.188)

U = ∩3i=1 Ui .

By the choice of U2 , Q1 , Q2 , and (2.177), if g ∈ U and if x ∈ Rn satisﬁes |x| ≤ K, then π g (x) = inf{U g (0, T, x, y) − T µ(g) + π g (y) :

Inﬁnite horizon autonomous problems

147

T ∈ [Q1 , Q2 ], y ∈ Rn , |y − hf | ≤ 8−1 δ}. By this relation, (2.187), (2.188), and the deﬁnition of U1 , δ (see (2.185) and (2.186)), for each g ∈ U and each x ∈ Rn satisfying |x| ≤ K, |π g (x) − inf{U f (0, T, x, y) − T µ(f ) + π f (hf ) : T ∈ [Q1 , Q2 ], y ∈ Rn , |y − hf | ≤ 8−1 δ}| ≤ 16−1 + 8−1 + 16−1 ≤ 4−1 . This implies that for each g1 , g2 ∈ U and each x ∈ Rn satisfying |x| ≤ K, |π g1 (x) − π g2 (x)| ≤ 2−1 . Therefore f is a continuity point of the mapping g → π g , g ∈ A. The theorem is proved. Proof of Theorem 4.1.2. Let ∈ (0, 1). Theorem 3.3.1 implies that there exists a natural number L such that if v : [0, ∞) → Rn is an (f )-good function, then for all large T , dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 2−8 .

(2.189)

It follows from Lemma 4.2.6 that there exist a neighborhood U1 of f in A with the weak topology and a positive number δ0 < 8−1

(2.190)

such that the following property holds: If g ∈ U1 , T ∈ [L, ∞) and if an a.c. function v : [0, T ] → Rn satisﬁes d(v(0), H(f )) ≤ δ0 , d(v(T ), H(f )) ≤ δ0 ,

(2.191)

σ g (0, T, v) ≤ l(g, Df + 4, T ) + δ0 , then for every S ∈ [0, T − L], dist(H(f ), {v(t) : t ∈ [S, S + L]}) ≤ .

(2.192)

Lemma 4.2.5 implies that there exists a neighborhood U2 of f in A with the weak topology such that the following property holds: If g ∈ U2 , h ∈ H(f ) and if a number T ≥ 1, then there exists an a.c. function v : [0, T ] → Rn such that v(0) = h, v(T ) ∈ H(f ), σ g (0, T, x) ≤ l(g, Df + 4, T ) + 8−1 δ0 .

(2.193)

148

TURNPIKE PROPERTIES

It is easy to see that there exists a positive number

such that

δ < 2−6 δ0

(2.194)

|π f (x) − π f (y)| ≤ 2−4 δ0

(2.195)

for each x, y ∈ Rn satisfying |x − y| ≤ δ and |x|, |y| ≤ Df + 4. By Theorem 4.1.1 there exists a neighborhood U3 of f in A with the weak topology such that |π f (x) − π g (x)| ≤ 2−4 δ0

(2.196)

for each g ∈ U3 and each x ∈ Rn satisfying |x| ≤ Df + 4. It follows from Theorem 3.1.3 that there are a neighborhood U4 of f in A with the weak topology and an integer L0 ≥ 1 such that for each g ∈ U4 and each (g)-good function v : [0, ∞) → Rn , dist(H(f ), {v(t) : t ∈ [T, T + L0 ]}) ≤ δ

(2.197)

for all large T . We may assume that L0 ≥ L. Put U = ∩4i=1 Ui . Assume that g ∈ U, v ∈ A(g), d(v(0), H(f )) ≤ δ.

(2.198)

It follows from (2.198) and Proposition 4.2.1 that v is a (g)-good function. By the deﬁnition of U4 and L0 there exists a number T0 such that (2.197) holds for each T ≥ T0 . Let T ≥ T0 + L. There exists τ ∈ [T + L0 , T + 2L0 ] such that |v(τ ) − hf | ≤ δ

(2.199)

(recall hf in (2.176)). We show that σ g (0, τ, v) ≤ l(g, Df + 4, τ ) + δ0 .

(2.200)

By (2.198), (2.199) and the choice of U3 (see (2.196)), l(g, Df + 4, τ ) ≤ σ g (0, τ, v) ≤σ ˜ g (0, τ, v) + 2−3 δ0 = τ µ(g) + 2−3 δ0 .

(2.201)

It follows from the choice of U2 (see (2.193)) that there exists an a.c. function u : [0, τ ] → Rn for which u(0), u(τ ) ∈ H(f ),

Inﬁnite horizon autonomous problems

149

σ g (0, τ, u) ≤ l(g, Df + 4, τ ) + 8−1 δ0 . By these relations, the representation formula (1.6) and the choice of U3 (see (2.196)), ˜ g (0, τ, u) l(g, Df + 4, τ ) + 8−1 δ0 ≥ σ −2−3 δ0 ≥ τ µ(g) − 2−3 δ0 . Together with (2.201) this implies (2.200). By (2.198)-(2.200) and the deﬁnition of U1 , (2.192) holds for all S ∈ [0, τ − L]. This completes the proof of the theorem. Proof of Theorem 4.1.3. Let ∈ (0, 1) and K > Df +4. Theorem 4.1.2 implies that there exist numbers δ ∈ (0, ), L > 0, and a neighborhood U1 of f in A with the weak topology such that the following property holds: If g ∈ U1 , v ∈ A(g) satisﬁes d(v(0), H(f )) ≤ δ and if T ∈ [0, ∞), then dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ .

(2.202)

It follows from Theorem 3.1.3 that there exist a neighborhood U2 of f in A with the weak topology and a natural number L0 such that the following property holds: For each g ∈ U2 and each (g)-good function v : [0, ∞) → Rn , dist(H(f ), {v(t) : t ∈ [T, T + L0 ]}) ≤ 8−1 δ

(2.203)

for all large T . By Lemma 4.2.1 there exist a neighborhood U3 of f in A with the weak topology and a natural number N ≥ 10 such that the following property holds: If g ∈ U3 , T1 ≥ 0, T2 ≥ T1 + N L0 and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes |v(Ti )| ≤ K + 8, i = 1, 2, Φg (T1 , T2 , v) ≤ 4,

(2.204)

then for each S ∈ [T1 , T2 − N L0 ], there exists an integer i0 ∈ [0, N − 8] such that dist(H(f ), {v(t) : t ∈ [T, T + L0 ]}) ≤ δ (2.205) for all T ∈ [S + i0 L0 , S + (i0 + 7)L0 ]. Put U = ∩3i=1 Ui , Q = N L0 .

(2.206)

Assume that g ∈ U, v ∈ A(g), |v(0)| ≤ K.

(2.207)

150

TURNPIKE PROPERTIES

Equation (2.207) and Proposition 4.2.1 imply that v is a (g)-good function. Therefore by the deﬁnition of U2 , |v(t)| ≤ K + 1 for all large t. It follows from this relation, (2.207), (2.206), and the deﬁnition of U3 , N that there exists τ ∈ [0, Q] for which d(v(τ ), H(f )) ≤ δ. By this relation, (2.207), and the deﬁnition of U1 , δ, L, the relation (2.202) holds for each T ≥ τ . This completes the proof of the theorem.

4.3.

Proof of Theorem 4.1.4

Lemma 4.3.1 Assume that f ∈ A, x : [0, ∞) → Rn is an (f )-good function and h ∈ Ω(x). Then there exists an a.c. function v : R1 → Ω(x) such that v ∈ B(f ), v(0) = h. Proof. Theorem 1.2.2 implies that the function x is bounded. It is not diﬃcult to see that the following property holds: (a) for each > 0 there exists T () > 0 such that σ ˜ f (T1 , T2 , x) − (T2 − T1 )µ(f ) ≤ for each T1 ≥ T (), T2 > T1 , There exists a sequence of numbers {Tp }∞ p=0 such that Tp+1 ≥ Tp + 1, p = 0, 1, . . . , x(Tp ) → h as p → ∞.

(3.1)

For every integer p ≥ 1 we set vp (t) = x(t + Tp ), t ∈ [−Tp , ∞).

(3.2)

By Proposition 1.3.5, the boundedness of x, (3.1), and (3.2) there exist 1 n a subsequence {vpj }∞ j=1 and an a.c. function v : R → R such that for each integer N ≥ 1, vpj (t) → v(t) as j → ∞ uniformly in [−N, N ], I f (−N, N, v) ≤ lim inf I f (−N, N, vpj ). j→∞

(3.3)

Equations (3.1)-(3.3) imply that v(0) = h and v(t) ∈ Ω(x), t ∈ R1 . It follows from property (a), (3.3), and (3.2) that v ∈ B(f ). The lemma is proven. Theorems 1.2.2 and 1.2.3 and Proposition 4.2.1 imply the following result.

Inﬁnite horizon autonomous problems

151

Lemma 4.3.2 Let f ∈ A. Then each function v ∈ B(f ) is bounded. Assertion (1) of Theorem 4.1.4 follows from Lemma 4.3.1. Assertion (2) of Theorem 4.1.4 follows from Lemma 4.3.2 and Theorem 4.1.3. Assertion (3) of Theorem 4.1.4 follows from Assertion (2) and Theorem 4.1.4. Lemma 4.3.1 implies the following result. Proposition 4.3.1 Assume that f ∈ A and there exists a compact set H(f ) ⊂ Rn such that for each v ∈ B(f ) the following relations hold: v(t) ∈ H(f ), t ∈ R1 , {y ∈ Rn : there exists a sequence {ti }∞ i=0 ⊂ [0, ∞) for which ti → ∞, v(ti ) → y as i → ∞} = H(f ). Then the integrand f has the asymptotic turnpike property with the turnpike H(f ).

Chapter 5 TURNPIKE FOR AUTONOMOUS PROBLEMS

In this chapter we continue to study the turnpike property for autonomous variational problems. For a class of smooth nonconvex integrands we improve the results of Chapter 3. We establish the turnpike property for a generic nonconvex integrand f (x, u). We show that for a generic f , any small > 0 and an extremal v : [0, T ] → Rn of the variational problem with large enough T , ﬁxed end points and the integrand f , for each τ ∈ [L1 , T − L1 ] the set {v(t) : t ∈ [τ, τ + L2 ]} is equal to a set H(f ) up to in the Hausdorﬀ metric. Here H(f ) ⊂ Rn is a compact set depending only on the integrand f and L1 > L2 > 0 are constants which depend only on and |v(0)|, |v(T )|.

5.1.

Main results

In this chapter we analyse the structure of optimal solutions of the variational problem T 0

f (z(t), z (t))dt → min, z(0 = x, z(T ) = y,

(P )

z : [0, T ] → Rn is an absolutely continuous function where T > 0, x, y ∈ Rn and f : R2n → R1 is an integrand. We say that an integrand f = f (x, u) ∈ C(R2n ) has the turnpike property if there exists a compact set H(f ) ⊂ Rn such that for each bounded set K ⊂ Rn and each > 0 there exist numbers L1 > L2 > 0 such that for each T ≥ 2L1 , each x, y ∈ K and an optimal solution v : [0, T ] → Rn for the variational problem (P) the relation dist(H(f ), {v(t) : t ∈ [τ, τ + L2 ]}) ≤

154

TURNPIKE PROPERTIES

holds for each τ ∈ [L1 , T − L1 ]. (Here dist(·, ·) is the Hausdorﬀ metric). Our goal is to show that the turnpike property is a general phenomenon which holds for a class of autonomous variational problems with vector-valued functions. We consider the complete metric space of ¯k (k is a nonnegative integer) described below and establish integrands N ¯k which is a countable intersection of open the existence of a set F ⊂ N ¯ everywhere dense sets in Nk and such that each integrand f ∈ F has the turnpike property. Moreover we show that the turnpike property holds for approximate solutions of variational problems with a generic integrand f and that the turnpike phenomenon is stable under small perturbations of a generic integrand f . In Chapter 3 we studied the weak version of this turnpike property for optimal solutions of the variational problem (P) with x, y ∈ Rn , large enough T and a generic integrand f belonging to the space of functions A. In this weak version of the turnpike property established in Chapter 3 for an optimal solution of the problem (P) with x, y ∈ Rn , large enough T and a generic integrand f ∈ A the relation dist(H(f ), {v(t) : t ∈ [τ, τ + L2 ]}) ≤ with L2 which depends on and |x|, |y| and a compact set H(f ) ⊂ Rn depending only on the integrand f , holds for each τ ∈ [0, T ] \ E where E ⊂ [0, T ] is a measurable subset such that the Lebesgue measure of E does not exceed a constant which depends on and on |x|, |y|. The turnpike property which is established in this chapter guarantees that we may take E = [0, L1 ] ∪ [T − L1 , T ] where L1 > 0 is a constant which depends on and |x|, |y|. The results of this chapter have been established in [100]. Denote by | · | the Euclidean norm in Rn . Let a > 0 be a constant and let ψ : [0, ∞) → [0, ∞) be an increasing function such that ψ(t) → ∞ as t → ∞. Consider the space of functions A introduced in Section 3.1. Recall that A is the set of continuous functions f : Rn × Rn → R1 which satisfy the following assumptions: A(i) for each x ∈ Rn the function f (x, ·) : Rn → R1 is convex; A(ii) f (x, y) ≥ max{ψ(|x|), ψ(|u|)|u|} − a for each (x, u) ∈ Rn × Rn ; A(iii) for each M, > 0 there exist Γ, δ > 0 such that |f (x1 , u1 ) − f (x2 , u2 )| ≤ max{f (x1 , u1 ), f (x2 , u2 )} for each u1 , u2 , x1 , x2 ∈ Rn which satisfy |xi | ≤ M, |ui | ≥ Γ, i = 1, 2 and max{|x1 − x2 |, |u1 − u2 |} ≤ δ.

155

Turnpike for autonomous problems

For the set A we consider the uniformity introduced in Section 3.1. This uniformity is determined by the following base: E(N, , λ) = {(f, g) ∈ A × A : |f (x, u) − g(x, u)| ≤ (u, x ∈ Rn , |x|, |u| ≤ N ), (|f (x, u)| + 1)(|g(x, u)| + 1)−1 ∈ [λ−1 , λ] (x, u ∈ Rn , |x| ≤ N )} where N > 0, > 0 and λ > 1 [37]. It is known that the uniform space A is metrizable and complete (see Section 3.1). We consider functionals of the form I f (T1 , T2 , x) =

T2 T1

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

(1.1)

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

(1.2)

It is easy to see that −∞ < U f (T1 , T2 , y, z) < ∞ for each f ∈ A, each y, z ∈ Rn and all numbers T1 , T2 satisfying 0 ≤ T1 < T2 . Let f ∈ A. For any a.c. function x : [0, ∞) → Rn we set J(x) = lim inf T −1 I f (0, T, x). T →∞

(1.3)

Of special interest is the minimal long-run average cost growth rate µ(f ) = inf{J(x) : x : [0, ∞) → Rn is an a.c. function}.

(1.4)

Clearly −∞ < µ(f ) < +∞. By Theorems 3.6.1 and 3.6.2, U f (0, T, x, y) = T µ(f ) + π f (x) − π f (y) + θTf (x, y),

(1.5)

x, y ∈ Rn , T ∈ (0, ∞), where π f : Rn → R1 is a continuous function and (T, x, y) → θTf (x, y) ∈ R1 is a continuous nonnegative function deﬁned for T > 0, x, y ∈ Rn , π f (x) = inf{lim inf [I f (0, T, v) − µ(f )T ] : v : [0, ∞) → Rn T →∞

is an a.c. function satisfying v(0) = x}, x ∈ Rn ,

(1.6)

156

TURNPIKE PROPERTIES

and for every T > 0, every x ∈ Rn there is y ∈ Rn satisfying θTf (x, y) = 0. Let f ∈ A. An a.c. function x : [0, ∞) → Rn is called an (f )-good function if the function φfx : T → I f (0, T, x) − µ(f )T , T ∈ (0, ∞) is bounded. By Theorem 3.6.3 for each f ∈ A and each z ∈ Rn there exists an (f )-good function v : [0, ∞) → Rn satisfying v(0) = z. We denote d(x, B) = inf{|x−y| : y ∈ B} for x ∈ Rn , B ⊂ Rn . Denote by dist(A, B) the Hausdorﬀ metric for two sets A ⊂ Rn , B ⊂ Rn . For every bounded a.c. function x : [0, ∞) → Rn deﬁne Ω(x) = {y ∈ Rn : there exists a sequence {ti }∞ i=0 ⊂ (0, ∞) for which ti → ∞, x(ti ) → y as i → ∞}.

(1.7)

We say that an integrand f ∈ A has the asymptotic turnpike property, or brieﬂy (ATP), if Ω(v2 ) = Ω(v2 ) for all (f )-good functions vi : [0, ∞) → Rn , i = 1, 2. By Theorem 3.1.1 there exists a set F ⊂ A which is a countable intersection of open everywhere dense subsets of A and such that each f ∈ F has (ATP). Assume that an integrand f ∈ A has the asymptotic turnpike property. Then Proposition 4.1.1 implies that there exists a compact set H(f ) ⊂ Rn such that Ω(v) = H(f ) for each (f )-good function v : [0, ∞) → Rn . We say that the set H(f ) is the turnpike of f . Denote by N the set of all functions f ∈ C 1 (R2n ) satisfying assumptions which ensure that each solution of (P) belongs to C 2 ([0, T ]; Rn ): ∂f /∂ui ∈ C 1 (R2n ) for i = 1, . . . , n; the matrix (∂ 2 f /∂ui ∂uj )(x, u), i, j = 1, . . . , n is positive deﬁnite for all (x, u) ∈ R2n ; f (x, u) ≥ max{ψ(|x|), ψ(|u|)|u|} − a for all x, u ∈ Rn × Rn ; there exist a number c0 > 1 and monotone increasing functions φi : [0, ∞) → [0, ∞), i = 0, 1, 2, such that φ0 (t)t−1 → ∞ as t → ∞, f (x, u) ≥ φ0 (c0 |u|) − φ1 (|x|), x, u ∈ Rn ; max{|∂f /∂xi (x, u)|, |∂f /∂ui (x, u)|} ≤ φ2 (|x|)(1 + φ0 (|u|)), x, u ∈ Rn , i = 1, . . . , n. It is easy to see that N ⊂ A. We will establish the following result. Theorem 5.1.1 Assume that an integrand f ∈ N has the asymptotic turnpike property and , K > 0. Then there exists a neighborhood U of f

Turnpike for autonomous problems

157

in A and numbers M > K, l0 > l > 0, δ > 0 such that for each g ∈ U, each T ≥ 2l0 and each a.c. function v : [0, T ] → Rn which satisﬁes |v(0)|, |v(T )| ≤ K, I g (0, T, v) ≤ U g (0, T, v(0), v(T )) + δ the relation |v(t)| ≤ M holds for all t ∈ [0, T ] and dist(H(f ), {v(t) : t ∈ [τ, τ + l]}) ≤

(1.8)

for each τ ∈ [l0 , T − l0 ]. Moreover if d(v(0), H(f )) ≤ δ, then (1.8) holds for each τ ∈ [0, T − l0 ] and if d(v(T ), H(f )) ≤ δ, then (1.8) holds for each τ ∈ [l0 , T − l]. Let k ≥ 1 be an integer. Denote by Ak the set of all integrands f ∈ A ∩ C k (R2n ). For p = (p1 , . . . , p2n ) ∈ {0, . . . , k}2n and f ∈ C k (R2n ) we set |p| =

2n i=1

p2n pi , Dp f = ∂ |p| f /∂y1p1 . . . ∂y2n .

For the set Ak we consider the uniformity which is determined by the following base: E(N, , λ) = {(f, g) ∈ Ak × Ak : |Dp f (x, u) − Dp g(x, u)| ≤ for each u, x ∈ Rn satisfying |x|, |u| ≤ N and each p ∈ {0, . . . , k}2n satisfying |p| ≤ k, |f (x, u) − g(x, u)| ≤ for each u, x ∈ Rn satisfying |x|, |u| ≤ N, (|f (x, u)| + 1)(|g(x, u)| + 1)−1 ∈ [λ−1 , λ] for each x, u ∈ Rn satisfying |x| ≤ N } where N > 0, > 0, λ > 1. It is easy to verify that the uniform space Ak is metrizable and complete. For each integer k ≥ 1 we deﬁne Nk = N ∩ Ak . Set A0 = A, N0 = N . ¯k the closure of Nk in Ak and Let k ≥ 0 be an integer. Denote by N ¯ consider the topological subspace Nk ⊂ Ak with the relative topology. We will establish the following result. Theorem 5.1.2 Let q ≥ 0 be an integer. Then there exists a set Fq ⊂ ¯q which is a countable intersection of open everywhere dense subsets N

158

TURNPIKE PROPERTIES

¯q and such that each f ∈ Fq has the asymptotic turnpike property of N and the following property: For each , K > 0 there exist a neighborhood U of f in A and numbers M > K, l0 > l > 0, δ > 0 such that for each g ∈ U, each T ≥ 2l0 and each a.c. function v : [0, T ] → Rn which satisﬁes |v(0)|, |v(T )| ≤ K, I g (0, T, v) ≤ U g (0, T, v(0), v(T )) + δ the relation |v(t)| ≤ M holds for all t ∈ [0, T ] and dist(H(f ), {v(t) : t ∈ [τ, τ + l]}) ≤

(1.9)

for each τ ∈ [l0 , T − l0 ]. Moreover if d(v(0), H(f )) ≤ δ, then (1.9) holds for each τ ∈ [0, T − l0 ] and if d(v(T ), H(f )) ≤ δ, then (1.9) holds for each τ ∈ [l0 , T − l]. Chapter 5 is organized as follows. Theorem 5.1.1 will be proved in Section 5.2. Section 5.3 contains the proof of Theorem 5.1.2 while Section 5.4 contains an example.

5.2.

Proof of Theorem 5.1.1

Assume that f ∈ N has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . For each a.c. function u : [τ1 , τ2 ] → Rn where τ1 ≥ 0, τ2 > τ1 and each r1 , r2 ∈ [τ1 , τ2 ] satisfying r1 < r2 we set σ(r1 , r2 , u) = I f (r1 , r2 , u) − π f (u(r1 )) + π f (u(r2 )) − (r2 − r1 )µ(f ), Φ(r1 , r2 , u) = I f (r1 , r2 , u) − U f (r1 , r2 , u(r1 ), u(r2 )).

(2.1)

Lemma 5.2.1 Assume that h ∈ H(f ). Then there exists an (f )-good function v : [0, ∞) → H(f ) such that v(0) = h and σ f (T1 , T2 , x) = 0 for each T1 ≥ 0, T2 > T1 . Proof. Consider any (f )-good function w : [0, ∞) → Rn . Since the integrand f has the asymptotic turnpike property with the turnpike H(f ) we have Ω(w) = H(f ). Theorem 1.2.2 implies that sup{|w(t)| : t ∈ [0, ∞)} < ∞. It is easy to see that the following property holds:

159

Turnpike for autonomous problems

(a) For each > 0 there exists T () > 0 such that σ f (T1 , T2 , w) ≤ for each T1 ≥ T (), T2 > T1 . There exists a sequence of numbers {Tp }∞ p=0 ⊂ [0, ∞) such that Tp+1 ≥ Tp + 1, p = 0, 1, . . . , w(Tp ) → h as p → ∞.

(2.2)

For every integer p ≥ 1 we set vp (t) = w(t + Tp ), t ∈ [0, ∞).

(2.3)

By Proposition 1.3.5, the boundedness of w, (2.3) and property (a) there n exist a subsequence {vpj }∞ j=1 and an a.c. function v : [0, ∞) → R such that for each integer N ≥ 1, vpj (t) → v(t) as j → ∞ uniformly in [0, N ], I f (0, N, v) ≤ lim inf I f (0, N, vpj ). j→∞

(2.4)

(2.2)-(2.4) imply that v(0) = h and v(t) ∈ H(f ), t ∈ [0, ∞). It follows from property (a), (2.3) and (2.4) that σ f (T1 , T2 , w) = 0 for each T1 ≥ 0, T2 > T1 . The lemma is proved. Lemma 5.2.1 implies that there is an (f )-good function v∗ : [0, ∞) → H(f ) such that for each T1 ≥ 0, T2 > T1 , I f (T1 , T2 , v∗ ) = µ(f )(T2 − T1 ) + π f (v∗ (T1 )) − π f (v∗ (T2 )).

(2.5)

By Proposition 2.5.1, v∗ ∈ C 2 ([0, ∞); Rn ).

(2.6)

Lemma 5.2.2 The function π f · v∗ ∈ C 1 ([0, ∞); R1 ). Proof. It follows from (2.5) that π f (v∗ (T )) = −I f (0, T, v∗ ) + µ(f )T + π f (v∗ (0)) for each T ≥ 0. Combined with (2.6) this equality implies the assertion of the lemma. For each τ ∈ [0, ∞) we put Pτ (t) = t(v∗ (τ ) − v∗ (1)), t ∈ R1 , ψ(τ ) = I f (0, 1, v∗ + Pτ ).

(2.7)

Lemma 5.2.3 ψ ∈ C 1 ([0, ∞); R1 ). Proof. For λ, t ∈ [0, ∞) put B(λ, t) = (v∗ (t) + Pλ (t), v∗ (t) + Pλ (t)).

(2.8)

160

TURNPIKE PROPERTIES

Let τ, h ∈ [0, ∞), τ = h and t ∈ [0, 1]. By (2.7), (2.8) there is λh (t) ∈ [min{h, τ }, max{h, τ }] such that (h − τ )−1 [f (B(h, t)) − f (B(τ, t))] = ∂f /∂x(B(λh (t), t))tv∗ (λh (t)) +∂f /∂u(B(λh (t), t))v∗ (λh (t)) → ∂f /∂x(B(τ, t))v∗ (τ )t + ∂f /∂u(B(τ, t))v∗ (τ ) as h → τ uniformly for all t ∈ [0, 1]. This implies that ψ ∈ C 1 ([0, ∞); R1 ). The lemma is proved. The next auxiliary result plays a crucial role in the proof of Theorem 5.1.1. Lemma 5.2.4 For each > 0 there exists a number q ≥ 8 such that the following property holds: If h1 , h2 ∈ H(f ), then there exists an a.c. function v : [0, q] → Rn such that v(0) = h1 , v(q) = h2 , (2.9) σ f (0, q, v) ≤ .

(2.10)

Proof. Deﬁne a function φ : [0, ∞) → R1 by φ(t) = ψ(t) − µ(f ) − π f (v∗ (0)) + π f (v∗ (t)), t ∈ [0, ∞).

(2.11)

It follows from (2.11), (2.7), Lemmas 5.2.2 and 5.2.3, (2.5) and the representation formula (see (1.5), (1.6)) that φ ∈ C 1 ([0, ∞); R1 ), φ(1) = 0, φ(t) ≥ 0, t ∈ [0, ∞).

(2.12)

It follows from Theorem 3.4.1 that there exists a sequence of positive numbers {δi }∞ i=0 such that δ0 ∈ (0, 8−1 ), δi+1 < δi , i = 0, 1, . . .

(2.13)

and that the following property holds: If an integer i ≥ 0 and if x1 , x2 , y1 , y2 ∈ H(f ) satisfy |xj − yj | ≤ δi , j = 1, 2, then |U f (0, 1, x1 , x2 ) − U f (0, 1, y1 , y2 )| ≤ 2−i−8 , |π f (xj ) − π f (yj )| ≤ 2−i−8 , j = 1, 2.

(2.14)

161

Turnpike for autonomous problems

By the deﬁnition of v∗ and Theorem 3.3.1 there exists an integer L ≥ 10 such that (2.15) dist(H(f ), {v∗ (t) : t ∈ [T, T + L]}) ≤ 4−1 δ0 for all T ∈ [0, ∞). Since Ω(v∗ ) = H(f ) and v∗ (0) ∈ H(f ) there exists a sequence of numbers {Tp }∞ p=1 such that Tp ≥ 2L + 8, |v∗ (0) − v∗ (Tp )| ≤ 2−8 δp , p = 1, 2, . . . ,

(2.16)

Fix a positive number 0 for which 0 < 2−8 L−1 .

(2.17)

It follows from (2.12) that there exists a positive number ∆ such that ∆ < 2−8 , |φ (t)| ≤ 2−1 0 , t ∈ [1 − ∆, 1 + ∆]. Choose an integer

N > 64(L + 1)∆−1

and set q=

N

Ti + 8L + 8.

(2.18) (2.19)

(2.20)

i=1

Let h1 , h2 ∈ H(f ). We will construct an a.c. function v : [0, q] → Rn satisfying (2.9) and (2.10). It follows from (2.15), which holds for each T ∈ [0, ∞), that there exist numbers t1 , t2 such that t1 ∈ [0, L], t2 ∈ [8, L + 8], |hj − v∗ (tj )| ≤ 4−1 δ0 , j = 1, 2. Set

∆0 = (N − 1)−1 (8L + 8 − (t2 − t1 )).

(2.21) (2.22)

(2.22), (2.21), (2.19) and (2.18) imply that ∆0 ∈ (0, ∆).

(2.23)

By Proposition 1.3.5, (2.16) and (2.21) there exists an a.c. function w0 : [0, T1 − t1 ] → Rn such that w0 (0) = h1 , w0 (t) = v∗ (t1 + t), t ∈ [1, T1 − t1 − 1], w0 (T1 − t1 ) = v∗ (0), (2.24) (2.25) Φf (τ, τ + 1, w0 ) = 0, τ = 0, T1 − t1 − 1. It follows from (2.5), (2.16), (2.24), (2.25) and the deﬁnitions of {δj }∞ j=0 that σ(0, T1 − t0 , w0 ) = σ(0, 1, w0 ) + σ(T1 − t1 − 1, T1 − t1 , w0 )

(2.26)

162

TURNPIKE PROPERTIES

= U f (0, 1, h1 , v∗ (t1 + 1)) − π f (h1 ) + π f (v∗ (t1 + 1)) − µ(f ) +U f (0, 1, v∗ (T1 − 1), v∗ (0)) − π f (v∗ (T1 − 1)) + π f (v∗ (0)) − µ(f ) ≤ 4 · 2−8 + U f (0, 1, v∗ (t1 ), v∗ (t1 + 1)) − π f (v∗ (t1 )) +π f (v∗ (t1 + 1)) − µ(f ) + U f (0, 1, v∗ (T1 − 1), v∗ (T1 )) −π f (v∗ (T1 − 1)) + π f (v∗ (T1 )) − µ(f ) ≤ 2−6 . Let k ≥ 1 be an integer. It follows from Proposition 1.3.5, (2.7), (2.16), (2.18) and (2.23) that there exists an a.c. function wk : [0, ∆0 + Tk+1 ] → Rn such that wk (t) = v∗ (t) + P1−∆0 (t), t ∈ [0, 1], wk (t) = v∗ (t − ∆0 ), t ∈ [1, ∆0 + Tk+1 − 1], wk (∆0 + Tk+1 ) = v∗ (0), f

Φ (∆0 + Tk+1 − 1, ∆0 + Tk+1 , wk ) = 0.

(2.27)

Equations (2.27) and (2.7) imply that wk (0) = v∗ (0).

(2.28)

We will estimate σ(0, Tk+1 + ∆0 , wk ). By (2.5), (2.24) and (2.27), σ(0, Tk+1 + ∆0 , wk ) = σ(0, 1, wk ) + σ(Tk+1 + ∆0 − 1, Tk+1 + ∆0 , wk ). (2.29) It follows from (2.7), (2.11), (2.24) and (2.27) that σ(0, 1, wk ) = φ(1 − ∆0 ).

(2.30)

By (2.30), (2.23), (2.18) and (2.12) σ(0, 1, wk ) ≤ 2−1 ∆0 0 .

(2.31)

By (2.27), (2.24), (2.16), the deﬁnition of v∗ , (2.5) and the deﬁnition of {δi }∞ i=0 (see (2.13) and (2.14)), σ(Tk+1 + ∆0 − 1, Tk+1 + ∆0 , wk ) = U f (0, 1, v∗ (Tk+1 − 1), v∗ (0)) − π f (v∗ (Tk+1 − 1)) + π f (v∗ (0)) − µ(f ) ≤ U f (0, 1, v∗ (Tk+1 − 1), v∗ (Tk+1 )) − π f (v∗ (Tk+1 − 1)) +π f (v∗ (Tk+1 )) − µ(f ) + 2 · 2−k−9 = 2−k−8 .

(2.32)

Combining (2.29), (2.31) and (2.32) we obtain that σ(0, Tk+1 + ∆0 , wk ) ≤ 2−1 0 ∆0 + 2−k−8 .

(2.33)

163

Turnpike for autonomous problems

It follows from Proposition 1.3.5 that there exists an a.c. function u0 : [0, t2 ] → Rn such that u0 (t) = v∗ (t), t ∈ [0, t2 − 1], u0 (t2 ) = h2 ,

(2.34)

Φf (t2 − 1, t2 , u0 ) = 0. By (2.5), (2.21), (2.24), (2.34), the deﬁnition of {δi }∞ i=0 (see (2.13), (2.14)) and the deﬁnition of v∗ , σ(0, t2 , u0 ) = σ(t2 − 1, t2 , u0 ) = U f (0, 1, v∗ (t2 − 1), h2 ) − π f (v∗ (t2 − 1)) + π f (h2 ) − µ(f ) ≤ U f (0, 1, v∗ (t2 − 1), v∗ (t2 )) − π f (v∗ (t2 − 1)) +π f (v∗ (t2 )) − µ(f ) + 2−7 ≤ 2−7 .

(2.35)

It follows from (2.20) and (2.22) that N −1

T1 − t1 +

(∆0 + Tk+1 ) + t2 = q.

(2.36)

k=1

By (2.36), (2.25), (2.27), (2.28) and (2.34) there exists an a.c. function v : [0, q] → Rn such that v(t) = w0 (t), t ∈ [0, T1 − t1 ],

v(t) = wk t − t∈

k

k

Ti + (k − 1)∆0 − t1

i=1

Ti + (k − 1)∆0 − t1 ,

i=1

v(t) = u0 t −

N

N

(2.37)

Ti + k∆0 − t1 , k = 1, . . . , N − 1,

i=1

t∈

k+1

,

Ti + (N − 1)∆0 − t1

i=1

,

Ti + (N − 1)∆0 − t1 , q .

i=1

(2.37), (2.25), (2.36) and (2.34) imply that v(0) = h1 , v(q) = h2 . By (2.17), (2.21), (2.22), (2.24), (2.26), (2.33), (2.35) and (2.37), σ f (0, q, v) = σ(0, T1 − t1 , w0 ) +

N −1 k=1

σ(0, Tk+1 + ∆0 , wk ) + σ(0, t2 , u0 )

164 ≤ 2−6 +

TURNPIKE PROPERTIES N −1

(2−1 0 ∆0 + 2−k−8 ) + 2−7 ≤ 2−5 + 2−1 (N − 1)0 ∆0

k=1

≤ 2−5 + 2−1 (9L + 16)0 ≤ 2−1 . This completes the proof of the lemma. Proof of Theorem 5.1.1. Let , K > 0. We may assume that < 1, K > sup{|h| : h ∈ H(f )} + 4. It follows from Theorem 1.2.3 that there exist a number M > K and a neighborhood U1 of f in A such that the following property holds: If g ∈ U1 , T1 ≥ 0, T2 ≥ T1 + 1 and if an a.c. function v : [T1 , T2 ] → Rn satisﬁes |v(Ti )| ≤ 2K + 4, i = 1, 2, Φg (T1 , T2 , v) ≤ 2, (2.38) then |v(t)| ≤ M, t ∈ [T1 , T2 ].

(2.39)

Theorem 3.4.1 implies that there exists δ1 ∈ (0, 8−1 )

(2.40)

such that the following property holds: If x1 , x2 , y1 , y2 ∈ Rn satisfy |xi |, |yi | ≤ 2M + 4 + 2 sup{|h| : h ∈ H(f )}, |xi − yi | ≤ 4δ1 , i = 1, 2, (2.41) then |U f (0, 1, x1 , x2 ) − U f (0, 1, y1 , y2 )| ≤ 2−8 , |π f (xi ) − π f (yi )| ≤ 2−8 , i = 1, 2.

(2.42)

By Theorem 3.3.1 there exists an integer l ≥ 1 such that for each (f )good function v : [0, ∞) → Rn the inequality dist(H(f ), {v(t) : t ∈ [T, T + l]}) ≤

(2.43)

is valid for all large T . By Lemma 3.8.4 there exists a positive number δ0 < 2−1 δ1

(2.44)

such that the following property holds: If T ∈ [l, ∞) and if an a.c. function v : [0, T ] → Rn satisﬁes d(v(0), H(f )) ≤ δ0 , d(v(T ), H(f )) ≤ δ0 ,

(2.45)

165

Turnpike for autonomous problems

σ f (0, T, v) ≤ δ0 ,

(2.46)

then for every S ∈ [0, T − l], dist(H(f ), {v(t) : t ∈ [S, S + l]}) ≤ .

(2.47)

By Theorem 3.4.1 there exists a positive number δ < 32−1 δ0

(2.48)

such that the following property holds: If x1 , x2 , y1 , y2 ∈ Rn satisfy |xi |, |yi | ≤ 2M + 4 + 2 sup{|h| : h ∈ H(f )}, |xi − yi | ≤ 4δ, i = 1, 2, (2.49) then |U f (0, 1, x1 , x2 ) − U f (0, 1, y1 , y2 )| ≤ 2−8 δ0 , |π f (xi ) − π f (yi )| ≤ 2−8 δ0 , i = 1, 2.

(2.50)

By Theorem 3.3.1 there exists an integer L ≥ 1 such that for each (f )good function v : [0, ∞) → Rn , dist(H(f ), {v(t) : t ∈ [T, T + L]}) ≤ 8−1 δ

(2.51)

for all large T . It follows from Lemma 3.8.2 that there exist a neighborhood U2 of f in A and a natural number N ≥ 10 such that the following property holds: If g ∈ U2 , S ∈ [0, ∞) and if an a.c. function x : [S, S + N L] → Rn satisﬁes |x(S)|, |x(S + N L)| ≤ 2M + 2, Φg (S, S + N L, x) ≤ 4,

(2.52)

then there exists an integer i0 ∈ [0, N − 8] such that the inequality dist(H(f ), {x(t) : t ∈ [T, T + L]}) ≤ δ

(2.53)

is true for all T ∈ [S + i0 L, S + (i0 + 7)L]. By Lemma 5.2.4 there exists a number q ≥ 8 such that for each h1 , h2 ∈ H(f ) there exists an a.c. function v : [0, q] → Rn which satisﬁes v(0) = h1 , v(q) = h2 , σ f (0, q, v) ≤ 8−1 δ.

(2.54)

It follows from Proposition 1.3.8 that there exists a neighborhood U3 of f in A such that the following property holds:

166

TURNPIKE PROPERTIES

If g ∈ U3 , T1 ≥ 0, T2 ∈ [T1 + 8−1 , T1 + 6N (q + l + L)] and if an a.c. function x : [T1 , T2 ] → Rn satisﬁes max{I f (T1 , T2 , x), I g (T1 , T2 , x)} ≤ 4 + 2 sup{|π f (h)| : h ∈ Rn , |h| ≤ sup{|z| : z ∈ H(f )} + 4}

then

+6|µ(f )|N (q + l + L),

(2.55)

|I f (T1 , T2 , x) − I g (T1 , T2 , x)| ≤ 4−1 δ.

(2.56)

By Proposition 1.3.9 there exists a neighborhood U4 of f in A such that |U f (0, 1, x1 , x2 ) − U g (0, 1, x1 , x2 )| ≤ 2−8 δ

(2.57)

for each g ∈ U4 and each x1 , x2 ∈ Rn satisfying |x1 |, |x2 | ≤ 2M + 4 + 2 sup{|z| : z ∈ H(f )}. Put l0 = 2l + 2q + 2N L + 6, U=

∩4i=1 Ui .

(2.58) (2.59)

Assume that g ∈ U, T ≥ 2l0 and an a.c. function v : [0, T ] → Rn satisﬁes |v(0)|, |v(T )| ≤ K, Φg (0, T, v) ≤ δ. (2.60) It follows from the deﬁnition of U1 (see (2.38), (2.39)) and (2.60) that |v(t)| ≤ M, t ∈ [0, T ].

(2.61)

Assume that there exist numbers S1 , S2 ∈ [0, T ] such that d(v(Si ), H(f )) ≤ δ, i = 1, 2, S2 − S1 ∈ [1 + l + q, 5N (L + l + q)]. (2.62) We will show that for each τ ∈ [S1 , S2 − l], dist(H(f ), {v(t) : t ∈ [τ, τ + l]}) ≤ .

(2.63)

Let us assume the converse. Then there exists a number τ such that τ ∈ [S1 , S2 − l], dist(H(f ), {v(t) : t ∈ [τ, τ + l]}) > .

(2.64)

By (2.48), (2.62), (2.64) and the deﬁnition of δ0 (see (2.44)-(2.47)), σ f (S1 , S2 , v) > δ0 .

(2.65)

167

Turnpike for autonomous problems

We show that I g (S1 , S2 , v) − (S2 − S1 )µ(f ) − π f (v(S1 )) + π f (v(S2 )) > δ0 /2. (2.66) Let us assume the converse. Then (2.62) implies that I g (S1 , S2 , v) ≤ 2 sup{|π f (z)| : z ∈ Rn , d(z, H(f )) ≤ 1} +|µ(f )|(S2 − S1 ) + 1. By this inequality, (2.62) and the choice of U3 (see (2.55), (2.56)), |I f (S1 , S2 , v) − I g (S1 , S2 , v)| ≤ 4−1 δ. Combined with (2.65) this inequality implies (2.66). The contradiction we have reached proves (2.66). By (2.62) there exist h1 , h2 ∈ H(f ) such that |v(Si ) − hi | ≤ δ, i = 1, 2.

(2.67)

It follows from Lemma 5.2.1 that there exists an (f )-good function

and

w0 : [0, ∞) → H(f ) such that w0 (0) = h1

(2.68)

σ f (t1 , t2 , w0 ) = 0

(2.69)

for each t1 ≥ 0, t2 > t1 . By the choice of q (see (2.54)), (2.62) and (2.68) there exists an a.c. function w1 : [0, q] → Rn such that w1 (0) = w0 (S2 − S1 − q), w1 (q) = h2 , σ f (0, q, w1 ) ≤ 8−1 δ.

(2.70)

It follows from Proposition 1.3.5, (2.62), (2.68) and (2.70) that there exists an a.c. function u : [0, T ] → Rn such that u(t) = v(t), t ∈ [0, S1 ] ∪ [S2 , T ], u(t) = w0 (t − S1 ), t ∈ [S1 + 1, S2 − q], u(t) = w1 (t − (S2 − q)), t ∈ [S2 − q, S2 − 1], I g (r, r + 1, u) = U g (0, 1, u(r), u(r + 1)), r = S1 , S2 − 1.

(2.71)

For each a.c. function y : [a, b] → Rn where a ≥ 0, b > a and each r1 , r2 ∈ [a, b] satisfying r1 ≤ r2 we set σ g (r1 , r2 , y) = I g (r1 , r2 , y)−(r2 −r1 )µ(f )−π f (y(r1 ))+π f (y(r2 )). (2.72)

168

TURNPIKE PROPERTIES

(2.60), (2.71) and (2.72) imply that δ ≥ I g (0, T, v) − I g (0, T, u) = σ g (0, T, v) − σ g (0, T, u) = σ g (S1 , S2 , v) − σ g (S1 , S2 , u).

(2.73)

It follows from (2.70), (2.68) and the choice of M (see (2.38), (2.39)), that |w1 (t)| ≤ M, t ∈ [0, q]. (2.74) By (2.70) there exists an a.c. function w ˜ : [S1 , S2 ] → Rn such that w(t) ˜ = w0 (t − S1 ), t ∈ [S1 , S2 − q],

(2.75)

w(t) ˜ = w1 (t − (S2 − q)), t ∈ [S2 − q, S2 ]. It follows from (2.73), (2.72), (2.66), (2.71) and (2.75) that δ ≥ 2−1 δ0 − σ g (S1 , S2 , u) = 2−1 δ0 − σ g (S1 , S2 , w) ˜

(2.76)

˜ − σ g (S1 , S1 + 1, u)] +[σ g (S1 , S1 + 1, w) +[σ g (S2 − 1, S2 , w) ˜ − σ g (S2 − 1, S2 , u)]. We will estimate σ g (S1 , S2 , w) ˜ and σ g (h, h + 1, w) ˜ − σ g (h, h + 1, u), h = S1 , S2 − 1. Let h ∈ {S1 , S2 − 1}. It follows from (2.70), (2.75), (2.71), (2.74), (2.68), (2.61), (2.62) and (2.67) that |w(h)|, ˜ |w(h ˜ + 1)|, |u(h)|, |u(h + 1)| ≤ M + sup{|z| : z ∈ H(f )}, |w(h) ˜ − u(h)|, |w(h ˜ + 1) − u(h + 1)| ≤ δ. By these inequalities, (2.71), (2.72), the choice of U4 (see (2.57)) and δ (see (2.49), (2.50), (2.48)), ˜ − σ g (h, h + 1, u) σ g (h, h + 1, w) ˜ w(h ˜ + 1)) − π f (w(h)) ˜ + π f (w(h ˜ + 1)) ≥ U g (0, 1, w(h), −[U g (0, 1, u(h), u(h + 1)) − π f (u(h)) + π f (u(h + 1))] ≥ U f (0, 1, w(h), ˜ w(h ˜ + 1)) − π f (w(h)) ˜ + π f (w(h ˜ + 1)) −[U f (0, 1, u(h), u(h + 1)) − π f (u(h)) + π f (u(h + 1))] − 2−7 δ ≥ −2−6 δ0 , h ∈ {S1 , S2 − 1}.

(2.77)

We will estimate σ g (S1 , S2 , w). ˜ It follows from (2.69), (2.70) and (2.75) that ˜ − (S2 − S1 )µ(f ) − π f (w(S ˜ 1 )) + π f (w(S ˜ 2 )) ≤ 8−1 δ. (2.78) I f (S1 , S2 , w)

Turnpike for autonomous problems

169

By this inequality, (2.62), (2.75), (2.68), (2.70) and the deﬁnition of U3 (see (2.55), (2.56)), ˜ − I g (S1 , S2 , w)| ˜ ≤ 4−1 δ. |I f (S1 , S2 , w) Combined with (2.78) and (2.72) this inequality implies that ˜ ≤ 3 · 8−1 . σ g (S1 , S2 , w) By this inequality, (2.76) and (2.77), δ ≥ 2−1 δ0 − 3 · 8−1 δ − 2−5 δ0 . This is contradictory to (2.48). The obtained contradiction proves that (2.63) holds for each τ ∈ [S1 , S2 − l]. Therefore we have shown that the following property holds: Property D. For each S1 , S2 ∈ [0, T ] which satisfy (2.62) relation (2.63) holds for each τ ∈ [S1 , S2 − l]. It follows from (2.60), (2.61) and the deﬁnition of U2 , N (see (2.52), (2.53)) that for each r0 ∈ [0, T − (1 + l + q + L(N + 2))] there exists a number r1 such that r1 − r0 ∈ [1 + l + q + 2L, 1 + l + q + L(N + 2)], d(v(r1 ), H(f )) ≤ δ. This implies that there exists a ﬁnite sequence of numbers {Si }Q i=1 ⊂ [0, T ] such that S0 = 0, Si+1 − Si ∈ [1 + l + q + 2L, 1 + l + q + L(N + 2)], i = 0, . . . , Q − 1, T − SQ ≤ 1 + l + q + L(N + 2), d(v(Si ), H(f )) ≤ δ, i = 1, . . . , Q. The assertion of the theorem follows from these relations and Property D.

5.3.

Proof of Theorem 5.1.2

Lemma 5.3.1 Let an integrand f ∈ N have the asymptotic turnpike property and let be a positive number. Then there exists a neighborhood U of f in A such that dist(Ω(v), H(f )) ≤ for each g ∈ U and each (g)-good function v : [0, ∞) → Rn .

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TURNPIKE PROPERTIES

Proof. It follows from Theorem 1.2.2 that there exist a positive number K and a neighborhood U1 of f in A such that lim sup |v(t)| < K t→∞

for each g ∈ U1 and each (g)-good function v : [0, ∞) → Rn . By Theorem 5.1.1 there exist a neighborhood U of f in A which satisﬁes U ⊂ U1 and numbers l0 > l > 0, δ > 0 such that the following property holds: If g ∈ U, T ≥ 2l0 and if an a.c. function v : [0, T ] → Rn satisﬁes |v(0)|, |v(T )| ≤ K, I g (0, T, v) ≤ U g (0, T, v(0), v(T )) + δ, then for each τ ∈ [l0 , T − l0 ], dist(H(f ), {v(t) : t ∈ [τ, τ + l]}) ≤ . Assume that g ∈ U and v : [0, ∞) → Rn is a (g)-good function. By Proposition 4.1.1 and the choice of U, U1 , K there exists a number T0 ≥ 0 such that |v(t)| ≤ K, t ∈ [T0 , ∞), I g (t1 , t2 , v) ≤ U g (t1 , t2 , v(t1 ), v(t2 )) + δ for each t1 ≥ T0 , t2 > t1 . It follows from these relations and the deﬁnition of U, l0 , l, δ that dist(H(f ), Ω(v)) ≤ . The lemma is proved. Construction of the set Fq . Suppose that q is a nonnegative integer. By Lemmas 3.7.1, 3.7.3 and 3.7.4 and Proposition 3.7.1 there exists a ¯q and such that set Eq ⊂ Nq which is an everywhere dense subset of N each integrand f ∈ Eq has the asymptotic turnpike property with the turnpike H(f ) ⊂ Rn . Therefore Ω(v) = H(f ) for each f ∈ Eq and each (f )-good function v : [0, ∞) → Rn . It follows from Theorem 5.1.1 and Lemma 5.3.1 that for each f ∈ Eq and each integer p ≥ 1 there exist an open neighborhood U(f, p) of f in A and numbers M (f, p) > p, l0 (f, p) > l(f, p) > 0, δ(f, p) ∈ (0, p−1 ) such that: dist(H(f ), Ω(v)) ≤ 4−1 δ(f, p) for each g ∈ U(f, p) and each (g)-good function v : [0, ∞) → Rn ; if g ∈ U(f, p), T ≥ 2l0 (f, p) and if an a.c. function v : [0, T ] → Rn satisﬁes |v(0)|, |v(T )| ≤ p, I g (0, T, v) ≤ U g (0, T, v(0), v(T )) + δ(f, p),

(3.1)

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171

then |v(t)| ≤ M (f, p), t ∈ [0, T ] and the following properties hold: (i) for each τ ∈ [l0 (f, p), T − l0 (f, p)], dist(H(f ), {v(t) : t ∈ [τ, τ + l(f, p)]}) ≤ p−1 ;

(3.2)

(ii) if d(v(0), H(f )) ≤ δ(f, p), then (3.2) holds for each τ ∈ [0, T − l0 (f, p)]; (iii) if d(v(T ), H(f )) ≤ δ(f, p), then (3.2) holds for each τ ∈ [l0 (f, p), T − l(f, p)]. We deﬁne ¯ Fq = [∩∞ p=1 ∪ {U(f, p) : f ∈ Eq }] ∩ Nq .

(3.3)

Clearly Fq is a countable intersection of open everywhere dense subsets ¯q . of N Assume that f ∈ Fq , , K > 0. Fix a natural number p such that p > 2K + 4 + 8−1 .

(3.4)

There exists G ∈ Eq such that f ∈ U(G, p).

(3.5)

It follows from (3.4), (3.5) and the deﬁnition of U(G, p), δ(G, p) that for each (f )-good function v : [0, ∞) → Rn , dist(H(G), Ω(v)) ≤ 4−1 δ(G, p) < (4p)−1 < 8−1 .

(3.6)

This implies that for each (f )-good function vi : [0, ∞) → Rn , i = 1, 2, dist(Ω(v1 ), Ω(v2 )) ≤ . Since is any positive number we conclude that f has the asymptotic turnpike property and there exists a compact set H(f ) ⊂ Rn such that Ω(w) = H(f ) for each (f )-good function w : [0, ∞) → Rn . It follows from (3.6) that dist(H(G), H(f )) ≤ 4−1 δ(G, p). (3.7) Set U = U(G, p), M = M (G, p), l0 = l0 (G, p), l = l(G, p), δ = 8−1 δ(G, p).

(3.8)

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TURNPIKE PROPERTIES

Assume that g ∈ U, T ≥ 2l0 and an a.c. function v : [0, T ] → Rn satisﬁes |v(0)|, |v(T )| ≤ K, I g (0, T, v) ≤ U g (0, T, v(0), v(T )) + δ.

(3.9)

It follows from (3.9), (3.8), (3.6), (3.4) and the deﬁnition of U(G, p), M (G, p), l0 (G, p), l(G, p), δ(G, p) that |v(t)| ≤ M, t ∈ [0, T ],

(3.10)

and properties (i)-(iii) hold with f = G. Together with (3.7), (3.8) and (3.4) this implies that dist(H(f ), {v(t) : t ∈ [τ, τ + l]}) ≤

(3.11)

for each τ ∈ [l0 , T − l0 ]; if d(v(0), H(f )) ≤ δ, then (3.11) holds for each τ ∈ [0, T − l0 ]. If d(v(T ), H(f )) ≤ δ, then (3.11) holds for each τ ∈ [l0 , T − l]. This completes the proof of the theorem.

5.4.

Examples

Fix a constant a > 0 and set ψ(t) = t, t ∈ [0, ∞). Consider the complete metric space A of integrands f : Rn × Rn → R1 deﬁned in Section 5.1. Example 1. Consider an integrand f (x, u) = |x|2 + |u|2 , x, u ∈ Rn . It is easy to see that f ∈ Nq for each integer q ≥ 0 if the constant a is large enough. We can show (see Section 3.12) that Ω(v) = {0} for every (f )-good function v : [0, ∞) → Rn . Therefore Theorem 5.1.1 holds with the integrand f . Example 2. Fix a number q > 0 and consider an integrand g(x, u) = q|x|2 |x − e|2 + |u|2 , x, u ∈ Rn , where e = (1, 1, . . . , 1). It is easy to see that g ∈ N if the constant a is large enough. Clearly f does not have the turnpike property.

Chapter 6 LINEAR PERIODIC CONTROL SYSTEMS

In this chapter we study the existence and asymptotic behavior of overtaking optimal trajectories for linear control systems with periodic convex integrands f : [0, ∞) × Rn × Rm → R1 . We extend the results obtained by Artstein and Leizarowitz for tracking periodic problems with quadratic integrands [3] and establish the existence and uniqueness of optimal trajectories on an inﬁnite horizon. The asymptotic behavior of ﬁnite time optimizers is examined.

6.1.

Main results

We consider a linear control system deﬁned by x (t) = Ax(t) + Bu(t), x(0) = x0 ,

(1.1)

where A and B are given matrices of dimensions n × n and n × m, x(t) ∈ Rn , u(t) ∈ Rm and the admissible controls are measurable functions. We assume that the linear system (1.1) is controllable and that the integrand f is a Borel measurable function. The performance of the above control system is measured on any ﬁnite interval [T1 , T2 ] by the integral functional I(T1 , T2 , x, u) =

T2 T1

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

(1.2)

Artstein and Leizarowitz [3] analyzed the existence and structure of solutions of the linear system (1.1) with an integrand f (t, x, u) = (x − Γ(t)) Q(x − Γ(t)) + u P u (t ∈ [0, ∞), x ∈ Rn , u ∈ Rm ), (1.3)

174

TURNPIKE PROPERTIES

where P is a given positive deﬁnite symmetric matrix, Q is a positive semideﬁnite symmetric matrix, the pair (A, Q) is observable and Γ : [0, ∞) → Rn is a measurable function satisfying Γ(t + T ) = Γ(t) (t ∈ [0, ∞)) for some constant T > 0. Artstein and Leizarowitz [3] showed the existence of a unique solution for the inﬁnite horizon tracking of the periodic trajectory Γ 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 this chapter we discuss an another approach which was developed in [111] to extend the results of [3] to a cost function f : [0, ∞) × Rn × Rn → R1 which satisﬁes Assumption A (i) f (t + τ, x, u) = f (t, x, u) (t ∈ [0, ∞), x ∈ Rn , u ∈ Rm ) for some constant τ > 0. (ii) For any t ∈ [0, ∞) the function f (t, ·, ·) : Rn × Rm → R1 is strictly convex. (iii) The function f is bounded on any bounded subset of [0, ∞) × Rn × Rm ; (iv) f (t, x, u) → ∞ as |x| → ∞ uniformly in (t, u) ∈ [0, ∞) × Rm ; (v) f (t, x, u)|u|−1 → ∞ as |u| → ∞ uniformly in (t, x) ∈ [0, ∞) × Rn . Assumption A implies that f is bounded below on [0, ∞) × Rn × Rm . We denote by | · | the Euclidean norm in Rn . For each r > 0 and each x ∈ Rn set B(r) = {y ∈ Rn : |y| ≤ r}, B(x, r) = {y ∈ Rn : |x − y| ≤ r}. In this chapter we assume that the integrand f satisﬁes Assumption A and prove the following results. Proposition 6.1.1 There exists a trajectory-control pair x∗ : [0, τ ] → Rn , u∗ : [0, τ ] → Rm which is the unique solution of the following variational problem: minimize I(0, τ, x, u) subject to x(0) = x(τ ). We show that the periodic trajectory {x∗ (t) : t ∈ [0, ∞)} is a turnpike for optimal solutions of our control problem. Put (1.4) µ = τ −1 I(0, τ, x∗ , u∗ ). The scalar µ is the minimal long-run average cost growth rate. The following results were obtained in [111].

Linear periodic control systems

175

Theorem 6.1.1 For any trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm either (i) I(0, T, x, u) − T µ → ∞ as T → ∞ or (ii) sup{|I(0, T, x, u) − T µ| : T > 0} < ∞. Moreover, in the case relation (ii) holds, then sup{|x(iτ + t) − x∗ (t)| : t ∈ [0, τ ]} → 0 as i → ∞ over the integers. We say that a trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → is good if

Rm

sup{|I(0, T, x, u) − T µ| : T > 0} < ∞. The second statement of Theorem 6.1.1 describes the asymptotic behavior of good trajectory-control pairs. ˜ : [0, ∞) → Rm We say that a trajectory-conrol pair x ˜ : [0, ∞) → Rn , u is overtaking optimal if ˜, u ˜) − I(0, T, x, u)] ≤ 0 lim sup[I(0, T, x T →∞

for each trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisfying x(0) = x ˜(0). Theorem 6.1.2 Let x0 ∈ Rn . Then there exists an overtaking optimal ˜ : [0, ∞) → Rm satisfying trajectory-control pair x ˜ : [0, ∞) → Rn , u x ˜(0) = x0 . Moreover, if a trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisﬁes x(0) = x0 , then there are a time T0 and > 0 such that I(0, T, x, u) ≥ I(0, T, x ˜, u ˜) + for all T ≥ T0 . Theorem 6.1.3 describes the limit behavior of overtaking optimal trajectories. Theorem 6.1.3 Let M, be positive numbers. Then there exists a natural number N such that for any overtaking optimal trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm which satisﬁes |x(0)| ≤ M the relation (1.5) sup{|x(iτ + t) − x∗ (t)| : t ∈ [0, τ ]} ≤ holds for all integers i ≥ N . Moreover, there exists a positive number δ such that for any overtaking optimal trajectory-control pair x : [0, ∞) →

176

TURNPIKE PROPERTIES

Rn , u : [0, ∞) → Rm satisfying |x(0) − x∗ (0)| ≤ δ, the relation (1.5) holds for all integers i ≥ 0. For each z ∈ Rn and T > 0 we set ∆(z, T ) = inf{I(0, T, x, u) : x : [0, T ] → Rn , u : [0, T ] → Rm is a trajectory-control pair satisfying x(0) = z}.

(1.6)

We will see that −∞ < ∆(z, T ) < ∞. Theorem 6.1.4 establishes the turnpike property for optimal trajectories with the turnpike x∗ (·). Theorem 6.1.4 Let M, > 0. Then there exists an integer N ≥ 1 and a positive number δ such that the following property holds: For each T > 2N τ and each trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → Rm which satisﬁes |x(0)| ≤ M,

(1.7)

I(0, T, x, u) ≤ ∆(x(0), T ) + δ,

(1.8)

[N, τ −1 T

− N ]. Moreover the inequality (1.5) holds for all integers i ∈ ∗ if |x(0) − x (0)| ≤ δ, then the inequality (1.5) holds for all integers i ∈ [0, τ −1 T − N ]. The chapter is organized as follows. Section 6.2 contains auxiliary results. In Section 6.3 we discuss discrete-time optimal control problems related to the continuous-time optimal control problems. Theorem 6.1.1 is proved in Section 6.4. Section 6.5 contains the proof of Theorem 6.1.2. Section 6.6 contains the proof of Theorem 6.1.3 while Theorem 6.1.4 is proved in Section 6.7.

6.2.

Preliminary results

We have the following result [41]. Proposition 6.2.1 For every y˜, z˜ ∈ Rn and every T > 0 there exists a solution x(·), y(·) of the following system: x = Ax + BB t y, y = x − At y with the boundary conditions x(0) = y˜, x(T ) = z˜ (where B t denotes the transpose of B).

177

Linear periodic control systems

Assumption A and Proposition 6.2.1 imply that −∞ < ∆(z, T ) < +∞

(2.1)

for each z ∈ Rn and T > 0. For each y, z ∈ Rn deﬁne v(y, z) = min I(0, τ, x, u)

(2.2)

subject to x(0) = y, x(τ ) = z;

(2.3)

here x(·) is the response to u(·). It follows from Proposition 6.2.1 and Assumption A that the function v is convex and −∞ < v(y, z) < ∞ for each y, z ∈ Rn .

(2.4)

Proposition 6.2.2 Let M1 > 0 and 0 < τ0 < τ1 . Then there exists a positive number M2 such that the following property holds: If T ∈ [τ0 , τ1 ] and if an a.c. trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes I(0, T, x, u) ≤ M1 ,

(2.5)

|x(t)| ∈ B(M2 ), t ∈ [0, T ].

(2.6)

then Proof. We may assume without loss of generality that the function f is nonnegative. Fix a positive number δ < min{8−1 τ0 , (2||A|| + 2)−1 }.

(2.7)

It follows from Assumption A that there exist a number c0 > 1 such that f (t, x, u) ≥ 8|u|(||B|| + 1) for each (t, x, u) ∈ [0, ∞) × Rn × Rm satisfying |u| ≥ c0

(2.8)

and a positive number h0 such that f (t, x, u) ≥ 4M1 δ −1 for each (t, x, u) ∈ [0, ∞) × Rn × Rm satisfying |x| ≥ h0 .

(2.9)

M2 > 2 + 2h0 + 2||B||δc0 + 2M1 .

(2.10)

Fix a number

178

TURNPIKE PROPERTIES

Let T ∈ [τ0 , τ1 ] and x : [0, T ] → Rn , u : [0, T ] → Rm be a trajectorycontrol pair which satisﬁes (2.5). We show that (2.6) is fulﬁlled. Assume the contrary. Then there exists t0 ∈ [0, T ] such that |x(t0 )| > M2 .

(2.11)

It follows from (2.5), (2.7) and (2.9) that there exists t1 ∈ [0, T ] for which (2.12) x(t1 ) ∈ B(h0 ), |t1 − t0 | ≤ δ. There exists a number t2 such that min{t0 , t1 } ≤ t2 ≤ max{t0 , t1 }, |x(t2 )| = max{|x(t)| : t ∈ [min{t0 , t1 }, max{t0 , t1 }]}.

(2.13)

(1.1), (2.12) and (2.13) imply that t t t 2 2 2 |x(t1 ) − x(t2 )| = x (t)dt ≤ ||A|| |x(t)|dt + ||B|| |u(t)|dt t1

t1

t 2 ≤ ||A|||x(t2 )|δ + ||B|| |u(t)|dt .

t1

t1

(2.14)

It follows from the inequalities (2.8), (2.5) and (2.12) that t 2 |u(t)|dt ≤ c0 |t1 − t2 | + (8||B|| + 8)−1 I(0, T, x, u) t1

≤ δc0 + (8||B| + 8)−1 M1 . It follows from this inequality, (2.7) and (2.12)-(2.14) that |x(t1 ) − x(t2 )| ≤ 2−1 |x(t2 )| + ||B||δc0 + M1 . Combined with (2.11), (2.12) and (2.13) this inequality implies that 2−1 M2 − h0 ≤ ||B||δc0 + M1 . This relation is contradictory to (2.10). The contradiction we have reached proves the proposition. Proposition 6.2.3 Let M1 , , τ0 , τ1 > 0 and let τ0 < τ1 . Then there exists a number δ > 0 such that the following property holds: If T ∈ [τ0 , τ1 ] and if a trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes (2.5), then |x(t1 ) − x(t2 )| ≤

179

Linear periodic control systems

for each t1 , t2 ∈ [0, T ] such that |t1 − t2 | ≤ δ. Proof. Let a number M2 > 0 be as guaranteed in Proposition 6.2.2. We may assume without loss of generality that the function f is nonnegative. Fix a large positive number c1 . It follows from Assumption A that there exists a number c2 > 0 such that the following property holds: If (t, x, u) ∈ [0, ∞) × Rn × Rm satisﬁes |u| ≥ c2 , then f (t, x, u) ≥ c1 |u|. Let T ∈ [τ0 , τ1 ] and x : [0, T ] → Rn , u : 0, T ] → Rm be a trajectorycontrol pair satisfying (2.5). Then (2.6) holds. Let t1 , t2 ∈ [0, T ], t1 < t2 . By (2.5) and the choice of c2 , t2 t1

|u(t)|dt ≤ c2 (t2 − t1 ) + c−1 1 M1 .

It follows from this inequality, (2.6) and (1.1) that t t2 2 x (t)dt ≤ ||A||M2 (t2 − t1 ) + ||B|| |u(t)|dt

|x(t1 ) − x(t2 )| =

t1

t1

≤ ||A||M2 (t2 − t1 ) + ||B||c2 (t2 − t1 ) + ||B||c−1 1 M1 . This relation implies the validity of the proposition. Proposition 6.2.4 Let M1 and T be positive numbers and let F be the set of all trajectory-control pairs x : [0, T ] → Rn , u : [0, T ] → Rm satisfying (2.5). Then for every sequence {(xi , ui )}∞ i=1 ⊂ F there exist and (x, u) ∈ F such that xik (t) → x(t) as a subsequence {(xik , uik )}∞ k=1 k → ∞ uniformly in [0, T ], xik → x as k → ∞ weakly in L1 (Rn ; (0, T )), and uik → u as k → ∞ weakly in L1 (Rm ; (0, T )). Proof. It follows from Proposition 6.2.2 that there is a positive number M2 such that the inequality (2.6) is valid for each trajectory-control pair (x, u) ∈ F. Assume that {(xi , ui )}∞ i=1 ⊂ F so that I(0, T, xi , ui ) ≤ M1 , xi (t) ∈ B(M2 ) (t ∈ [0, T ], i = 1, 2, . . .).

(2.15)

The boundedness below of f , (2.15) and Assumption A imply that the sequence of functions {ui } is equiabsolutely integrable on [0, T ]. There1 m fore there exists a subsequence {(xik , uik )}∞ k=1 and u ∈ L (R ; (0, T )) such that uik → u as k → ∞ weakly in L1 (Rm ; (0, T )).

(2.16)

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TURNPIKE PROPERTIES

By (2.15), (2.16) and (1.1) we may assume that there exists a function h ∈ L1 (Rn ; (0, T )) such that xik → h as k → ∞ weakly in L1 (Rn ; (0, T )).

(2.17)

We may also assume that limk→∞ xik (0) exists, and for t ∈ (0, T ] we put t

x(t) = lim xik (0) + k→∞

0

h(s)ds.

(2.18)

It is easy to see that xik (t) → x(t) as k → ∞ for any t ∈ [0, T ]. By Proposition 6.2.3 xik (t) → x(t) as k → ∞ uniformly in [0, T ]. We need to show that (x, u) ∈ F. It follows from Assumption A that the set F is convex. Therefore there exists a sequence {(yi , vi )}∞ i=1 ⊂ F for which (2.19) yi (t) → x(t) as i → ∞ uniformly in [0, T ] and yi (t) → h(t), vi (t) → u(t) as i → ∞ a.e. in [0, T ].

(2.20)

(1.1), (2.18), (2.19) and (2.20) imply that (x, u) is a trajectory-control pair. By Assumption A and Fatou’s lemma, (x, u) satisﬁes (2.5). This completes the proof of the proposition. Corollary 6.2.1 Let x1 , x2 ∈ Rn . Then there is a unique trajectorycontrol pair x : [0, τ ] → Rn , u : [o, τ ] → Rm such that x(0) = x1 , x(τ ) = x2 and I(0, τ, x, u) = v(x1 , x2 ). Corollary 6.2.1 and Assumption A imply that the function v is strictly convex. It follows from Proposition 6.2.2 that v(y, z) → ∞ as |y| + |z| → ∞. Put

(2.21)

D = {(z, z) : z ∈ Rn }

and denote by (z ∗ , z ∗ ) a unique minimum of v on D. Corollary 6.2.1 now implies Proposition 6.1.1 with x∗ (0) = z ∗ , µ = τ −1 v(z ∗ , z ∗ ).

(2.22)

Proposition 6.2.5 There exists p ∈ Rn such that the function θ : Rn × Rn → R1 deﬁned by θ(y, z) = v(y, z) − v(z ∗ , z ∗ ) − p (y − z), y, z ∈ Rn

(2.23)

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Linear periodic control systems

is strictly convex and has the following property: θ(y, z) > 0 if (y, z) = (z ∗ , z ∗ ), and θ(z ∗ , z ∗ ) = 0. Proof. For any function φ : B → R1 set epi(φ) = {(r, x) : x ∈ B, r ≥ φ(x)}. Let

Ω = (epi(v)) ∪ {(z, z, v(z ∗ , z ∗ )) : z ∈ Rn }

and let w(y, z) be the function whose epigraph is conv(Ω) (namely, the closed convex hull of Ω). It is not diﬃcult to see that the function w : Rn × Rn → R1 is convex, −∞ < w(y, z) ≤ v(y, z) (y, z ∈ Rn ) and

w(z, z) ≥ v(z ∗ , z ∗ ) for all z ∈ Rn .

There exists a subgradient vector (p1 , p2 ) of w at (z ∗ , z ∗ ) for which w(y, z) ≥ w(z ∗ , z ∗ ) + (p1 , p2 ) ((y − z) − (z ∗ , z ∗ )) (y, z ∈ Rn ). Since w(z, z) = v(z ∗ , z ∗ ) for each z ∈ Rn we have p2 = −p1 . This equality implies that v(y, z) ≥ w(y, z) ≥ v(z ∗ , z ∗ ) + p1 (y − z) (y, z ∈ Rn ). This completes the proof of the proposition. Proposition 6.2.4 and the uniqueness in Corollary 6.2.1 imply the following result. Proposition 6.2.6 Let > 0. Then there exists δ > 0 such that if a trajectory-control pair x : [0, τ ] → Rn , u : [0, τ ] → Rn satisﬁes x(0), x(τ ) ∈ B(z ∗ , δ), I(0, τ, x, u) ≤ v(z ∗ , z ∗ ) + δ, then

(2.24)

|x(t) − x∗ (t)| ≤ (t ∈ [0, τ ]).

Proposition 6.2.7 For each positive number there exists δ > 0 such that the following property holds: If a trajectory-control pair x : [0, τ ] → Rn , u : [0, τ ] → Rm satisﬁes (2.24), then for all T ∈ (0, τ ], |I(0, T, x, u) − I(0, T, x∗ , u∗ )| ≤ .

182

TURNPIKE PROPERTIES

Proof. Assume the converse. Then there exist > 0, a sequence of numbers {Ti }∞ i=0 ⊂ (0, τ ] and a sequence of trajectory-control pairs xi : [0, τ ] → Rn , ui : [0, τ ] → Rm , i = 0, 1, . . . , such that

|xi (0) − z ∗ | + |xi (τ ) − z ∗ | → 0, I(0, τ, xi , ui ) − v(z ∗ , z ∗ ) → 0 as i → ∞,

|I(0, Ti , xi , ui ) − I(0, Ti , x∗ , u∗ )| ≥ , i = 0, 1, 2, . . . .

(2.25) (2.26)

We may assume that the limit T = lim Ti i→∞

(2.27)

exists and one of the relations below holds: (a) I(0, Ti , xi , ui ) − I(0, Ti , x∗ , u∗ ) ≥ for all natural numbers i; (b) I(0, Ti , xi , ui ) − I(0, Ti , x∗ , u∗ ) ≤ − for all natural numbers i. If relation (a) holds we deﬁne E = [T, τ ], Ei = [Ti , τ ], i = 0, 1, 2, . . . .

(2.28)

Otherwise we set E = [0, T ], Ei = [0, Ti ], i = 0, 1, 2, . . . .

(2.29)

In view of (2.25) we may assume that Ei

f (t, xi (t), ui (t))dt −

Ei

f (t, x∗ (t), u∗ (t))dt

≤ −2−1 for all integers i ≥ 1.

(2.30)

By Assumption A and (2.25) the sequence of the functions {ui }∞ i=0 is equiabsolutely integrable on [0, τ ]. In view of Corollary 6.2.1, Proposition 6.2.4 and (2.25) we may assume without loss of generality that xi (t) → x∗ (t) as i → ∞ uniformly in [0, τ ], xi → (x∗ ) as i → ∞ weakly in L1 (Rn ; (0, τ )) and ui → u∗ as i → ∞ weakly in L1 (Rm ; (0, τ )). Since the function f (t, ·, ·) is convex for any t ≥ 0 we conclude that there exists a sequence of trajectory-control pairs yi : [0, τ ] → Rn , wi : [0, τ ] → Rm , i = 0, 1, 2, . . .

183

Linear periodic control systems

such that

|yi (0) − z ∗ | + |yi (τ ) − z ∗ | → 0, I(0, τ, yi , wi ) − v(z ∗ , z ∗ ) → 0 as i → ∞,

(2.31)

yi (t) → x∗ (t) as i → ∞ uniformly in [0, τ ], τ 0

Ei

|wi (t) − u∗ (t)|dt → 0 as i → ∞,

f (t, yi (t), wi (t))dt −

Ei

(2.32)

f (t, x∗ (t), u∗ (t))dt

≤ −2−1 for all integers i ≥ 1.

(2.33)

We may assume by extracting a subsequence and re-indexing that wi (t) → u∗ (t) as i → ∞ a.e. in [0, τ ].

(2.34)

For a set F ⊂ [0, τ ] we deﬁne λF : [0, τ ] → R1 as follows: λF (t) = 1 (t ∈ F ), λF (t) = 0 (t ∈ [0, τ ] \ F ). By Fatou’s lemma, (2.32)-(2.34) and Assumption A, τ 0

∗

∗

f (t, x (t), u (t))λE (t)dt ≤ lim sup

τ

i→∞

≤ lim sup i→∞

Ei

∗

∗

−1

f (t, x (t), u (t))dt − 2

f (t, yi (t), wi (t))λEi (t)dt

0

=

E

f (t, x∗ (t), u∗ (t))dt − 2−1 .

The contradiction obtained proves the proposition.

6.3.

Discrete-time control systems

Let K be a compact metric space and let w : K ×K → R1 be bounded and lower semicontinuous. We deﬁne a(w) = sup{w(x, y) : x, y ∈ K}, b(w) = inf{w(x, y) : x, y ∈ K},

µ(w) = inf lim inf N N →∞

−1

N −1

w(zi , zi+1 ) :

{zi }∞ i=0

⊂K .

(3.1)

i=0

The following two results were established in [39] when K was a compact set in Rn but their proofs remain in force when K is any compact metric space.

184

TURNPIKE PROPERTIES

Proposition 6.3.1 1. For each natural number N and each sequence {zi }N i=0 ⊂ K, N −1

[w(zi , zi+1 ) − µ(w)] ≥ b(w) − a(w).

i=0

2. For every sequence {zi }∞ i=0 ⊂ K, either N −1

[w(zi , zi+1 ) − µ(w)] → ∞ as N → ∞

i=0

or

N −1

sup |

[w(zi , zi+1 ) − µ(w)]| : N = 1, 2, . . .

< ∞.

i=0

3. For every initial value z0 there is a sequence {zi }∞ i=0 ⊂ K such that −1 N [w(zi , zi+1 ) − µ(w)] ≤ 4|a(v) − b(v)| (N = 1, 2, . . .). i=0

Proposition 6.3.2 Let w : K × K → R1 be a continuous function. Deﬁne

N −1

w

[w(zi , zi+1 ) − µ(v)] :

π (x) = inf lim inf N →∞

{zi }∞ i=0

⊂ K, z0 = x ,

i=0

θw (x, y) = w(x, y) − µ(w) + π w (y) − π w (x) for x, y ∈ K. Then π w and θw are continuous functions, θw (x, y) ≥ 0 for all x, y ∈ K and for every x ∈ K there is y ∈ K for which θw (x, y) = 0. Consider a continuous function w : Rn × Rn → R1 satisfying w(x, y) → ∞ as |x| + |y| → ∞. Let x ∈ Rn . Deﬁne µ(w) = inf{lim inf N −1 N →∞

N −1

n w(zk , zk+1 ) : {zk }∞ k=0 ⊂ R , z0 = x}.

k=0

(3.2)

185

Linear periodic control systems

By Propositions 6.3.1 and 3.5.1, µ(w) is independent of the initial point x0 . For x, y ∈ Rn we set

π w (x) = inf lim inf N →∞

N −1

n [w(zi , zi+1 ) − µ(v)] : {zi }∞ i=0 ⊂ R , z0 = x ,

i=0

w

w

w

θ (x, y) = w(x, y) − µ(w) + π (y) − π (x).

(3.3) (3.4)

We have the following result. Proposition 6.3.3 Let M1 , M2 > 0, inf{w(x, y) : x, y ∈ Rn , |x| + |y| ≥ M1 } > |w(0, 0)| + 1.

(3.5)

Then there exists a natural number N > 2 such that for each natural number q ≥ N and any sequence {xk }qk=0 ⊂ Rn the following properties hold: 1. If {k ∈ {0, . . . , q} : xk ∈ B(M1 )} = {0, q} and y0 = x0 , yq = xq , yk = 0 (k = 1, . . . , q − 1), then q−1

[w(xk .xk+1 ) − w(yk .yk+1 )] ≥ M2 ;

(3.6)

k=0

2. If {k ∈ {0, . . . , q} : xk ∈ B(M1 )} = {0} and y0 = x0 , yk = 0 (k = 1, . . . , q), then (3.6) is true. Proposition 6.3.4 Assume that M1 , M2 > 0 and (3.5) holds. Then there exists a number M3 > M1 + M2 such that for each natural number q and each sequence {xk }qk=0 ⊂ Rn the following properties hold: 1. If x0 , xq ∈ B(M1 ), max{|xk | : k = 0, . . . , q} > M3 , (3.7) then there is a sequence {yk }qk=0 ⊂ Rn such that yi = zi , i = 0, q, and (3.6) holds. 2. If (3.8) x0 ∈ B(M1 ), max{|xk | : k = 0, . . . , q} > M3 , then there is a sequence {yk }qk=0 ⊂ Rn such that y0 = z0 and (3.6) holds. Proof. Let an integer N > 6 be as guaranteed in Proposition 6.3.3. Fix a large number M3 > M1 + M2 . We prove Assertion 1. Assume

186

TURNPIKE PROPERTIES

that {xk }qk=0 ⊂ Rn satisﬁes (3.7). Then there is j ∈ {0, . . . , q} such that |xj | > M3 . Set i1 = max{i ∈ {0, . . . , j} : |xi | ≤ M1 }, i2 = min{i ∈ {j, . . . q} : |xi | ≤ M1 }. If i2 − i1 ≥ N then the validity of Assertion 1 follows from the deﬁnition of N and Proposition 6.3.3. If i2 − i1 < N we set yi = xi , i ∈ {0, . . . , i1 } ∪ {i2 , . . . , q}, yi = 0, i = i1 + 1, . . . , i2 − 1 (3.9) and it is easy to see that (3.6) holds if the constant M3 is large enough. We will prove Assertion 2. Assume that {xi }qi=0 ⊂ Rn satisﬁes (3.8). Then there is j ∈ {1, . . . , q} such that |xj | > M3 . Set i1 = max{i ∈ {0, . . . , j} : |xi | ≤ M1 }. If |xi | > M1 for i = j, . . . , q we set yi = xi , i = 0, . . . , i1 , yi = 0, i = i1 + 1, . . . , q. Otherwise we set i2 = min{k ∈ {j, . . . , q} : |xk | ≤ M1 } and deﬁne {yk }qk=0 by (3.9). It is easy to verify that in both cases (3.6) holds. This completes the proof of the proposition. Propositions 3.5.4 and 6.3.4 imply the following result. n Proposition 6.3.5 There is M > 0 such that if {xk }∞ k=0 ⊂ R and if the sequence N ∞

[w(xk , xk+1 ) − µ(w)]

k=0

N =0

is bounded, then lim supk→∞ |xk | ≤ M .

6.4.

Proof of Theorem 6.1.1

Proposition 6.4.1 µ(v) = µτ (see (1.4), (2.2) and (3.2)). Proof. In view of (2.22), µ(v) ≤ µτ . It follows from Propositions 6.3.1 n and 3.5.2 that there exists a sequence {zi }∞ i=0 ⊂ R such that N [v(zi , zi+1 ) − µ(v)] : N = 0, 1, . . . < ∞. i=0

187

Linear periodic control systems

By Proposition 6.2.5 and (2.22), for each natural number N , N −1

[v(zi , zi+1 ) − µ(v)] =

i=0

N −1

θ(zi , zi+1 ) + p (z0 − zN ) + N (τ µ − µ(v)).

i=0

Proposition 3.5.4 implies that sup{|zi | : i = 0, 1, . . .} < ∞. Therefore the sequence sup{N (µτ − µ(v))}∞ N =1 is bounded from above. This completes the proof of the proposition. Proof of Theorem 6.1.1 Assumption A implies that relation (i) holds if and only if I(0, iτ, x, u) − iτ µ → ∞ as i → ∞ (4.1) for i integer, and relation (ii) holds if and only if sup{|I(0, iτ, x, u) − iτ µ| : i = 1, 2, . . .} < ∞.

(4.2)

It is easy to see that (4.2) holds if and only if ∞

[I(τ i, τ (i + 1), x, u) − v(x(iτ ), x((i + 1)τ ))] < ∞

(4.3)

i=0

and sup

N −1

[v(x(iτ ), x((i + 1)τ )) − µτ ] : N = 1, 2, . . .

< ∞.

(4.4)

i=0

It is easy to see that (4.1) is not fulﬁlled if and only if relations (4.3) and (4.4) hold. Therefore either condition (i) or (ii) is valid. Assume that relation (ii) holds so that (4.3) and (4.4) are satisﬁed. Propositions 3.5.4 and 6.4.1 imply that the sequence {x(iτ )}∞ i=0 is bounded. Combined with (4.3), (4.4), (2.22) and Proposition 6.2.5 this implies that ∞

θ(x(iτ ), x((i + 1)τ )) < ∞, x(iτ ) → z ∗ as i → ∞.

(4.5)

i=0

The ﬁnal assertion in the theorem now follows from (4.3), (4.5) and Proposition 6.2.6.

188

TURNPIKE PROPERTIES

6.5.

Proof of Theorem 6.1.2

Let x0 ∈ Rn . By Propositions 6.3.4 and 6.3.5 there exists a ball n D ⊂ Rn such that for every sequence {zi }∞ i=0 ⊂ R not included in D with z0 = x0 there exists a sequence {xi }∞ i=0 ⊂ D with s0 = z0 such that N

v(si , si+1 ) ≤

i=0

N

v(zi , zi+1 ) − 1 for all large N.

i=0

Denote by A the set of all sequences {zi }∞ i=0 ⊂ D such that z0 = x0 , lim inf N →∞

N

[v(zi , zi+1 ) − µτ ] < ∞

i=0

(see (1.4), (2.2), (3.2) and Proposition 6.4.1). It is easy to see that v

π (x0 ) = inf lim inf

N −1

N →∞

[v(zi , zi+1 ) − µτ ] :

{zi }∞ i=0

∈A .

(5.1)

i=0

It follows from (2.22) and Propositions 6.4.1 and 6.2.5 that for each {zi }∞ i=0 ∈ A, ∞

θ(zi , zi+1 ) < ∞, zi → z ∗ as i → ∞.

(5.2)

i=0

Put F ({zi }∞ i=0 ) =

∞

θ(zi , zi+1 ) where {zi }∞ i=0 ∈ A.

i=0

By Proposition 6.2.5 the function θ is strictly convex. This implies that the function F has a unique minimizer which we denote by {yi }∞ i=0 ∈ A. The proof of Theorem 6.1.2 will be based on the following auxiliary result. Lemma 6.5.1 π v (x0 ) = lim inf N →∞

N

[v(yi , yi+1 ) − µτ ] = lim

N →∞

i=0

N

[v(yi , yi+1 − µτ ] (5.3)

i=0

n and if a sequence {zi }∞ i=0 ⊂ R satisﬁes ∞ z0 = x0 , {zi }∞ i=0 = {yi }i=0 ,

(5.4)

189

Linear periodic control systems

then there is a natural number N0 such that N

inf{

[v(zi , zi+1 ) − v(yi , yi+1 )] : N ≥ N0 } > 0.

(5.5)

i=0

A,

Proof. (5.2), (2.22) and Proposition 6.2.5 imply that for each {zi }∞ i=0 ∈

lim

N →∞

N

N

[v(zi , zi+1 ) − τ µ] = lim [ N →∞

i=0

=

∞

θ(zi , zi+1 ) + p (x0 − zN )]

i=0

θ(zi , zi+1 ) + p (x0 − z ∗ ).

(5.6)

i=0

(5.1) and (5.6) imply (5.3). n Assume that a sequence {zi }∞ i=0 ⊂ R satisﬁes (5.4). We show that there exists a natural number N0 such that (5.5) holds. In view of our ∞ choice of D we may assume that {zi }∞ i=0 ∈ A. Since {yi }i=0 is a unique minimizer of the function F it follows from (5.6) that lim

N →∞

N

[v(zi , zi+1 ) − v(yi , yi+1 )] =

i=0

∞

θ(zi , zi+1 )

i=0

−

∞

θ(yi , yi+1 ) > 0.

i=0

This completes the proof of the lemma. By Corollary 6.2.1 there exists a trajectory-control pair x˜ : [0, ∞) → ˜ : [0, ∞) → Rm such that for i = 0, 1, . . ., Rn , u x ˜(iτ ) = yi , I(iτ, (i + 1)τ, x ˜, u ˜) = v(˜ x(iτ ), x ˜((i + 1)τ )).

(5.7)

Assume that x : [0, ∞) → Rn , u : [0, ∞) → Rm is a trajectory-control pair satisfying x, u ˜). (5.8) x(0) = x0 , (x, u) = (˜ We show that there is a time T0 such that inf{I(0, T, x, u) − I(0, T, x ˜, u ˜) : T ∈ [T0 , ∞)} > 0.

(5.9)

By the choice of x ˜,˜ u, Lemma 6.5.1 and Theorem 6.1.1, sup{|I(0, T, x ˜, u ˜) − µT | : T ∈ [0, ∞)} < ∞.

(5.10)

190

TURNPIKE PROPERTIES

In view of Theorem 6.1.1 we may assume without loss of generality that sup{|I(0, T, x, u) − µT | : T ∈ [0, ∞)} < ∞,

(5.11)

sup{|x(iτ + t) − x∗ (t)| : t ∈ [0, τ ]} → 0 as i → ∞ over the integers. (5.12) These relations imply that I(iτ, (i + 1)τ, x, u) − v(z ∗ , z ∗ ) → 0 as i → ∞.

(5.13)

We show that there exist > 0 and a natural number N0 such that I(0, N τ, x, u) − I(0, N τ, x ˜, u ˜) ≥ 2

(5.14)

for all integers N ≥ N0 . ˜(i0 τ ) for some natural number i0 , then the existence of If x(i0 τ ) = x > 0 and an integer N0 ≥ 1 such that (5.14) holds for all integers N ≥ ˜. If x(iτ ) = x ˜(iτ ) N0 follows from Lemma 6.5.1 and the deﬁnition of x˜, u for all natural numbers i, then this existence follows from (5.7), (5.8), and Corollary 6.2.1. The validity of the theorem now follows from (5.14), (5.7), (5.12), (5.13) and Proposition 6.2.7.

6.6.

Proof of Theorem 6.1.3

The following auxiliary result shows that if a sequence {xi }qi=p is an approximate solution of the related discrete-time optimal control problem and q − p is large enough, then xj and xj+1 are close to z ∗ for a certain j ∈ {p, . . . , q − 1}. Lemma 6.6.1 Let M1 , M2 , be positive numbers. Then there exists a natural number N ≥ 4 such that the following property holds: n If a natural number Q ≥ N , if a sequence {xk }Q k=0 ⊂ R satisﬁes xk ∈ B(M1 ) for k = 0, . . . , Q, Q−1

v(xk .xk+1 )

k=0

≤ inf{

Q−1 k=0

n v(yk , yk+1 ) : {yk }Q k=0 ⊂ R , y0 = x0 , yQ = xQ } + M2 , (6.1)

Linear periodic control systems

191

and if integers p, q ∈ {0, . . . , Q} satisfy q − p ≥ N , then there exists an integer j ∈ {p, . . . , q − 1} such that xj , xj+1 ∈ B(z ∗ , ).

(6.2)

Proof. Fix a large integer N ≥ 4. Suppose that an integer Q ≥ N , n a sequence {xk }Q k=0 ⊂ R satisﬁes (6.1), and integers p, q ∈ {0, . . . , Q} satisfy q −p ≥ N . We show that there exists an integer j ∈ {p, . . . , q −1} satisfying (6.2). Assume the converse and set yi = xi (i ∈ {0, . . . , p} ∪ {q, . . . , Q}), yi = z ∗ (i ∈ {p + 1, . . . , q − 1}). By (6.1) and Proposition 6.2.5, M2 ≥

Q−1

[v(xk , xk+1 ) − v(yk , yk+1 )]

k=0

=

q−1

θ(xk , xk+1 ) − θ(xp , z ∗ ) − θ(z ∗ , xq )

k=p

≥ N inf{θ(z1 , z2 ) : z1 , z2 ∈ B(M1 ), |z1 − z ∗ | + |z2 − z ∗ | ≥ } −2 sup{θ(z1 , z2 ) : z1 , z2 ∈ B(M1 + |z ∗ |)}. When N is suﬃciently large we obtain a contradiction which proves the lemma. Theorem 6.1.1, Propositions 6.2.2 and 6.3.4 and Corollary 6.2.1 imply the following result. Lemma 6.6.2 For each M0 > 0 there exists a number M1 > 0 such that the following property holds: If an overtaking optimal trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisﬁes |x(0)| ≤ M0 , then |x(t)| ≤ M1 for all t ∈ [0, ∞). Lemmas 6.6.1 and 6.6.2 and Corollary 6.2.1 imply the following result which shows that an overtaking optimal trajectory-control pair (x, u) reaches a neighborhood of z ∗ during a period of time which depends only on |x(0)|. Lemma 6.6.3 For each M, δ > 0 there exists an integer N ≥ 1 such that the following property holds: If an overtaking optimal trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisﬁes |x(0)| ≤ M , then there exists an integer i ∈ [0, N ] such that |x(iτ ) − z ∗ | ≤ δ.

192

TURNPIKE PROPERTIES

The next lemma establishes the turnpike property for an overtaking optimal trajectory-control pair (x, u) such that x(0) belongs to a suﬃciently small neighborhood of z ∗ . Lemma 6.6.4 For each > 0 there exists a positive number δ such that the following property holds: If an overtaking optimal trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisﬁes (6.3) |x(0) − z ∗ | ≤ δ, then for all integers i ≥ 0, sup{|x(iτ + t) − x∗ (t)| : t ∈ [0, τ ]} ≤ .

(6.4)

Proof. Let > 0. It follows from Proposition 6.2.6 that there exists a number δ0 ∈ (0, 1) such that the following property holds: If a trajectory-control pair x : [0, τ ] → Rn , u : [0, τ ] → Rm satisﬁes |x(0)−z ∗ | ≤ δ0 , |x(τ )−z ∗ | ≤ δ0 , I(0, τ, x, u)−v(x(0), x(τ )) ≤ δ0 , (6.5) then

|x(t) − x∗ (t)| ≤ (t ∈ [0, τ ]).

(6.6)

There exists a number M1 > 0 such that Lemma 6.6.2 holds with M0 = |z ∗ | + 1. Fix a small constant δ > 0. Let x : [0, ∞) → Rn , u : [0, ∞) → Rm be an overtaking optimal trajectory-control pair satisfying (6.3). By Corollary 6.2.1 there exists a trajectory-control pair y : [0, ∞) → Rn , w : [0, ∞) → Rm such that y(0) = x(0), y(iτ ) = z ∗ for i = 1, 2, . . . , I(iτ, (i + 1)τ, y, w) = v(y(iτ ), y((i + 1)τ )) for i = 0, 1, . . . . By Theorem 6.1.1 x(iτ ) → z ∗ as i → ∞. It is easy to see that 0 ≥ lim sup[I(0, kτ, x, u) − I(0, kτ, y, w)] k→∞

≥

∞

[I(kτ, (k + 1)τ, x, u) − v(x(kτ ), x((k + 1)τ ))]

k=0

+

∞

θ(x(kτ ), x((k + 1)τ )) − sup{θ(z, z ∗ ) : z ∈ RN , |z − z ∗ | ≤ δ}.

k=0

When the constant δ is suﬃciently small it follows from this relation and the deﬁnition of δ0 that (6.4) holds for all integers i ≥ 0. The lemma is proved. Theorem 6.1.3 now follows from Lemmas 6.6.3 and 6.6.4.

193

Linear periodic control systems

6.7.

Proof of Theorem 6.1.4

Proposition 6.3.4, the boundedness below of f and Corollary 6.2.1 imply the following result. Lemma 6.7.1 For each pair of positive numbers M0 , M1 there exists M2 > M1 such that the following property holds: If N is a natural number, T ∈ [N τ, (N + 1)τ ) and if a trajectorycontrol pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes |x(0)| ≤ M0 , I(0, T, x, u) ≤ ∆(x(0), T ) + M1 , then

N −1 k=0

v(x(kτ ), x((k + 1)τ )) ≤

N −1

v(yk .yk+1 ) + M2

k=0

n for each sequence {yk }N k=0 ⊂ R which satisﬁes y0 = y(0).

Proposition 6.3.4 and Lemma 6.7.1 imply the following result. Lemma 6.7.2 For each positive number M0 there exists M > M0 such that if a trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes T ≥ τ, |x(0)| ≤ M0 , I(0, T, x, u) ≤ ∆(x(0), T ) + 1,

(7.1)

then |x(iτ )| ≤ M for all integers i ∈ [0, τ −1 T ]. Lemmas 6.6.1, 6.7.1 and 6.7.2 imply the following result. Lemma 6.7.3 For each M0 , > 0 there exists a natural number N > 4 such that the following property holds: If T ≥ N τ , a trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes (7.1) and if a pair of integers p, q ∈ [0, τ −1 T ] satisfy q − p ≥ N , then there is an integer j ∈ {p, . . . , q − 1} such that max{|x(jτ ) − z ∗ |, |x((j + 1)τ ) − z ∗ |} ≤ . The next auxiliary result establishes the turnpike property for approximate optimal trajectory-control pairs (x, u) on ﬁnite intervals such that x(0) belongs to a suﬃciently small neighborhood of z ∗ . Lemma 6.7.4 For each > 0 there exist a natural number N and a positive number δ > 0 such that the following property holds: If T ≥ N τ and if a trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → m R satisﬁes I(0, T, x, u) ≤ ∆(x(0), T ) + δ, |x(0) − z ∗ | ≤ δ,

(7.2)

194

TURNPIKE PROPERTIES

then for all integers i ∈ [0, τ −1 T − N ], sup{|x(iτ + t) − x∗ (t)| : t ∈ [0, τ ]} ≤ .

(7.3)

Proof. Let > 0. It follows from Proposition 6.2.6 that there exists δ0 ∈ (0, 1) such that the following property holds: If a trajectory-control pair x : [0, τ ] → Rn , u : [0, τ ] → Rm satisﬁes |x(0) − z ∗ |, |x(τ ) − z ∗ | ≤ δ0 , I(0, τ, x, u) − v(x(0), x(τ )) ≤ δ0 , then for all t ∈ [0, τ ],

|x(t) − x∗ (t)| ≤ .

There exists a number M > |z ∗ | + 4 such that Lemma 6.7.2 holds with M0 = |z ∗ | + 4. By Proposition 6.2.5 there exists a positive number δ1 such that δ1 < min{1, , δ0 }, δ1 < 4−1 min{θ(z1 , z2 ) : z1 , z2 ∈ Rn , |z1 |, |z2 | ≤ M, |z1 − z ∗ | + |z2 − z ∗ | ≥ δ0 }.

(7.4)

Fix a positive number δ such that δ < 8−1 δ1 , sup{θ(z1 , z2 ) : z1 , z2 ∈ Rn , |z1 − z ∗ | + |z2 − z ∗ | ≤ δ} < 8−1 δ1 . (7.5) There exists an integer N0 > 4 such that Lemma 6.7.3 holds with M0 = |z ∗ | + 4, = δ, and N = N0 . Fix an integer N > 2N0 + 4.

(7.6)

Let T > N τ and let x : [0, T ] → Rn , u : [0, T ] → Rm be a trajectorycontrol pair satisfying (7.2). There exists an integer q ≥ N such that q ≤ T τ −1 < q + 1.

(7.7)

It follows from Lemma 6.7.3 and the deﬁnition of N0 that there is an integer j ∈ {q − N0 , . . . , q − 1} such that max{|x(jτ ) − z ∗ |, |x((j + 1)τ ) − z ∗ | ≤ δ.

(7.8)

By Corollary 6.2.1 there exists a trajectory-control pair y : [0, T ] → Rn , w : [0, T ] → Rm such that y(t) = x(t), w(t) = u(t) (t ∈ [jτ, T ]), y(0) = x(0), y(iτ ) = z ∗ (i = 1, . . . , j − 1), I(iτ, (i + 1)τ, y, w) = v(y(iτ ), y((i + 1)τ )) (i = 0, . . . , j − 1).

(7.9)

195

Linear periodic control systems

It follows from (7.2), (7.9) and Proposition 6.2.5 that δ ≥ I(0, T, x, u) − I(0, T, y, w) =

j−1

[I(iτ, (i + 1)τ, x, u) − v(x(iτ ), x((i + 1)τ ))]

i=0

+

j−1

θ(x(iτ ), x((i + 1)τ )) − θ(x(0), z ∗ ) − θ(z ∗ , x(jτ )).

i=0

Combined with (7.2), (7.8) and (7.5) this implies that for i = 0, . . . , j −1, sup{I(iτ, (i + 1)τ, x, u) − v(x(iτ ), x((i + 1)τ )), θ(x(iτ ), x((i + 1)τ ))} ≤ δ + 2 sup{θ(z1 , z2 ) : z1 , z2 ∈ Rn , |z1 − z ∗ | + |z2 − z ∗ | ≤ δ} ≤ 2−1 δ.

(7.10)

It follows from Lemma 6.7.2, the choice of M , and (7.2) that |x(iτ )| ≤ M (i = 0, . . . , q).

(7.11)

Equations (7.4), (7.10) and (7.11) imply that |x(iτ ) − z ∗ | ≤ δ0 for i = 0, . . . , j.

(7.12)

It is not diﬃcult to see that j > T τ −1 − 1 − N0 ≥ T τ −1 + 3 − N . It follows from (7.10), (7.12), and the deﬁnition of δ0 that |x(iτ + t) − x∗ (t)| ≤ (t ∈ [0, τ ], i = 0, . . . , j − 1). This completes the proof of the lemma. Theorem 6.1.4 now follows from Lemmas 6.7.3 and 6.7.4.

Chapter 7 LINEAR SYSTEMS WITH NONPERIODIC INTEGRANDS

In this chapter we analyze the existence and structure of optimal trajectories of linear control systems with nonperiodic convex integrands f : [0, ∞) × Rn × Rm → R1 , and extend the results of Chapter 6 established for linear periodic control systems.

7.1.

Main results

We consider a linear control system deﬁned by x (t) = Ax(t) + Bu(t), x(0) = x0 ,

(1.1)

where A and B are given matrices of dimensions n × n and n × m, x(t) ∈ Rn , u(t) ∈ Rm and the admissible controls are measurable functions. The performance of the above control system is measured on any ﬁnite interval [T1 , T2 ] by the integral functional I(T1 , T2 , x, u) =

T2 T1

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

(1.2)

where f : [0, ∞) × Rn × Rm → R1 is a given integrand. We assume that the linear system (1.1) is controllable and that the integrand f is a Borel measurable function. In this chapter we extend the results of Chapter 6 to a cost function f : [0, ∞) × Rn × Rm → R1 which satisﬁes the following assumptions: (A1) (i) (uniform strict convexity; see [82]). For any t ∈ [0, ∞) the function f (t, ·, ·) : Rn × Rm → R1 is strictly convex and moreover, for any > 0

198

TURNPIKE PROPERTIES

there exists δ() > 0 such that f (t, 2−1 (x1 + x2 ), 2−1 (u1 + u2 )) ≤ 2−1 [f (t, x1 , u1 ) + f (t, x2 , u2 )] − δ() for each t ≥ 0, each x1 , x2 ∈ Rn and each u1 , u2 ∈ Rn satisfying |x1 − x2 | + |u1 − u2 | ≥ ; (ii) f (t, x, u)|u|−1 → ∞ as |u| → ∞ uniformly in (t, x) ∈ [0, ∞) × Rn ; (iii) the function f is bounded on any bounded subset of [0, ∞)×Rn × Rm ; (iv) f (t, x, u) → ∞ as |x| → ∞ uniformly in (t, u) ∈ [0, ∞) × Rm ; (v) for any (x, u) ∈ Rn × Rm the function f (·, x, u) : [0, ∞) → R1 is bounded; (vi) the function f is bounded below on [0, ∞) × Rn × Rm ; (A2) For each M, > 0 there exist Γ, δ > 0 such that if t ≥ 0, x1 , x2 ∈ Rn and if u1 , u2 ∈ Rm satisfy |x1 | ≤ M, |u1 | ≥ Γ, |x1 − x2 | + |u1 − u2 | ≤ δ, then |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ (f (t, x1 , u1 ) + 1); (A3) For each M, > 0 there exists δ > 0 such that the following property holds: If t ≥ 0, x1 , x2 ∈ Rn and if u1 , u2 ∈ Rm satisfy |xi |ui | ≤ M (i = 1, 2), |x1 − x2 | + |u1 − u2 | ≤ δ, then |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ . We denote by | · | the Euclidean norm in Rn . For each r > 0 and each x ∈ Rn set B(r) = {y ∈ Rn : |y| ≤ r}, B(x, r) = {y ∈ Rn : |x − y| ≤ r}. Remark 1. It is not diﬃcult to see that if λ is a positive number and if Assumptions (A1)-(A3) hold with f = fi , i = 1, 2, where f1 , f2 : [0, ∞) × Rn × Rm → R1 are measurable functions, then Assumptions (A1)-(A3) hold with f = λf1 and f = f1 + f2 . Remark 2. It is not diﬃcult to see that Assumptions (A1)-(A3) hold with a function f : [0, ∞) × Rn × Rm → R1 deﬁned by f (t, x, u) = g(t)(x − Γ(t)) Q(x − Γ(t)) +h(t)u P u + H(t) (t ∈ [0, ∞), x ∈ Rn , u ∈ Rm ),

Linear systems with nonperiodic integrands

199

where Γ : [0, ∞) → Rn is a measurable and bounded on [0, ∞) function, P, Q are positive deﬁnite symmetric matrices, H, h, g : [0, ∞) → R1 are measurable bounded functions such that inf{g(t) : t ∈ [0, ∞)} > 0, inf{h(t) : t ∈ [0, ∞)} > 0. The optimal control problem with the autonomous plant (1.1) and this integrand f was studied in [3]. Remark 3. Assume that h : [0, ∞) → R1 is a bounded measurable function such that inf{h(t) : t ∈ [0, ∞)} > 0 and that a strictly convex function g = g(x, u) ∈ C 1 (Rn+m ) has the following properties: g(x, u) ≥ max{ψ(x), ψ(|u|)|u|}, max{|∂g/∂x(x, u)|, |∂g/∂u(x, u)|} ≤ ψ0 (|x|)(1 + ψ|u|)|u|, x ∈ Rn , u ∈ Rm , where ψ : [0, ∞) → (0, ∞), ψ0 : (0, ∞) → [0, ∞) are monotone increasing functions, ψ(t) → ∞ as t → ∞; for any > 0 there exists δ() > 0 such that if x1 , x2 ∈ Rn and if u1 , u2 ∈ Rm satisfy |x1 − x2 | + |u1 − u2 | ≥ , then g(2−1 (x1 + x2 ), 2−1 (u1 + u2 )) ≤ 2−1 [g(x1 , u1 ) + g(x2 , u2 )] − δ(). It is not diﬃcult to see that the integrand f (t, x, u) = h(t)g(x, u), t ∈ [0, ∞), x ∈ Rn , u ∈ Rm satisﬁes (A1), (A2) and (A3) and f is not taken from Remarks 1 and 2. In this chapter we prove the following results which were obtained in [112]. Theorem 7.1.1 Assume that (A1) holds and x0 ∈ Rn . Then there exists a trajectory-control pair x∗ : [0, ∞) → Rn , u∗ : [0, ∞) → Rm such that x∗ (0) = x0 , the function x∗ is bounded on [0, ∞) and for any trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm either (i) I(0, T, x, u) − I(0, T, x∗ , u∗ ) → ∞ as T → ∞ or (ii) sup{|I(0, T, x, u) − I(0, T, x∗ , u∗ )| : T > 0} < ∞. Moreover, if the relation (ii) is true, then lim (x(t) − x∗ (t)) = 0.

t→∞

We say that a trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → is good if it satisﬁes relation (ii) of Theorem 7.1.1.

Rm

200

TURNPIKE PROPERTIES

We say that a trajectory-control pair x ¯ : [0, ∞) → Rn , u ¯ : [0, ∞) → m R is overtaking optimal if ¯, u ¯) − I(0, T, x, u)] ≤ 0 lim sup[I(0, T, x T →∞

for any trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm such that x(0) = x ¯(0). Theorem 7.1.2 Let (A1)-(A3) hold and let x0 ∈ Rn . Then there exists an overtaking optimal trajectory-control pair x ¯ : [0, ∞) → Rn , m ¯(0) = x0 . Moreover if a trajectoryu ¯ : [0, ∞) → R such that x control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisﬁes x(0) = x0 and (x, u) = (¯ x, u ¯), then there are a time T0 and > 0 such that for all T ≥ T0 , I(0, T, x, u) ≥ I(0, T, x ¯, u ¯) + . Theorem 7.1.3 describes the limit behavior of overtaking optimal trajectories. Theorem 7.1.3 Let (A1)-(A3) hold, M, be positive numbers and let (x∗ , u∗ ) be an overtaking optimal trajectory-control pair. Then there exists a number T0 > 0 such that |x(t) − x∗ (t)| ≤ , t ∈ [T0 , ∞) for each overtaking optimal trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm which satisﬁes |x(0)| ≤ M . Moreover, there exists a positive number δ such that if T ≥ 0 and if an overtaking optimal trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisfy |x(T ) − x∗ (T )| ≤ δ, |x(0)| ≤ M, then |x(t) − x∗ (t)| ≤ (t ∈ [T, ∞)). For each z ∈ Rn and T > 0 we set ∆(z, T ) = inf{I(0, T, x, u) : x : [0, T ] → Rn , u : [0, T ] → Rm is a trajectory-control pair satisfying x(0) = z}. We will see that −∞ < ∆(z, T ) < ∞. Theorem 7.1.4 establishes the turnpike property for optimal trajectories with the turnpike x∗ (·).

Linear systems with nonperiodic integrands

201

Theorem 7.1.4 Assume that (A1)-(A3) hold, M, are positive numbers and (x∗ , u∗ ) is an overtaking optimal trajectory-control pair. Then there exist numbers T0 , δ > 0 such that: (1) If T > 2T0 and if a trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes |x(0)| ≤ M , I(0, T, x, u) ≤ ∆(x(0), T ) + δ,

(1.3)

then for all t ∈ [T0 , T − T0 ], |x(t) − x∗ (t)| ≤ .

(1.4)

(2) If s ≥ 0, T ≥ s+T0 and if a trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes (1.3) and |x(0)| ≤ M, |x(s) − x∗ (s)| ≤ δ, then (1.4) holds for all t ∈ [s, T − T0 ]. Chapter 7 is organized as follows. Section 7.2 contains auxiliary results. A discrete-time control system related to our optimal control problem is discussed in Section 7.3. Theorem 7.1.1 is proved in Section 7.4. Section 5 contains the proof of Theorem 7.1.2. Theorems 7.1.3 and 7.1.4 are proved in Section 7.6.

7.2.

Preliminary results

Suppose that (A1) holds. We have the following result (see Proposition 2.3 of [41]). Proposition 7.2.1 For every y˜, z˜ ∈ Rn and every T > 0 there exists a unique solution x(·), y(·) of the following system: x = Ax + BB t y, y = x − At y, with the boundary conditions x(0) = y˜, x(T ) = z˜ (where B t denotes the transpose of B). Let s ≥ 0 and τ > 0. Consider the function vτs (y, z) deﬁned on Rn ×Rn by

vτs (y, z) = min I(s, s + τ, x, u)

(2.1)

subject to x(s) = y, x(τ + s) = z;

(2.2)

here x(·) is the response to u(·).

202

TURNPIKE PROPERTIES

It follows from Proposition 7.2.1 and Assumption (A1) that the function vτs is convex and −∞ < vτs (y, z) < ∞ for each y, z ∈ Rn .

(2.3)

The following Propositions 7.2.2, 7.2.3 and 7.2.4 can be established analogously to Propositions 6.2.2, 6.2.3 and 6.2.4 respectively. Proposition 7.2.2 Let M1 > 0 and 0 < τ0 < τ1 . Then there exists a number M2 > 0 such that the following property holds: If numbers t1 , t2 satisfy 0 ≤ t1 < t2 , t2 −t1 ∈ [τ0 , τ1 ] and if a trajectorycontrol pair x : [t1 , t2 ] → Rn , u : [t1 , t2 ] → Rm satisﬁes I(t1 , t2 , x, u) ≤ M1 ,

(2.4)

then |x(t)| ≤ M2 , t ∈ [t1 , t2 ]. Proposition 7.2.3 Let M1 , , τ0 , τ1 > 0 and 0 < τ0 < τ1 . Then there exists a positive number δ such that the following property holds: If 0 ≤ T1 < T2 , T2 − T1 ∈ [τ0 , τ1 ] and if a trajectory-control pair x : [T1 , T2 ] → Rn , u : [T1 , T2 ] → Rm satisﬁes I(T1 , T2 , x, u) ≤ M1 , then |x(t1 ) − x(t2 )| ≤ for each t1 , t2 ∈ [T1 , T2 ] such that |t1 − t2 | ≤ δ. Proposition 7.2.4 Let M1 > 0, 0 ≤ t1 < t2 and let F be the set of all trajectory-control pairs x : [t1 , t2 ] → Rn , u : [t1 , t2 ] → Rm satisfying (2.4). Then for every sequence {(xi , ui )}∞ i=1 ⊂ F there exist a subseand (x, u) ∈ F such that xik (t) → x(t) as k → ∞ quence {(xik , uik )}∞ k=1 uniformly in [t1 , t2 ], xik → x as k → ∞ weakly in L1 ((t1 , t2 ); Rn ) and uik → u as k → ∞ weakly in L1 ((t1 , t2 ); Rm ). Corollary 7.2.1 Let t ≥ 0, s > 0 and let x1 , x2 ∈ Rn . Then there exists a unique trajectory-control pair x : [t, t + s] → Rn , u : [t, t + s] → Rm such that x(t) = x1 , x(t + s) = x2 and I(t, t + s, x, u) = vst (x1 , x2 ).

Linear systems with nonperiodic integrands

7.3.

203

Discrete-time control systems

In this section we establish some useful properties of discrete-time control systems with cost function vτs . Assume that (A1) holds. Proposition 7.2.2 implies the following result. Proposition 7.3.1 For each pair of positive numbers τ and M1 there exists a positive number M2 such that if s ∈ [0, ∞) and if x1 , x2 ∈ Rn satisfy |x1 | + |x2 | ≥ M2 , then vτs (x1 , x2 ) ≥ M1 . Proposition 7.3.2 For each τ, M1 > 0 the set {vτs (x1 , x2 ) : s ∈ [0, ∞), x1 , x2 ∈ B(M1 )} is bounded from above. Proof. Let τ, M1 > 0. Deﬁne a function f0 : Rn × Rm → R1 by f0 (x, u) = sup{f (t, x, u) : t ∈ [0, ∞)} for x ∈ Rn and u ∈ Rm . It follows from Assumption (A1) that the function f0 is well deﬁned and convex. For y, z ∈ Rn set τ

v˜(y, z) = inf

0

f0 (x(t), u(t))dt subject to x(0) = y, x(τ ) = z;

here x(·) is the response to u(·). By (A1), Proposition 7.2.1 and the convexity of f0 , the function v˜ : Rn × Rn → R1 is well deﬁned and convex. The continuity of v˜ implies the validity of the proposition. Analogously to Proposition 6.3.4 we can establish the following result. Proposition 7.3.3 Let τ, M1 , M2 be positive numbers such that inf{vτs (y, z) : s ∈ [0, ∞), y, z ∈ Rn , |y| + |z| ≥ M2 } > 2 sup{|vτs (0, 0)| : ; s ∈ [0, ∞)} + 1.

(3.1)

Then there exists a number M3 > M2 such that for every pair of integers 2 ⊂ Rn the q1 , q2 satisfying 0 ≤ q1 < q2 and every sequence {zk }qk=q 1 following properties hold: If zq1 , zq2 ∈ B(M2 ), max{|zk | : k = q1 , . . . , q2 } > M3 ,

204

TURNPIKE PROPERTIES

2 then there is a sequence {yk }qk=q ⊂ Rn such that yqi = zqi , i = 1, 2 and 1

q 2 −1

[vττ k (zk , zk+1 ) − vττ k (yk , yk+1 )] ≤ M1 ;

(3.2)

k=q1

(2) if zq1 ∈ B(M2 ) and sup{|zk | : k = q1 , . . . , q2 } > M3 , then there is 2 ⊂ Rn such that yq1 = zq1 and (3.2) holds. a sequence {yk }qk=q 1

7.4.

Proof of Theorem 7.1.1

Let x0 ∈ Rn . Fix a positive number τ . It follows from Proposition 7.2.4 that for any natural number k there exists a trajectory-control pair xk : [0, τ k] → Rn , uk : [0, τk ] → Rm such that xk (0) = x0 and I(0, τ k, xk , uk ) = ∆(x0 , τ k). By Corollary 7.2.1 and Proposition 7.3.3, sup{|xk (iτ )| : k = 1, 2, . . . , i = 0, . . . , k} < ∞.

(4.1)

There exists a subsequence of trajectory-control pairs {(xkj , ukj )}∞ j=1 such that for any integer i ≥ 0 there exists x∗i = lim xkj (iτ ). j→∞

(4.2)

In view of Corollary 7.2.1 there is a trajectory-control pair x∗ : [0, ∞) → Rn , u∗ : [0, ∞) → Rm such that for each integer i ≥ 0, x∗ (iτ ) = x∗i , I(iτ, (i + 1)τ, x∗ , u∗ ) = vττ i (x∗i , x∗i+1 ).

(4.3)

r0 = sup{|xk (iτ )| : k = 1, 2, . . . , i = 0, . . . , k}.

(4.4)

Put It follows from Assumption (A1), (4.1), (4.2), (4.3), Proposition 7.2.2 and 7.3.2 that sup{|I(τ i, τ (i + 1), x∗ , u∗ )| : i = 0, 1, . . .} < ∞, sup{|x∗ (t)| : t ∈ [0, ∞)} < ∞.

(4.5)

Thus we have constructed the trajectory-control pair (x∗ , u∗ ). We will show that (x∗ , u∗ ) has the properties described in Theorem 7.1.1. In order to meet this goal we consider the analogous properties for the related discrete-time control systems.

205

Linear systems with nonperiodic integrands

Lemma 7.4.1 For each S0 > 0 there exists S1 > S0 + 8 such that the following property holds: 2 ⊂ Rn satisﬁes If q1 ≥ 0, q2 > q1 are integers and if a sequence {yi }qi=q 1 yq1 ∈ B(S0 ), then q 2 −1

[vττ i (x∗i , x∗i+1 ) − vττ i (yi , yi+1 )] ≤ S1 .

(4.6)

i=q1

Proof. Let S0 > 0. We may assume without loss of generality that (3.1) holds with M2 = S0 . There exists a number S2 > S0 such that Proposition 7.3.3 holds with M1 = 1, M2 = S0 and M3 = S2 . Assumption (A1) and Proposition 7.3.2 imply that there exists a number S1 such that 6|vτσ (h1 , h2 )| < S1 − S0 − 9 for each σ ∈ [0, ∞), each h1 , h2 ∈ Rn satisfying |h1 | + |h2 | ≤ 2r0 + 2S2 . Let 0 ≤ q1 < q2 and let the sequence B(S0 ). We will show that (4.6) holds. Let us assume the converse. Then

2 {yi }qi=q 1

⊂

Rn

q 2 −1

[vττ i (x∗i , x∗i+1 ) − vττ i (yi , yi+1 )] > S1 .

(4.7)

satisfy yq1 ∈

(4.8)

i=q1

We may assume without loss of generality that q 2 −1

q 2 −1

vττ i (yi , yi+1 ) = inf{

i=q1

vττ i (zi , zi+1 ) :

i=q1 2 {zi }qi=q ⊂ Rn and zq1 = yq1 }. 1

In view of Proposition 7.3.3 which holds with M1 = 1, M2 = S0 , M3 = S2 , (4.9) yi ∈ B(S2 ) for all i = q1 , . . . , q2 . There exists a natural number k > q2 + 1 such that

q2 iτ ∗ ∗ iτ [vτ (xi , xi+1 ) − vτ (xk (iτ ), xk ((i + 1)τ ))] ≤ 1. i=q1

It is easy to see that k−1 i=0

vτiτ (xk (iτ ), xk ((i + 1)τ ))

(4.10)

206

TURNPIKE PROPERTIES

=

k−1

inf{vτiτ (zi , zi+1 ) : {zi }ki=0 ⊂ Rn and z0 = x0 }.

(4.11)

i=0

Set ai = xk (iτ ), i ∈ {0, . . . , q1 } ∪ {q2 + 1, . . . , k}, ai = yi , i = q1 + 1, . . . , q2 . (4.12) It follows from the deﬁnition of {ai }ki=0 , (4.11) with zi = ai (i = 0, . . . , k), (4.10) and (4.8) that 0≥

k−1

[vττ i (xk (iτ ), xk ((i + 1)τ )) − vττ i (ai , ai+1 )]

i=0

≥ −1 + S1 + vτq2 τ (x∗q2 , x∗q2 +1 ) + vτq1 τ (yq1 , yq1 +1 ) −vτq1 τ (aq1 , aq1 +1 ) − vτq2 τ (aq2 , aq2 +1 ).

Using this relation, (4.12), (4.4), (4.2) and (4.9) we obtain an estimation for S1 which is contradictory to (4.7). The obtained contradiction proves the lemma. The next auxiliary result establishes a useful property for the discretetime control system which implies the boundedness of any good trajectory. Lemma 7.4.2 There exists a positive number S0 such that if a sequence n {yi }∞ i=0 ⊂ R satisﬁes (4.13) lim sup |yi | > S0 , i→∞

then N −1

[vτiτ (yi , yi+1 ) − vττ i (x∗i , x∗i+1 )] → ∞ as N → ∞.

(4.14)

j=0

Proof. It follows from Propositions 7.3.1 and 7.3.2 that there exists a positive number S1 such that the inequality (3.1) holds with M2 = S1 and (4.14) is valid for each sequence {yi }∞ i=0 satisfying lim inf |yi | > S1 . i→∞

There exist numbers S0 > S1 + 1 and Q > S0 + 8 such that Proposition 7.3.3 holds with M1 = 4, M2 = S1 + 1, M3 = S0 and Lemma 7.4.1 holds with S1 = Q. n Assume that a sequence {yi }∞ i=0 ⊂ R satisﬁes (4.13). We will establish (4.14). In view of the choice of S1 we may assume that lim inf |yi | ≤ S1 . j→∞

(4.15)

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207

(4.13) and (4.15) imply that there exists a subsequence {yik }∞ k=1 such that 0 < i1 , yik ∈ B(S1 + 1), sup{|yi | : j = ik , . . . , ik+1 } > S0 (k = 1, 2, . . .). (4.16) By Proposition 7.3.3 which holds with M1 = 4, M2 = S1 + 1, M3 = S0 , k +1 ⊂ Rn such for any natural number k there exists a sequence {zj }ij=i k that ik+1 −1

zj = yj (j ∈ {ik , ik+1 }),

[vττ j (yj , yj+1 ) − vττ j (zj , zj+1 )] ≥ 4. (4.17)

j=ik

Relation (4.14) now follows from (4.17), (4.16) and Lemma 7.4.1 applied with S1 = Q. The lemma is proved. The next lemma follows from (4.3), Lemma 7.4.1, Assumption (A1), and the deﬁnition of (x∗ , u∗ ). Lemma 7.4.3 For each positive number S0 there exists a positive number S1 such that the following property holds: If an integer q ≥ 0, T ∈ (qτ, ∞) and if a trajectory-control pair x : [qτ, T ] → Rn , u : [qτ, T ] → Rm satisﬁes |x(qτ )| ∈ B(S0 ), then I(qτ, T, x∗ , u∗ ) ≤ I(qτ, T, x, u) + S1 . The next auxiliary result shows that a trajectory-control pair (x, u) is not good if the function |x(t)| is not bounded at inﬁnity by S1 . Its proof is based on Lemmas 7.4.2 and 7.4.3 and Proposition 7.2.2. Lemma 7.4.4 There is a positive number S1 such that if a trajectorycontrol pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisﬁes lim sup |x(t)| > S1 ,

(4.18)

I(0, T, x, u) − I(0, T, x∗ , u∗ ) → ∞ as T → ∞.

(4.19)

t→∞

then

Proof. It follows from Lemma 7.4.2 that there exists a positive number S0 such that the following property holds: n If a sequence {yi }∞ i=0 ⊂ R satisﬁes (4.13), then relation (4.14) holds. Lemma 7.4.3, Proposition 7.2.2 and (4.5) imply that there exists a number S1 > S0 + 1 such that the following property holds: If a trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisﬁes (4.18) and (4.20) lim sup |x(iτ )| ≤ S0 where i is an integer, i→∞

208

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then relation (4.19) holds. Suppose that x : [0, ∞) → Rn , u : [0, ∞) → Rm is a trajectory-control pair which satisﬁes (4.18). Then (4.19) follows if (4.20) holds. Otherwise (4.19) follows from Lemma 7.4.2, Assumption (A1), and the deﬁnition of (x∗ , u∗ ). The lemma is proved. The next lemma establishes the ﬁrst part of Theorem 7.1.1. Lemma 7.4.5 Let x : [0, ∞) → Rn , u : [0, ∞) → Rm be a trajectorycontrol pair. Then either (A) I(0, T, x, u) − I(0, T, x∗ , u∗ ) → ∞ as T → ∞ or (B) sup{|I(0, T, x, u) − I(0, T, x∗ , u∗ )| : T ∈ (0, ∞)} < ∞. Proof. In view of Lemma 7.4.4 we may assume without loss of generality that the function x : [0, ∞) → Rn is bounded. Lemma 7.4.3 implies that there exists a positive number S1 such that I(qτ, T, x∗ , u∗ ) ≤ I(qτ, T, x, u) + S1

(4.21)

for each natural number q and each number T > qτ . Assume that (B) does not hold. By (4.5) and Assumption (A1) the sequence {|I(0, N τ, x, u) − I(0, N τ, x∗ , u∗ )|}∞ N =0 is not bounded. Combined with (4.21) this implies (A). The lemma is proved. The next auxiliary result proves the second part of Theorem 7.1.1. Lemma 7.4.6 Assume that x : [0, ∞) → Rn , u : [0, ∞) → Rm is a trajectory-control pair such that limt→∞ sup |x(t) − x∗ (t)| > 0. Then lim I(0, T, x, u) − I(0, T, x∗ , u∗ ) = ∞.

T →∞

(4.22)

Proof. In view of Lemma 7.4.4 we may assume without loss of generality that the function x : [0, ∞) → Rn is bounded. Put = 2−1 lim sup |x(t) − x∗ (t)|, t→∞

(4.23)

y(t) = 2−1 (x(t) + x∗ (t)), w(t) = 2−1 (u(t) + u∗ (t)) (t ∈ [0, ∞)). (4.24) In view of the boundedness of x, the deﬁnition of (x∗ , u∗ ), and Lemmas 7.4.3 and 7.4.5 we may assume that the sequence {I(iτ, (i+1)τ, x, u)}∞ i=0 is bounded from above. Then by (4.23) and Proposition 7.2.3 there exist

Linear systems with nonperiodic integrands

209

a positive number δ < 1 and a sequence of nonnegative numbers {Ti }∞ i=0 such that Ti+1 − Ti ≥ 10, i = 0, 1, . . . , |x(t) − x∗ (t)| ≥ 2−1 (t ∈ [Ti − δ, Ti + δ], i = 0, 1, 2, . . .). It follows from these relations, (4.24) and Assumption (A1) that 2−1 [I(0, T, x, u) + I(0, T, x∗ , u∗ )] − I(0, T, y, w) → ∞ as T → ∞. (4.25) By the boundedness of x, (4.24), (4.5) and Lemma 7.4.3, the function T → I(0, T, y, w) − I(0, T, x∗ , u∗ ), T ∈ (0, ∞) is bounded from below. Combined with (4.25) this implies (4.22). The lemma is proved. Theorem 7.1.1 now follows from (4.5) and Lemmas 7.4.3, 7.4.5 and 7.4.6.

7.5.

Proof of Theorem 7.1.2

Assume that (A1)-(A3) hold. We begin with the following auxiliary result which establishes a useful continuity property of the function vτs . It should be mentioned that the number δ in the statement of Lemma 7.5.1 depends only on τ , and M and does not depend on s. Lemma 7.5.1 For each M, τ, > 0 there exists a positive number δ such that the following property holds: If s ∈ [0, ∞) and if y1 , y2 , z1 , z2 ∈ Rn satisfy |zi |, |yi | ∈ B(M ), i = 1, 2, max{|y1 − y2 |, |z1 − z2 |} ≤ δ, then

|vτs (y1 , z1 ) − vτs (y2 , z2 )| ≤ .

(5.1) (5.2)

Proof. Let M, τ, be positive numbers. It follows from Proposition 7.2.1 that for each y˜, z˜ ∈ Rn there exists a unique solution x(·), y(·) of the following system: (x , y )t = C((x, y)t ),

(5.3)

with the boundary constraints x(0) = y˜, x(τ ) = z˜ and C(x, y)t = (Ax + BB t y, x − At y)t

(5.4)

210

TURNPIKE PROPERTIES

(here I : Rn → Rn is the identity operator, Iz = z, z ∈ Rn ). For any initial value (x0 , y0 ) ∈ Rn × Rn there exists a unique solution of (5.3) satisfying (x(s), y(s))t = esC (x0 , y0 )s , s ∈ R1 . Clearly for each y˜, z˜ ∈ Rn there exists a unique vector D(˜ y , z˜) ∈ Rn such that the function y , D(˜ y , z˜))t )t , s ∈ R1 (x(s), y(s)) = (esC (˜ satisﬁes (5.3) with the boundary constraints x(0) = y˜, x(τ ) = z˜. It is easy to see that D : Rn × Rn → R1 is a linear operator. Set M0 = sup{|vτs (y, z)| : y, z ∈ B(M + 1), s ∈ [0, ∞)}.

(5.5)

Assumption (A1) and Proposition 7.3.2 imply that M0 is ﬁnite. It follows from Proposition 7.2.2 that there exists a positive number M1 such that the following property holds: If s ≥ 0 and if a trajectory-control pair x : [s, s + τ ] → Rn , u : [s, s + τ ] → Rm satisﬁes I(s, s + τ, x, u) ≤ 4M0 + 1, then x(t) ∈ B(M1 ), t ∈ [s, s + τ ].

(5.6)

Choose numbers c1 , δ1 > 0 such that f (t, x, u) ≥ −c1 , ((t, x, u) ∈ R2n+1 ), 4δ(2τ + c1 + c1 τ + M0 ) ≤ . (5.7) It follows from Assumption (A2) that there exist a positive number Γ0 and δ2 ∈ (0, 8−1 ) such that the following property holds: If t ≥ 0, if x1 ∈ B(M1 ), x2 ∈ Rn and if u1 , u2 ∈ Rm satisfy |u1 | ≥ Γ0 , |x1 − x2 | + |u1 − u2 | ≤ δ2 ,

(5.8)

|f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ δ1 (f (t, x1 , u1 ) + 1).

(5.9)

then It follows from Assumption (A3) that there exists δ3 ∈ (0, 4−1 min{δ1 , δ2 })

(5.10)

such that the following property holds: If t ≥ 0, x1 , x2 ∈ Rn and if u1 , u2 ∈ Rm satisfy |xi |, |ui | ≤ Γ0 + M1 + 1, i = 1, 2, |x1 − x2 | + |u1 − u2 | ≤ δ3 ,

(5.11)

then |f (t, x1 , u1 ) − f (t, x2 , u2 )| ≤ δ1 .

(5.12)

Linear systems with nonperiodic integrands

There exists a number

δ ∈ (0, 8−1 δ3 )

211 (5.13)

such that for each y, z ∈ B(δ), (1 + ||B||)|esC (y, D(y, z))t | ≤ 2−1 δ3 (s ∈ [0, τ ]).

(5.14)

Assume that s ∈ [0, ∞) and y1 , y2 , z1 , z2 ∈ Rn satisfy (5.1). By Corollary 7.2.1 there exists a trajectory-control pair x1 : [s, s + τ ] → Rn , u1 : [s, s + τ ] → Rm such that x1 (s) = y1 , x1 (s + τ ) = z1 , I(s, s + τ, x1 , u1 ) = vτs (y1 , z1 ).

(5.15)

By (5.1), (5.15), (5.5) and the choice of M1 , |x1 (t)| ∈ B(M1 ), t ∈ [s, s + τ ].

(5.16)

Deﬁne functions h1 , h2 : R1 → Rn by (h1 (r), h2 (r))t = erC (y2 − y1 , D(y2 − y1 , z2 − z1 ))t , r ∈ R1 .

(5.17)

Put x2 (t) = x1 (t)+h1 (t−s), u2 (t) = u1 (t)+B t h2 (t−s) (t ∈ [s, s+τ ]). (5.18) By the deﬁnition of D1 , D2 , (5.17), (5.3), (5.4) and (5.15), (x2 , u2 ) is a trajectory-control pair, x2 (s) = y2 , x2 (s + τ ) = z2 . (5.19) It follows from (5.1), (5.17), (5.19) and the choice of δ that |x2 (t) − x1 (t)|, |u2 (t) − u1 (t)| ≤ 2−1 δ3 (t ∈ [s, s + τ ]).

(5.20)

By (5.1), (5.15), (5.7) and (5.5) for any measurable set E ⊂ [s, s + τ ],

E

f (t, x1 (t), u1 (t))dt ≤ M0 + c1 τ.

(5.21)

Put E1 = {t ∈ [s, s + τ ] : |u1 (t)| ≥ Γ0 }, E2 = [s, s + τ ] \ E1 . We will estimate separately Ei

|f (t, x1 (t), u1 (t)) − f (t, x2 (t), u2 (t))|dt, i = 1, 2.

It follows from (5.16), (5.20), the deﬁnition of Γ0 , δ2 and (5.21) that

E1

|f (t, x1 (t), u1 (t)) − f (t, x2 (t), u2 (t))|dt

212

TURNPIKE PROPERTIES

≤ δ1 τ + δ1

E1

f (t, x1 (t), u1 (t))dt

(5.22)

≤ δ1 τ + c1 τ δ1 + δ1 M0 . By (5.16), (5.20) and the deﬁnition of δ3 ,

E2

|f (t, x1 (t), u1 (t)) − f (t, x2 (t), u2 (t))|dt ≤ δ1 τ.

This relation, (5.22), (5.7), (5.15) and (5.19) imply that vτs (y2 , z2 ) ≤ vτs (y1 , z1 ) + . This completes the proof of the lemma. Let x0 ∈ Rn , τ > 0. Consider the trajectory-control pairs x∗ : [0, ∞) → Rn , u∗ : [0, ∞) → Rm , xk : [0, τ k] → Rn , uk : [0, τ k] → Rm , k = 1, 2, . . . deﬁned in Section 7.4 (see (4.1)-(4.3)). We will show that (x∗ , u∗ ) is a unique overtaking optimal trajectorycontrol pair which satisﬁes the initial condition x∗ (0) = x0 . The following lemma is one of the important gradients in our proof. Lemma 7.5.2 Let 0 ≤ τ1 < τ2 and let x : [τ1 , τ2 ] → Rn , u : [τ1 , τ2 ] → Rm be a trajectory-control pair such that x(τ1 ) = x∗ (τi ) (i = 1, 2), (x, u) = (x∗ , u∗ )|[τ1 ,τ2 ]

(5.23)

where (x∗ , u∗ )|[τ1 ,τ2 ] is the restriction of (x∗ , u∗ ) to [τ1 , τ2 ]. Then I(τ1 , τ2 , x, u) > I(τ1 , τ2 , x∗ , u∗ ). Proof. By Assumption (A1) and (5.23) it is suﬃcient to show that I(τ1 , τ2 , x, u) ≥ I(τ1 , τ2 , x∗ , u∗ ). Let us assume the converse. Fix an integer q0 and a number such that ∈ (0, 8−1 [I(τ1 , τ2 , x∗ , u∗ ) − I(τ1 , τ2 , x, u)]), q0 > τ −1 τ2 + 5.

(5.24)

By (4.1), (4.2), (4.3) and the continuity of vτs : Rn × Rn → R1 , there exists a natural number k > 2q0 + 4 such that |vττ i (x∗ (iτ ), x∗ ((i + 1)τ )) − vττ i (xk (iτ ), xk ((i + 1)τ ))| ≤ (2q0 + 1)−1 (i = 0, . . . , 2q0 + 1), |vτq0 τ (x∗ (q0 τ ), xk ((q0 +1)τ ))−vτq0 τ (xk (q0 τ ), xk ((q0 +1)τ ))|

(5.25) ≤ (2q0 +1)−1 .

213

Linear systems with nonperiodic integrands

It follows from Corollary 7.2.1, (5.23) and (5.24) that there exists a trajectory-control pair y : [0, τ k] → Rn , w : [0, τ k] → Rm such that y(t) = x∗ (t), w(t) = u∗ (t) (t ∈ [0, τ1 ] ∪ [τ2 , q0 τ ]), y(t) = x(t), w(t) = u(t) (t ∈ [τ1 , τ2 ]), y(t) = xk (t), w(t) = uk (t) (t ∈ [(q0 + 1)τ, kτ ]),

(5.26)

I(q0 τ, (q0 + 1)τ, y, w) = vτq0 τ (y(q0 τ ), y((q0 + 1)τ )). In view of the deﬁnition of xk , uk , y and w, I(0, τ k, y, w) ≥ I(0, τ k, xk , uk ). On the other hand by (5.26), (4.3), (5.25), (5.24) and the deﬁnition of (xk , uk ), I(0, τ k, y, w) − I(0, τ k, xk , uk ) = I(0, τ q0 , y, w) − I(0, τ q0 , x∗ , u∗ ) +vτq0 τ (x∗ (q0 τ ), xk ((q0 + 1)τ )) + I(0, τ q0 , x∗ , u∗ ) − I(0, τ q0 , xk , uk ) −vτq0 τ (xk (q0 τ ), xk ((q0 + 1)τ )) = I(τ1 , τ2 , x, u) − I(τ1 , τ2 , x∗ , u∗ ) +

q 0 −1

[vττ i (x∗ (iτ ), x∗ ((i + 1)τ )) − vττ i (xk (iτ ), xk ((i + 1)τ ))]

i=0 q0 τ ∗ +vτ (x (q0 τ ), xk ((q0

+ 1)τ )) − vτq0 τ (xk (q0 τ ), xk ((q0 + 1)τ ))

≤ I(τ1 , τ2 , x, u) − I(τ1 , τ2 , x∗ , u∗ ) + < −. The obtained contradiction proves the lemma. The following auxiliary result shows that (x∗ , u∗ ) is an overtaking optimal trajectory-control pair. Lemma 7.5.3 Let x : [0, ∞) → Rn , u : [0, ∞) → Rm be a trajectorycontrol pair such that x(0) = x0 . Then lim sup[I(0, T, x∗ , u∗ ) − I(0, T, x, u)] ≤ 0. T →∞

Proof. Let us assume the converse. Put = 2−1 lim sup[I(0, T, x∗ , u∗ ) − I(0, T, x, u)]. T →∞

(5.27)

Lemma 7.4.6 implies that lim (x(t) − x∗ (t)) = 0.

t→∞

(5.28)

214 Put

TURNPIKE PROPERTIES

M0 = sup{|x(t)| + |x∗ (t)| : t ∈ [0, ∞)}.

(5.29)

By (5.28) and (4.5), M0 is ﬁnite. It follows from Lemma 7.5.1 that there exists a positive number δ0 such that the following property holds: If s ≥ 0, y1 , y2 , z1 , z2 ∈ B(M0 + 1) and if max{|y1 − y2 |, |z1 − z2 |} ≤ δ0 , then

|vτs (y1 , z1 ) − vτs (y2 , z2 )| ≤ 16−1 .

(5.30)

(5.28) implies that there exists a natural number N0 such that |x(t) − x∗ (t)| ≤ 8−1 δ0 for each t ≥ N0 .

(5.31)

It follows from (5.27) that there exists a number T0 > 2N0 + 2 such that I(0, T0 , x∗ , u∗ ) − I(0, T0 , x, u) ≥ .

(5.32)

Corollary 7.2.1 implies that there exists a trajectory-control pair y : [0, ∞) → Rn , w : [0, ∞) → Rm such that y(t) = x(t), w(t) = u(t)

(t ∈ [0, T0 ]),

y(t) = x∗ (t), w(t) = u∗ (t) (t ∈ [T0 + τ, ∞)), I(T0 , T0 + τ, y, w) = vττ0 (x(T0 ), x∗ (T0 + τ )). By Lemma 7.5.2, I(T0 , T0 + τ, x∗ , u∗ ) = vτT0 (x∗ (T0 ), x∗ (T0 + τ )).

(5.33)

It follows from the deﬁnition of (y, w), (5.32), and (5.33) that I(0, T0 + τ, y, w) − I(0, T0 , x∗ , u∗ ) ≤ I(0, T0 , x, u) − I(0, T0 , x∗ , u∗ ) +vτT0 (x(T0 ), x∗ (T0 + τ )) − vτT0 (x∗ (T0 ), x∗ (T0 + τ )) ≤ − + vτT0 (x(T0 ), x∗ (T0 + τ )) − vτT0 (x∗ (T0 ), x∗ (T0 + τ )). It follows from this relation, (5.29), (5.30), (5.31), and the deﬁnition of δ0 that I(0, T0 + τ, y, w) − I(0, T0 + τ, x∗ , u∗ ) ≤ −2−1 . This is contradictory to Lemma 7.5.2. The obtained contradiction proves the lemma.

Linear systems with nonperiodic integrands

215

Proof of Theorem 7.1.2. Let x : [0, ∞) → Rn , u : [0, ∞) → Rm be a trajectory-control pair such that x(0) = x0 , (x, u) = (x∗ , u∗ ).

(5.34)

We set y(t) = 2−1 (x(t) + x∗ (t)), w(t) = 2−1 (u(t) + u∗ (t)) (t ∈ [0, ∞)). (5.35) Clearly, (y, w) is a trajectory-control pair satisfying y(0) = x0 . By Assumption (A1) (i) and (5.34) there exists a number γ > 0 such that for all T , I(0, T, y, w) ≤ 2−1 I(0, T, x, u) + 2−1 I(0, T, x∗ , u∗ ) − γ.

(5.36)

By Lemma 7.5.3, lim sup[I(0, T, x∗ , u∗ ) − I(0, T, y, w)] ≤ 0. T →∞

Together with (5.36) this implies the validity of the theorem.

7.6.

Proofs of Theorems 7.1.3 and 7.1.4

In this section we establish the turnpike property of optimal trajectory-control pairs. Assume that (A1)-(A3) hold, x0 ∈ Rn , τ > 0 and consider the trajectory-control pairs x∗ : [0, ∞) → Rn , u∗ : [0, ∞) → Rm , xk : [0, τ k] → Rn , uk : [0, τ k] → Rm (k = 1, 2, . . .) deﬁned in Section 7.4 (see (4.1)-(4.3)). It was shown in Section 7.5 that (x∗ , u∗ ) is an overtaking optimal trajectory-control pair. We begin with the following auxiliary result which shows that all overtaking optimal trajectory-control pairs starting from a given bounded set stay in a certain ball. Lemma 7.6.1 For each M0 > 0 there exists a positive number M1 such that the following property holds: If an overtaking optimal trajectory-control pair x : [0, ∞) → Rn , u : [0, ∞) → Rm satisﬁes x(0) ∈ B(M0 ), then x(t) ∈ B(M1 ) for all t ∈ [0, ∞). Proof. Let M0 be a positive number. It follows from Lemma 7.4.4 that there exists a positive number S1 such that for any overtaking optimal trajectory-control pair (x, u), lim sup |x(t)| ≤ S1 . t→∞

216

TURNPIKE PROPERTIES

Combined with Proposition 7.3.3 this inequality implies that there exists a number S2 > 0 such that the following property holds: If an overtaking optimal trajectory-control pair (x, u) satisﬁes x(0) ∈ B(M0 ), then x(τ i) ∈ B(S2 ) for all integers i ≥ 0. The validity of the lemma now follows from Propositions 7.2.2 and 7.3.2. Propositions 7.3.2, 7.3.3 and Assumption (A1) imply the following result. Lemma 7.6.2 For each pair of positive numbers M0 , M1 there exists M2 > M1 such that the following property holds: If N is a natural number, T ∈ [N τ, (N + 1)τ ), if a trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes x(0) ∈ B(M0 ), I(0, T, x, u) ≤ ∆(x(0), T ) + M1

(6.1)

n and if a sequence {yk }N k=0 ⊂ R satisﬁes y0 = x(0), then N −1 k=0

vτkτ (x(kτ ), x((k + 1)τ )) ≤

N −1

vτkτ (yk , yk+1 ) + M2 .

k=0

Propositions 7.3.3, 7.3.2, 7.2.2 and Lemma 7.6.2 imply the following result which shows that for an approximate optimal trajectory-control pair (x, u) satisfying (6.1) the function |x(t)| is bounded by a constant which depends only on M0 , M1 and does not depend on T . Lemma 7.6.3 For each pair of positive numbers M0 , M1 there exists M2 > 0 such that the following property holds: If N is a natural number, T ∈ [N τ, (N + 1)τ ) and if a trajectorycontrol pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes (6.1), then x(t) ∈ B(M2 ) for all t ∈ [0, N τ ]. The next lemma is an important tool in our proof. It shows that if an approximate optimal trajectory-control pair (x, u) is deﬁned on an interval [0, T ] with a suﬃciently large T , then x(t) is close to x∗ (t) at some point t ∈ [0, T ]. Lemma 7.6.4 Let , M0 , M1 > 0. Then there is a natural number N0 such that the following property holds: If T ≥ N0 τ and if a trajectory-control pair x : [0, T ] → Rn , u : [0, T ] → Rm satisﬁes x(t) ∈ B(M0 ), t ∈ [0, T ], I(0, T, x, u) ≤ vT0 (x(0), x(T )) + M1 , then

inf{|x(t) − x∗ (t)| : t ∈ [s, s + N0 τ ]} ≤

(6.2) (6.3)

Linear systems with nonperiodic integrands

217

for each number s satisfying 0 ≤ s ≤ T − τ N0 . Proof. We may assume that x∗ (t) ∈ B(M0 ), t ∈ [0, ∞).

(6.4)

Assumption (A1) and Proposition 7.3.2 imply that there exists a number M2 > sup{|vτs (z1 , z2 )| : z1 , z2 ∈ B(2M0 ), s ∈ [0, ∞)} + 1.

(6.5)

It follows from Lemma 7.4.3 that there exists a positive number S1 such that the following property holds: If an integer q ≥ 0, T > qτ and if a trajectory-control pair x : [qτ, T ] → Rn , u : [qτ, T ] → Rm satisﬁes x(qτ ) ∈ B(M0 ), then I(qτ, T, x∗ , u∗ ) ≤ I(qτ, T, x, u) + S1 .

(6.6)

It follows from Assumption (A1) that there exists a positive number δ0 such that the following property holds: If t ∈ [0, ∞) and if x1 , x2 ∈ Rn , u1 , u2 ∈ Rm satisfy |x1 − x2 | + |u1 − u2 | ≥ 8−1 , then 2−1 f (t, x1 , u1 ) + 2−1 f (t, x2 , u2 ) − f (t, 2−1 (x1 + x2 ), 2−1 (u1 + u2 )) ≥ δ0 . (6.7) Choose a natural number N0 ≥ 4 + (δ0 τ )−1 (4M2 + 2M1 + S1 + 1).

(6.8)

Let T ≥ N0 τ , 0 ≤ s ≤ T −τ N0 and let x : [0, T ] → Rn , u : [0, T ] → Rm be a trajectory-control pair satisfying (6.2). We will show that (6.3) holds. Let us assume the converse. Then |x(t) − x∗ (t)| > for all t ∈ [s, s + N0 τ ].

(6.9)

By Corollary 7.2.1 there exists a trajectory-control pair x1 : [0, T ] → u1 : [0, T ] → Rm such that

Rn ,

x1 (0) = x(0), x1 (t) = x∗ (t), u1 (t) = u∗ (t) (t ∈ [τ, T − τ ]), x1 (T ) = x(T ), I(jτ, (j + 1)τ, x1 , u1 ) = vτjτ (x1 (jτ ), x1 ((j + 1)τ )), j = 0, T − τ. (6.10)

218

TURNPIKE PROPERTIES

Since (x∗ , u∗ ) is an overtaking optimal trajectory-control pair it follows from (6.10), (6.5), (6.2), and (6.4) that |I(0, T, x∗ , u∗ ) − I(0, T, x1 , u1 )| ≤ 4M2 .

(6.11)

In view of (6.2) and the choice of S1 , I(0, T, x∗ , u∗ ) ≤ I(0, T, x, u) + S1 .

(6.12)

Deﬁne a trajectory-control pair x2 : [0, T ] → Rn , u2 : [0, T ] → Rm as follows: x2 (t) = 2−1 (x1 (t)+x(t)), u2 (t) = 2−1 (u1 (t)+u(t)) for t ∈ [0, T ]. (6.13) It follows from (6.2), (6.13) and (6.10) that I(0, T, x, u) ≤ I(0, T, x2 , u2 ) + M1 .

(6.14)

By (6.13), (6.10), (6.9), Assumption (A1) and the choice of δ0 , I(0, T, x2 , u2 ) ≤ 2−1 I(0, T, x, u)+2−1 I(0, T, x1 , u1 )−δ0 (N0 −2)τ. (6.15) (6.11) and (6.12) imply that I(0, T, x1 , u1 ) ≤ I(0, T, x, u) + 4M2 + S1 . Combining this relation and (6.14) and (6.15) we obtain a relation which is contradictory to (6.8). The obtained contradiction proves the lemma. The following auxiliary result is another important ingredient in our proof. It establishes that if x(Ti ) is close enough to x∗ (Ti ), i = 1, 2, then an approximate optimal trajectory-control pair (x, u) deﬁned on an interval [T1 , T2 ] is close to x∗ at any point of [T1 , T2 ]. Lemma 7.6.5 For each , M0 > 0 there exists a positive number δ such that the following property holds: If T1 ≥ 0, T2 ≥ T1 + 3τ and if a trajectory-control pair x : [T1 , T2 ] → Rn , u : [T1 , T2 ] → Rm satisﬁes x(t) ∈ B(M0 ), t ∈ [T1 , T2 ], |x(Ti ) − x∗ (T1 )| ≤ δ, i = 1, 2,

then

I(T1 , T2 , x, u) ≤ vTT21−T1 (x(T1 ), x(T2 )) + δ,

(6.16)

|x(t) − x∗ (t)| ≤ for all t ∈ [T1 , T2 ].

(6.17)

Proof. Let , M0 be positive numbers. We may assume that x∗ (t) ∈ B(M0 ) for all t ∈ [0, ∞).

(6.18)

219

Linear systems with nonperiodic integrands

Assumption (A1) and Proposition 7.3.2 imply that there exists a number s M1 > sup{|v2τ (z1 , z2 )| : s ∈ [0, ∞), z1 , z2 ∈ Rn , |z1 | + |z2 | ≤ 2M0 + 1}. (6.19) It follows from Proposition 7.2.3 that there exists a number ∆ ∈ (0, 4−1 τ ) such that the following property holds: If s ≥ 0 and if a trajectory-control pair x : [s, s + 2τ ] → Rn , u : [s, s + 2τ ] → Rm satisﬁes

I(s, s + 2τ, x, u) ≤ 2M1 + 2, then

(6.20)

|x(t1 ) − x(t2 )| ≤ 8−1

for each t1 , t2 ∈ [s, s + 2τ ] such that |t1 − t2 | ≤ ∆. By Assumption (A1) (i) there is a positive number δ1 such that the following property holds: If t ≥ 0 and if x1 , x2 ∈ Rn and u1 , u2 ∈ Rm satisfy |x1 − x2 | + |u1 − u2 | ≥ 8−1 , then 2−1 f (t, x1 , u1 ) + 2−1 f (t, x2 , u2 ) − f (t, 2−1 (x1 + x2 ), 2−1 (u1 + u2 )) ≥ δ1 . (6.21) It follows from Lemma 7.5.1 that there exists a positive number δ < min{16−1 , 16−1 , 16−1 δ1 ∆}

(6.22)

such that the following property holds: If s ≥ 0 and if y1 , y2 , z1 , z2 ∈ Rn satisfy zi , yi ∈ B(M0 + 1) (i = 1, 2), max{|y1 − y2 |, |z1 − z2 |} ≤ δ, then

s s (y1 , z1 ) − v∆ (y2 , z2 )| ≤ 16−1 δ1 ∆. |v∆

(6.23) (6.24)

Rn ,

u : [T1 , T2 ] → Rm Let T1 ≥ 0, T2 ≥ T1 + 3τ and let x : [T1 , T2 ] → be a trajectory-control pair which satisﬁes (6.16). We will show that (6.17) holds. Let us assume the converse. Then there exists a number t0 such that t0 ∈ [T1 , T2 ], |x(t0 ) − x∗ (t0 )| > .

(6.25)

Since (x∗ , u∗ ) is an overtaking optimal trajectory-control pair it follows from (6.18) and (6.19) that for any s ∈ [0, ∞), s (x∗ (s), x∗ (s + 2τ ))| ≤ M1 . |I(s, s + 2τ, x∗ , u∗ )| = |v2τ

(6.26)

220

TURNPIKE PROPERTIES

By (6.16), (6.22) and (6.19), s I(s, s + 2τ, x, u) ≤ v2τ (x(s), x(s + 2τ )) + 1 ≤ M1 + 1

(6.27)

for any s ∈ [T1 , T2 − 2τ ]. It follows from (6.25)-(6.27) and the choice of ∆ that the following property holds: If t ∈ [T1 , T2 ] satisﬁes |t0 − t1 | ≤ ∆, then |x(t0 ) − x(t)| ≤ 8−1 , |x∗ (t0 ) − x∗ (t)| ≤ 8−1 , |x(t) − x∗ (t)| ≥ 3 · 4−1 . Combined with (6.16) and (6.22) this property implies that T1 < t0 − ∆, t0 + ∆ < T2 , |x(t) − x∗ (t)| ≥ 3 · 4−1 ,

(6.28)

t ∈ [t0 − ∆, t0 + ∆]. It follows from Corollary 7.2.1 that there exist trajectory-control pairs x1 : [0, ∞) → Rn , u1 : [0, ∞) → Rm and x2 : [T1 , T2 ] → Rn , u2 : [T1 , T2 ] → Rm which satisfy x1 (t) = x∗ (t), u1 (t) = u∗ (t) (t ∈ [0, T1 ] ∪ [T2 , ∞)), x1 (t) = x(t), u1 (t) = u(t) (t ∈ [T1 + ∆, T2 − ∆]), a (x1 (a), x1 (a + ∆)) (a ∈ {T1 , T2 − ∆}), (6.29) I(a, a + ∆, x1 , u1 ) ≤ v∆

x2 (Ti ) = x(Ti ), i = 1, 2, x2 (t) = x∗ (t), u2 (t) = u∗ (t) (t ∈ [T1 + ∆, T2 − ∆]), a (x2 (a), x2 (a + ∆)) (a ∈ {T1 , T2 − ∆}). (6.30) I(a, a + ∆, x2 , u2 ) ≤ v∆

It follows from (6.30), (6.29), (6.16) and Lemma 7.5.2 that I(T1 , T2 , x1 , u1 ) + I(T1 , T2 , x2 , u2 ) − I(T1 , T2 , x∗ , u∗ ) − I(T1 , T2 , x, u) T1 ∗ T1 ≤ [v∆ (x (T1 ), x(T1 + ∆)) − v∆ (x(T1 ), x(T1 + ∆))] T1 T1 ∗ (x(T1 ), x∗ (T1 + ∆)) − v∆ (x (T1 ), x∗ (T1 + ∆))] +[v∆ T2 −∆ T2 −∆ (x(T2 − ∆), x∗ (T2 )) − v∆ (x(T2 − ∆), x(T2 ))] +[v∆ T2 −∆ ∗ T2 −∆ ∗ (x (T2 − ∆), x(T2 )) − v∆ (x (T2 − ∆), x∗ (T2 ))]. +[v∆

(6.31)

By (6.31), (6.16), (6.18) and the deﬁnition of δ (see (6.22)-(6.24)), I(T1 , T2 , x1 , u1 ) + I(T1 , T2 , x2 , u2 ) −I(T1 , T2 , x∗ , u∗ ) − I(T1 , T2 , x, u) ≤ 4−1 δ1 ∆.

(6.32)

Linear systems with nonperiodic integrands

221

Deﬁne trajectory-control pairs yi : [T1 , T2 ] → Rn , wi : [T1 , T2 ] → Rm for i = 1, 2 as follows: y1 (t) = 2−1 (x1 (t) + x∗ (t)), w1 (t) = 2−1 (u1 (t) + u∗ (t)), (t ∈ [T1 , T2 ]), y2 (t) = 2−1 (x2 (t) + x(t)), w2 (t) = 2−1 (u2 (t) + u(t)), t ∈ [T1 , T2 ]. (6.33) By (6.28) there exists a number d such that t0 ∈ [d, d + ∆] ⊂ [T1 + ∆, T2 − ∆].

(6.34)

It follows from (6.33), (6.29), (6.30), (6.28), (6.34) and the deﬁnition of δ1 that for i = 1, 2, I(d, d + ∆, yi , wi ) ≤ 2−1 I(d, d + ∆, x∗ , u∗ ) + 2−1 I(d, d + ∆, x, u) − δ1 ∆. (6.35) (6.33), (6.29), (6.30), (6.16), Assumption (A1) and (6.32) imply that −δ ≤ I(T1 , T2 , y1 , w1 ) + I(T1 , T2 , y2 , w2 ) −I(T1 , T2 , x∗ , w∗ ) − I(T1 , T2 , x, w) ≤ −2δ1 ∆ + 2−1 [I(T1 , T2 , x1 , u1 ) + I(T1 , T2 , x2 , u2 ) −I(T1 , T2 , x∗ , w∗ ) − I(T1 , T2 , x, w)] ≤ −2δ1 ∆ + 8−1 δ1 ∆ ≤ −δ1 ∆. This is contradictory to (6.22). The obtained contradiction proves the lemma. Theorem 7.1.3 now follows from Lemmas 7.6.1, 7.6.4 and 7.6.5. Theorem 7.1.4 now follows from Lemmas 7.6.3, 7.6.4, 7.6.5.

Chapter 8 DISCRETE-TIME CONTROL SYSTEMS

In this chapter we study the structure of “approximate” solutions for a discrete-time control system determined by a sequence of continuous functions vi : X × X → R1 , i = 0, ±1, ±2, . . . where X is a complete metric space. We show that for a generic sequence of functions {vi }∞ i=−∞ there exists a sequence {yi }∞ i=−∞ ⊂ X (the “turnpike”) such that the 2 following properties hold: (i) {yi }ki=k is an optimal solution for any 1 ﬁnite interval [k1 , k2 ]; (ii) given > 0, each “approximate” solution on an interval [k1 , k2 ] with suﬃciently large k2 −k1 is within of the turnpike for all i ∈ {L + k1 , . . . , k2 − L} where L is a constant which depends only on .

8.1.

Convex inﬁnite dimensional control systems

Let X be a Banach space, || · || be the norm on X, and let K ⊂ X be a closed convex bounded set. Denote by A the set of all bounded convex functions v : K × K → R1 which satisfy the following assumption: (A) For each positive number there is a positive number δ such that if x1 , x2 , y1 , y2 ∈ K satisfy ||xi − yi || ≤ δ, i = 1, 2, then |v(x1 , x2 ) − v(y1 , y2 )| ≤ . We equip the space A with the metric ρ deﬁned by ρ(u, v) = sup{|v(x, y) − u(x, y)| : x, y ∈ K},

u, v ∈ A.

224

TURNPIKE PROPERTIES

It is easy to see that the metric space A is complete. We study the structure of “approximate” solutions of optimization problems n−1

v(xi , xi+1 ) → min,

(P )

i=0

{xi }ni=0 ⊂ K,

x0 = y, xn = z

where v ∈ A, y, z ∈ K and n is a natural number. The interest in these discrete-time optimal problems stems from the study of various optimization problems which can be reduced to this framework, e.g., continuous-time control systems which are represented by ordinary diﬀerential equations whose cost integrand contains a discounting factor [39], the inﬁnite-horizon control problem of minimizing T 0 L(z, z )dt as T → ∞ [42] and the analysis of a long slender bar of a polymeric material under tension [20, 44, 90-92]. Similar optimization problems are also considered in mathematical economics [47, 48, 56-61, 69, 76, 77]. In [99] we show that for a generic function v ∈ A there exists yv ∈ K such that the following turnpike property holds: For all large enough n and each y, z ∈ K an “approximate” solution {xi }ni=0 of problem (P) is contained in a small neighborhood of yv for all i ∈ {N, . . . , n − N } where N is a constant which depends on the neighborhood and does not depend on n. In almost all studies of discrete time control systems the turnpike property was considered for a single cost function v and a space of states K which was a compact convex set in a ﬁnite dimensional space. In these studies the compactness of K plays an important role. Speciﬁcally for the optimization problems considered in [99] if a function v has the turnpike property then as we will see later, its turnpike yv is a unique solution of the optimization problem v(x, x) → min,

x ∈ K.

The existence of a solution of this problem is guaranteed only if K satisﬁes some compactness assumptions. To obtain the uniqueness of the solution we need additional assumptions on v such as its strict convexity. In [99], instead of considering the turnpike property for a single cost function v, we investigate it for the space of all such functions equipped with some natural metric, and show that this property holds for most of these functions. This allows us to establish the turnpike property without compactness assumptions on the space of states and assumptions on functions themselves.

225

Discrete-time control systems

For each v ∈ A, each pair of integers m1 , m2 > m1 and each y1 , y2 ∈ K we deﬁne m 2 −1

σ(v, m1 , m2 ) = inf{

i=m1

2 v(zi , zi+1 ) : {zi }m i=m1 ⊂ K},

σ(v, m1 , m2 , y1 , y2 ) = inf{

m 2 −1

v(zi , zi+1 ) :

i=m1 2 {zi }m i=m1 ⊂ K, zm1 = y1 , zm2 = y2 },

and the minimal growth rate µ(v) = inf{lim inf N −1 N →∞

N −1

v(zi , zi+1 ) : {zi }∞ i=0 ⊂ K}.

i=0

For each x ∈ K and each positive number r set B(x, r) = {y ∈ K : ||x − y|| ≤ r}. We establish the existence of a set F ⊂ A which is a countable intersection of open everywhere dense subsets of A and such that the following two theorems are valid. The ﬁrst theorem shows that for v ∈ F the minimization problem v(z, z) → min, z ∈ K has a unique solution yv while the second theorem establishes the turnpike property for v with the turnpike yv . Theorem 8.1.1 For each v ∈ F there exists a unique yv ∈ K such that v(yv , yv ) = µ(v) and the following assertion holds: For each positive number there exist a neighborhood U of v in A and a positive number δ such that if u ∈ U and if y ∈ K satisﬁes u(y, y) ≤ µ(u) + δ, then ||y − yv || ≤ . Theorem 8.1.2 Let w ∈ F and be a positive number. Then there exist a neighborhood U of w in A, δ ∈ (0, ) and a natural number N such that the following property holds: If u ∈ U, a natural number n ≥ 2N and if a sequence {xi }ni=0 ⊂ K satisﬁes n−1 i=0

u(xi , xi+1 ) ≤ σ(u, 0, n, x0 , xn ) + δ,

226

TURNPIKE PROPERTIES

then there exist τ1 ∈ {0, . . . , N } and τ2 ∈ {n − N, . . . , n} such that ||xt − yw || ≤ ,

t = τ1 , . . . , τ2 .

Moreover, if ||x0 − yw || ≤ δ, then τ1 = 0, and if ||yw − xn || ≤ δ, then τ2 = n. Theorems 8.1.1 and 8.1.2 were obtained in [99]. In Section 8.2 we prove three auxiliary lemmas. Theorems 8.1.1 and 8.1.2 are proved in Section 8.3.

8.2.

Preliminary results

Set D0 = sup{||x|| : x ∈ K}.

(2.1)

For each bounded function u : K × K → R1 we set ||u|| = sup{|u(x, y)| : x, y ∈ K}.

(2.2)

Proposition 8.2.1 For each v ∈ A µ(v) = inf{v(z, z) : z ∈ K}. Proof. Let v ∈ A. Evidently inf{v(x, x) : x ∈ K} ≥ µ(v).

(2.3)

It is not diﬃcult to see that for each natural number m ≥ 1, σ(v, 0, m) ≤ mµ(v).

(2.4)

Let > 0. Since the function v is uniformly continuous there exists δ ∈ (0, ) such that the following property holds: If x1 , x2 , y1 , y2 ∈ K satisfy ||xi − yi || ≤ δ, i = 1, 2, then |v(x1 , x2 ) − v(y1 , y2 )| ≤ .

(2.5)

Fix a natural number m such that 8m−1 (D0 + 1) ≤ δ. Consider a sequence {yi }m i=0 ⊂ K such that m−1 i=0

v(yi , yi+1 ) ≤ σ(v, 0, m) + δ.

(2.6)

227

Discrete-time control systems

Put z0 = m−1

m−1

yi ,

z1 = m−1

i=0

m

yi .

i=1

It is not diﬃcult to see that ||z0 − z1 || ≤ 2m−1 D0 < δ

(2.7)

and v(z0 , z1 ) ≤ m−1

m−1

v(yi , yi+1 ) ≤ m−1 [σ(v, 0, m) + δ].

(2.8)

i=0

It follows from (2.7) and the choice of δ (see (2.5)) that |v(z0 , z0 ) − v(z0 , z1 )| ≤ . Combined with (2.8) and (2.4) this inequality implies that v(z0 , z0 ) ≤ 2 + m−1 σ(v, 0, m) ≤ µ(v) + 2. Since is an arbitrary positive number we conclude that inf{v(z, z) : z ∈ K} ≤ µ(v). This completes the proof of the proposition. The next auxiliary result shows that for v ∈ A and > 0 we can ﬁnd u ∈ A which is close to v in A such that all approximate solutions of the minimization problem u(y, y, ) → min, y ∈ K are contained in a small ball with radius . Proposition 8.2.2 For each v ∈ A and each ∈ (0, 1) there exist δ ∈ (0, ), u ∈ A and z0 ∈ K such that 0 ≤ u(x, y) − v(x, y) ≤ ,

x, y ∈ K,

µ(v) + δ ≥ v(z0 , z0 )

(2.9)

and that the following property holds: If y ∈ K satisﬁes u(y, y) ≤ µ(u) + δ, then ||y − z0 || ≤ . Proof. Let v ∈ A and ∈ (0, 1). Choose numbers δ, γ such that γ ∈ (0, (8D0 + 4)−1 ),

δ ∈ (0, 8−1 γ).

Proposition 8.2.1 implies that there is z0 ∈ K such that v(z0 , z0 ) < µ(v) + δ.

(2.10)

228

TURNPIKE PROPERTIES

Deﬁne a function u : K × K → R1 by u(x, y) = v(x, y) + γ(||x − z0 || + ||y − z0 ||),

x, y ∈ K.

(2.11)

It is not diﬃcult to see that u ∈ A and (2.9) is valid. Assume that y ∈ K and u(y, y) ≤ µ(u) + δ.

(2.12)

By Proposition 8.2.1, (2.11), (2.12), (2.10) and the choice of z0 , µ(v) ≤ µ(u) ≤ u(z0 , z0 ) = v(z0 , z0 ) ≤ µ(v) + δ, 2γ||y − z0 || + v(y, y) = u(y, y) ≤ µ(u) + δ ≤ µ(v) + 2δ ≤ v(y, y) + 2δ,

||y − z0 || ≤ δγ −1 < .

This completes the proof of the proposition. The next result shows that for a generic v ∈ A the minimization problem v(z, z) → min, z ∈ K has a unique solution. Proposition 8.2.3 There exists a set F0 which is a countable intersection of open everywhere dense subsets of A and such that for each v ∈ F0 there exists a unique yv ∈ K which satisﬁes v(yv , yv ) = µ(v) and that the following property holds: For each positive number there exist a neighborhood U of v in A and a positive number δ such that if u ∈ U and if y ∈ K satisﬁes u(y, y) ≤ µ(u) + δ, then ||y − yv || ≤ . Proof. Let w ∈ A and i be a natural number. It follows from Proposition 8.2.2 that there exist a positive number δ(w, i) < 4−i , a function u(w,i) ∈ A and z(w, i) ∈ K such that 0 ≤ u(w,i) (x, y) − w(x, y) ≤ 4−i ,

x, y ∈ K,

w(z(w, i), z(w, i)) ≤ µ(w) + δ(w, i) and the following property holds: If z ∈ K satisﬁes u(w,i) (z, z) ≤ µ(u(w,i) ) + δ(w, i), then z ∈ B(z(w, i), 4−i ).

(2.13)

229

Discrete-time control systems

Put

U(w, i) = {u ∈ A : ρ(u, u(w,i) ) < 8−1 δ(w, i)}.

(2.14)

Assume that u ∈ U(w, i) and z ∈ K satisﬁes u(z, z) ≤ µ(u) + 8−1 δ(w, i). Then it follows from (2.14) that u(w,i) (z, z) ≤ u(z, z) + 8−1 δ(w, i) ≤ µ(u) + 4−1 δ(w, i) ≤ µ(u(w,i) ) + 2−1 δ(w, i) and in view of the choice of δ(w, i), u(w,i) and z(w, i), z ∈ B(z(w, i), 4−i ). Thus we have shown that the following property holds: (a) If u ∈ U(w, i) and if z ∈ K satisﬁes u(z, z) ≤ µ(u) + 8−1 δ(w, i), then z ∈ B(z(w, i), 4−i ). Set F0 = ∩∞ q=1 ∪ {U(w, i) : w ∈ A,

i = q, q + 1, . . .}.

It is easy to see that F0 is a countable intersection of open everywhere dense subsets of A. Assume that v ∈ F0 . By Proposition 8.2.1 there exists a sequence {xj }∞ j=1 ⊂ K such that lim v(xj , xj ) = µ(v).

j→∞

(2.15)

By property (a) and the deﬁnition of F0 , {xj }∞ j=1 is a Cauchy sequence. Since K is a closed subset of the Banach space X and the function v is continuous we obtain that there exists limj→∞ xj and v( lim xj , lim xj ) = µ(v). j→∞

j→∞

Since any sequence {xj }∞ j=1 ⊂ K satisfying (2.15), converges in K, we conclude that there exists a unique yv ∈ K such that v(yv , yv ) = µ(v). Let > 0. Choose a natural number q such that 2−q < 8−1 .

(2.16)

In view of the deﬁnition of F0 there are w ∈ A and a natural number i ≥ q such that v ∈ U(w, i). By property (a), z(w, i) ∈ B(yv , 4−i ).

(2.17)

230

TURNPIKE PROPERTIES

(2.17), property (a) and (2.16) imply that if u ∈ U(w, i) and if y ∈ K satisﬁes u(y, y) ≤ µ(u) + 8−1 δ(w, i), then y ∈ B(yv , ) holds. This completes the proof of the proposition.

8.3.

Proofs of Theorems 8.1.1 and 8.1.2

Let the set F0 be as guaranteed in Proposition 8.2.3. For each w ∈ F0 there exists a unique yw ∈ K such that w(yw , yw )= µ(w).

(3.1)

˜ ∈ A which is close to w in A and Now for any w ∈ F0 we construct w has the turnpike property. Let v ∈ F0 , γ ∈ (0, 1). Deﬁne vγ (x, y) = v(x, y) + γ(||x − yv || + ||y − yv ||),

x, y ∈ K.

(3.2)

Clearly vγ ∈ A. We show that vγ has the turnpike property. We begin with the following lemma which establishes that for any approximate solution {xi }ni=0 with large enough n there is j ∈ {0, . . . , n − 1} such that xj and xj+1 are close enough to yv . Lemma 8.3.1 For each ∈ (0, 1) there exists a natural number n such that the following property holds: If a sequence {xi }ni=0 ⊂ K satisﬁes n−1

vγ (xi , xi+1 ) ≤ σ(vγ , 0, n, x0 , xn ) + 4,

(3.3)

i=0

then there exists j ∈ {0, . . . , n − 1} such that xj , xj+1 ∈ B(yv , ).

(3.4)

Proof. Let ∈ (0, 1). Choose a natural number n > (γ)−1 (5 + 4(||vγ || + ||v||)) + 4.

(3.5)

We show that nµ(v) ≤ σ(v, 0, n) + 2||v||. Let ∆ be a positive number. There exists a sequence n−1 i=0

v(zi , zi+1 ) ≤ σ(v, 0, n) + ∆.

(3.6) {zi }ni=0

such that

231

Discrete-time control systems

Then it follows from Proposition 8.2.1 and the convexity of v that nµ(v) ≤ nv(n−1

n−1

zi , n−1

i=0

n−1

zi )

i=0

n−2

= nv(n−1 [

(zi , zi+1 ) + (zn−1 , z0 )])

i=0

≤

n−2

v(zi , zi+1 ) + v(zn−1 , z0 ) ≤

i=0

n−1

v(zi , zi+1 ) + 2||v||

i=0

≤ σ(v, 0, n) + 2|v|| + ∆. Since ∆ is an arbitrary positive number we obtain the inequality (3.6). Assume that {xi }ni=0 ⊂ K satisﬁes (3.3). Put yi = xi ,

i = 0, n,

yi = yv ,

i = 1, . . . , n − 1.

(3.7)

By (3.2), (3.3), (3.7), (3.1), (3.6) and (3.5), σ(v, 0, n) + γ

≤

n−1

n−1

n−1

i=0

i=0

(||xi − yv || + ||xi+1 − yv ||) ≤

vγ (xi , xi+1 )

vγ (yi , yi+1 ) + 4 ≤ 4||vγ || + 4 + nv(yv , yv )

i=0

= 4 + 4||vγ || + nµ(v) ≤ 4 + 4||vγ || + σ(v, 0, n) + 2||v||, min{||xi − yv || + ||xi+1 − yv || : i = 0, . . . , n − 1} ≤ (nγ)−1 (4 + 4||vγ || + 2||v||) < . This completes the proof of the lemma. Lemma 8.3.1 implies the following auxiliary result which shows that the convergence property established in Lemma 8.3.1 for vγ also holds for approximate solutions with the cost function u belonging to a small neighborhood of vγ in A. Lemma 8.3.2 For each ∈ (0, 1) there exist a neighborhood U of vγ in A and a natural number n such that the following property holds: If u ∈ U and if a sequence {xi }ni=0 ⊂ K satisﬁes n−1 i=0

u(xi , xi+1 ) ≤ σ(u, 0, n, x0 , xn ) + 3,

232

TURNPIKE PROPERTIES

there is j ∈ {0, . . . , n − 1} for which xj , xj+1 ∈ B(yv , ). The next auxiliary result is an important tool in our proof. It shows that if x0 , xn are close to yv , then the approximate solution {xi }ni=0 is close to yv for all i = 0, . . . , n. Lemma 8.3.3 For each ∈ (0, 1) there exists δ ∈ (0, ) such that the following property holds: If n is a natural number and if a sequence {xi }ni=0 ⊂ K satisﬁes n−1

x0 , xn ∈ B(yv , δ),

vγ (xi , xi+1 ) ≤ σ(vγ , 0, n, x0 , xn ) + δ,

(3.8)

i=0

then xi ∈ B(yv , ),

i = 0, . . . , n.

(3.9)

Proof. We show that for each natural number m, σ(v, 0, m, yv , yv ) = mµ(v).

(3.10)

Let m be a natural number. Evidently σ(v, 0, m, yv , yv ) ≤ µ(v). Assume that {zi }m i=0 ⊂ K and z0 , zm = yv . Then it follows from the convexity of v and Proposition 8.2.1 that m−1

m−1

v(zi , zi+1 ) ≥ v(m−1

i=0

= v(m−1

m−1

(zi , zi+1 ))

i=0 m−1

(zi , zi )) ≥ µ(v).

i=0

Therefore the equality (3.10) is true. Let ∈ (0, 1). Fix δ0 ∈ (0, 8−1 γ).

(3.11)

Since the function v is uniformly continuous there exists a positive number δ < 2−1 δ0 such that the following property holds: If x1 , x2 , y1 , y2 ∈ K satisfy ||xi − yi || ≤ δ, i = 1, 2, then |v(x1 , x2 ) − v(y1 , y2 )| ≤ 64−1 δ0 .

(3.12)

Assume that n is a natural number and a sequence {xi }ni=0 ⊂ K satisﬁes (3.8). We will show that (3.9) is valid.

233

Discrete-time control systems

Let us assume the converse. Then n ≥ 2 and there exists j ∈ {1, . . . , n − 1} such that ||xj − yv || > . (3.13) Set zi = xi ,

i = 0, n,

i = 1, . . . , n − 1,

zi = yv ,

(3.14)

hi = yv , i = 0, n, hi = xi , i = 1, . . . , n − 1. By (3.2), (3.13), (3.8), (3.14) and (3.1), γ +

n−1

v(xi , xi+1 ) ≤

i=0

n−1

vγ (xi , xi+1 ) ≤

i=0

n−1

vγ (zi , zi+1 ) + δ

(3.15)

i=0

= δ + vγ (x0 , yv ) + vγ (yv , xn ) + (n − 2)µ(v). It follows from (3.8), (3.1) and the choice of δ (see (3.12)) that |v(xi , yv ) − µ(v)|,

|v(yv , xi ) − µ(v)| ≤ 64−1 δ0 ,

|v(yv , x1 ) − v(x0 , x1 )| ≤ 64−1 δ0 ,

i = 0, n,

(3.16)

|v(xn−1 , xn ) − v(xn−1 , yv )| ≤ 64−1 δ0 .

By (3.16), (3.8), (3.15), (3.10), (3.14) and (3.2), γ +

n−1

v(hi , hi+1 ) ≤ γ +

i=0

n−1

v(xi , xi+1 ) + 32−1 δ0

i=0

≤ 32−1 δ0 + δ + (n − 2)µ(v) + vγ (x0 , yv ) + vγ (yv , xn ) ≤ 32−1 δ0 + δ + µ(v)n + 32−1 δ0 + 2γδ ≤ µ(v)n + 16−1 δ0 +3δ = σ(v, 0, n, yv , yv ) + 16−1 δ0 + 3δ. Combined with (3.14) this implies that γ ≤ 4δ0 . This is contradictory to (3.11). The obtained contradiction proves the lemma. The next lemma shows that the convergence property established in Lemma 8.3.3 for vγ also holds for all u ∈ A which are close to vγ . Lemma 8.3.4 For each ∈ (0, 1) there exist δ ∈ (0, ), a neighborhood U of vγ in A and a natural number N such that the following property holds: If u ∈ U, a natural number n ≥ 2N and if a sequence {xi }ni=0 ⊂ K satisﬁes n−1 i=0

u(xi , xi+1 ) ≤ σ(u, 0, n, x0 , xn ) + δ,

(3.17)

234

TURNPIKE PROPERTIES

then there exist τ1 ∈ {0, . . . , N }, τ2 ∈ {−N + n, n} such that xi ∈ B(yv , ),

t = τ1 , . . . , τ2 ,

(3.18)

and moreover if x0 ∈ B(yv , δ), then τ1 = 0, and if xn ∈ B(yv , δ) then τ2 = n. Proof. Let ∈ (0, 1). It follows from Lemma 8.3.3 that there exists δ ∈ (0, 4−1 ) such that the following property holds: If n is a natural number and if a sequence {xi }ni=0 ⊂ K satisﬁes xi ∈ B(yv , 4δ), i = 0, n,

n−1

vγ (xi , xi+1 ) ≤ σ(vγ , 0, n, x0 , xn ) + 4δ,

i=0

(3.19) then xi ∈ B(yv , ),

i = 0, . . . , n.

(3.20)

It follows from Lemma 8.3.2 that there exist a natural number N and a neighborhood U1 of vγ in A such that the following property holds: If u ∈ U1 and if a sequence {xi }N i=0 ⊂ K satisﬁes N −1

u(xi , xi+1 ) ≤ σ(u, 0, N, x0 , xN ) + 3,

(3.21)

i=0

then there is j ∈ {0, . . . N − 1} such that

Deﬁne

xj , xj+1 ∈ B(yv , δ).

(3.22)

U = {u ∈ U1 : ρ(u, vγ ) ≤ (16N )−1 δ}.

(3.23)

Assume that u ∈ U, a natural number n ≥ 2N and a sequence {xi }ni=0 ⊂ K satisﬁes (3.17). It follows from (3.17) and the deﬁnition of U1 , N (see (3.21), (3.22)) that there exist integers τ1 , τ2 such that: τ1 ∈ {0, . . . , N },

τ2 ∈ {n − N, . . . , n},

xτi ∈ B(yv , δ), i = 1, 2; (3.24) if x0 ∈ B(yv , δ), then τ1 = 0; if xn ∈ B(yv , δ), then τ2 = n. We will show that (3.18) is valid. Let us assume the converse. Then there is an integer s ∈ (τ1 , τ2 ) for which ||xs − yv || > .

(3.25)

It follows from (3.17), (3.24) and the deﬁnition of U1 , N (see (3.21), (3.22)) that there exist integers t1 , t2 such that sup{τ1 , s − N } ≤ t1 < s,

s < t2 ≤ inf{τ2 , s + N },

235

Discrete-time control systems

xti ∈ B(yv , δ), i = 1, 2.

(3.26)

In view of (3.23), (3.26) and (3.17), t 2 −1 i=t1

vγ (xi , xi+1 ) ≤ σ(vγ , t1 , t2 , xt1 , xt2 ) + 2δ.

(3.27)

By (3.26), (3.27) and the choice of δ (see (3.19), (3.20)), ||xt − yt || ≤ ,

t = t1 , . . . , t2 .

This is contradictory to (3.25). The obtained contradiction proves that (3.18) is valid. This completes the proof of the lemma. Clearly the set {vγ : v ∈ F0 , γ ∈ (0, 1)} is everywhere dense in A. Now we are ready to construct the set F. Let v ∈ F0 , γ ∈ (0, 1) and let j be a natural number. There exist a natural number N (v, γ, j), an open neighborhood U0 (v, γ, j) of vγ in A and a number δ(v, γ, j) ∈ (0, 2−j ) such that Lemma 8.3.4 holds with v, γ, = 2−j , δ = δ(v, γ, j), U = U0 (v, γ, j), N = N (v, γ, j). There are an open neighborhood U(v, γ, j) of vγ in A and a natural number N1 (v, γ, j) such that U(v, γ, j) ⊂ U0 (v, γ, j) and Lemma 8.3.2 holds with v, γ, U = U(v, γ, j), n = N1 (v, γ, j), = 4−j δ(v, γ, j). Deﬁne F = [∩∞ q=1 ∪ {U(v, γ, j) : v ∈ F0 , γ ∈ (0, 1), j = q, q + 1, . . .}] ∩ F0 . Clearly F is a countable intersection of open everywhere dense subsets of A. It is easy to see that Theorem 8.1.1 follows from Proposition 8.2.3 and the deﬁnition of F. Proof of Theorem 8.1.2. Let w ∈ F, > 0. We may assume that < 1. Choose a natural number q such that 64 · 2−q < .

(3.28)

There exist v ∈ F0 , γ ∈ (0, 1) and a natural number j ≥ q such that w ∈ U(v, γ, j).

(3.29)

In view of (3.29), Lemma 8.3.2 which holds with U = U(v, γ, j), n = N1 (v, γ, j), = 4−j δ(v, γ, j), v, γ, and the equality σ(w, 0, N1 (v, γ, j), yw , yw ) = N1 (v, γ, j)µ(w) we have

yv ∈ B(yw , 4−j δ(v, γ, j)).

(3.30)

236

TURNPIKE PROPERTIES

Put U = U(v, γ, j),

N = N (v, γ, j),

δ = 4−j δ(v, γ, j).

(3.31)

Assume that u ∈ U, a natural number n ≥ 2N and a sequence {xi }ni=0 ⊂ K satisﬁes n−1

u(xi , xi+1 ) ≤ σ(u, 0, n, x0 , xn ) + δ.

(3.32)

i=0

By (3.32), (3.31), the choice of U0 (v, γ, j), N (v, γ, j), δ(v, γ, j) and Lemma 8.3.4, there exist τ1 ∈ {0, . . . , N }, τ2 ∈ {n − N, . . . , n} such that xi ∈ B(yv , 2−j ), t = τ1 , . . . , τ2 . Moreover if x0 ∈ B(yv , δ(v, γ, j)), then τ1 = 0, and if xn ∈ B(yv , δ(v, γ, j)), then τ2 = n. Combined with (3.30), (3.28), (3.31) this implies that: xi ∈ B(yw , ),

i = τ1 . . . , τ2 ;

if x0 ∈ B(yw , δ), then x0 ∈ B(yv , δ(v, γ, j)) and τ1 = 0; if xn ∈ B(yw , δ), then xn ∈ B(yv , δ(v, γ, j)) and τ2 = n. This completes the proof of the theorem.

8.4.

Nonautonomous control systems in metric spaces

Let X be a complete metric space and let d(·, ·) be the metric on X. We equip the set X × X with a metric d1 (·, ·) deﬁned by d1 ((x1 , x2 ), (y1 , y2 )) = d(x1 , y1 ) + d(x2 , y2 ),

x1 , x2 , y1 , y2 ∈ X.

Denote by M the set of all sequences of functions v = {vi }∞ i=−∞ which satisfy the following assumptions: A(i) vj : X × X → R1 is a continuous function for each integer j; A(ii) (uniform boundedness) sup{|vj (x, y)| :

x, y ∈ X, j = 0, ±1, ±2, . . .} < ∞;

237

Discrete-time control systems

A(iii) (uniform continuity) for each positive number there exists a positive number δ such that if x1 , x2 , y1 , y2 ∈ X satisfy d(x1 , x2 ),

d(y1 , y2 ) ≤ δ,

then |vj (x1 , y1 ) − vj (x2 , y2 )| ≤ for each integer j. Such a sequence of functions {vi }∞ i=−∞ ∈ M will occasionally be denoted by a boldface v (similarly {ui }∞ i=−∞ will be denoted by u, etc.) We endow the set M with the metric ρ(v, w) = sup{|vi (x, y) − wi (x, y)| : x, y ∈ X, i = 0, ±1, ±2, . . .},

v, w ∈ M.

(4.1)

Clearly the metric space (M, ρ) is complete. In this chapter we investigate the structure of “approximate” solutions of the optimization problem k 2 −1

vi (xi , xi+1 ) → min,

i=k1

2 {xi }ki=k ⊂ X, 1

xk1 = y, xk2 = z

(P )

where v = {vi }∞ i=−∞ ∈ M, y, z ∈ X and k2 > k1 are integers. For each v ∈ M, each pair of integers m2 > m1 and each y1 , y2 ∈ X we deﬁne m 2 −1

σ(v, m1 , m2 ) = inf{

vi (zi , zi+1 ) :

i=m1 m 2 −1

σ(v, m1 , m2 , y1 , y2 ) = inf{

2 {zi }m i=m1 ⊂ X},

(4.2)

vi (zi , zi+1 ) :

i=m1 2 {zi }m i=m1 ⊂ X, zm1 = y1 , zm2 = y2 }.

If the space of states X is compact, then the problem (P) has a solution for each v ∈ M, y, z ∈ X and each pair of integers k2 > k1 . For the noncompact space X the existence of solutions of the problem (P) is not guaranteed and in this situation we consider δ-approximate solutions. Let v ∈ M, y, z ∈ X, k2 > k1 be integers and let δ be a positive 2 ⊂ X which satisﬁes xk1 = y, xk2 = z is number. A sequence {xi }ki=k 1 called a δ-approximate solution of the problem (P) if k 2 −1 i=k1

vi (xi , xi+1 ) ≤ σ(v, k1 , k2 , y, z) + δ.

238

TURNPIKE PROPERTIES

The following optimality criterion for inﬁnite horizon problems was introduced by Aubry and Le Daeron [6]. ∞ Let v = {vi }∞ i=−∞ ∈ M. We say that a sequence {xi }i=−∞ is (v)minimal if m 2 −1 i=m1

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

(4.3)

for each pair of integers m1 < m2 . If the space of states X is compact, then a (v)-minimal sequence can be constructed as a limit of a sequence of optimal solutions on ﬁnite intervals. For the noncompact space X the problem is more diﬃcult and less understood. We say that a sequence v = {vi }∞ i=−∞ ∈ M has the turnpike property if there is a (v)-minimal sequence {xvi }∞ i=−∞ ⊂ X which satisﬁes the following condition: For each > 0 there exist a positive number δ and a natural number 2 of the problem (P) L such that each δ-approximate solution {xi }ki=k 1 with y, z ∈ X, and k2 ≥ k1 + 2L satisﬁes d(xi , xvi ) ≤ ,

i = k1 + L, . . . , k2 − L.

(4.4)

In this chapter we prove the following result. Theorem 8.4.1 There exists a set F ⊂ M which is a countable intersection of open everywhere dense sets in M such that each v ∈ F satisﬁes the following conditions: (a) there is a unique (v)-minimal sequence {xvi }∞ i=−∞ ⊂ X; (b) for each positive number there exist δ ∈ (0, ), a natural number N and a neighborhood U of v in M such that the following property holds: 2 ⊂X If w ∈ U , k1 , k2 > k1 +2N are integers and if a sequence {yi }ki=k 1 satisﬁes k2 i=k1

then

wi (yi , yi+1 ) ≤ σ(w, k1 , k2 , yk1 , yk2 ) + δ,

d(yi , xvi ) ≤ ,

i = k1 + L1 , . . . , k2 − L2 ,

where the integers L1 , L2 ∈ [0, N ], and moreover, if d(yk1 , xvk1 ) ≤ δ, then L1 = 0, and if d(yk2 , xvk2 ) ≤ δ, then L2 = 0. Theorem 8.4.1 was obtained in [104]. In Section 8.5 we construct a set F0 ⊂ M which is a countable intersection of open everywhere dense subsets of M such that for each v ∈ F0

239

Discrete-time control systems

there is a v-minimal sequence {xvi }∞ i=−∞ . In Section 8.6 we construct a set F ⊂ F0 which is a countable intersection of open everywhere dense subsets of M such that for each v ∈ F the turnpike property holds with the turnpike {xvi }∞ i=−∞ .

8.5.

An auxiliary result

For each v ∈ M we deﬁne ||v|| = sup{|vi (x, y)| : x, y ∈ X, i = 0, ±1, ±2, . . .}. Denote by Card(A) the cardinality of a set A, by Z the set of all integers and by N the set of all natural numbers. In this section we will establish the following result. Proposition 8.5.1 There exists a set F0 ⊂ M which is a countable intersection of open everywhere dense sets in M such that for each v ∈ F0 there is a (v)-minimal sequence {xvi }∞ i=−∞ ⊂ X. Construction of the set F0 . For each v ∈ M and each natural number q we deﬁne a number n(v,q) (v, q) > 0, an integer n(v, q) ≥ 1, a sequence {zi (v, q)}i=−n(v,q) ⊂ X,

u(v,q) ∈ M, a positive number γ(v, q) and an open neighborhood U (v, q) of u(v,q) in M. Let v = {vi }∞ i=−∞ ∈ M and let q ∈ N. It follows from Assumption A(iii) that there exists a number (v, q) ∈ (0, 2−20 q −2 ) ∩ {i−1 : i ∈ N}

(5.1)

such that the following property holds: If x1 , x2 , y1 , y2 ∈ X satisfy d(x1 , x2 ), d(y1 , y2 ) ≤ 4(v, q), then for each j ∈ Z, |vj (x1 , y1 ) − vj (x2 , y2 )| ≤ 2−8 q −2 .

(5.2)

Choose a natural number n(v, q) > 4 · 642 (8||v|| + 8)q(v, q)−1 .

(5.3)

n(v,q)

Consider a sequence {zi (v, q)}i=−n(v,q) ⊂ X such that n(v,q)−1

vi (zi (v, q), zi+1 (v, q)) ≤ σ(v, −n(v, q), n(v, q)) + 2−10 q −2 .

i=−n(v,q)

(5.4)

240

TURNPIKE PROPERTIES (v,q) ∞ }i=−∞

Deﬁne u(v,q) = {ui

(v,q)

ui (v,q)

ui

= vi ,

by

i ∈ Z \ [−n(v, q), n(v, q) − 1],

(5.5)

(x, y) = vi (x, y) + (4q)−1 min{d(x, zi (v, q)) + d(y, zi+1 (v, q)), 1}, x, y ∈ X, i ∈ Z ∩ [−n(v, q), n(v, q) − 1].

Evidently u(v,q) ∈ M. Choose a number γ(v, q) ∈ (0, 2−1 (64n(v, q))−1 (v, q)) and put

(5.6)

ρ(w, u(v,q) ) < γ(v, q)}.

U (v, q) = {w ∈ M :

(5.7)

Deﬁne F0 = ∩∞ m=1 ∪ {U (v, q) : v ∈ M and q ∈ N ∩ [m, ∞)}.

(5.8)

It is easy to see that F0 is a countable intersection of open everywhere dense sets in M. We preface the proof of Proposition 8.5.1 with the following auxiliary n(v,q) lemma which shows that any approximate solution {zi }i=−n(v,q) with (v,q) ∞ }i=−∞

respect to {ui

n(v,q)

is close enough to {zi (v, q)}i=−n(v,q) . n(v,q)

Lemma 8.5.1 Let v ∈ M, q ∈ N and let a sequence {zi }i=−n(v,q) ⊂ X satisfy n(v,q)−1

(v,q)

ui

(zi , zi+1 ) ≤ σ(u(v,q) , −n(v, q), n(v, q)) + 32−1 q −2 . (5.9)

i=−n(v,q)

Then d(zi , zi (v, q)) ≤ (4q)−1 ,

i = −n(v, q), . . . , n(v, q).

(5.10)

Proof. It follows from (5.4), (5.5) and (5.9) that n(v,q)−1

n(v,q)−1

−1

vi (zi , zi+1 ) + (4q)

i=−n(v,q)

inf{1, d(zi , zi (v, q))

i=−n(v,q)

+d(zi+1 , zi+1 (v, q))} n(v,q)−1

=

i=−n(v,q)

(v,q)

ui

(zi , zi+1 ) ≤ 32−1 q −2 + σ(u(v,q) − n(v, q), n(v, q))

241

Discrete-time control systems n(v,q)−1 −1 −2

≤ 32

q

+

(v,q)

ui

(zi (v, q), zi+1 (v, q))

i=−n(v,q) n(v,q)−1 −1 −2

= 32

q

+

vi (zi (v, q), zi+1 (v, q))

i=−n(v,q) n(v,q)−1

≤ 32−1 q −2 +

vi (zi , zi+1 ) + 2−10 q −2 .

i=−n(v,q)

This imples (5.10). The lemma is proved. The next auxiliary result gives an important estimation of the cardinality of the set of all integers i such that zi is not close enough to n(v,q) zi (v, q) where {zi }i=−n(v,q) is an approximate solution with respect to (v,q) ∞ }i=−∞ .

{ui

n(v,q)

Lemma 8.5.2 Let v ∈ M, q ∈ N and let {zi }i=−n(v,q) ⊂ X satisfy n(v,q)−1

(v,q)

ui

(zi , zi+1 ) ≤ σ(u(v,q) , −n(v, q), n(v, q)) + 8(||v|| + 2).

i=−n(v,q)

(5.11) Then Card{i ∈ {−n(v, q), . . . , n(v, q)−1} :

d(zi , zi (v, q))+d(zi+1 , zi+1 (v, q)) (5.12) −1 ≥ (v, q)} ≤ (8||v|| + 3)4q(v, q) .

Proof. It follows from (5.5), (5.11) and (5.4) that n(v,q)−1

n(v,q)−1

vi (zi , zi+1 ) + (4q)−1

i=−n(v,q)

inf{1, d(zi , zi (v, q))

i=−n(v,q) n(v,q)−1

+d(zi+1 , zi+1 (v, q))} =

(v,q)

ui

(zi , zi+1 )

i=−n(v,q)

≤ σ(u(v,q) , −n(v, q), n(v, q)) + 8(||v|| + 2) n(v,q)−1

≤

i=−n(v,q)

(v,q)

ui

(zi (v, q), zi+1 (v, q)) + 8(||v|| + 2)

242

TURNPIKE PROPERTIES n(v,q)−1

≤

vi (zi (v, q), zi+1 (v, q)) + 8(||v|| + 2)

i=−n(v,q) n(v,q)−1

≤

vi (zi , zi+1 ) + 2−10 q −2 + 8(||v|| + 2).

i=−n(v,q)

This implies (5.12). The lemma is proved. Lemma 8.5.1, (5.1), (5.6) and (5.7) imply the following result. Lemma 8.5.3 For each v ∈ M, each q ∈ N, each w ∈ U (v, q) and each n(v,q) {zi }i=−n(v,q) ⊂ X which satisfy n(v,q)

wi (zi , zi+1 ) ≤ σ(w, −n(v, q), n(v, q)) + 32−1 q −2 − 16−1 (v, q),

i=−n(v,q)

the inequality (5.10) holds. Lemma 8.5.2, (5.1), (5.6), (5.7) and (5.5) imply the following result which shows that the estimation obtained in Lemma 8.5.2 for (v,q) ∞ }i=−∞

{ui

also holds for any w which is close enough to u(v,q) in M. Lemma 8.5.4 For each v ∈ M, each q ∈ N, each w ∈ U (v, q) and each n(v,q) {zi }i=−n(v,q) ⊂ X which satisfy n(v,q)−1

wi (zi , zi+1 ) ≤ σ(w, −n(v, q), n(v, q)) + 8||w|| + 13,

i=−n(v,q)

the inequality (5.12) holds. The next lemma plays a crucial role in our proof. It shows that two approximate solutions with respect to w ∈ U (v, q) are close for all i belonging to a certain subinterval of [−n(v, q), n(v, q)] with a large length. Lemma 8.5.5 Let v ∈ M, q ≥ 1 be an integer and let w ∈ U (v, q). n(v,q) Assume that a sequence {zi }i=−n(v,q) ⊂ X satisﬁes n(v,q)−1

i=−n(v,q)

wi (zi , zi+1 ) ≤ σ(w, −n(v, q), n(v, q)) + 64−2 q −2 ,

(5.13)

243

Discrete-time control systems

a natural number m > n(v, q) and that a sequence {yi }m i=−m ⊂ X satisﬁes m−1

wi (yi , yi+1 ) ≤ σ(w, −m, m) + 64−2 q −2 .

i=−m

Then d(zi , yi ) ≤ (2q)−1 ,

i ∈ {−n(v, q) + 2 + 8q(8||v|| + 3)(v, q)−1 , . . . , (5.14) n(v, q) − 2 − 8q(8||v|| + 3)(v, q)−1 }.

Proof. We show that n(v,q)−1

wi (yi , yi+1 ) ≤ σ(w, −n(v, q), n(v, q)) + 4||w|| + 1.

(5.15)

i ∈ {−m, . . . , −n(v, q) − 1} ∪ {n(v, q) + 1, . . . , m},

(5.16)

i=−n(v,q)

Set xi = yi ,

xi = zi , i ∈ {−n(v, q), . . . , n(v, q)}. In view of (5.16) and (5.13), 64−2 q −2 ≥

m−1

wi (yi , yi+1 ) −

i=−m n(v,q)−1

≥

m−1

wi (xi , xi+1 )

i=−m n(v,q)−1

wi (yi , yi+1 ) −

i=−n(v,q)

wi (zi , zi+1 ) − 4||w||.

i=−n(v,q)

Combined with (5.13) this inequality implies (5.15). It follows from (5.13), (5.15) and Lemma 8.5.4 that Card{i ∈ {−n(v, q), . . . , n(v, q) − 1} :

d(zi , zi (v, q))

(5.17)

+d(zi+1 , zi+1 (v, q)) ≥ (v, q)} ≤ (8||v|| + 3)4q(v, q))−1 , Card{i ∈ {−n(v, q), . . . , n(v, q) − 1} :

d(yi , zi (v, q))

+d(yi+1 , zi+1 (v, q)) ≥ (v, q)} ≤ (8||v|| + 3)4q(v, q))−1 . By (5.17) and (5.3) there exist integers

for which

j1 , j2 ∈ [−n(v, q) + 1, n(v, q) − 1]

(5.18)

j1 ≤ −n(v, q) + 2 + 8q(8||v|| + 3)(v, q)−1 ,

(5.19)

244

TURNPIKE PROPERTIES

j2 ≥ n(v, q) − 2 − 8q(8||v|| + 3)(v, q)−1 , d(yjp , zjp (v, q)), d(zjp , zjp (v, q)) ≤ (v, q),

p = 1, 2.

n(v,q)

Deﬁne sequences {hi }i=−n(v,q) ⊂ X and {si }m i=−m ⊂ X by hi = zi ,

i ∈ {−n(v, q), . . . , j1 − 1} ∪ {j2 + 1, . . . , n(v, q)},

hi = yi , i ∈ {j1 , . . . , j2 },

(5.20)

si = yi , i ∈ {−m, . . . , j1 −1}∪{j2 +1, . . . , m},

si = zi ,

i ∈ {j1 , . . . , j2 }.

We estimate n(v,q)−1

i=−n(v,q)

n(v,q)−1

wi (hi , hi+1 ) −

wi (zi , zi+1 ).

i=−n(v,q)

In view of the choice of (v, q) (see (5.1), (5.2)), (5.20), (5.19) and (5.18), |vp (hp , hp+1 ) − vp (zp , zp+1 )|, |vp (sp , sp+1 ) − vp (yp , yp+1 )|

(5.21)

≤ 2−8 q −2 , p = j1 − 1, j2 . (5.18), (5.20) and (5.19) imply that for p = j1 − 1, j2 , | inf{1, d(hp , zp (v, q)) + d(hp+1 , zp+1 (v, q))} − inf{1, d(zp , zp (v, q)) + d(zp+1 , zp+1 (v, q))}| ≤ |d(hp , zp (v, q)) + d(hp+1 , zp+1 (v, q)) −d(zp , zp (v, q)) − d(zp+1 , zp+1 (v, q))| ≤ d(zp , hp ) + d(hp+1 , zp+1 ) < 2(v, q), | inf{1, d(sp , zp (v, q)) + d(sp+1 , zp+1 (v, q))} − inf{1, d(yp , zp (v, q)) + d(yp+1 , zp+1 (v, q))}| ≤ |d(sp , zp (v, q)) + d(sp+1 , zp+1 (v, q)) −d(yp , zp (v, q)) − d(yp+1 , zp+1 (v, q))| ≤ d(sp , yp ) + d(sp+1 , yp+1 ) < 2(v, q). Combined with (5.5) and (5.21) these inequalities imply that for p = j1 − 1, j2 , (hp , hp+1 ) − u(v,q) |u(v,q) (zp , zp+1 )|, p p

|up(v,q) (sp , sp+1 )

(yp , yp+1 )| ≤ 2−8 q −2 + (4q)−1 2(v, q). −u(v,q) p

245

Discrete-time control systems

It follows from this inequality, (5.3), (5.6) and (5.7) that for p = j1 −1, j2 , |wp (hp , hp+1 ) − wp (zp , zp+1 )|,

|wp (sp , sp+1 ) − wp (yp , yp+1 )|

(5.22)

≤ 2−8 q −2 + (v, q)[(2q)−1 + (32q)−1 ]. (5.20), (5.13) and (5.22) imply that m−1

64−2 q −2 ≥

i=−m j2

= j 2 −1

wi (si , si+1 )

i=−m j2

wi (yi , yi+1 ) −

i=j1 −1

=

m−1

wi (yi , yi+1 ) −

wi (si , si+1 )

i=j1 −1

wi (yi , yi+1 ) −

i=j1

j 2 −1 i=j1

wi (zi , zi+1 ) + wj1 −1 (yj1 −1 , yj1 )

+wj2 (yj2 , yj2 +1 ) − wj1 −1 (sj1 −1 , sj1 ) − wj2 (sj2 , sj2 +1 ) ≥

j 2 −1

j 2 −1

wi (yi , yi+1 ) −

i=j1

wi (zi , zi+1 )

i=j1

−2[2−8 q −2 + (v, q)((2q)−1 + (32q)−1 )], j 2 −1

wi (yi , yi+1 ) −

i=j1

j 2 −1

wi (zi , zi+1 )

i=j1

≤ 64−2 q −2 + 2−7 q −2 + (v, q)(q −1 + (16q)−1 ). By (5.23), (5.22) and (5.20), n(v,q)−1

n(v,q)−1

i=−n(v,q)

=

j2

=

i=j1

wi (zi , zi+1 )

i=−n(v,q)

wi (hi , hi+1 ) −

i=j1 −1 j 2 −1

wi (hi , hi+1 ) −

wi (yi , yi+1 ) −

j2

wi (zi , zi+1 )

i=j1 −1 j 2 −1 i=j1

wi (zi , zi+1 ) + wj1 −1 (hj1 −1 , hj1 )

+wj2 (hj2 , hj2 +1 ) − wj1 −1 (zj1 −1 , zj1 ) − wj2 (zj2 , zj2 +1 ) ≤ 64−2 q −2 + 2−7 q −2 + (v, q)(q −1 + (16q)−1 ) +2[2−8 q −2 + (v, q)((2q)−1 + (32q)−1 )]

(5.23)

246

TURNPIKE PROPERTIES

≤ 64−2 q −2 + 2−6 q −2 + (v, q)(2q −1 + (8q)−1 ) and

n(v,q)−1

n(v,q)−1

wi (hi , hi+1 ) −

i=−n(v,q)

wi (zi , zi+1 )

i=−n(v,q)

≤ 64−2 q −2 + 2−6 q −2 + (v, q)(2q −1 + (8q)−1 ). It follows from this inequality, (5.13) and (5.1) that n(v,q)−1

wi (hi , hi+1 ) ≤ σ(w, −n(v, q), n(v, q))

i=−n(v,q)

+2 · 64−2 q −2 + 2−6 q −2 + (v, q)(2q −1 + (8q)−1 ) ≤ σ(w, −n(v, q), n(v, q)) + 32−1 q −2 − 16−1 (v, q). In view of this inequality and Lemma 8.5.3, d(hi , zi (v, q)) ≤ (4q)−1 ,

i = −n(v, q), . . . , n(v, q).

Combined with (5.19) and (5.20) this inequality implies that d(yi , zi (v, q)) ≤ (4q)−1 ,

i = −n(v, q) + 2

(5.24)

+8q(8||v|| + 3)(v, q)−1 , . . . , n(v, q) − 2 − 8q(8||v|| + 3)(v, q)−1 . By (5.13) and Lemma 8.5.3 d(zi , zi (v, q)) ≤ (4q)−1 ,

i = −n(v, q), . . . , n(v, q).

Combined with (5.24) this implies (5.14). The lemma is proved. The next auxiliary result follows from Lemma 8.5.5. Lemma 8.5.6 For each w ∈ F0 , each ∈ (0, 1) and each k ∈ N there exist 0 ∈ (0, ) and a natural number k0 > k such that the following property holds: mp ⊂ X, p = 1, 2 satisﬁes If m1 , m2 ≥ k0 are integers and if {zip }i=−m p mp −1

p wi (zip , zi+1 ) ≤ σ(w, −mp , mp ) + 0 , p = 1, 2,

(5.25)

i=−mp

then

d(zi1 , zi2 ) ≤ ,

i = −k, . . . , k.

(5.26)

247

Discrete-time control systems

Proof. Let w ∈ F0 , ∈ (0, 1) and let k ∈ N. Choose q0 ∈ N such that q0 > 8k + 8 + 8−1 . (5.27) In view of (5.8) there exist q ∈ N and v ∈ M such that q ≥ q0 and w ∈ U (v, q).

(5.28)

Put k0 = n(v, q) + 1 + k,

0 = 64−2 q −2 .

(5.29)

n(v,q)

Consider a sequence {xi }i=−n(v,q) ⊂ X such that n(v,q)−1

wi (xi , xi+1 ) ≤ σ(w, −n(v, q), n(v, q)) + 64−2 q −2 .

(5.30)

i=−n(v,q) m

p Let m1 , m2 ≥ k0 be integers and let sequences {zip }i=−m ⊂ X, p = p 1, 2 satisfy (5.25). By Lemma 8.5.5, (5.28), (5.30), (5.29) and (5.25),

d(xi , zip ) ≤ (2q)−1 , i ∈ {−n(v, q) + 2 + 8q(8||v|| + 3)(v, q)−1 , . . . , n(v, q) − 2 − 8q(8||v|| + 3)(v, q)−1 },

p = 1, 2.

Combined with (5.27) and (5.3) this inequality implies that d(zi1 , zi2 ) ≤ q −1 ≤ for any integer i ∈ [−2−1 n(v, q), 2−1 n(v, q)] ⊃ [−k, k]. This completes the proof of the lemma. Proof of Proposition 8.5.1. Let w ∈ F0 . By Lemma 8.5.6 there exist a strictly increasing sequence of natural numbers {sk }∞ k=1 and a strictly such that decreasing sequence of positive numbers {δk }∞ k=1 δk < k −1 ,

sk > k, k = 1, 2, . . .

(5.31)

and that for each k ∈ N the following property holds: mp If integers m1 , m2 ≥ sk and if sequences {zip }i=−m ⊂ X, p = 1, 2 p satisfy mp −1

p wi (zip , zi+1 ) ≤ σ(w, −mp , mp ) + δk ,

i=−mp

then d(zi1 , zi2 ) ≤ (2k)−1 ,

i = −k, . . . , k.

p = 1, 2,

248

TURNPIKE PROPERTIES

k For any k ∈ N we ﬁx a sequence {yik }si=−s ⊂ X such that k

s k −1

k wi (yik , yi+1 ) ≤ σ(w, −sk , sk ) + δk .

(5.32)

i=−sk

It is not diﬃcult to see that for each i ∈ N there exists xi = limk→∞ yik . Assume that k1 , k2 > k1 are integers. It follows from (5.32), (5.31) that k 2 −1

wi (xi , xi+1 ) = lim

i=k1

k→∞

k 2 −1

k wi (yik , yi+1 )

i=k1

≤ lim sup σ(w, k1 , k2 , ykk1 , ykk2 ) = σ(w, k1 , k2 , xk1 , xk2 ). k→∞

This completes the proof of the proposition.

8.6.

Proof of Theorem 8.4.1

It follows from Proposition 8.5.1 there exists a set F0 ⊂ M which is a countable intersection of open everywhere dense sets in M such that for each v ∈ F0 there is a (v)-minimal sequence {xvi }∞ i=−∞ ⊂ X satisfying k 2 −1 i=k1

vi (xvi , xvi+1 ) = σ(v, k1 , k2 , xvk1 , xvk2 )

(6.1)

for each pair of integers k1 < k2 . For each v ∈ F0 and each r ∈ (0, 1] deﬁne vr = {vir }∞ i=−∞ as vir (x, y) = vi (x, y) + r inf{d(x, xvi ), 1},

(6.2)

x, y ∈ X, i = 0, ±1, . . . . Clearly vr ∈ M. We will see that vr has the turnpike property with the turnpike {xvi }∞ i=−∞ . We will preface the proof of Theorem 8.4.1 with the following auxiliary lemmas. Lemma 8.6.1 For each v ∈ F0 and each pair of integers k2 > k1 , k 2 −1 i=k1

vi (xvi , xvi+1 ) ≤ σ(v, k1 , k2 ) + 1 + 4||v||.

249

Discrete-time control systems

Proof. Let v ∈ F0 and k2 > k1 be integers. We may assume that 2 ⊂ X such that k2 − k1 ≥ 3. Consider a sequence {yi }ki=k 1 k 2 −1

vi (yi , yi+1 ) ≤ σ(v, k1 , k2 ) + 8−1 .

i=k1

Set

zi = xvi , i = k1 , k2 ,

zi = yi , i = k1 + 1, . . . , k2 − 1.

It is not diﬃcult to see that k 2 −1

vi (xvi , xvi+1 ) ≤

i=k1

k 2 −1

vi (zi , zi+1 ) ≤

i=k1

k 2 −1

vi (yi , yi+1 )

i=k1

+4||v|| ≤ σ(v, k1 , k2 ) + 4||v|| + 8−1 . The lemma is proved. The following auxiliary result shows that any approximate solution r {yi }p+N i=p with respect to v and with large enough N is close to the turnpike for some i ∈ [p, p + N ]. Lemma 8.6.2 Assume that v ∈ F0 , r ∈ (0, 1], δ ∈ (0, 1) and M > 0. Then there exists an integer N ≥ 4 such that for each integer p and each sequence {yi }p+N i=p ⊂ X satisfying p+N −1

vir (yi , yi+1 ) ≤ σ(vr , p, p + N, yp , yp+N ) + M

(6.3)

i=p

the following relation holds: inf{d(yi , xvi ) :

i ∈ {p, . . . , p + N }} < δ.

(6.4)

Proof. Choose a natural number N ≥ 6 + (M + 8||v|| + 8)(δr)−1 .

(6.5)

Assume that p ∈ Z, {yi }p+N i=p ⊂ X and (6.3) holds. To prove the lemma it is suﬃcient to show that (6.4) holds. Assume the contrary. Then d(yi , xvi ) ≥ δ,

i = p, . . . , p + N.

(6.6)

Deﬁne {zi }p+N i=p ⊂ X by zi = yi , i = p, p + N,

zi = xvi , i = p + 1, . . . , p + N − 1.

(6.7)

250

TURNPIKE PROPERTIES

By (6.2), (6.3), (6.6) and (6.7), p+N −1

vir (zi , zi+1 ) + M ≥ σ(vr , p, p + N, yp , yp+N ) + M

i=p

≥

p+N −1

vir (yi , yi+1 ) ≥

p+N −1

i=p

vi (yi , yi+1 ) + rδN.

i=p

It follows from this inequality, (6.7), Lemma 8.6.1 and (6.2) that rδN + σ(v, p, p + N ) ≤ M +

p+N −1

vir (zi , zi+1 ) ≤ M

i=p

+

p+N −1

vir (xvi , xvi+1 ) + 4(||v|| + 1) ≤ M + 4||v|| + 4

i=p

+σ(v, p, p + N ) + 1 + 4||v||. This is contradictrory to (6.5). The obtained contradiction proves the lemma. Lemma 8.6.2 implies the following result which shows that the property established in Lemma 8.6.2 for vr also holds for all w belonging to a small neighborhood of vr in M. Lemma 8.6.3 For each v ∈ F0 , each r ∈ (0, 1], each δ ∈ (0, 1) and each positive number M there exist a natural number N ≥ 4 and a neighborhood U of vr in M such that the following property holds: If w ∈ U , p ∈ Z and if a sequence {yi }p+N i=p ⊂ X satisﬁes p+N −1

wi (yi , yi+1 ) ≤ σ(w, p, p + N, yp , yp+N ) + M,

i=p

then the inequality (6.4) holds. 2 The next lemma shows that if an approximate solution {yi }ki=k with 1 r respect to v is close to the turnpike for i = k1 , k2 , then it is close to the turnpike for all i = k1 , . . . , k2 .

Lemma 8.6.4 Let v ∈ F0 , r ∈ (0, 1] and ∈ (0, 1). Then there exists δ ∈ (0, ) such that the following property holds: 2 If k1 , k2 ≥ k1 + 2 are integers and if a sequence {yi }ki=k ⊂ X satisﬁes 1 d(yi , xvi ) ≤ δ,

i = k1 , k2 ,

(6.8)

251

Discrete-time control systems k 2 −1 i=k1

then

vir (yi , yi+1 ) ≤ σ(vr , k1 , k2 , yk1 , yk2 ) + δ, d(yi , xvi ) ≤ ,

i = k1 , . . . , k2 .

(6.9)

Proof. Choose a positive number 0 < 8−1 r.

(6.10)

It follows from Assumption A(iii) that there exists a positive number δ < 8−1 0 such that the following property holds: If i is an integer and if x1 , x2 , y1 , y2 ∈ X satisfy d(x1 , y1 ) + d(x2 , y2 ) ≤ 8δ,

(6.11)

then |vi (x1 , x2 ) − vi (y1 , y2 )|, |vir (x1 , x2 ) − vir (y1 , y2 )| ≤ 64−1 0 .

(6.12)

2 Assume that k1 and k2 ≥ k1 +2 are integers and {yi }ki=k ⊂ X satisﬁes 1 (6.8). To prove the lemma it is suﬃcient to show that (6.9) holds. Let us assume the converse. Then

sup{d(yi , xvi ) : i = k1 + 1, . . . , k2 − 1} > .

(6.13)

Put zi = xvi , i = k1 , k2 ,

zi = yi , i = k1 + 1, . . . , k2 − 1.

(6.14)

By (6.8) and the choice of δ (see (6.11), (6.12)), |σ(vr , k1 , k2 , yk1 , yk2 ) − σ(vr , k1 , k2 , xvk1 , xvk2 )| ≤ 16−1 0 .

(6.15)

It follows from (6.14), (6.8) and the choice of δ (see (6.11), (6.12)) that |

k 2 −1 i=k1

vir (yi , yi+1 ) −

k 2 −1

vir (zi , zi+1 )| ≤ 16−1 0 .

i=k1

In view of this inequality, (6.15), (6.8), (6.1) and (6.2), k 2 −1 i=k1

vir (zi , zi+1 ) ≤ 8−1 0 + δ + σ(vr , k1 , k2 , xvk1 , xvk2 ) = 8−1 0 + δ + σ(v, k1 , k2 , xvk1 , xvk2 ).

252

TURNPIKE PROPERTIES

Combined with (6.2), (6.13) and (6.14) this relation implies that σ(v, k1 , k2 , xvk1 , xvk2 ) + δ + 8−1 0 ≥

≥

k 2 −1

vi (zi , zi+1 ) + r,

k 2 −1

vir (zi , zi+1 )

i=k1

r ≤ δ + 8−1 0 ≤ 4−1 0 .

i=k1

This is contradictory to (6.10). The obtained contradiction proves the lemma. The following lemma shows that the property established in Lemma 8.6.4 for vr also holds for all w belonging to a small neighborhood of vr in M. Lemma 8.6.5 For each v ∈ F0 , each r ∈ (0, 1] and each ∈ (0, 1) there exist δ ∈ (0, ) and a neighborhood U of vr in M such that the following property holds: 2 ⊂X If w ∈ U , k1 , k2 ≥ k1 + 2 are integers and if a sequence {yi }ki=k 1 satisﬁes (6.16) d(yi , xvi ) ≤ δ, i = k1 , k2 , k 2 −1 i=k1

then

wi (yi , yi+1 ) ≤ σ(w, k1 , k2 , yk1 , yk2 ) + δ, d(yi , xvi ) ≤ ,

i = k1 , . . . , k2 .

(6.17)

Proof. Let v ∈ F0 , r ∈ (0, 1] and ∈ (0, 1). It follows from Lemma 8.6.4 that there exists (6.18) δ ∈ (0, 4−1 ) such that the following property holds: 2 ⊂ X satisﬁes If k1 , k2 ≥ k1 + 2 are integers and if a sequence {yi }ki=k 1 d(yi , xvi ) ≤ 4δ, k 2 −1 i=k1

i = k1 , k2 ,

(6.19)

vir (yi , yi+1 ) ≤ σ(vr , k1 , k2 , yk1 , yk2 ) + 4δ,

then the inequality (6.17) holds. It follows from Lemma 8.6.3 that there exist a neighborhood U1 of vr in M and a natural number N ≥ 4 such that the following property holds:

253

Discrete-time control systems

If w ∈ U1 , p is an integer and if a sequence {yi }p+N i=p ⊂ X satisﬁes p+N −1

wi (yi , yi+1 ) ≤ σ(w, p, p + N, yp , yp+N ) + 8,

(6.20)

i=p

then

inf{d(yi , xvi ) :

i = p, . . . , p + N } < δ.

U = {w ∈ U1 :

ρ(w, vr ) < δ(16N )−1 }.

Deﬁne

(6.21) (6.22) 2 {yi }ki=k 1

Assume that w ∈ U , k1 , k2 ≥ k1 + 2 are integers and ⊂ X satisﬁes (6.16). To prove the lemma it is suﬃcient to show that (6.17) holds. Assume the contrary. Then there is an integer

such that

i0 ∈ [k1 + 1, k2 − 1]

(6.23)

d(yi0 , xvi0 ) > .

(6.24)

It follows from (6.16) and the deﬁnition of U1 , N (see (6.20), (6.21)) that there exist integers i1 , i2 ∈ [k1 , k2 ] for which i0 − N ≤ i1 < i0 < i2 ≤ i0 + N,

d(yip , xvip ) ≤ δ, p = 1, 2.

(6.25)

In view of (6.22), (6.16) and (6.25), i 2 −1 i=i1

vir (yi , yi+1 ) ≤ σ(vr , i1 , i2 , yi1 , yi2 ) + 2δ.

It follows from this inequality, (6.25) and the choice of δ (see (6.18), (6.19)) that d(yi , xvi ) ≤ , i = i1 , . . . , i2 . This is contradictory to (6.24). The obtained contradiction proves the lemma. Lemma 8.6.3 and 8.6.5 imply the following result. Lemma 8.6.6 Let v ∈ F0 , r ∈ (0, 1] and ∈ (0, 1). Then there exist a neighborhood U of vr in M, a positive number δ < and a natural number N such that the following property holds: 2 ⊂X If w ∈ U , k1 , k2 ≥ k1 +2N are integers and if a sequence {yi }ki=k 1 satisﬁes k 2 −1 i=k1

wi (yi , yi+1 ) ≤ σ(w, k1 , k2 , yk1 , yk2 ) + δ,

254

TURNPIKE PROPERTIES

then

d(yi , xvi ) ≤ ,

i = L1 + k1 , . . . , k2 − L2

where integers L1 , L2 ∈ [0, N ], and moreover, if d(yk1 , xvk1 ) ≤ δ, then L1 = 0, and if d(yk2 , xvk2 ) ≤ δ, then L2 = 0. Completion of the proof of Theorem 8.4.1. For each v ∈ F0 , each r ∈ (0, 1] and each natural number i there exist an open neighborhood U (v, r, i) of vr in M, numbers δ(v, r, i) ∈ (0, (2i)−1 ),

δ0 (v, r, i) ∈ (0, 2−1 δ(v, r, i))

(6.26)

and natural numbers N (v, r, i), N0 (v, r, i) such that Lemma 8.6.6 holds with U = U (v, r, i), = (2i)−1 , N = N (v, r, i), δ = δ(v, r, i) and also holds with = 2−1 δ(v, r, i), δ = δ0 (v, r, i), N = N0 (v, r, i), U = U (v, r, i). Deﬁne F = F0 ∩ ∩∞ i=1 ∪ {U (v, r, i) : v ∈ F0 , r ∈ (0, 1]}.

(6.27)

Clearly F is a countable intersection of open everywhere dense sets in M. Let w ∈ F. We show that properties (a) and (b) hold. Let ∈ (0, 1). Choose a natural number i > 8−1 + 8.

(6.28)

It follows from (6.27) that there exists v ∈ F0 and r ∈ (0, 1] such that w ∈ U (v, r, i).

(6.29)

Assume that {yi }∞ i=−∞ ⊂ X is a (w)-minimal sequence. Then for each pair of integers k2 > k1 , k 2 −1 i=k1

wi (yi , yi+1 ) = σ(w, k1 , k2 , yk1 , yk2 ).

(6.30)

∞ By (6.30), the choice of {xw i }i=−∞ ⊂ X (see (6.1)), Lemma 8.6.6 and the choice of U (v, r, i), δ0 (v, r, i), δ(v, r, i) and N0 (v, r, i),

d(yj , xvj ),

v −1 d(xw j , xj ) ≤ 2 δ(v, r, i),

j = 0, ±1, . . . .

(6.31)

(6.31), (6.26) and (6.28) imply that d(yj , xw j ) ≤ ,

j = 0, ±1, ±2, . . . .

Since is any positive number from the interval (0, 1), we conclude that yj = xw j for any integer j. Therefore property (a) holds.

255

Discrete-time control systems

Assume that u ∈ U (v, r, i), k1 , k2 > k1 + 2N (v, r, i) are integers, {yi }kk21 ⊂ X and k 2 −1 i=k1

ui (yi , yi+1 ) ≤ σ(u, k1 , k2 , uk1 , uk2 ) + δ(v, r, i).

(6.32)

It follows from (6.32), the deﬁnition of U (v, r, i), N (v, r, i), δ(v, r, i) and Lemma 8.6.6 that d(yj , xvj ) ≤ (2i)−1 ,

j = k1 + L1 , . . . , k2 − L2 ,

where integers L1 , L2 ∈ [0, N (v, r, i)], and if d(yk1 , xvk1 ) ≤ δ(v, r, i), then L1 = 0, and if d(yk2 , xvk2 ) ≤ δ(v, r, i), then L2 = 0. Together with (6.31), (6.26) and (6.28) this implies that d(yj , xw j ) ≤ ,

j = k1 + L1 , . . . , k2 − L2 ,

w and if d(yk1 , xw k1 ) ≤ δ0 (v, r, i), then L1 = 0, and if d(yk2 , xk2 ) ≤ δ0 (v, r, i), then L2 = 0. Therefore property (b) holds. This completes the proof of the theorem.

Chapter 9 CONTROL PROBLEMS IN HILBERT SPACES

In this chapter we analyze the structure of optimal solutions for inﬁnite-dimensional optimal continuous-time control problems. We show that an optimal trajectory deﬁned on an interval [0, τ ] is contained in a small neighborhood of the optimal steady-state in the weak topology for all t ∈ [0, τ ] \ E, where E ⊂ [0, τ ] is a measurable set such that the Lebesgue measure of E does not exceed a constant which depends only on the neighborhood of the optimal steady-state and does not depend on τ . Moreover, we show that the set E is a ﬁnite union of intervals and their number does not exceed a constant which depends only on the neighborhood.

9.1.

Main results

We consider a system described by the following input-output relationship: T

x(t) = S(t)x0 +

0

S(t − s)Bu(s)ds, t ∈ I,

(1.1)

where I is either [0, ∞) or [0, T ] (0 ≤ T < ∞), E and F are separable Hilbert spaces, x0 ∈ E, {S(t) : t ≥ 0} is a strongly continuous semigroup on E with generator A, u(·) ∈ L2loc (I; F ), the space of all strongly measurable functions u(·) : I → F , which are square-integrable on every ﬁnite interval ∆ ⊂ I, and B : F → E is a bounded linear operator. Thus x(·) is the mild solution of the state equation x (t) = Ax(t) + Bu(t), t ∈ I,

(1.2)

258

TURNPIKE PROPERTIES

x(0) = x0 ,

(1.3)

where A is a possibly unbounded, closed, and densely deﬁned operator in E. In addition we know (see [7]) that although a mild solution need not be absolutely continuous it does satisfy the following mild diﬀerential equation for any y ∈ D(A∗ ): (d/dt) < x(t), y >=< x(t), A∗ y > + < Bu(t), y > a.e. t ∈ I, lim < x(t), y >=< x0 , y >,

t→0+

(1.4) (1.5)

where A∗ is the adjoint operator associated with A, with domain D(A∗ ). We assume that x(t) ∈ X, t ∈ I where X is a convex and closed subset of E

(1.6)

and u(t) ∈ U (x(t)) ⊂ F, t ∈ I where U (·) : X → 2F is a point to set mapping which is convex valued and satisﬁes αU (x1 ) + (1 − α)U (x2 ) ⊂ U (αx1 + (1 − α)x2 ), x1 , x2 ∈ X, α ∈ [0, 1], if un → u, xn → x as n → ∞ in the weak topology, un ∈ U (xn ), n = 1, 2, . . . , then u ∈ U (x).

(1.7)

The performance of the control system is measured on any ﬁnite interval [T1 , T2 ] by the integral functional I(T1 , T2 , x, u) =

T2 T1

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

(1.8)

where f : E × F → R1 is a convex function. We assume that f is lower semicontinuous on E × F and that there exist positive numbers K1 and K such that f (x, u) ≥ K(||x||2 + ||u||2 ) for each x ∈ E, u ∈ F satisfying ||x||2 + ||u||2 > K1 .

(1.9)

A function x : I → E where I is either [0, ∞) or [0, T ] (T > 0) is called a trajectory if there exists u(·) ∈ L2loc (I; F ) (referred to as a control) such that the pair (x, u) satisﬁes (1.1) and x(t) ∈ X, u(t) ∈ U (x(t)), t ∈ I.

(1.10)

259

Control problems in Hilbert spaces

: [0, ∞) → E, u : [0, ∞) → F is overtaking A trajectory-control pair x (resp. weakly overtaking) optimal if for any other trajectory-control pair (0), x : [0, ∞) → E, u : [0, ∞) → F satisfying x(0) = x , u ) − I(0, T, x, u)] ≤ 0 lim sup[I(0, t, x t→∞

, u ) − I(0, T, x, u)] ≤ 0). ( resp. lim inf [I(0, t, x t→∞

(1.11)

Assume the following. Assumption 1. The optimal steady state problem (OSSP) Min f (x, u) over all (x, u) ∈ E × F satisfying 0 =< x, A∗ y > + < Bu, y > for any y ∈ D(A∗ ) x ∈ X, u ∈ U (x), has a solution (¯ x, u ¯) with x ¯ uniquely deﬁned. It is not diﬃcult to see that the OSSP is a convex programming problem in a Hilbert space. Therefore there exists p¯ ∈ D(A∗ ) such that (see [30, 73]) (1.12) f (¯ x, u ¯) ≤ f (x, u)− < x, A∗ p¯ > − < Bu, p¯ > for every x ∈ X and u ∈ U (x). Deﬁne a function L : E × F → [0, ∞) by L(x, u) = f (x, u) − f (¯ x, u ¯)− < x, A∗ p¯ > − < Bu, p¯ > if x ∈ X and u ∈ U (x), L(x, u) = ∞ otherwise.

(1.13)

Then we have L(¯ x, u ¯) = 0. Since L diﬀers from f through an aﬃne function of x and u, it still satisﬁes the growth property (1.9) with f replaced by L. Let I be either [0, ∞) or [0, T ] (T > 0), x : I → E, u : I → F be a trajectory-control pair, and T1 , .T2 ∈ I, T1 < T2 . We deﬁne IL (T1 , T2 , x, u) =

T2 T1

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

(1.14)

For a trajectory-control pair x : [0, ∞) → E, u : [0, ∞) → F we deﬁne IL (0, ∞, x, u) =

∞ 0

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

For each T > 0 and each z ∈ E we deﬁne σ(z, T ) = inf{I(0, T, x, u) : x : [0, T ] → E, u : [0, T ] → F is a trajectory-control pair, x(0) = z}. In this chapter we prove the following results.

(1.15)

260

TURNPIKE PROPERTIES

Theorem 9.1.1 Suppose that Assumption 1 holds and x : [0, ∞) → E, u : [0, ∞) → F is a trajectory-control pair. Then either (i) sup{|I(0, T, x, u) − T f (¯ x, u ¯)| : T ∈ (0, ∞)} < ∞ or (ii) I(0, T, x, u) − T f (¯ x, u ¯) → ∞ as T → ∞. Moreover (i) holds if and only if IL (0, ∞, x, u) < ∞. The next theorem establishes that ||x(t)|| is bounded by some constant ∆ > 0 for all t ∈ [0, T ] if (x, u) : [0, T ] → E × F is an approximate optimal trajectory-control pair and T is large enough. Note that ∆ does not depend on T . Theorem 9.1.2 Suppose that Assumption 1 holds and r1 , r2 , r3 > 0. Then there exist positive numbers ∆, r such that if T > 0 and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes the following conditions: (a) ||x(0)|| ≤ r2 , I(0, T, x, u) ≤ σ(x(0), T ) + r3 ; (b) there is a trajectory-control pair y : [0, ∞) → E, v : [0, ∞) → F satisfying y(0) = x(0), IL (0, ∞, y, v) ≤ r1 , then ||x(t)|| ≤ ∆, t ∈ [0, T ], IL (0, T, x, u) ≤ r. The following result is an extension of Theorem 1 of [18] to optimal trajectories deﬁned on ﬁnite intervals. It shows that if (x, u) : [0, T ] → E × F is an optimal trajectory-control pair, then the average of x on any subinterval of [0, T ] with a suﬃciently large length belongs to a small neighborhood of x ¯ in the weak topology. Theorem 9.1.3 Suppose that Assumption 1 holds, r1 , r2 , r3 > 0, and V is a neighborhood of x ¯ in the weak topology. Then there exists a number l > 0 such that the following property holds: If T ≥ l and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes conditions (a) and (b) from Theorem 9.1.2, then (T2 − T1 )−1

T2 T1

x(t)dt ∈ V

for each T1 , T2 ∈ [0, T ] satisfying T2 − T1 ≥ l. Denote by F the set of all trajectory-control pairs x : [0, ∞) → E, u : [0, ∞) → F such that L(x(t), u(t)) = 0 a.e. on [0, ∞).

(1.16)

Control problems in Hilbert spaces

261

We say that F has property G if for any neighborhood V of x ¯ in the weak topology there exists a number tv > 0 such that x(t) ∈ V for each t ≥ tv and each trajectory-control pair (x, u) ∈ F. This property appears in [18] and corresponds to property (S) in [40]. We establish the following result which describes the structure of optimal solutions on ﬁnite intervals. Theorem 9.1.4 Suppose that Assumptiom 1 holds and F has property ¯ in G. Let r1 , r2 , r3 be positive numbers and let V be a neighborhood of x the weak topology. Then there exist a natural number Q and a positive number l such that the following property holds: If T > 0 and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes conditions (a) and (b) from Theorem 9.1.2, then there exists a sequence of intervals [bj , cj ], j = 1, . . . , q such that 1 ≤ q ≤ Q, 0 < cj − bj ≤ l, j = 1, . . . , q, and x(t) ∈ V for each t ∈ [0, T ] \ ∪qj=1 [bj , cj ]. The following result is a generalization of Theorem 4 in [18] which establishes the existence of an overtaking optimal solution in the subclass of bounded trajectories. Theorem 9.1.5 Suppose that Assumption 1 holds and F has property G. Let x ˜ : [0, ∞) → E, u ˜ : [0, ∞) → F be a trajectory-control pair ˜, u ˜) < ∞. Then there exists an overtaking optimal satisfying IL (0, ∞, x trajectory-control pair x∗ : [0, ∞) → E, u∗ : [0, ∞) → F such that ˜(0). x∗ (0) = x The results of this chapter were obtained in [105]. Chapter 9 is organized as follows. Section 9.2 contains four simple auxiliary results. Theorems 9.1.1-9.1.3 are proved in Section 9.3. Section 9.4 contains the proofs of Theorems 9.1.4 and 9.1.5. In Section 9.5 we discuss systems with distributed parameters and boundary controls.

9.2.

Preliminary results

Denote by N the set of all natural numbers. For each positive number r set BE (r) = {x ∈ E : ||x|| ≤ r},

262

TURNPIKE PROPERTIES

BF (r) = {x ∈ F : ||x|| ≤ r}. In the sequel we use the following lemmas which can be established in a straightforward manner. Lemma 9.2.1 The set {||S(t)|| : t ∈ [0, 1]} is bounded. Lemma 9.2.2 Let −∞ < T1 < T2 < ∞, u ∈ L1 ([T1 , T2 ]; F ). Then ||

T2 T1

u(t)dt|| ≤

T2 T1

||u(t)dt.

Lemma 9.2.3 Let −∞ < T1 < T2 < ∞, D ⊂ E × F be a closed convex subset, x ∈ L1 ([T1 , T2 ]; E), y ∈ L1 ([T1 , T2 ]; F ), and (x(t), y(t)) ∈ D, t ∈ [T1 , T2 ]. Then ((T2 − T1 )−1

T2 T1

x(t)dt, (T2 − T1 )−1

T2 T1

y(t)dy) ∈ D.

Lemma 9.2.4 Suppose that Assumption 1 holds. Then for any T > 0 the function φ : (x(·), u(·)) →

T 0

L(x(t), u(t))dt

is convex and weakly lower semicontinuous on L2 ([0, T ]; E) × L2 ([0, T ]; F ).

9.3.

Proof of Theorems 9.1.1-9.1.3

In this section we assume that Assumption 1 holds. Lemma 9.3.1 There exist positive numbers K2 , K3 such that f (x, u), L(x, u) ≥ K2 (||x||2 + ||u||2 ) − K3 for all (x, u) ∈ E × F. (3.1) Lemma 9.3.1 follows from equations (1.9), (1.12), (1.13), convexity and lower semicontinuity of f . (1.13) and (1.2) imply the following auxiliary result. Lemma 9.3.2 For each T1 ≥ 0, T2 > T1 , T ≥ T2 and each trajectorycontrol pair x : [0, T ] → E, u : [0, T ] → F , I(T1 , T2 , x, u) = (T2 −T1 )f (¯ x, u ¯)+IL (T1 , T2 , x, u)+ < p¯, x(T2 )−x(T1 ) > .

263

Control problems in Hilbert spaces

The following auxiliary result shows that for any approximate optimal with respect to the functional IL (0, T, ·, ·) trajectory-control pair (x, u) : [0, T ] → E × F with large enough T the function ||x(t)|| is bounded by some constant which does not depend on T . Lemma 9.3.3 For each pair of positive numbers r, l there exists a positive number N such that the following property holds: If a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes T ≥ l, x(0) ∈ BE (r), IL (0, T, x, u) ≤ r,

(3.2)

then x(t) ∈ BE (N ) for all t ∈ [0, T ]. Proof. Let r, l > 0. Choose numbers δ ∈ (0, 8−1 min{1, l}), h > 1 + r,

(3.3)

δ(K2 h − K3 ) ≥ 20r

(3.4)

such that (see (3.1)) and choose a number N > (1 + sup{||S(t)|| : t ∈ [0, 1]})(h + 1)(h + 1 + ||B||)(1 + K2−1 (r + K3 )). (3.5) Assume that T ≥ l and x : [0, T ] → E, u : [0, T ] → F is a trajectorycontrol pair which satisﬁes (3.2). We show that x(t) ∈ BE (N ) for all t ∈ [0, T ]. Assume that there exists t0 ∈ [0, T ] for which ||x(t0 || > N.

(3.6)

It follows from (3.1)-(3.6) that there is a number t1 such that t1 ∈ [max{0, t0 − δ}, t0 ], x(t1 ) ∈ BE (h).

(3.7)

By (3.3) and (3.7),

||x(t0 )|| = S(t0 )x(0) +

t0 0

S(t0 − s)Bu(s)ds

t1 ≤ S(t0 − t1 ) S(t1 )x(0) + S(t1 − s)Bu(s)ds 0 t 0 + S(t0 − s)Bu(s)ds t1 t0

≤ sup{||S(τ )|| : τ ∈ [0, 1]} h + ||B|| 1 +

t1

||u(s)||2 ds

.

264

TURNPIKE PROPERTIES

Combined with (3.6), (3.1) and (3.2) this implies that N < sup{||S(τ )|| : τ ∈ [0, 1]}(h + ||B||(1 + (r + K3 )K2−1 )). This is contradictory to (3.5). This completes the proof of the lemma. Proof of Theorem 9.1.1. Assume that x : [0, ∞) → E, u : [0, ∞) → F is a trajectory-control pair such that IL (0, ∞, x, u) < ∞. Lemma 9.3.3 implies that the function x(t) is bounded. Combined with Lemma 9.3.2 this implies relation (i). Assume that IL (0, ∞, x, u) = ∞. We show that relation (ii) holds. There exists ∆ > ||x(0)|| such that f (w, p) ≥ 8 + |f (¯ x, u ¯)| for each w ∈ BE (∆) and each p ∈ F.

(3.8)

τ (T ) = sup{t ∈ [0, T ] : x(t) ∈ BE (∆)}.

(3.9)

For any T > 0 put

We may assume without loss of generality that τ (T ) → ∞ as T → ∞. It follows from Lemma 9.3.2, (3.8), and (3.9) that for any T > 0, I(0, T, x, u) − T f (¯ x, u ¯) ≥ I(0, τ (T ), x, u) − τ (T )f (¯ x, u ¯) ≥ IL (0, τ (T ), x, u)+ < p¯, x(τ (T )) − x(0) > p||∆ → ∞ as T → ∞. ≥ IL (0, τ (T ), x, u) − 2||¯ This completes the proof of the theorem. In order to prove Theorem 9.1.2 we need the following lemma which is its weakened version. Instead of the inequality ||x(t)|| ≤ ∆ for all t ∈ [0, T ] in Theorem 9.1.2 the lemma guarantees that ||x(t)|| ≤ ∆ for some t ∈ [0, T ]. Lemma 9.3.4 Let r1 , r2 , r3 be positive numbers and let ∆ > r2 and f (w, p) ≥ 4 + |f (¯ x, u ¯)| for each w ∈ E satisfying ||w|| ≥ ∆ and each p ∈ F.

(3.10)

Then there exists T∆ > 1 such that the following property holds: If T ≥ T∆ and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes conditions (a) and (b) of Theorem 9.1.2, then inf{||x(t)|| : t ∈ [T − T∆ , T ]} ≤ ∆.

(3.11)

Control problems in Hilbert spaces

265

Proof. It follows from Lemma 9.3.3 that there exists a positive number N1 such that the following property holds: If T ≥ 1 and if a trajectory-control pair y : [0, T ] → E, v : [0, T ] → F satisﬁes (3.12) y(0) ∈ BE (r1 + r2 ), IL (0, T, y, v) ≤ r1 + r2 , then y(t) ∈ BE (N1 ), t ∈ [0, T ].

(3.13)

p||∆ + 2||¯ p||(N1 + r2 ). T∆ > r3 + r1 + 1 + 2||¯

(3.14)

Choose a number

Assume that T ≥ T∆ and a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes conditions (a) and (b) of Theorem 9.1.2. There exists a trajectory-control pair y : [0, ∞) → E, v : [0, ∞) → F such that y(0) = x(0), IL (0, ∞, y, v) ≤ r1 .

(3.15)

It follows from the choice of N1 , condition (a) of Theorem 9.1.2, and (3.15) that (3.16) y(t) ∈ BE (N1 ), t ∈ [0, ∞). (3.15), (3.16) and Lemma 9.3.2 imply that p||N1 . I(0, T, y, v) ≤ T f (¯ x, u ¯) + r1 + 2||¯

(3.17)

τ = sup{t ∈ [0, T ] : x(t) ∈ BE (∆)}.

(3.18)

Put In view of (3.18), (3.10), and Lemma 9.3.2, I(0, T, x, u) ≥ (T − τ )[4 + |f (˜ x, u ˜)|] + τ f (¯ x, u ¯) − 2||¯ p||∆.

(3.19)

Since (x, u) satisﬁes condition (a) of Theorem 9.1.2 it follows from (3.15) and (3.17) that p||∆ − r1 − 2||¯ p||N1 . r3 ≥ I(0, T, x, u) − I(0, T, y, v) ≥ 4(T − τ ) − 2||¯ It follows from this inequality and (3.14) that T − τ < T∆ . This completes the proof of the lemma. The next lemma plays a crucial role in the proof of Theorem 9.1.2. Its proof is based on Lemmas 9.3.2-9.3.4. This lemma can be considered as a weakened version of Theorem 9.1.2. Instead of the inequalities of Theorem 9.1.2 the lemma establishes analogous inequalities but with larger bounds.

266

TURNPIKE PROPERTIES

Lemma 9.3.5 Assume that r1 , r2 , r3 > 0, a positive number ∆ satisﬁes (3.10) and a number T∆ > 1 is as guaranteed in Lemma 9.3.4. Then there exist Q > ∆, h > 0 such that the following property holds: If T ≥ T∆ + 1 and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes conditions (a) and (b) of Theorem 9.1.2, then x(t) ∈ BE (Q) for all t ∈ [0, T ] and IL (0, T, x, u) ≤ h.

(3.20)

Proof. It follows from Lemma 9.3.3 that there exists ∆0 > ∆ such that the following property holds: If T ≥ 1 and if a trajectory-control pair y : [0, T ] → E, v : [0, T ] → F satisﬁes y(0) ∈ BE (r2 + 1), IL (0, T, y, v) ≤ r1 + r3 + 2 + 2||¯ p||(∆ + r2 ), (3.21) then y(t) ∈ BE (∆0 ) for all t ∈ [0, T ]. It follows from Lemma 9.3.3 that there exists ∆1 < ∆0 such that the following property holds: If T ≥ 1 and if a trajectory-control pair y : [0, T ] → E, v : [0, T ] → F satisﬁes y(0) ∈ BE (r2 +1), IL (0, T, y, v) ≤ r1 +r3 +4+2||¯ p||(2∆+∆0 +r2 ), (3.22) then y(t) ∈ BE (∆1 ) for all t ∈ [0, T ]. Choose numbers Q > ∆1 + 2 + K2−1 (||B|| + 1)(sup{||S(t)|| : t ∈ [0, T∆ }) ×[2 + ||¯ p||(2∆ + ∆0 + r2 ) + (K2 + K3 )T∆ + K2 ∆1 + r3 + r1 +T∆ |f (¯ x, u ¯)], h > 1 + r3 + r1 + ||¯ p||(∆0 + r2 + 2Q).

(3.23)

Assume that T ≥ T∆ + 1 and that x : [0, T ] → E, u : [0, T ] → F is a trajectory-control pair which satisﬁes conditions (a) and (b) of Theorem 9.1.2. Then there exists a trajectory-control pair y : [0, ∞) → E, v : [0, ∞) → F such that y(0) = x(0), IL (0, ∞, y, v) ≤ r1 .

(3.24)

It follows from (3.24) and the choice of ∆0 that y(t) ∈ BE (∆0 ), t ∈ [0, ∞).

(3.25)

(3.24), condition (a) of Theorem 9.1.2, (3.25), and Lemma 9.3.2 imply that I(0, T, y, v) ≤ T f (¯ x, u ¯) + r1 + ||¯ p||(∆0 + r2 ). (3.26)

267

Control problems in Hilbert spaces

Since the trajectory-control pair (x, u) satisﬁes condition (a) of Theorem 9.1.2 it follows from (3.26) and (3.24) that x, u ¯)+r1 +||¯ p||(∆0 +r2 ). (3.27) I(0, T, x, u) ≤ I(0, T, y, v)+r3 ≤ r3 +T f (¯ In view of Lemma 9.3.4 and the choice of T∆ there exists a number t1 such that t1 ∈ [T − T∆ , T ], x(t1 ) ∈ BE (∆), ||x(t)|| ≥ ∆, t ∈ [t1 , T ].

(3.28)

By (3.27), (3.28), and (3.10), I(0, t1 , x, u) = I(0, T, x, u) − I(t1 , T, x, u) p||(∆0 + r2 ) + t1 f (¯ x, u ¯). ≤ r3 + r1 + ||¯

(3.29)

It follows from Lemma 9.3.2, (3.28) and (3.10) that I(0, t1 , x, u) ≥ t1 f (¯ x, u ¯) + JL (0, t1 , x, u) − 2||¯ p||∆.

(3.30)

(3.29) and (3.30) imply that p||(2∆ + ∆0 + r2 ). IL (0, t1 , x, u) ≤ r3 + r1 + ||¯

(3.31)

By the choice of ∆1 , (3.31) and (3.28), x(t) ∈ BE (∆1 ) for all t ∈ [0, t1 ].

(3.32)

We show that x(t) ∈ BE (Q) for all t ∈ [t1 , T ]. Assume the contrary. Then there exists a number t2 such that t2 ∈ (t1 , T ] and ||x(t2 )|| > Q.

(3.33)

(3.33) and (3.28) imply that t2 − t1 ∈ (0, T∆ ], x(t2 ) = S(t2 )x(0) +

= S(t2 − t1 ) S(t1 )x(0) +

t1 0

t2 0

(3.34)

S(t2 − s)Bu(s)ds

S(t1 − s)Bu(s)ds +

= S(t2 − t1 )x(t1 ) +

t2 t1

t2 t1

S(t2 − s)Bu(s)ds

S(t2 − s)Bu(s)ds.

(3.35)

By (3.1), t2 t1

||u(s)||ds ≤

t2 t1

[1 + K2−1 (f (x(s), u(s)) + K3 )]ds.

(3.36)

268

TURNPIKE PROPERTIES

It follows from (3.34), (3.33), (3.32), and (3.35) that sup{||S(t)|| : t ∈ [0, T∆ ]}||B||

t2 t1

||u(s)||ds

≥ Q − ∆1 sup{||S(t)|| : t ∈ [0, T∆ ]}. Combined with (3.36) and (3.34) this inequality implies that t2 t1

f (x(s), u(s))ds ≥

t2 t1

||u(s)||ds − (1 + K2−1 K3 )T∆ K2

≥ −(K2 + K3 )T∆ + K2 Q(||B|| + 1)−1 ×(sup{||S(t)|| : t ∈ [0, T∆ ]})−1 − K2 ∆1 (||B|| + 1)−1 .

(3.37)

By (3.33), (3.28), (3.10), (3.30) and (3.37), I(0, T, x, u) ≥ I(0, t1 , x, u) + I(t1 , t2 , x, u) ≥ t1 f (¯ x, u ¯) − 2||¯ p||∆ − (K2 + K3 )T∆ − K2 ∆1 +K2 Q(||B|| + 1)−1 (sup{||S(t)|| : t ∈ [0, T∆ ]})−1 . It follows from this inequality, (3.27) and (3.28) that K2 Q(||B|| + 1)−1 (sup{||S(t)|| : t ∈ [0, T∆ ]})−1 ≤ 2||¯ p||∆ + (K2 + K3 )T∆ + K2 ∆1 + r3 + T∆ |f (¯ x, u ¯)| + r1 + ||¯ p||(∆0 + r2 ). This is contradictory to (3.23). The obtained contradiction proves that x(t) ∈ BE (Q) for all t ∈ [t1 , T ]. Combined with (3.32) and (3.23) this implies that x(t) ∈ BE (Q) for all t ∈ [0, T ]. It follows from this inequality, Lemma 9.3.2, (3.27) and (3.23) that IL (0, T, x, u) + T f (¯ x, u ¯) − 2||¯ p||Q ≤ I(0, T, x, u) ≤ r3 + T f (¯ x, u ¯) + r1 + ||¯ p||(∆0 + r2 ), IL (0, T, x, u) ≤ h. This completes the proof of the lemma. Proof of Theorem 9.1.2. We may assume that x ¯ ∈ BE (r2 ). It follows from Lemma 9.3.5 that there exist positive numbers T0 , Q, h such that the following property holds:

269

Control problems in Hilbert spaces

If T ≥ T0 and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes conditions (a) and (b) of Theorem 9.1.2, then x(t) ∈ BE (Q) for all t ∈ [0, T ], IL (0, T, x, u) ≤ h.

(3.38)

Lemma 9.3.3 implies that there exists a positive number N0 such that the following property holds: If T ≥ 1 and if a trajectory-control pair y : [0, T ] → E, v : [0, T ] → F satisﬁes (3.39) y(0) ∈ BE (r2 ), IL (0, T, y, v) ≤ r1 , then y(t) ∈ BE (N0 ) for all t ∈ [0, T ]. Choose numbers ∆ > Q + 1 + 2r2 + [(r3 + r1 + 2||¯ p||N0 + T0 (|f (¯ x, u ¯)| + K3 + K2 ))K2−1 +r2 (1 + ||B||)−1 ](1 + ||B||) sup{||S(t)|| : t ∈ [0, T0 ]},

(3.40)

p||(∆ + N0 ) + 2T0 |f (¯ x, u ¯)|. r > h + 1 + r3 + r1 + 2||¯ Assume that T > 0 and x : [0, T ] → E, u : [0, T ] → F is a trajectorycontrol pair which satisﬁes conditions (a) and (b) of Theorem 9.1.2. In view of (3.40) and the choice of T0 , Q, h we may assume that T < T0 .

(3.41)

Assume that there exists t0 ∈ [0, T ] such that ||x(t0 )|| > ∆. By this inequality, (3.41), and condition (a) of Theorem 9.1.2, t0 ∆ − r2 < ||x(t0 ) − x(0)|| = S(t0 )x(0) + S(t0 − s)Bu(s)ds − x(0) 0

≤ r2 (1 + sup{||S(t)|| : t ∈ [0, T0 ]}) +||B|| sup{||S(t)|| : t ∈ [0, T0 ]}

t 0

||u(s)||ds.

It follows from this inequality, (3.1) and (3.41) that T 0

f (x(t), u(t))dt ≥ −K3 T0 + K2 −K3 T0 − K2 T0 + K2

t0 0

t0 0

||u(t)||2 dt

||u(t)||dt

≥ −T0 (K3 + K2 ) + K2 [(∆ − 2r2 )(1 + ||B||)−1

270

TURNPIKE PROPERTIES

×(sup{||S(t)|| : t ∈ [0, T0 ]})−1 − r2 (1 + ||B||)−1 ].

(3.42)

Since the trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes condition (b) of Theorem 9.1.2 there exists a trajectory-control pair y : [0, ∞) → E, v : [0, ∞) → F such that y(0) = x(0), IL (0, ∞, y, v) ≤ r1 .

(3.43)

By the choice of N0 (see (3.39)) and (3.43), y(t) ∈ BE (N0 ), t ∈ [0, ∞). It follows from this inclusion, (3.43), (3.41) and Lemma 9.3.2 that I(0, T, y, v) ≤ T0 |f (¯ x, u ¯)| + r1 + 2||¯ p||N0 . Combined with (3.43) and condition (a) of Theorem 9.1.2 this inequality implies that x, u ¯)| + r3 + r1 + 2||¯ p||N0 . (3.44) I(0, T, x, u) ≤ I(0, T, y, v) + r3 ≤ T0 |f (¯ (3.44) and (3.42) imply that ∆ ≤ [(r3 + r1 + 2||¯ p||N0 + T0 (|f (¯ x, u ¯)| + K2 + K3 ))K2−1 + r2 (1 + ||B||)−1 ] ×(1 + ||B||) sup{||S(t)|| : t ∈ [0, T0 ]} + 2r2 . This is contradictory to (3.40). The obtained contradiction proves that x(t) ∈ BE (∆) for all t ∈ [0, T ]. By this inequality, Lemma 9.3.2, (3.41), (3.44) and (3.40), IL (0, T, x, u) ≤ I(0, T, x, u) + T0 |f (¯ x, u ¯)| + 2||¯ p||∆ ≤ 2T0 |f (¯ x, u ¯)| + r1 + r3 + 2||¯ p||(∆ + N0 ) ≤ r. This completes the proof of the theorem. We preface the proof of Theorem 9.1.3 with the following auxiliary result. Lemma 9.3.6 For each pair of positive numbers r1 , r2 and each neighborhood V of x ¯ in the weak topology there exists a number l > 0 such that the following property holds: If T > l and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes (3.45) x(0) ∈ BE (r2 ), IL (0, T, x, u) ≤ r1 ,

271

Control problems in Hilbert spaces

then T

−1

T

x(t)dt ∈ V.

0

(3.46)

¯ in the weak Proof. Let r1 , r2 > 0 and let V be a neighborhood of x topology. Assume that the lemma is not true. Then there exists a sequence of trajectory-control pairs xj : [0, Tj ] → E, uj : [0, Tj ] → F , j ∈ N such that Tj < Tj+1 , j ∈ N, Tj → ∞ as j → ∞, xj (0) ≤ BE (r2 ), IL (0, Tj , xj , uj ) ≤ r1 ,

Tj

Tj−1

xj (t)dt ∈ V, j ∈ N.

0

(3.47)

Lemma 9.3.3 and (3.47) imply that there exists a positive number ∆0 such that (3.48) xj (t) ∈ BE (∆0 ) for all t ∈ [0, Tj ], j ∈ N. Assume that j ∈ N and y ∈ D(A∗ ). Integrating the state equation in the mild form (1.4) with x = xj , u = uj , we have

Tj−1 < xj (Tj ) − xj (0), y >=

+ Tj−1 B

0

Tj

Tj−1

Tj 0

xj (t)dt, A∗ y

ui (t)dt , y .

(3.49)

By (3.48), (3.47) and (3.1) the set

Ω=

Tj−1

Tj 0

xj (t)dt, Tj−1

Tj 0

uj (t)dt

: j∈N

is bounded. In view of Lemma 9.2.3, (1.6) and (1.7), ¯ ⊂ {(x, u) : x ∈ X, u ∈ U (x)}, Ω

(3.50)

¯ is the closure of Ω in the weak topology. where Ω Let (x∗ , u∗ ) be a cluster point of the set Ω. By (3.48), (3.49), and (3.50), (x∗ , u∗ ) is an admissible point for the OSSP. It follows from Lemma 9.3.2, (3.47) and (3.48) that for j ∈ N, x, u ¯) + r1 + 2||¯ p||∆0 . I(0, Tj , xj , uj ) ≤ Tj f (¯ Combined with the lower semicontinuity of f and and Jensen’s inequality this inequality implies that x, u ¯). f (x∗ , u∗ ) ≤ f (¯

272

TURNPIKE PROPERTIES

Since (x∗ , u∗ ) is an admissible point for the OSSP it follows from As¯. This is contradictory to (3.47), The contradicsumption 1 that x∗ = x tion we have reached proves the lemma. Proof of Theorem 9.1.3. It follows from Theorem 9.1.2 that there exist positive numbers ∆1 , ∆2 such that the following property holds: If T > 0 and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes conditions (a) and (b) of Theorem 9.1.2, then x(t) ∈ BE (∆1 ) for all t ∈ [0, T ] and IL (0, T, x, u) ≤ ∆2 .

(3.51)

It follows from Lemma 9.3.6 that there exists a positive number l such that the following property holds: If T ≥ l and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes x(0) ∈ BE (∆2 + ∆1 + r1 + r2 ), IL (0, T, x, u) ≤ ∆2 + ∆1 + r1 + r2 , (3.52) then T −1

T 0

x(t)dt ∈ V.

Let T ≥ l, T1 , T2 ∈ [0, T ], T2 − T1 ≥ l and let x : [0, T ] → E, u : [0, T ] → F be a trajectory-control pair which satisﬁes conditions (a) and (b) of Theorem 9.1.2. Therefore (3.51) holds. Deﬁne ˜(t) = u(t + T1 ), t ∈ [0, T2 − T1 ]. x ˜(t) = x(t + T1 ), u For t ∈ [0, T2 − T1 ] we have

t+T

x ˜(t) = x(t + T1 ) = S(t + T1 )x(0) +

= S(t) S(T1 )x(0) + +

t+T1 T1

T1 0

0

(3.53)

S(t + T1 − s)Bu(s)ds

S(T1 − s)Bu(s)ds

S(t + T1 − s)Bu(s)ds

= S(t)˜ x(0) +

t 0

S(t − s)B u ˜(s)ds.

By this relation and (3.53), x ˜ : [0, T2 − T1 ] → E, u ˜ : [0, T2 − T1 ] → F is a trajectory-control pair. It follows from (3.51), (3.53), (3.52) and the deﬁnition of l that (T2 − T1 )−1

T2 T1

x(t)dt = (T2 − T1 )−1

This completes the proof of the theorem.

T2 −T1 0

x ˜(t)dt ∈ V.

273

Control problems in Hilbert spaces

9.4.

Proof of Theorems 9.1.4 and 9.1.5

In this section we assume that Assumption 1 holds and the set F has property G. We begin with the following auxiliary result which shows that if (x, u) : [0, T ] → E × F is an approximate optimal trajectory-control pair with respect to the functional IL (0, T, ·, ·) and if T is large enough, then x(t) belongs to a small neighborhood of x ¯ in the weak topology for most t ∈ [0, T ]. Proposition 9.4.1 For each neighborhood V of x ¯ in the weak topolgy there exist positive numbers δ, l such that the following property holds: If T ≥ 2l and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes IL (0, T, x, u) ≤ δ, then x(t) ∈ V, t ∈ [l, T − l]. Proof. Let V be a neighborhood of x ¯ in the weak topology. Assume that the proposition is not true. Then there exist a sequence of numbers {tj }∞ j=1 and a sequence of trajectory-control pairs xj : [0, Tj ] → E, uj : [0, Tj ] → F , j ∈ N such that Tj ≥ 8j, IL (0, Tj , xj , uj ) ≤ j −1 , tj ∈ [4j, Tj − 4j], xj (tj ) ∈ V, j ∈ N.

(4.1)

Lemma 9.3.1 implies that there exists a positive number r0 such that L(x, u) ≥ 8 for each (x, u) ∈ E × F satisfying ||x|| ≥ r0 .

(4.2)

In view of Lemma 9.3.3 there exists a positive number r1 such that the following property holds: If T ≥ 1 and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes (4.3) x(0) ∈ BE (r0 ), IL (0, T, x, u) ≤ 1, then x(t) ∈ BE (r1 ), t ∈ [0, T ].

(4.4)

There exists a neighborhood V0 of 0 in the weak topology such that x ¯ + 3V0 ⊂ V.

(4.5)

It follows from property G that there exists τv > 0 such that for each trajectory-control pair (x, u) ∈ F, x(t) ∈ V + x ¯ for each t ≥ τv .

(4.6)

274

TURNPIKE PROPERTIES

(4.1) and (4.2) imply that for any j ∈ N there exists a number hj which satisﬁes hj ∈ [0, 1], x(hj ) ∈ BE (r0 ). (4.7) For j ∈ N we deﬁne ˜j (t) = uj (t + hj ), t ∈ [0, Tj − hj ]. x ˜j (t) = xj (t + hj ), u

(4.8)

It is easy to verify that x ˜j : [0, Tj − hj ] → E, u ˜j : [0, Tj − hj ] → F is a trajectory-control pair for any j ∈ N. Therefore by (4.1), (4.7), (4.8) and the deﬁnition of r1 (see (4.3), (4.4)), xj (t) ∈ BE (r1 ), t ∈ [1, Tj ], j ∈ N.

(4.9)

For any integer j ≥ τv we deﬁne x ˜j (t) = xj (t+tj −τv ), u ˜j (t) = uj (t+tj −τv ), t ∈ [0, Tj −tj +τv ]. (4.10) It is easy to verify that x ˜j : [0, Tj − tj + τv ] → E, u ˜j : [0, Tj − tj + τv ] → F is a trajectory-control pair for any j ≥ τv . By (4.1) and (4.10), x ˜j (τv ) ∈ V for any integer j ≥ τv .

(4.11)

By (4.10), (4.9) and (4.1), x ˜j (t) ∈ BE (r1 ) for each t ∈ [0, Tj − tj + τv ] and each integer j ≥ τv , (4.12) −1 ˜j , u ˜j ) ≤ j for each integer j ≥ τv . (4.13) IL (0, Tj − tj + τv , x In view of (4.13) and (3.1) for any positive T the sequence {˜ uj (·)} is 2 bounded in L ([0, T ]; F ). Thus, extracting a subsequence if necessary, we may suppose that there exist x∗0 ∈ E and u∗ ∈ L2loc ([0, ∞); F ) such that (4.14) x ˜j (0) → x∗0 as j → ∞ weakly in E, u ˜j → u∗ as j → ∞ weakly in L2 ([0, T ]; F ) for any T > 0.

(4.15)

(4.12), (4.14), and (4.15) imply that there is a function x∗ : [0, ∞) → E such that (x∗ , u∗ ) is a trajectory-control pair deﬁned on [0, ∞), x∗ (0) = x∗0 , x ˜j (t) → x∗ (t) as j → ∞ weakly in E for any t ∈ [0, ∞), x ˜j → x∗ as j → ∞ weakly in L2 ([0, T ]; E) for any T > 0.

(4.16)

Control problems in Hilbert spaces

275

By (4.16), (4.15), (4.13) and Lemma 9.2.4, IL (0, T, x∗ , u∗ ) = 0 for any T > 0. Since the function L is nonnegative this equality implies that L(x∗ (t), u∗ (t)) = 0 a.e. on [0, ∞). Hence (x∗ , u∗ ) ∈ F, and by the deﬁnition of τv (see (4.6)), ¯. x∗ (τv ) ∈ V0 + x It follows from this relation, (4.10), (4.16) and (4.5) that for j suﬃciently large we have ˜j (τv ) ∈ x ¯ + 2V0 ⊂ V, xj (tj ) = x which contradicts (4.1). The obtained contradiction proves the proposition. Corollary 9.4.1 Assume that x : [0, ∞) → E, u : [0, ∞) → F is a trajectory-control pair such that IL (0, ∞, x, u) < ∞. Then x(t) converges weakly to x ¯ as t → ∞. Proof of Theorem 9.1.4. It follows from Theorem 9.1.2 that there exist positive numbers ∆, r such that the following properties hold: If T > 0 and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes conditions (a) and (b) (see the statement of Theorem 9.1.2), then x(t) ∈ BE (∆) for all t ∈ [0, T ] and IL (0, T, x, u) ≤ r.

(4.17)

Proposition 9.4.1 implies that there exist positive numbers δ, l0 such that the following property holds: If T ≥ 2l0 and if a trajectory-control pair x : [0, T ] → E, u : [0, T ] → F satisﬁes IL (0, T, x, u) ≤ δ, then x(t) ∈ V, t ∈ [l0 , T − l0 ].

(4.18)

Choose Q ∈ N and a number l > 0 such that Q > 4rδ −1 + 8, l > 1 + 2l0 .

(4.19)

Assume that T > 0 and x : [0.T ] → E, u : [0, T ] → F is a trajectorycontrol pair which satisﬁes conditions (a) and (b) of Theorem 9.1.2. Then (4.17) follows from the deﬁnition of ∆, r. There exists a sequence of numbers {tj }N j=0 such that t0 = 0, tj < tj+1 , j = 0, . . . , N − 1, tN = T,

276

TURNPIKE PROPERTIES

IL (tj , tj+1 , x, u) = δ if 0 ≤ j < N − 1, IL (tN −1 , T, x, u) ≤ δ.

(4.20)

(4.17) and (4.20) imply that N ≤ rδ −1 + 1.

(4.21)

A = {j ∈ {0, . . . , N − 1} : tj+1 − tj ≥ l}.

(4.22)

Put Let j ∈ A and deﬁne xj (t) = x(t + j), uj (t) = u(t + tj ), t ∈ [0, tj+1 − tj ].

(4.23)

Clearly xj : [0, tj+1 − tj ] → E, uj : [0, tj+1 − tj ] → F is a trajectorycontrol pair and IL (0, tj+1 − tj , xj , uj ) ≤ δ. In view of this inequality, (4.22), (4.19) and the choice of δ, l0 (see (4.18)), xj (t) ∈ V, t ∈ [l0 , tj+1 − tj − l0 ]. Combined with (4.23) this relation implies that x(t) ∈ V, t ∈ [tj + l0 , tj+1 − l0 ], j ∈ A. This completes the proof of the theorem. We have the following result (see Lemma 2 of [18]). Proposition 9.4.2 Assume that z ∈ E and there exists a trajectorycontrol pair x ˜ : [0, ∞) → E, u ˜ : [0, ∞) → F satisfying IL (0, ∞, x ˜, u ˜) < ∞, x ˜(0) = z. Then there exists a trajectory-control pair x∗ : [0, ∞) → E, u∗ : [0, ∞) → F such that x∗ (0) = z, IL (0, ∞, x∗ , u∗ ) ≤ IL (0, ∞, x, u) for each trajectory-control pair x : [0, ∞) → E, u : [0, ∞) → F satisfying x(0) = z. Proof of Theorem 9.1.5. It follows from Proposition 9.4.2 that there exists a trajectory-control pair x∗ : [0, ∞) → E, u∗ : [0, ∞) → F such that ˜(0) x∗ (0) = x and

IL (0, ∞, x∗ , u∗ ) ≤ IL (0, ∞, x, u)

(4.24)

for each trajectory-control pair x : [0, ∞) → E, u : [0, ∞) → F satisfying x(0) = x ˜(0). Evidently IL (0, ∞, x∗ , u∗ ) < ∞.

(4.25)

277

Control problems in Hilbert spaces

Let x : [0, ∞) → E, u : [0, ∞) → F be a trajectory-control pair such that x(0) = x∗ (0). We show that lim sup[I(0, T, x∗ , u∗ ) − I(0, T, x, u)] ≤ 0. T →∞

In view of (4.25) and Theorem 9.1.1 we may assume that IL (0, ∞, x, u) < ∞.

(4.26)

(4.25), (4.26) and Corollary 9.4.1 imply that ¯, x(t) → x ¯ as t → ∞ in the weak topology. x∗ (t) → x It follows from this relation, (4.24) and Lemma 9.3.2 that lim sup[I(0, T, x∗ , u∗ ) − I(0, T, x, u)] T →∞

= lim sup[IL (0, T, x∗ , u∗ ) − IL (0, T, x, u)+ < p¯, x∗ (t) − x(T ) >] ≤ 0. T →∞

This completes the proof of the theorem.

9.5.

Systems with distributed and boundary controls

We extend Theorems 9.1.1-9.1.5 to the cases of inﬁnite-dimensional control problems with distributed and boundary controls considered in Section 4 of [18]. We consider the following control system:

with initial condition

x (t) = σx(t) + B1 u(t),

(5.1)

γx(t) = B2 u2 (t),

(5.2)

x(0) = x0 ,

(5.3)

and the following additional constraints: x(t) ∈ X ⊂ E1 , uj (t) ∈ Uj (x(t)) ⊂ Fj , j = 1, 2, t ∈ I,

(5.4)

where I is either [0, ∞) or [0, T ] (T > 0), E1 , E2 , F1 and F2 are separable Hilbert spaces, x0 ∈ E1 , σ is a closed linear and densely deﬁned operator on E1 , γ is a linear operator (the boundary operator) with domain in

278

TURNPIKE PROPERTIES

E1 and range in E2 , and Bi : Fi → Ei , i = 1, 2, are linear continuous operators. The sets X and Uj (x), j = 1, 2, x ∈ X satisfy the assumptions (1.6) and (1.7). A solution to the system (5.1)-(5.3) satisﬁes the following input-output relation: t

x(t) = S(t)x0 + −A

0

t 0

S(t − s)(B1 u1 (s) + σBu2 (s))ds S(t − s)Bu2 (s)ds,

(5.5)

where the operators A, B and S(·) are deﬁned in the assumptions below. Assumption 2. We assume that D(σ) ⊂ D(γ) (here D(·) denotes the domain) and that the restriction of γ to D(σ) is continuous with respect to the graph norm of D(σ). Assumption 3. The operator A : D(A) ⊂ E1 → E1 deﬁned by Ay = σy, y ∈ D(A) = {y ∈ D(σ) : γy = 0},

(5.6)

is the inﬁnitesimal generator of a strongly continuous semigroup {S(t) : t ≥ 0} on E1 . Assumption 4. There exists a linear continuous operator B : F2 → E1 such that σB ∈ L(F2 , E1 ), γ(Bv) = B2 v for all v ∈ F2 , ||Bv||E1 ≤ c||B2 ||E2 for all v ∈ F2 .

(5.7)

Assumption 5. For each t ≥ 0 and v ∈ L2loc ([0, ∞); F2 ), t 0

S(t − s)Bv(s)ds ∈ D(A)

(5.8)

and there exists δ ∈ L2loc (0, ∞) such that ||AS(t)B|| ≤ δ(t) a.e.,

(5.9)

for each t ≥ 0, each h ∈ [0, 1] we have t+h t

δ(s)ds ≤ K,

(5.10)

where K is a constant. Under the above hypotheses, the expression (5.5) is well deﬁned for uj (·) ∈ L2loc ([0, ∞); Fj ), j = 1, 2, and agrees with the formulation presented in [8].

279

Control problems in Hilbert spaces

The performance of the system is evaluated by the cost functional J(T1 , T2 , x, u1 , u2 ) =

T2 T1

f (x(t), u1 (t), u2 (t))dt,

(5.11)

where f : E1 ×F1 ×F2 → R1 is a convex lower semicontinuous functional which satisﬁes the following coercivity assumption: there exist K1 > 0 and K > 0 such that f (x, u1 , u2 ) ≥ K(||x||2 + ||u1 ||2 + ||u2 ||2 ) for each x ∈ E1 , u1 ∈ F1 , u2 ∈ F2 satisfying ||x||2 + ||u1 ||2 + ||u2 ||2 > K1 .

(5.12)

Assume the following. Assumption 6. The optimal steady state problem (OSSP) consisting of Min f (x, u1 , u2 ) subject to 0 =< x − Bu2 , A∗ z > + < B1 u1 + σBu2 , z > for all z ∈ D(A∗ ) x ∈ X, u1 ∈ U1 (x), u2 ∈ U2 (x)

(5.13)

¯2 ). has a unique solution (¯ x, u ¯1 , u Once again the convexity assumptions we have imposed allow us to deﬁne the nonnegative convex lower semicontinuous functional L : E1 × F1 × F2 → R1 by L(x, u1 , u2 ) = f (x, u1 , u2 ) − f (¯ x, u ¯1 , u ¯2 ) − < x − Bu2 , A∗ p¯ > − < B1 u1 + σBu2 , p¯ > if x ∈ X and uj ∈ Uj (x), j = 1, 2, L(x, u) = ∞ otherwise,

(5.14)

where p¯ ∈ D(A∗ ). A function x : I → E1 where I is either [0, ∞) or [0, T ] (T > 0) is called a trajectory if there exists uj (·) ∈ L2loc (I; Fj ), j = 1, 2 (referred to as a control) such that the pair (x, u1 , u2 ) satisﬁes (5.5) for all t ∈ I and (5.4). Let I be either [0, ∞) or [0, T ] (T > 0), x : I → E1 , uj : I → Fj , j = 1, 2 be a trajectory-control pair, and T1 , T2 ∈ I, T1 < T2 . We deﬁne JL (T1 , T2 , x, u1 , u2 ) =

T2 T1

L(x(t), u1 (t), u2 (t))dt.

(5.15)

280

TURNPIKE PROPERTIES

For a trajectory-control pair x : [0, ∞) → E1 , uj : [0, ∞) → Fj , j = 1, 2 we deﬁne JL (0, ∞, x, u1 , u2 ) =

∞

L(x(t), u1 (t), u2 (t))dt.

0

For each T > 0 and each z ∈ E1 we deﬁne σ(z, T ) = inf{J(0, T, x, u1 , u2 ) : x : [0, T ] → E1 , uj : [0, T ] → Fj , j = 1, 2 is a trajectory-control pair, x(0) = z}.

(5.16)

The proof of the following theorems is identical to the proof of Theorems 9.1.1-9.1.5. Theorem 9.5.1 Suppose that Assumptions 2-6 hold and x : [0, ∞) → E1 , uj : [0, ∞) → Fj , j = 1, 2 is a trajectory-control pair. Then one of the following relations holds: x, u ¯1 , u ¯2 )| : T ∈ (0, ∞)} < ∞. (i) sup{|J(0, T, x, u1 , u2 ) − T f (¯ x, u ¯1 , u ¯2 ) → ∞ as T → ∞. (ii) J(0, T, x, u1 , u2 ) − T f (¯ Moreover (i) holds if and only if JL (0, ∞, x, u1 , u2 ) < ∞. Theorem 9.5.2 Suppose that Assumptions 2-6 hold and r1 , r2 , r3 are positive numbers. Then there exist ∆, r > 0 such that ||x(t)|| ≤ ∆, t ∈ [0, T ], JL (0, T, x, u1 , u2 ) ≤ r for each T > 0 and each trajectory-control pair x : [0, T ] → E1 , uj : [0, T ] → Fj , j = 1, 2 which has the following properties: (a) ||x(0)|| ≤ r2 , J(0, T, x, u1 , u2 ) ≤ σ(x(0), T ) + r3 ; (b) there is a trajectory-control pair y : [0, ∞) → E1 , vj : [0, ∞) → Fj , j = 1, 2, satisfying y(0) = x(0), JL (0, ∞, y, v1 , v2 ) ≤ r1 . Theorem 9.5.3 Suppose that Assumptions 2-6 hold, r1 , r2 , r3 are positive numbers, and V is a neighborhood of x ¯ in the weak topology. Then there exists a number l > 0 such that −1

(T2 − T1 )

T2 T1

x(t)dt ∈ V

for each T ≥ l, each trajectory-control pair x : [0, T ] → E1 , uj : [0, T ] → Fj , j = 1, 2, which has properties (a) and (b) from Theorem 9.5.2 and each T1 , T2 ∈ [0, T ] satisfying T2 − T1 ≥ l.

Control problems in Hilbert spaces

281

Denote by F the set of all trajectory-control pairs x : [0, ∞) → E1 , uj : [0, ∞) → Fj , j = 1, 2 such that L(x(t), u1 (t), u2 (t)) = 0 a.e. on [0, ∞). We say that F has property G if for any neighborhood V of x ¯ in the weak topology there exists a number tv > 0 such that x(t) ∈ V for each t ≥ tv and each trajectory-control pair (x, u1 , u2 ) ∈ F. Theorem 9.5.4 Suppose that Assumptions 2-6 hold and F has property ¯ in G. Let r1 , r2 , r3 be positive numbers and let V be a neighborhood of x the weak topology. Then there exist an integer Q ≥ 1 and a number l > 0 such that for each T > 0 and each trajectory-control pair x : [0, T ] → E1 , uj : [0, T ] → Fj , j = 1, 2, which has properties (a) and (b) from Theorem 9.5.2, there exists a sequence of intervals [bj , cj ], j = 1, . . . , q such that 1 ≤ q ≤ Q, 0 < cj − bj ≤ l, j = 1, . . . , q, and x(t) ∈ V for each t ∈ [0, T ] \ ∪qj=1 [bj , cj ]. Theorem 9.5.5 Suppose that Assumptions 2-6 hold and F has property G. Let x ˜ : [0, ∞) → E1 , u ˜j : [0, ∞) → Fj , j = 1, 2, be a trajectory-control ˜, u ˜1 , u ˜2 ) < ∞. Then there exists an overtaking pair satisfying JL (0, ∞, x optimal trajectory-control pair x∗ : [0, ∞) → E1 , u∗j : [0, ∞) → Fj , ˜(0). j = 1, 2 such that x∗ (0) = x

Chapter 10 A CLASS OF DIFFERENTIAL INCLUSIONS

In this chapter we study the turnpike property for a class of diﬀerential inclusions of the form x (t) ∈ b(x(t)) − x(t) arising in economic dynamics where b belongs to a space of convex processes (superlinear set-valued mappings). This class of set-valued mappings was introduced by Rubinov in [76] and studied in [83, 84, 86, 96]. We establish the existence of an open everywhere dense subset E of the space of set-valued mappings such that each mapping from E has the turnpike property.

10.1.

Main result

n be the cone of the elements of the Euclidean space Let R+

Rn = {x = (x1 , . . . , xn ) : xi ∈ R1 , i = 1, . . . , n} n . We that have nonnegative coordinates and let Kn be the interior of R+ n . Let I : Rn → Rn suppose that the space Rn is ordered by the cone R+ n i be the identity operator and let ||x|| = sup{|x | : i = 1, . . . , n} for each x ∈ Rn . Denote by < ·, · > the scalar product in Rn . For each matrix V = (v ij ) of a dimension m × n we set

||V || = sup{|v ij | : i = 1, . . . , m, j = 1, . . . , n}. We will identify a linear operator A : Rm → Rn with its matrix in the standard basis. For each metric space Q denote by Π(Q) the collection of nonempty compact subsets of Q endowed with the Hausdorﬀ metric dist(·, ·).

284

TURNPIKE PROPERTIES

n → Π(Rn ) is called a convex process if for each A mapping b : R+ + n and each λ ∈ (0, ∞), x, y ∈ R+

b(x + y) ⊃ b(x) + b(y) and b(λx) = λb(x). n → Π(Rn ) denote by Gr(b) the graph of b. For each mapping b : R+ + n n A mapping b : R+ → Π(R+ ) is called normal if n n n b(x) = (b(x) − R+ ) ∩ R+ for each x ∈ R+ .

Note that convex processes are set-valued analogs of linear mappings [48, 71, 75, 76]. n → Π(Rn ) with a closed Consider a normal convex process b : R+ + graph such that b(0) = {0}. The von Neumann growth rate of the mapping b is a number α(b) deﬁned by 1 n α(b) = sup{β ∈ R+ : there is y ∈ R+ \ {0} for which βy ∈ b(y)} (1.1)

(see [48, 75]). n \{0} It is not diﬃcult to see that α(b) ∈ [0, ∞) and there exists X ∈ R+ such that α(b)X ∈ b(X). Assume that b(1, 1, . . . , 1) ∩ Kn = ∅. Denote by b the dual mapping n → Π(Rn ) deﬁned as b : R+ + n n b (f ) = {g ∈ R+ :< f, x >≥< g, y > for each x ∈ R+ , y ∈ b(x)}. (1.2)

There exists a generalized equilibrium state (α(b), (X, α(b)X), p) where n X, p ∈ R+ , ||X||, ||p|| = 1, α(b)X ∈ b(X), p ∈ b (α(b)p).

(1.3)

n → Π(Rn ) be a normal convex process with a closed graph Let b : R+ + such that b(0) = {0}. n where I is either [T, ∞)(T ≥ 0) or [T , T ](0 ≤ A function u : I → R+ 1 2 T1 < T2 ) will be called a trajectory of the model Z(b) generated by the mapping b, if u is absolutely continuous (a.c.) and satisﬁes the relation

u (t) ∈ b(u(t)) − u(t), a.e. t ∈ I.

(1.4)

Assume that b(1, 1, . . . , 1) ∩ Kn = ∅. Let 0 ≤ T1 < T2 , u : [T1 , T2 ] → n be an be a trajectory of the model Z(b) and let f : [T1 , T2 ] → R+ a.c. function which satisﬁes

n R+

f (t) ∈ f (t) − b∗ (f (t)),

(1.5)

n n b∗ (g) = {h ∈ R+ : g ∈ b (h)}, g ∈ R+ .

(1.6)

a.e. t ∈ [T1 , T2 ] where

285

A class of diﬀerential inclusions

It is easy to verify that the function t →< f (t), u(t) >, t ∈ [T1 , T2 ] is monotone decreasing. Assume that there exists a generalized equilibrium state (α(b), (X, α(b)X), p) with X, p ∈ Kn such that (1.3) holds. It is easy to see that the function t → exp((α(b) − 1)t)X, t ∈ [0, ∞) is a trajectory of Z(b), the function f (t) = exp((1 − α(b))t)p satisﬁes (1.5) for a.e. t ∈ [0, ∞), and for each n of the model Z(b) the function exp((1 − trajectory u : [0, ∞) → R+ α(b))t) < u(t), p >, t ∈ [0, ∞) is monotone decreasing. We say that a trajectory u : [0, ∞) → Rn of the model Z(b) has the von Neumann growth rate if lim exp((1 − α(b))t) < p, u(t) >> 0.

(1.7)

t→∞

It follows from Propositions 0.1 and 1.1 of [86] that for each x ∈ Kn there exists a trajectory u : [0, ∞) → Rn which has the von Neumann growth rate and satisﬁes u(0) = x. We say that the mapping b has the turnpike property if for each n of Z(b) which has the von Neumann growth trajectory u : [0, ∞) → R+ rate, there exists a positive number λ such that lim exp((1 − α(b))t)u(t) = λX.

t→∞

Most studies which are concerned with the turnpike property for convex processes assume the strict convexity of their graphs. In this chapter we establish the turnpike property without such assumptions. Let m, n be natural numbers such that m ≤ n. Denote by L0 the set of all matrices V = (v ij ) of a dimension m×n such that v ij ∈ [0, 1] for each i = 1, . . . , m, j = 1, . . . , n. For each V ∈ L0 and each i ∈ {1, . . . , m} denote by V i a diagonal matrix of a dimension n × n with diagonal elements v i1 , . . . , v in . n → Π(Rn ), i = Assume that V ∈ L0 and a = (a1 , . . . , am ), ai : R+ + 1, . . . , m are normal convex processes such that Gr(ai ) is a closed set and ai (0) = {0}, i = 1, . . . , m. Consider the mapping n m n m Q(a, V ) : (R+ ) → Π((R+ ) )

deﬁned as n m n Q(a, V )(x) = {y = (y 1 , . . . , y m ) ∈ (R+ ) : y i ∈ R+ , n , i = 1, . . . , m, y i ≤ V i xi + di , di ∈ R+

m i=1

di ∈

m i=1

ai (xi )}

(1.8)

286

TURNPIKE PROPERTIES

n )m , xi = (xi1 , . . . , xin ) ∈ Rn , i = 1, . . . , m). (here x = (x1 , . . . , xm ) ∈ (R+ + It is easy to see that Q(a, V ) is a normal convex process, the graph of Q(a, V ) is closed and Q(a, V )(0) = {0}. Set

α(a, V ) = α(Q(a, V )).

(1.9)

Assume that Is ⊂ {1, . . . , n}, Is = ∅, s = 1, . . . , m, ∪{Is : s = 1, . . . , m} = {1, . . . , n}, (1.10)

∅ for each k ∈ {1, . . . , m}, Ik \ ∪{Is : s ∈ {1, . . . , m} \ {k}} = P ⊂ {(i, j) : i = 1, . . . , m, j = 1, . . . , n}. Let L be the set of all matrices V ∈ L0 such that (i, j) ∈ P. Denote by M the set of all mappings

v ij

(1.11) = 0 for each

n m n m a = (a1 , . . . , am ) : (R+ ) → (Π(R+ )) n → such that for each s ∈ {1, . . . , m} the normal convex process as : R+ n ) has the following properties: Π(R+ (i) the graph of as is closed, as (0) = {0} and n as (R+ ) ⊂ {x = (x1 , . . . , xn ) ∈ Rn : xi = 0 for each i ∈ {1, . . . , n} \ Is };

(ii) for each x, g ∈ Kn the optimization problem < g, z >→ max, z ∈ as (x) has a unique solution; n \ {λx : λ ∈ [0, ∞)}, (iii) for each g, x1 ∈ Kn and each x2 ∈ R+ 1 sup{< g, z >: z ∈ as (x1 ) + as (x2 )} < sup{< g, z >: z ∈ as (x1 + x2 )}. Our interest in a set-valued mapping Q(a, V ) with Q ∈ M and V ∈ L stems from the work by Rubinov [76] who studied an analogous class of set-valued mappings describing multisector models of economic dynamics. Let a = (a1 , . . . , am ) ∈ M, V ∈ L. The mapping Q(a, V ) describes the model of economic dynamics with m sectors and n products such that for each i ∈ {1, . . . , m}, the production process of the ith sector is described by the pair (ai , V i ). Rubinov [76] (see also [86]) studied a special case of this model given in the example below, such that m = n and each sector produces one product. For this case the production

287

A class of diﬀerential inclusions

process of the ith sector is described by the pair (φi , V i ), where φi : n → R1 is a superlinear function. Rubinov [76] studied the discreteR+ time model. In this chapter we consider a generalization of his model. For the discrete-time generalization of the model by Rubinov a state of n m the economy at time t = 0, 1, . . . is a vector xt = (x1t , . . . , xm t ) ∈ (R+ ) i i1 in n where xt = (xt , . . . , xt ) ∈ R+ is the state of the ith sector of the economy, i = 1, . . . , m. During the time interval [t, t + 1] the ith sector produces a vector of products yi ∈ ai (xi ). It also has a vector of the old products in the amount V i xi . At time t + 1 the newly produced vector m of products i=1 yi is distributed between m sectors. We assume that the sth sector produces only products with indices i ∈ Is (see (1.10) and (i)) which is nonempty. Relations (1.10) mean that each product is produced by some sector and that for each sector there is a product which is produced only by this sector. In this chapter we study the continuous-time version of the model. l → Π(Rl ) which For each pair of normal convex processes b1 , b2 : R+ + have closed graphs and satisfy bi (0) = {0}, i = 1, 2 we set d(b1 , b2 ) = dist({(x, y) ∈ Gr(b1 ), ||x|| ≤ 1},

(1.12)

{(x, y) ∈ Gr(b2 ), ||x|| ≤ 1}). For the set M we consider the product topology generated by the metric d(·, ·). It is easy to verify that the mapping (a, V ) → Q(a, V ), a ∈ M, V ∈ L is continuous. In this chapter we will establish the following result. Theorem 10.1.1 There exists an open everywhere dense set E ⊂ M×L such that each (a, V ) ∈ E satisﬁes the following conditions: There is a generalized equilibrium state (α(a, V ), (X, α(a, V )X), p) such that α(a, V ) > ||V ||, X, p ∈ (Kn )m ; the mapping Q(a, V ) has the turnpike property. Example. Assume that m = n, P = ∅, Is = {s}, s = 1, . . . , n and consider the spaces M, L deﬁned above. Let e1 , . . . , en be the standard basis in Rn . It is easy to see that (a1 , . . . , an ) ∈ M if and only if there n → R1 , i = 1, . . . , n such that for each exist continuous functions φi : R+ + i ∈ {1, . . . , n} the following properties hold: n ; ai (x) = [0, φi (x)]ei , x ∈ R+ n , λ ∈ [0.∞); φi (λx) = λφi (x), φi (x + y) ≥ φi (x) + φi (y), x, y ∈ R+ n \ {λx : λ ∈ [0, ∞)} the inequality for each x1 ∈ Kn and each x2 ∈ R+ 1 i i + φ (x2 ) < φ (x1 + x2 ) holds.

φi (x1 )

288

TURNPIKE PROPERTIES

In [86] we studied the turnpike property for the space M×L described in this example. Theorem 10.1.1 which is a generalization of Theorem 3.1 of [86] was obtained in [96]. Chapter 10 is organized as follows. In Section 10.2 we study a generalized equilibrium state of the model Q(a, V ) and obtain useful auxiliary results. In Section 10.3 we obtain a suﬃcient condition for the turnpike property. This condition is some property of a generalized equilibrium state of the model. Section 10.4 contains a number of auxiliary results. Theorem 10.1.1 is proved in Section 10.5.

10.2.

Preliminary results

We have the following result (see Proposition 1.2 of [86]). n → Π(Rn ) be a normal convex process Proposition 10.2.1 Let b : R+ + such that Gr(b) is closed, b(0) = {0}, b(1, 1, . . . , 1) ∩ Kn = ∅ and let (α(b), (X, α(b)X), p) be a generalized equilibrium state such that X, p ∈ n is a trajectory of the model Z(b) Kn . Assume that x : [0, ∞) → R+ which has the von Neumann growth rate and put n Ω = {y ∈ R+ : there exists a sequence {ti }∞ i=1 ⊂ (0, ∞) such that

ti → ∞, exp((1 − α(b))ti )x(ti ) → y as i → ∞}. n such Then for each y0 ∈ Ω there exists an a.c. function y : R1 → R+ that y(0) = y0 ,

exp((1 − α(b))t)y(t) ∈ Ω, t ∈ R1 , y (t) ∈ b(y(t)) − y(t) a.e. t ∈ R1 . Denote by Card(B) the cardinality of a set B. Consider the spaces M and L deﬁned in Section 10.1. For each g = n )m where g i = (g i1 , . . . , g in ) ∈ Rn , i = 1, . . . , m we (g 1 , . . . , g m ) ∈ (R+ deﬁne (2.1) r(g, j) = sup{g sj : s = 1, . . . , m}, j = 1, . . . , n, r(g) = (r(g, 1), . . . , r(g, n)). Let V ∈ L, a = (a1 , . . . , am ) ∈ M. Consider the dual mapping n )m n )m → 2(R+ Q(a, V ) : (R+ \ {∅} which is deﬁned as n m Q(a, V ) (f ) = {g ∈ (R+ ) :< f, x >≥< g, y > for each n m n m x ∈ (R+ ) , y ∈ Q(a, V )(x)}, f = (f 1 , . . . , f m ) ∈ (R+ ) .

(2.2)

289

A class of diﬀerential inclusions

Analogously to Propositions 1 and 2 in [76] we can establish the following results which describe some useful relations between Q(a, V ) and its dual mapping. n )m . Then g ∈ Q(a, V ) (f ) if and only if Lemma 10.2.1 Let f, g ∈ (R+

< f i − V i g i , u >≥ sup{< r(g), z >: z ∈ ai (u)} n and each i ∈ {1, . . . , m}. for each u ∈ R+ n )m , g ∈ Q(a, V ) (f ), y ∈ Q(a, V )(x), Lemma 10.2.2 Let f, g, x, y ∈ (R+ i

i i

i

i

y ≤ V x + d , d ≥ 0,

m i=1

i

d =

m

zi

i=1

where z i ∈ ai (xi ), i = 1, . . . , m. Then < f, x >= < g, y > if and only if for each i ∈ {1, . . . m} the following relations hold: (a) < g i , di + V i xi − y i >= 0; (b) < r(g) − g i , di >= 0; (c) < r(g), z i >= sup{< r(g), u >: u ∈ ai (xi )}; (d) < f i − V i g i , xi >= sup{< r(g), u >: u ∈ ai (xi )}. Lemmas 10.2.1 and 10.2.2 imply the following auxiliary result which gives an important property of a generalized equilibrium state. Lemma 10.2.3 Let α = α(a, V ) > ||V ||, X ∈ (Kn )m , α(a, V )X ∈ n )m \ {0}. Then p ∈ Q(a, V ) (αp) if and only if the Q(a, V )X, p ∈ (R+ following relations hold: n \ {0}; (a) p = (r, . . . , r) where r ∈ R+ n (b) < (αIn − V i )r, u >≥ sup{< r, d >: d ∈ ai (u)} for each u ∈ R+ and each i ∈ {1, . . . , m}; (c) < (αIn − V i )r, X i >= sup{< r, d >: d ∈ ai (X i )}, i = 1, . . . , m. n → R1 Let e1 , . . . , en be the standard basis in Rn . A function ψ : R+ + is called a CES-function if n

ψ(x) = γ(

i=1

n βi (xi )ρ )1/ρ , x ∈ R+

(2.3)

where γ > 0, ρ ∈ (−∞, 1) \ {0}, β = (β1 , . . . , βn ) ∈ Kn , ni=1 βi = 1. The vector (γ, β, ρ)(ψ) = (γ, β, ρ) is called the coordinates of the CESfunction ψ. In this chapter we consider only CES-functions ψ for which ρ(ψ) > 0. Let (a, V ) ∈ M × L and let (α(a, V ), (X, α(a, V )X), p)

290

TURNPIKE PROPERTIES

be its generalized equilibrium state. In view of Lemma 10.2.3 it is important that this generalized equilibrium state satisﬁes α(a, V ) > ||V || and that all coordinates of X and p are positive. In general these conditions do not hold. In the sequel for any (a, V ) ∈ M × L we construct (˜ a, V˜ ) ∈ M × L which is close to (a, V ) in M × L and for which these properties hold (see Lemmas 10.2.4-10.2.6). n → R1 , i = 1, . . . , n be CESLet (a, V ) ∈ M × L and let φi : R+ + s n functions. We deﬁne a ⊕ φ = (a ⊕ φ)s=1 by (as ⊕ φ)(x) = as (x) +

[0, φi (x)]ei ,

(2.4)

i∈Is n x ∈ R+ , s = 1, . . . , m.

It is easy to see that a ⊕ φ ∈ M. n → R1 , i = 1, . . . , n Lemma 10.2.4 Let (a, V ) ∈ M × L and let φi : R+ + be CES-functions. Then α(a ⊕ φ, V ) > ||V ||.

Proof. There exist p ∈ {1, . . . , m}, q ∈ {1, . . . , n} such that v pq = ||V ||. It is easy to see that if q ∈ Ip then α(a ⊕ φ, V ) > ||V ||. Assume that q ∈ Ip . Then there exists s ∈ {1, . . . , m} \ {p} for which q ∈ Is . n )m as Fix any r ∈ Ip . Clearly r = q. We deﬁne X ∈ (R+ X j = 0, j ∈ {1, . . . , m} \ {s, p}, X p = ep , X s = 2−1 φr (X p )er . It is easy to verify that there exists a number α > ||V || for which αX ∈ Q(a ⊕ φ, V ). The lemma is proved. n → R1 , i = 1, . . . , n Lemma 10.2.5 Let (a, V ) ∈ M × L and let φi : R+ + be CES-functions. Assume that n m ) \ {0}, X ∈ (R+

α(a ⊕ φ, V )(X) ∈ Q(a ⊕ φ, V )(X).

(2.5)

Then X i > 0, i = 1, . . . , m. Proof. Assume the contrary. Then there exists i ∈ {1, . . . , m} for which X i = 0. By Lemma 10.2.4, α(a ⊕ φ, V ) > ||V ||.

(2.6)

Set E1 = {i ∈ {1, . . . , m} : X i > 0}, E2 = {1, . . . , m} \ E1 .

(2.7)

n , i = 1, . . . , m such We have that E1 = ∅, E2 = ∅. There exist di ∈ R+ that (2.8) α(a ⊕ φ, V )X i ≤ V i X i + di , i = 1, . . . , m,

291

A class of diﬀerential inclusions m

i

d ∈

i=1

m

(ai ⊕ φ)(X i ).

i=1

We may assume that

di = 0, i ∈ E2 .

(2.9)

By (2.8), (2.9) and (2.4) for each j ∈ E1 there exist n such that z j ∈ R+ φi (X j )ei , zj ≤

yj

∈

aj (X j )

and

(2.10)

i∈Ij

di =

i∈E1

(y j + z j ).

(2.11)

j∈E1

Fix k ∈ E2 . There exists p ∈ Ik \ ∪{Ij : j ∈ {1, . . . , m} \ {k}}.

(2.12)

It follows from (2.6) and (2.12) that X ip = 0, i ∈ E1 .

(2.13)

For all small enough h > 0 we set Xhi = 0, i ∈ E2 \ {k}, Xhi = X i + hep , i ∈ E1 .

(2.14)

By (2.13) and (2.14) there exist constants ρ, c0 ∈ (0, 1) such that for all small enough h > 0, each j ∈ E1 and each r ∈ Ij ,

We set

φr (Xhj ) ≥ φr (X j ) + c0 hρ .

(2.15)

c1 = min{φp (ej ) : j = 1, . . . , n}.

(2.16)

For all small enough h > 0 we deﬁne Xhkr = [4(α(a ⊕ φ, V ) + 1)]−1 c0 hρ if

(2.17)

otherwise Xhkr = 0.

r ∈ ∪{Ij : j ∈ E1 },

(2.6), (2.8), (2.9), (2.10) and (2.11) imply that

[α(a ⊕ φ, V )X i − V i X i ] ≤

i∈E1

i∈E1

di ≤

(y i + z i )

(2.18)

i∈E1

n ⊂ {x ∈ R+ : xi = 0 for each i ∈ {1, . . . , n} \ ∪{Ij : j ∈ E1 }}.

It is easy to see that for all small enough h > 0, [α(a ⊕ φ, V ) + 1]Xhk ≤ 4−1

[

j∈E1 r∈Ij

(φr (Xhj ) − φr (X j ))er ]

(2.19)

292

TURNPIKE PROPERTIES

≤ [4α(a ⊕ φ, V ) + 4][m

(Xhi − X i )] ≤ φp (Xhk )ep .

i∈E1

For all small enough h > 0 we deﬁne yhj = 0, j ∈ E2 , zhj = 0, j ∈ E2 \ {k},

(2.20)

zhk = φp (Xhk )ep , yhj = y j , i ∈ E1 , zhj = z j +

i∈Ij

(φi (Xhj ) − φi (X j ))ei , j ∈ E1 ,

djh = 0, j ∈ E2 \ {k}, dkh = 4−1 djh = dj +4−1

m−1 [

s∈E1

(

j∈E1 r∈Ij

(φr (Xhj ) − φr (X j ))er ), (2.21)

(φr (Xhs )−φr (X s ))er ]+m−1 φp (Xhk )ep , j ∈ E1 .

r∈Is

By (2.20), (2.21), (2.14), (2.8), (2.10), (2.11) and (2.4) for all small enough h > 0, yhj , zhj , djh

n R+ ,

∈

j = 1, . . . , m,

m j

m

j=1

j=1

dh ≤

(yhj + zhj ),

(2.22)

yhj + zhj ∈ (aj ⊕ φ)(Xhj ), j = 1, . . . , m. By (2.14), (2.17), (2.21) and (2.19) for each small enough h > 0, dih + V i Xhi ≥ [α(a ⊕ φ, V ) + 1)]Xhi , i ∈ E2 .

(2.23)

It follows from (2.21), (2.14), (2.8) and (2.19) that for each small enough h > 0 and each j ∈ E1 , djh + V j Xhj ≥ dj + V j X j + m−1 φp (Xhk )ep

+(4m)−1

[

(2.24)

(φr (Xhs )−φr (X s ))er ] ≥ α(a⊕φ, V )X j +m−1 φp (Xhk )ep

s∈E1 r∈Is

+(4m)−1

(φr (Xhs ) − φr (X s ))er ] ≥ α(a ⊕ φ, V )X j

[

s∈E1 r∈Is

+4(α(a ⊕ φ, V ) + 1)(Xhj − X j ) +(4m)−1

[

(φr (Xhs ) − φr (X s ))er ].

s∈E1 r∈Is

(2.6), (2.18) and (2.14) imply that for each small enough h > 0 there exists a number λh > 0 which satisﬁes

[

s∈E1 r∈Is

(φr (Xhs ) − φr (X s ))er ] ≥ λh X i , i ∈ E1 .

293

A class of diﬀerential inclusions

Together with (2.23), (2.24), (2.22) this implies that for each small enough h > 0 there exists a number αh > α(a ⊕ φ, V ) such that djh + V j Xhj ≥ αh Xhj , j ∈ {1, . . . , m}, αh Xh ∈ Q(a ⊕ φ, V )(Xh ). This is contradictory to the deﬁnition of α(a ⊕ φ, V ). The obtained contradiction proves the lemma. n → R1 , i = 1, . . . , n be Lemma 10.2.6 Let (a, V ) ∈ M × L, φi : R+ + n m CES-functions and let X ∈ (R+ ) \ {0},

α(a ⊕ φ, V )X ∈ Q(a ⊕ φ, V )(X).

(2.25)

n )m \ Then α(a ⊕ φ, V ) > ||V ||, X ∈ (Kn )m . Moreover for each p ∈ (R+ {0} satisfying p ∈ Q(a ⊕ φ, V ) (α(a ⊕ φ, V )p) there exists r ∈ Kn such that p = (r, . . . , r).

Proof. By Lemmas 10.2.4 and 10.2.5, α(a ⊕ φ, V ) > ||V ||, X i > 0, i = 1, . . . m.

(2.26)

Assume that X ∈ (Kn )m . Then there exist k ∈ {1, . . . , m}, j, j1 ∈ {1, . . . , n} such that X kj = 0, X kj1 > 0. n )m . For all small ˆ h ∈ (R+ For all small enough h > 0 we deﬁne X enough h > 0 we set Xhk = X k + hej . There exist constants c1 > 0, ρ ∈ (0, 1) such that for each small enough h > 0 and each i ∈ {1, . . . , m}, φi (Xhk ) ≥ φi (X k ) + c1 hρ . There are two cases: (i) j ∈ Ik ; (ii) j ∈ Ik . In the case (i) we set Xhs = X s , s ∈ {1, . . . , m} \ {k}. Consider the case (ii). There exists q ∈ {1, . . . , m} \ {k} for which j ∈ Iq . We set Xhs = X s , s ∈ {1, . . . , m} \ {q, k}, Xhqi = X qi , i ∈ {1, . . . , m} \ Ik , Xhqi = X qi + (α(a ⊕ φ, V ))−1 c1 hρ , i ∈ Ik . In the case (ii) there exists a constant c2 ∈ (0, c1 ) such that φj (Xhq ) ≥ φj (X q ) + c2 hρ . In both cases we set ˆ hk = Xhk + (2α(a ⊕ φ, V ))−1 c2 hρ ej . ˆ hs = Xhs , s ∈ {1, . . . , m} \ {k}, X X It is easy to verify that for all small enough h > 0, ˆ h ∈ Q(a ⊕ φ, V )(Xh ). α(a ⊕ φ, V )X

(2.27)

Since for all small enough h > 0, Card{(i, j) ∈ {1, . . . , m} × {1, . . . , n} : Xhij > 0}

(2.28)

294

TURNPIKE PROPERTIES

> Card{(i, j) ∈ {1, . . . , m} × {1, . . . , n} : X ij > 0} n )m \ {0} satisfying (2.25) we conclude that and X is any vector from (R+ there exists X0 ∈ (Kn )m for which α(a ⊕ φ, V )X0 ∈ Q(a ⊕ φ, V )(X0 ). n )m \ {0} satisfying There exists p ∈ (R+

p ∈ Q(a ⊕ φ, V ) (α(a ⊕ φ, V )p).

(2.29)

n \ {0} such that By Lemma 10.2.3 there exists r ∈ R+

< (α(a ⊕ φ, V )In − V s )r, u >

p = (r, . . . , r),

≥ sup{< r, d >: d ∈ as (u)} +

i∈Is

< (α(a ⊕ φ, V )In − V +

i∈Is

s

)r, X0s

(2.30)

n ri φi (u), s = 1, . . . , m, u ∈ R+ ,

>= sup{< r, d >: d ∈ as (X0s )}

ri φi (X0s ), s = 1, . . . , m.

Since the functions φi , i = 1, . . . n are strictly monotone increasing these relations imply that r ∈ Kn . It follows from this relation, (2.29), ˆ h that for all small enough h > 0, (2.27), (2.30) and the deﬁnition of X ˆh > < α(a ⊕ φ, V )p, Xh >≥< p, α(a ⊕ φ, V )X >< p, α(a ⊕ φ, V )Xh > . The obtained contradiction proves the lemma.

10.3.

Suﬃcient condition for the turnpike property

We need the following result established by Leizarowitz [38]. Proposition 10.3.1 Let G ⊂ Rn × Rn be a convex compact set, domG = {x ∈ Rn : there exists y ∈ Rn such that (x, y) ∈ G}, G(x) = {y ∈ Rn : (x, y) ∈ G}, x ∈ domG. Assume that the following properties hold: (i) There exists a unique Y ∈ domG such that 0 ∈ G(Y ); (ii) if a, b ∈ Rn , a scalar α = 0 and the function z(t) = Y +cos(αt)a+ sin(αt)b, t ∈ [0, ∞) satisﬁes z (t) ∈ G(z(t))

(3.1)

295

A class of diﬀerential inclusions

almost everywhere in [0, ∞) then a, b = 0. Denote by E the set of all a.c. functions z : [0, ∞) → domG which satisfy (3.1) a.e. in [0, ∞). Then for each > 0 there exists T > 0 such that |z(t) − Y | ≤ for each z(·) ∈ E and each t ≥ T . Denote by C the set of complex numbers, for each linear operator M : Cq → Ck denote by M the dual operator and denote by σ(M ) the spectrum of M . For each x = (x1 , . . . , xs ) ∈ Rs , y = (y 1 , . . . , y q ) ∈ Rq we set [x, y] = (x1 , . . . , xs , y 1 , . . . , y q ) ∈ Rs+q . (3.2) For each matrix M = (mij ), N = (nij ) of a dimension s × q we deﬁne σ(M, N ) = {θ ∈ C : there exists λ ∈ Cs \ {0}

(3.3)

for which M θλ = N λ} and denote by M ∗N the matrix B = (bij ) of the dimension s×q such that bij = mij nij , i = 1, . . . , s, j = 1, . . . , q. For each Y = (Y 1 , . . . , Y m ) ∈ (Rn )m denote by Yˆ = (ˆ y ij ) the matrix of the dimension m × n for which ij ij yˆ = Y , i = 1, . . . , m, j = 1, . . . , n. In this section we assume that V ∈ L, a = (a1 , . . . , am ) ∈ M, Q = Q(a, V ), α = α(a, V ) > ||V ||, (3.4) X ∈ (Kn )m , r ∈ Kn , p = (r, . . . , r), αX ∈ Q(X), p ∈ Q (αp), < p, X >= 1, Z i ∈ ai (X i ), i = 1, . . . , m,

m

((αIn − V i )X i − Z i ) = 0.

i=1

The next simple but important lemma describes the set of all pairs n )m such that y ∈ Q(x) and < αpx >= (p, y >. x, y ∈ (R+ Lemma 10.3.1 Assume that n m n ) , y ∈ Q(x), di ∈ R+ , i = 1, . . . , m, x ∈ (R+

z i ∈ ai (xi ), i = 1, . . . , m, y i ≤ V i xi + di , i = 1, . . . , m, m

(di − z i ) = 0, < αp, x >=< p, y > .

i=1

Then there exist numbers λi ≥ 0, i = 1, . . . , m such that [xi , z i ] = i = 1, . . . , m.

λi [X i , Z i ],

296

TURNPIKE PROPERTIES

Proof. It follows from Lemma 10.2.2 that y i = V i xi + di , < (αIn − V i )r, xi >=< r, z i > = sup{< r, u >: u ∈ ai (xi )}, i = 1, . . . , m. Combined with Lemma 10.2.3 and property (ii) (see Section 10.1) this implies the validity of the lemma. The following auxiliary result establishes a suﬃcient condition for the turnpike property. It is important that this condition depends only on α, X, V, Z. The proof of this lemma is based on Lemma 10.3.1 and Proposiion 10.3.1. Lemma 10.3.2 Assume that ˆ −V ∗X ˆ − Z) ˆ λ = 0} = {γ(1, 1, . . . , 1) : γ ∈ C}, (3.5) {λ ∈ Cn : (αX ˆ , (V ∗ X ˆ + Z) ˆ ) : Reθ = α} = {α}. {θ ∈ σ(X Then the mapping Q has the turnpike property. n )m be a trajectory of the model Z(Q) Proof. Let x : [0, ∞) → (R+ which has the von Neumann growth rate. We will show that

lim exp((1 − α)t)x(t) = λX with some λ > 0.

t→∞

We may assume without loss of generality that lim exp((1 − α)t) < x(t), p >= 1.

(3.6)

t→∞

Put

n m Ω = {y ∈ (R+ ) :< y, p >= 1},

G = {(y, z) ∈ Ω × Rn : z ∈ Q(y) − αy}, Ω0 = {y ∈

n m (R+ )

: there exists a sequence

{tj }∞ j=1

(3.7) ⊂ (0, ∞)

such that tj → ∞, exp((1 − α)tj )x(tj ) → y as j → ∞}. Clearly G is a compact convex set. It follows from (3.4), (3.7), (3.5) and Lemma 10.3.1 that (Y, 0) ∈ G if and only if Y = X. Let a = (a1 , . . . , am ), b = (b1 , . . . , bm ) ∈ (Rn )m , β = 0 and let

Assume that

z(t) = X + cos(βt)a + sin(βt)b, t ∈ [0, ∞).

(3.8)

(z(t), z (t)) ∈ G for each t ∈ [0, ∞).

(3.9)

297

A class of diﬀerential inclusions

We will show that a, b = 0. It follows from (3.8), (3.9) and (3.7) that X + cos(βt)a + sin(βt)b ∈ Ω, t ∈ [0, ∞),

(3.10)

α(X + cos(βt)a + sin(βt)b) + sin(βt)(−β)a + cos(βt)βb ∈ Q(X + cos(βt)a + sin(βt)b), t ∈ [0, ∞), < p, − sin(βt)βa + cos(βt)b >=< p, z (t) >= 0,

(3.11)

< p, αX + (cos βt)(αa + βb) + sin(βt)(αb − βa) >= α. By (3.10), (3.11) and Lemma 10.3.1 there exist numbers γai , γbi , i = 1, . . . , m such that bi = γbi X i , ai = γai X i , i = 1, . . . , m, m

[αX i + (cos βt)(αγai + βγbi )X i + (sin βt)(αγbi X i − βγai X i )

(3.12) (3.13)

i=1

−V i (X i + cos(βt)γai X i + sin(βt)γbi X i )] =

m

(1 + cos(βt)γai + (sin βt)γbi )Z i

i=1

for each t ∈ [0, ∞). (3.4) and (3.13) which holds for each t ∈ [0, ∞) imply that m

[(cos(βt)(αγai + βγbi ) + sin(βt)(αγbi − βγai ))X i

(3.14)

i=1

−V i (cos(βt)γai +sin(βt)γbi )X i ] =

m

(cos(βt)γai +sin(βt)γbi )Z i , t ∈ [0, ∞).

i=1

This implies that m

(αγai + βγbi )X i =

i=1

γai (V i X i + Z i ),

i=1

m

(αγbi

i=1

m

−

βγai )X i

=

m

γbi (V i X i + Z i ).

i=1

By (3.15), ˆ ((α + iβ)γ) = (V ∗ X ˆ + Z) ˆ γ, X where

γ = (γb1 , . . . , γbm ) + i(γa1 , . . . , γam ).

(3.15)

298

TURNPIKE PROPERTIES

Combined with (3.5), (3.3) and (3.12) this implies that a, b = 0. Therefore the set G has properties (i) and (ii) stated in Proposition 10.3.1. It follows from Proposition 10.3.1 that for each a.c. function z : R1 → Ω which satisﬁes (3.16) (z(t), z (t)) ∈ G a.e. in R1 the following relation holds: z(t) = X, t ∈ R1 .

(3.17)

Consider any y0 ∈ Ω0 . In view of Proposition 10.2.1 there exists an n such that a.c. function y : R1 → R+ y(0) = y0 , exp((1 − α)t)y(t) ∈ Ω0 , t ∈ R1 and

y (t) ∈ Q(y(t)) − y(t) a.e. t ∈ R1 .

It is easy to see that (3.16) holds a.e. in R1 for z(t) = exp((1−α)t)y(t). This implies (3.17) and the equality y0 = X. Therefore Ω0 = {X}. The lemma is proved.

10.4.

Preliminary lemmas

The main result of this section is Lemma 10.4.6 which is an important tool in the proof of Theorem 10.1.1. With any (a, V ) ∈ M × L we associate a generalized equilibrium state (α, (X, αX), p) and a vector Z as in Section 10.3. We consider the set K which consists of all vectors (α, X, V, Z) which are associated with all (a, V ) ∈ M × L. Let (a, V ) ∈ M × L and let a vector (α, X, V, Z) ∈ K be associated with (a, V ). V , Z) ∈ K which is close to X, Lemma 10.4.6 shows that there is (α, (α, X, V, Z) and which satisﬁes the condition (3.5) in Lemma 10.3.2. , V ) such that an element of K In Section 10.5 we construct a pair (a V , Z). Then by Lemma 10.3.2 (a , V ) is (α, X, , V ) has associated with (a the turnpike property. Let G be a metric space with a metric d(·, ·). For each x ∈ G and each h > 0 we set B(x, h) = {y ∈ G : d(x, y) < h}. Let A = (aij ), B = (bij ) be matrices of a dimension m × n, aij , bij ∈ C, i = 1, . . . , m, j = 1, . . . , n. Consider the function λ → λA − B , λ ∈ C. For each subset F ⊂ {1, . . . , n} satisfying Card(F ) = m denote

299

A class of diﬀerential inclusions

by PF (A, B) the polynomial such that for each λ ∈ C the number PF (A, B)(λ) is the determinant of the matrix of the dimension m × m with rows (λa1j − b1j , . . . , λamj − bmj ), j ∈ F . Denote by P (A, B) the greatest common divizor of all polynomials PF (A, B) where F ⊂ {1, . . . , n}, Card(F ) = m. It is easy to see that λ ∈ σ(A , B ) if and only if P (A, B)(λ) = 0. Remark. If polynomials P1 , . . . , Ps = 0, then their greatest common divizor is 0. Otherwise their greatest common divizor is a polynomial whose leading coeﬃcient is equal to 1. s Let P (A, B)(λ) = i=1 (λ − λi )mi , λ ∈ C where s ≥ 1, λi = λj for each i, j ∈ {1, . . . , s} satisfying i = j. Set m(A, B) =

s

mi ≤ m.

i=1

If P (A, B) = 1 we set m(A, B) = 0. Let g be a polynomial. By deg(g) we denote the degree of g. Lemma 10.4.1 Let A0 , B0 be matrices of the dimension m × n, m(A0 , B0 ) ≥ 1, m(A0 ,B0 )

P (A0 , B0 )(λ) =

(λ − λ0i ), λ ∈ C

i=1

where

λ0i

∈ C, i = 1, . . . , m(A0 , B0 ). Assume that , M > 0, |λ0i | < M, i = 1, . . . , m(A0 , B0 ), B(λ0i , ) ∩ B(λ0j , ) = ∅

for each i, j ∈ {1, . . . , m(A0 , B0 )} satisfying λ0i = λ0j . Then there exists δ > 0 such that for each (A, B) ∈ B((A0 , B0 ), δ) the polynomial P (A, B) = 0 and one of the following conditions holds: (a) P (A, B)(λ) = 1, λ ∈ C; (b) 1 ≤ m(A, B) ≤ m and there exist numbers λi ∈ C, i = 1, . . . , m(A, B), an integer s ∈ {0, . . . , m(A, B)} and a bijection τ : {1, . . . , m(A0 , B0 )} → {1, . . . , m(A0 , B0 )} such that m(A,B)

P (A, B)(λ) =

i=1

(λ − λi ), λ ∈ C,

300

TURNPIKE PROPERTIES

|λi − λ0τ (i) | < , 1 ≤ i ≤ s, |λi | ≥ M if s + 1 ≤ i ≤ m(A, B). Proof. We deﬁne E = {F ⊂ {1, . . . , n} : Card(F ) = m, PF (A0 , B0 ) = 0}. We will prove the following Assertion. Assume that m1 ≥ 1 is an integer, At , Bt , t = 1, 2 . . . are matrices of the dimension m × n, At → A0 , Bt → B0 as t → ∞, Pt is the greatest common divizor of the polynomials PF (At , Bt ), F ∈ E, t = 1, 2, . . . , Pt = 0, Pt = Qt Q1t whereQt , Q1t are polynomials, deg(Qt ) = t t 1 m1 , deg(Q1t ) ≥ 0, t = 1, 2 . . ., Qt = m i=1 (λ−λi ), |λi | ≤ M, i = 1, . . . , m1 , t t = 1, 2, . . ., λi → Λi as t → ∞, i = 1, . . . , m1 . Then m1 ≤ m(A0 , B0 ) and there exists a bijection τ : {1, . . . , m(A0 , B0 )} → {1, . . . , m(A0 , B0 )} such that Λi = λ0τ (i) , i = 1, . . . , m1 . Proof of the assertion. Consider any F ∈ E. For each t ∈ {1, 2, . . .} there exists a polynomial QFt such that deg(QFt ) ≥ 0, PF (At , Bt ) = Qt Q1t QFt .

(4.1)

It is easy to see that PF (At , Bt ) → PF (A0 , B0 ), Qt (λ) →

m1

(λ − Λi ), λ ∈ C as t → ∞. (4.2)

i=1

This implies that the set of all coeﬃcients of the polynomials Q1t QFt , t = 1, 2, . . . is bounded. Therefore we may assume that there exists a polyˆ for which nomial Q ˆ as t → ∞. (4.3) Q1t QFt → Q m1 ˆ Then by (4.1)-(4.3) PF (A0 , B0 )(λ) = i=1 (λ − Λi )Q, λ ∈ C. Since m1 F is any element of E we conclude that i=1 (λ − Λi ) is a divizor of P (A0 , B0 ). This completes the proof of the assertion.

Assume that the lemma is wrong. Then there exist sequences of ma∞ trices {At }∞ t=1 , {Bt }t=1 of the dimension m × n such that: (i) At → A0 , Bt → B0 as t → ∞ and for each integer t ≥ 1 the following relations hold: P (At , Bt ) = 0, m(At , Bt ) ≥ 1, m(At ,Bt )

P (At , Bt )(λ) =

i=1

(λ − λti ), λ ∈ C;

301

A class of diﬀerential inclusions

(ii) |λti | ≥ M if and only if st + 1 ≤ i ≤ m(At , Bt ) with some st ∈ {0, . . . , m(At , Bt )}; (iii) for each bijection τ : {1, . . . , m(A0 , B0 )} → {1, . . . , m(A0 , B0 )} the relation st ≤ m(A0 , B0 ) implies that |λti − λ0τ (i) | ≥ for some natural number i ≤ st . We may assume without loss of generality that there exist an integer m1 ≥ 1 and numbers Λi ∈ C, i = 1, . . . , m1 for which st = m1 , t = 1, 2, . . . , λti → Λi as t → ∞, i = 1, . . . , m1 .

(4.4)

For each integer t ≥ 1 we set Qt (λ) =

m1

(λ − λti ), λ ∈ C

(4.5)

i=1

and denote by Pt the greatest common divizor of polynomials PF (At , Bt ), F ∈ E. For each integer t ≥ 1 there exists a polynomial Q1t for which deg(Q1t ) ≥ 0, Pt = Qt Q1t .

(4.6)

We may assume without loss of generality that Pt = 0, t = 1, 2 . . .. It follows from the Assertion, conditions (i),(ii), the deﬁnition of Pt , t = 1, 2, . . . and (4.4)-(4.6) that m1 ≤ m(A0 , B0 ) and there exists a bijection τ : {1, . . . , m(A0 , B0 )} → {1, . . . , m(A0 , B0 )} for which Λi = λ0τ (i) , i = 1, . . . , m1 . Together with (4.4) this implies that for all large enough t, |λti − λ0τ (i) | ≤ 4−1 , i = 1, . . . , m1 . This is contradictory to condition (iii). proves the lemma.

The obtained contradiction

We deﬁne n m ) : z ij = 0 (i ∈ {1, . . . , m}, j ∈ Ii )}, K = {z ∈ (R+

(4.7)

A = {(α, X, V, Z) ∈ K1 × (Kn )m × L × K : ˆ −V ∗X ˆ − Z) ˆ (1, 1, . . . , 1) = 0}. (αX

(4.8)

We consider the topological subspace A ⊂ R1 × (Rn )m × L × (Rn )m with the relative topology.

302

TURNPIKE PROPERTIES

Lemma 10.4.2 Let > 0, (α0 , X0 , V0 , Z0 ) ∈ A, ˆ 0 − Zˆ0 ) θ = 0} = {λ(1, 1, . . . , 1) : λ ∈ C}. ˆ 0 − V0 ∗ X {θ ∈ Cm : (α0 X Assume that there exists r0 ∈ Kn such that ˆ 0 − Zˆ0 )r0 = 0. ˆ 0 − V0 ∗ X ||r0 || = 1, (α0 X

(4.9)

Then there exists a neighborhood U of (α0 , X0 , V0 , Z0 ) in A such that for each (α, X, V, Z) ∈ U there exists r ∈ Kn which has the following properties: ˆ −V ∗X ˆ − Z)r ˆ = 0, ||r|| = 1, ||r − r || < , (αX

(4.10)

ˆ −V ∗X ˆ − Z) ˆ θ = 0} = {λ(1, 1, . . . , 1) : λ ∈ C}. (4.11) {θ ∈ Cm : (αX ˆ 0 − Zˆ0 is m − 1 there ˆ 0 − V0 ∗ X Proof. Since the rank of the matrix α0 X ˆ ˆ exists a submatrix B of the matrix α0 X0 − V0 ∗ X0 − Zˆ0 such that the dimension of B is (m − 1) × (m − 1) and the determinant of B is not zero. Set ˆ 0 − Zˆ0 belongs to B}, ˆ 0 − V0 ∗ X I(st) = {i ∈ {1, . . . , m} : ith row of α0 X I(sl) = {i ∈ {1, . . . , n} : ˆ 0 − Zˆ0 belongs to B}, ˆ 0 − V0 ∗ X ith column of α0 X It is easy to see that there exists a neighborhood U0 of the (α0 , X0 , V0 , Z0 ) in A such that for each (α, X, V, Z) ∈ U0 the determinant of the submaˆ −V ∗X ˆ − Z) ˆ which has a dimension (m − 1) × (m − 1) and is trix of (αX generated by the sets I(st), I(sl) is not zero and (4.11) holds. We deﬁne E1 = {x ∈ Rm : xs = 0 for s ∈ {1, . . . , m} \ I(st)},

(4.12)

E2 = {x ∈ Rm : xs = 0 for s ∈ I(st)}, E3 = {x ∈ Rn : xs = 0 for s ∈ {1, . . . , m} \ I(sl)}, E4 = {x ∈ Rn : xs = 0 for s ∈ I(sl)}. Let Pi : Rm → Ei , i = 1, 2, Pi : Rn → Ei , i = 3, 4 be linear projectors. We have by (4.9) that ˆ 0 − Zˆ0 ]P3 r0 ˆ 0 − V0 ∗ X P1 [α0 X

(4.13)

303

A class of diﬀerential inclusions

ˆ 0 − Zˆ0 ]P4 r0 . ˆ 0 − V0 ∗ X = −P1 [α0 X For each (α, X, V, Z) ∈ A we denote by π(α, X, V, Z) the restriction ˆ −V ∗X ˆ − Z) ˆ to E3 . It is easy to see that of P1 (αX π(α, X, V, Z)(E3 ) ⊂ E1 , the operator π(α0 , X0 , V0 , Z0 ) : E3 → E1 is invertible.

(4.14)

There exists a neighborhood U1 ⊂ U0 of (α0 , X0 , V0 , Z0 ) in A such that for each (α, X, V, Z) ∈ U1 the operator π(α, X, V, Z) is invertible. For (α, X, V, Z) ∈ U1 we set ˆ −V ∗X ˆ − Z)P ˆ 4 r0 ] + P4 r0 ∈ Rn . r(α, X, V, Z) = −π(α, X, V, Z)−1 [P1 (αX (4.15) By (4.12)-(4.15), ˆ −V ∗X ˆ − Z)r(α, ˆ X, V, Z) = 0. r(α0 , X0 , V0 , Z0 ) = r0 , P1 (αX

(4.16)

ˆ −V ∗X ˆ − Zˆ is a linear combination Since each row of the matrix αX ij ˆ ˆ ˆ of rows ((αX − V ∗ X − Z) , j = 1, . . . , n), i ∈ I(st) we conclude that ˆ −V ∗X ˆ − Z)r(α, ˆ (αX X, V, Z) = 0. The assertion of the lemma now follows from the continuity of the function (α, V, X, Z) → r(α, V, X, Z), (α, V, X, Z) ∈ U1 . ˆ V ∗X ˆ + Z) ˆ = 0. Clearly P (X, ˆ V ∗X ˆ+ Let (α, X, V, Z) ∈ A, P (X, ˆ ˆ ˆ ˆ Z)(α) = 0. There exist λ1 , . . . , λk ∈ C such that k = m(X, V ∗ X + Z), ˆ V ∗X ˆ + Z)(λ) ˆ P (X, =

k

(λ − λi ), λ ∈ C.

i=1

Set m(α, X, V, Z) = Card{i ∈ {1, . . . , k} : Reλi = α}.

(4.17)

Lemma 10.4.3 Let (α0 , X0 , V0 , Z0 ) ∈ A, α0 > ||V0 || and U be a neighborhood of (α0 , X0 , V0 , Z0 ) in A. Then there exists (α0 , X, V, Z) ∈ U ∩ A such that the vectors (X i1 , . . . , X in ), i = 1, . . . , m are linearly independent. Proof. It is easy to see that for each γ > 0 there exists X(γ) ∈ (Kn )m such that ||X(γ) − X0 || ≤ γ, (X i1 (γ), . . . , X in (γ)), i = 1, . . . , m are linearly independent, ˆ 0 ) (1, 1, . . . , 1) ∈ Kn , ˆ 0 − V0 ∗ X Zˆ0 (1, 1, . . . , 1) = (α0 X ˆ ( γ)) (1, 1, . . . , 1) → ˆ (α0 X(γ) − V0 ∗ X

304

TURNPIKE PROPERTIES

ˆ 0 ) (1, 1, . . . , 1) as γ → 0. ˆ 0 − V0 ∗ X (α0 X There exists γ0 > 0 such that for each γ ∈ (0, γ0 ), ˆ 0 ) (1, 1, . . . , 1) ≤ (α0 X(γ) ˆ ˆ 0 − V0 ∗ X ˆ 2−1 (α0 X (1, 1, . . . , 1) − V0 ∗ X(γ))

ˆ 0 ) (1, 1, . . . , 1). ˆ 0 − V0 ∗ X ≤ 2(α0 X It is easy to see that for each γ ∈ (0, γ0 ) there exists a diagonal matrix D(γ) = (dij (γ)), i, j = 1, . . . , n such that ˆ ˆ 0 ) (1, 1, . . . , 1)] = (α0 X(γ) ˆ 0 − V0 ∗ X ˆ D(γ)[(α0 X (1, 1, . . . , 1), − V0 ∗ X(γ))

dii (γ) ∈ [2−1 , 2], i = 1, . . . , n. n )m as For each γ ∈ (0, γ0 ) we deﬁne Z(γ) ∈ (R+

Z i (γ) = D(γ)Z0i , i = 1, . . . , m, Z(γ) = (Z 1 (γ), . . . , Z m (γ)). Clearly Z(γ) ∈ K, γ ∈ (0, γ0 ). It is easy to see that for each γ ∈ (0, γ0 ) the relation (α0 , X(γ), V0 , Z(γ)) ∈ A holds and (X(γ), Z(γ)) → (X0 , Z0 ) as γ → 0. The lemma is proved. Lemma 10.4.4 Assume that (α0 , X0 , V0 , Z0 ) ∈ A, X0i , i = 1, . . . , m are linearly independent. Then there are M > 0 and a neighborhood U of (α0 , X0 , V0 , Z0 ) in A such that for each (α, X, V, Z) ∈ U and each ˆ , (V ∗ X ˆ + Z) ˆ ) the relation |λ| ≤ M holds. λ ∈ σ(X Proof. Let us assume the converse. Then there exist sequences ∞ {(αt , Xt , Vt , Zt )}∞ t=1 ∈ A, {λt }t=1 ⊂ C such that (αt , Xt , Vt , Zt ) → (α0 , X0 , V0 , Z0 ), |λt | → ∞ as t → ∞, ˆ , (Vt ∗ X ˆ t + Zˆt ) ), t = 1, 2, . . . . λt ∈ σ(X t For each integer t ≥ 1 there exists θt ∈ Cn \ {0} such that ||θt || = 1, ˆ t + Zˆt ) )θt = 0. ˆ t − (Vt ∗ X (λt X We may assume without loss of generality that there exists θ0 ∈ Cn \ {0} for which θt → θ0 as t → ∞. This implies that ˆ 0 θ0 = lim X ˆ t + Zˆ t ) θt = 0. ˆ t θt = lim λ−1 (Vt ∗ X X t t→∞

t→∞

This is contradictory to the linear independence of X0i , i = 1, . . . , m. The obtained contradiction proves the lemma. Lemma 10.4.5 Assume that (α0 , X0 , V0 , Z0 ) ∈ A, X0i , i = 1, . . . , m are ˆ 0 , V0 ∗ X ˆ 0 + Zˆ0 ) = 0, linearly independent, > 0, ||V0 || < 1, α0 , P (X

305

A class of diﬀerential inclusions

ˆ , (V0 ∗ X ˆ 0 + Zˆ0 ) ), λ0 = α0 , Re(λ0 ) = α0 . Then there exists λ01 ∈ σ(X 0 1 1 (α0 , X, V, Z) ∈ A ∩ B((α0 , X0 , V0 , Z0 ), ) for which m(α0 , X, V, Z) < m(α0 , X0 , V0 , Z0 ), X i , i = 1, . . . , m are linearly independent. ˆ V ∗ Proof. We may assume without loss of generality that P (X, ˆ + Z) ˆ = 0 and X i , i = 1, . . . , m are linearly independent for each X (α, X, V, Z) ∈ A ∩ B((α0 , X0 , V0 , Z0 ), ). There are integers k ≥ 1, mi ≥ 1, i = 1, . . . , k, numbers λi ∈ C, i = 1, . . . , k such that ˆ 0 , V0 ∗ X ˆ 0 + Zˆ0 )(λ) = P (X

k

(λ − λ0i )mi , λ ∈ C,

(4.18)

i=0

λ00 = α0 , λ0i = λ0j for each i, j ∈ {0, . . . , k} satisfying i = j. We may assume that there exists an integer k0 ∈ {1, . . . , k} such that Re(λ0i ) = αi if and only if 0 ≤ i ≤ k0 .

(4.19)

There exists δ ∈ (0, ) such that B(λ0i , 8δ) ∩ B(λ0j , 8δ) = ∅ for each i, j ∈ {0, . . . , k}

(4.20)

satisfying i = j, B(λ0i , 8δ) ∩ {y ∈ C : Rey = α0 } = ∅

(4.21)

for each i ∈ {1, . . . , k} satisfying i > k0 . It follows from Lemma 10.4.4 that there exist a neighborhood U1 of (α0 , X0 , V0 , Z0 ) in A and a number M ≥ 8(α0 + + 1 +

k

|λ0i |)

(4.22)

i=0

ˆ , (V ∗ X ˆ + Z) ˆ ) the such that for each (α, X, V, Z) ∈ U1 , each λ ∈ σ(X −1 relation |λ| ≤ 2 M holds. By Lemma 10.4.1 there exists 0 ∈ (0, ) such that for each (α, X, V, Z) ∈ A ∩ B((α0 , X0 , V0 , Z0 ), 0 ) the following properties hold: ˆ V ∗X ˆ + Z)) ˆ ≥ 1, (i) deg(P (X, q(i+1)−1 ˆ V ∗X ˆ + Z)(λ) ˆ P (X, = si=0 ( j=q(i) (λ − λj ))Q(λ), where 0 ≤ s ≤ k, q(0) = 0, q(i + 1) > q(i), i = 0, . . . , s, Q is a polynomial; (ii) if deg(Q) ≥ 1 then for each root z of Q the relation |z| ≥ M holds; (iii) there exists a bijection τ : {0, . . . , k} → {0, . . . , k} such that q(i + 1) − q(i) ≤ m(τ (i)), |λ0τ (i) − λq(i)+j | ≤ δ, 0 ≤ j < q(i + 1) − q(i) − 1, i = 0, . . . , s.

(4.23)

306

TURNPIKE PROPERTIES

It follows from the deﬁnition of U1 , M that for each (α, X, V, Z) ∈ U1 ∩ B((α0 , X0 , V0 , Z0 ), ), properties (i), (ii), (iii) hold with Q(λ) = 1, λ ∈ C. For each u = f + ig ∈ Cm with f, g ∈ Rm we set Reu = f, Imu = g, u ¯ = f − ig.

(4.24)

There exists θ ∈ Cm \ {0} for which ˆ 0 λ01 θ = (V0 ∗ X ˆ 0 + Zˆ0 ) θ. X

(4.25)

ˆ θ = 0. Since vectors X0i , i = 1, . . . , m are linearly independent we have X 0 0 1 0 1 There are two cases: 1) λ1 ∈ R ; 2) λ1 ∈ R . In the ﬁrst case we can assume that θ ∈ Rm \ {0} and vectors θ, (1, 1, . . . , 1) are linearly independent. Consider the case 2) and set f = Reθ, g = Imθ. We will show that vectors f, g, (1, 1 . . . , 1) are linearly independent. Assume the contrary. Then there exist c1 , c2 , c3 ∈ C such that |c1 | + |c2 | + |c3 | = 0, c1 f + c2 g + c3 (1, 1, . . . , 1) = 0.

(4.26)

¯0 ˆ ˆ ˆλ X 0 1 (f − ig) = (X0 + V0 ∗ Z0 ) (f − ig)

(4.27)

It is easy to see that

and vectors f +ig, f −ig are linearly independent over C. Together with (4.26) this implies that there exist c4 , c5 ∈ C for which ¯ (1, 1, . . . , 1) = c4 θ + c5 θ.

(4.28)

Set (4.29) E = {γ1 θ + γ2 θ¯ : γ1 , γ2 ∈ C} ˆ 0 + Zˆ0 ) to E. ˆ , (V0 ∗ X and denote by A, B the restriction of operators X 0 0 0 ¯ ∈ σ(A, B), P (A , B ) = 0, By (4.8), (4.25), (4.27) and (4.29), α0 , λ1 , λ 1 σ(A, B) = C. Then ˆ 0 + Zˆ0 ) ) = C, P (X ˆ 0 , V0 ∗ X ˆ 0 + Zˆ0 ) = 0. ˆ 0 , (V0 ∗ X σ(X This is contradictory to (4.18).The obtained contradiction proves that vectors f, g, (1, 1, . . . , 1) are linearly independent. Let (4.30) 1 ∈ (0, 0 ), γ ∈ (0, 1), 2|1 − γ|α0 < δ. We will show that in both cases if 1 − γ is small enough then there exists a matrix ∆ = (∆ij ), i = 1, . . . , m, j = 1, . . . , n for which ||∆|| < 1 ,

(4.31)

A class of diﬀerential inclusions

ˆ 0 + ∆) − V0 ∗ X ˆ 0 − Zˆ0 ) (1, 1, . . . , 1) = 0, (α0 (X

307 (4.32)

ˆ 0 + ∆) − (V0 ∗ X ˆ 0 + Zˆ0 ) )(θ) = 0. (γλ01 (X It follows from (4.8), (4.25) that (4.32) holds if and only if ˆ θ. ∆ (1, 1, . . . , 1) = 0, (γ∆ )θ = (1 − γ)X 0 It is easy to see that (4.32) holds if and only if ˆ 0i , Reθ >, (4.33) < ∆i , (1, 1, . . . , 1) >= 0, < ∆i , Reθ >= γ −1 (1 − γ) < X ˆ i , Imθ >, i = 1, . . . , n < ∆i , Imθ >= γ −1 (1 − γ) < X 0 ˆ i = (X 1i , . . . , X mi ), i = 1, . . . , n). (here ∆i = (∆1i , . . . , ∆mi ), X 0 0 0 Since in the case (1) θ, (1, 1, . . . , 1) are linearly independent and in the case (2) f, g, (1, 1, . . . , 1) are linearly independent we conclude that there exists a neighborhood W of 1 in R1 such that for each γ ∈ W there exist ∆i (γ) ∈ Rm , i = 1, . . . , n such that (4.33) holds with ∆i = ∆i (γ) for i = 1, . . . , m and moreover ∆i (γ) → 0 as γ → 1 for i = 1, . . . , n.

(4.34)

For each γ ∈ W we deﬁne a matrix ∆(γ) of the dimension m × n which has columns ∆1 (γ), . . . , ∆n (γ). Clearly for each γ ∈ W (4.32) holds with ∆ = ∆(γ). For each γ ∈ W there exists X(γ) ∈ (Rn )m such that ˆ ˆ 0 + ∆(γ). X(γ) =X

(4.35)

By (4.34) we may assume without loss of generality that for each γ ∈ W , ||∆(γ)|| < 1 , X(γ) ∈ (Kn )m .

(4.36)

For γ ∈ W, i = 1, . . . , m, j = 1, . . . , n we set v ij (γ) = v0ij X0ij (X ij (γ))−1 .

(4.37)

By (4.36), (4.34) and (4.35) we may assume without loss of generality that (4.38) V (γ) ∈ L, ||V (γ)|| < α0 for each γ ∈ W. Since for each γ ∈ W (4.32) holds with ∆ = ∆(γ) it follows from (4.32), (4.35), (4.37), (4.8), (4.36), (4.38) that for each γ ∈ W the following relations hold: ˆ ˆ − V (γ) ∗ X(γ) − Zˆ0 ) (1, 1, . . . , 1) = 0, (α0 X(γ)

(4.39)

308

TURNPIKE PROPERTIES

ˆ ˆ − (V (γ) ∗ X(γ) (γλ01 X(γ) + Zˆ0 ) )θ = 0, (α0 , X(γ), V (γ), Z0 ) ∈ A. By (4.34), (4.35), (4.37), (4.39) there exists a neighborhood W1 ⊂ W of 1 in R1 such that W1 ⊂ (1 − δM −1 , 1 + δM −1 ),

(4.40)

(α0 , X(γ), V (γ), Z0 ) ∈ U1 ∩ B((α0 , X0 , V0 , Z0 ), 0 ) for each γ ∈ W1 . It follows from the deﬁnition of 0 , properties (i), (ii), (iii) which holds for each (α, X, V, Z) ∈ U1 ∩ B((α0 , X0 , V0 , Z0 ), ) with Q ≡ 1, (4.40) and (4.22) that for each γ ∈ W1 , m(α0 , X(γ), V (γ), Z0 ) < m(α0 , X0 , V0 , Z0 ). This completes the proof of the lemma. The next auxiliary result plays a crucial role in the proof of Theorem 10.1.1. Lemma 10.4.6 Assume that (α0 , X0 , V0 , Z0 ) ∈ A, > 0, ||V0 || < 1, α0 , ˆ 0 + Zˆ0 ) = 0, ˆ 0 , V0 ∗ X P (X U0 is an open neighborhood of (α0 , X0 , V0 , Z0 ) in A, ˆ 0 − V0 ∗ X ˆ 0 − Zˆ0 )r0 = 0, r0 ∈ Kn , ||r0 || = 1, (αX ˆ 0 − Zˆ0 ) θ = 0} = {λ(1, 1, . . . , 1) : λ ∈ C1 }. ˆ 0 − V0 ∗ X {θ ∈ Cm : (α0 X Then there exists an open nonempty set U1 ⊂ U0 such that for each (α, X, V, Z) ∈ U1 the following properties hold: ˆ V ∗X ˆ + Z) ˆ = 0, there exists r ∈ Kn for which (a) ||V || < α, 1, P (X, ˆ −V ∗X ˆ − Z)r ˆ = 0; ||r|| = 1, ||r − r0 || ≤ , (αX ˆ −V ∗X ˆ − Z) ˆ θ = 0} (b) {θ ∈ Cm : (αX = {λ(1, 1, . . . , 1) : λ ∈ C1 }; ˆ + Z) ˆ ) ∩ {z ∈ C : Rez = α} = {α}. ˆ , (V ∗ X (c) σ(X Proof. By Lemma 10.4.2 we may assume that for each (α, X, V, Z) ∈ U0 properties (a),(b) hold. By Lemma 10.4.3 there exists an open set

309

A class of diﬀerential inclusions

U2 ⊂ U0 such that for each (α, X, V, Z) ∈ U2 the vectors X i , i = 1, . . . , m are linearly independent. There exists (α1 , X1 , V1 , Z1 ) ∈ U2 such that m(α1 , X1 , V1 , Z1 ) = inf{m(α, X, V, Z) : (α, X, V, Z) ∈ U2 }.

(4.41)

By Lemma 10.4.4 there is an open neighborhood U3 ⊂ U2 of (α1 , X1 , V1 , Z1 ) in A and a number M > 0 such that for each (α, X, V, Z) ∈ U3 and each ˆ , (V ∗ X ˆ + Z) ˆ ) the relation |λ| ≤ 2−1 M holds. λ ∈ σ(X There exist integers k ≥ 1, mi ≥ 1, i = 1, . . . , k and numbers λ1i ∈ C, i = 1, . . . , k such that ˆ 1 , V1 ∗ X ˆ 1 + Zˆ1 )(λ) = P (X

k

(λ − λ1i )mi , λ ∈ C,

i=1

λ11 = α1 , λ1i = λ1j for each i, j ∈ {1, . . . , k} satisfying i = j. It follows from Lemma 10.4.5 and (4.41) that Reλ1i = α1 , i ∈ {1, . . . , k} \ {1}.

(4.42)

There exists 1 ∈ (0, 2−1 ) such that B(λ1i , 41 ) ∩ B(λ1j , 41 ) = ∅

(4.43)

for each i, j ∈ {1, . . . , k} satisfying i = j, |Reλ1i − α1 | ≥ 81 , i ∈ {1, . . . , k} \ {1}.

(4.44)

By Lemma 10.4.1, the deﬁnition of U3 , M, 1 (see (4.43),(4.44)) there exists an open neighborhood U1 of (α1 , X1 , V1 , Z1 ) in A such that U1 ⊂ U3 ∩ B((α1 , X1 , V1 , Z1 ), 4−1 1 )

(4.45)

and for each (α, X, V, Z) ∈ U1 the following properties hold: q(i+1)−1 ˆ V ∗X ˆ + Z)(λ) ˆ (d) P (X, = si=1 j=q(i) (λ − λj ), λ ∈ C where 1 ≤ s ≤ k, q(1) = 1, q(i + 1) > q(i), i = 1, . . . , s; (e) there exists a bijection τ : {1, . . . , k} → {1, . . . , k} such that |λ1τ (i) − λq(i)+j | ≤ 1 , 0 ≤ j < q(i+1)−q(i)−1, i = 1, . . . , s, q(i+1)−q(i) ≤ mτ (i) , i = 1, . . . , s. Let (α, X, V, Z) ∈ U1 . We will show that m(α, X, V, Z) = m(α1 , X1 , V1 , Z1 ). It is suﬃcient to show that m(α, X, V, Z) ≤ m(α1 , X1 , V1 , Z1 ). Let numbers s, q(i), i = 1, . . . , s and a bijection τ be as guaranteed in properties

310

TURNPIKE PROPERTIES

ˆ V ∗X ˆ + Z)(α) ˆ (d), (e). By (4.8) P (X, = 0. It follows from properties (d),(e) that there exists i0 ∈ {1, . . . , k} for which |λ1τ (i0 ) − α| ≤ 1 .

(4.46)

(4.43), (4.45), (4.46) and (4.44) imply that |α − α1 | ≤ 4−1 1 , λ1τ (i0 ) = α1 , τ (i0 ) = 1.

(4.47)

Assume that i ∈ {1, . . . , s}, an integer t ∈ {q(i), . . . , q(i + 1) − 1} and Reλt = α. By this relation and property (e), (4.47) |Reλ1τ (i) − α| ≤ |λ1τ (i) − λt | ≤ 1 , |Reλ1τ (i) − α1 | ≤ 21 . Together with (4.44) and (4.47) this implies that τ (i) = 1, i = i0 , {t ∈ {1, . . . , q(s+1)−1} : Reλt = α} ⊂ [q(i0 ), q(i0 + 1) − 1]. It follows from these relations, (4.17), property (e), (4.47) and (4.42) that m(α, X, V, Z) ≤ q(i0 + 1) − q(i0 ) ≤ mτ (i0 ) = m1 = m(α1 , X1 , V1 , Z1 ). Therefore m(α, X, V, Z) = m(α1 , X1 , V1 , Z1 ) for each (α, X, V, Z) ∈ U1 .

(4.48)

Assume that there exists (α2 , X2 , V2 , Z2 ) ∈ U1 which does not have ˆ , (V2 ∗ X ˆ 2 + Zˆ2 ) ) for which property (c). Then there exists λ ∈ σ(X 2 λ = α2 , Reλ = α2 . By Lemma 10.4.5 there exists (α3 , X3 , V3 , Z3 ) ∈ U1 such that m(α3 , X3 , V3 , Z3 ) < m(α2 , X2 , V2 , Z2 ) = m(α1 , X1 , V1 , Z1 ) (see (4.48)). This is contradictory to (4.41). The obtained contradiction proves that each (α, X, V, Z) ∈ U1 has property (c). This completes the proof of the lemma.

10.5.

Proof of Theorem 10.1.1

The next lemma plays an important role in the proof of Theorem 10.1.1. It allows us to construct an open everywhere dense subset of M×L such that for each pair (a, V ) belonging to this subset the suﬃcient condition for the turnpike property holds. Lemma 10.5.1 Assume that a0 ∈ M, V0 ∈ L, α(a0 , V0 ) > ||V0 ||, n m {x ∈ (R+ ) \ {0} : α(a0 , V0 )x ∈ Q(a0 , V0 )(x)} ⊂ (Kn )m , n )m \ {0} : p ∈ Q(a , V ) (α(a , V )p)} ⊂ (K )m . {p ∈ (R+ 0 0 0 0 n Then there exists an open neighborhood U of (a0 , V0 ) in M × L such that for each (a, V ) ∈ U the following properties hold: n )m \ {0} : p ∈ Q(a, V ) (α(a, V )p)} ⊂ (a) α(a, V ) > ||V ||, {p ∈ (R+ m (Kn ) ;

311

A class of diﬀerential inclusions

(b) there exists X(a, V ) ∈ (Kn )m such that ||X(a, V )|| = 1, n m ) : α(a, V )x ∈ Q(a, V )(x)} = {λX(a, V ) : λ ∈ [0, ∞)}; {x ∈ (R+

(c) there exists a unique vector Z(a, V ) = (Z 1 (a, V ), . . . , Z m (a, V )) ∈ n m i i i (R + ) such that Z (a, V ) ∈ a (X (a, V )), i = 1, . . . , m, m i i i i=1 [(α(a, V )In − V )X (a, V ) − Z (a, V )] = 0; m ˆ ˆ V ) − Z(a, ˆ V )) θ = 0} (d) {θ ∈ R : (α(a, V )X(a, V ) − V ∗ X(a, 1 = {λ(1, 1, . . . , 1) : λ ∈ R }; (e) there exists r(a, V ) ∈ Kn such that ||r(a, V )|| = 1, ˆ V ) − V ∗ X(a, ˆ V ) − Z(a, ˆ V ))r(a, V ) = 0, (α(a, V )X(a, (r(a, V ), . . . , r(a, V )) ∈ Q(a, V ) (α(a, V )(r(a, V ), . . . , r(a, V )). Moreover the functions (a, V ) → α(a, V ), X(a, V ), Z(a, V ), ((a, V ) ∈ U ) are continuous. Proof. Clearly there exists an open neighborhood U0 of (a0 , V0 ) in M × L such that for each (a, V ) ∈ U0 property (a) holds, n m ) \ {0} : α(a, V )(x) ∈ Q(a, V )(x)} ⊂ (Kn )m {x ∈ (R+

(5.1)

and moreover a function (a, V ) → α(a, V ), (a, V ) ∈ U0 is continuous. Let (a, V ) ∈ U0 , X ∈ (Kn )m , ||X|| = 1, α = α(a, V ), αX ∈ Q(a, V )(X).

(5.2)

There exist Z i ∈ ai (X i ), i = 1, . . . , m such that m

((αIn − V i )X i − Z i )) = 0.

(5.3)

ˆ −V ∗X ˆ − Z) ˆ θ = 0 (αX

(5.4)

i=1

Therefore

with θ = (1, 1, . . . , 1). Assume that equation (5.4) has a solution which belongs to Rm \ m \ {λ(1, 1, . . . , 1) : λ ∈ R1 }. Then equation (5.4) has a solution θ ∈ R+ ({0} ∪ Km ). This implies that α(θ1 X 1 , . . . , θi X i , . . . , θm X m ) ∈ Q(a, V )(θ1 X 1 , . . . , θi X i , . . . , θm X m ) m and (θi X i )m i=1 ∈ (Kn ) . This contradicts (5.1). The contradiction we have reached proves that {λ(1, 1, . . . , 1) : λ ∈ R1 } is the set of all θ ∈ Rm which satisfy (5.4).

312

TURNPIKE PROPERTIES

n )m satisfy Let p, Y ∈ (R+

||p|| = 1, αY ∈ Q(a, V )(Y ), p ∈ Q(a, V ) (αp).

(5.5)

By Lemma 10.2.3, (5.5) and property (a), there exists r ∈ Kn for which p = (r, . . . , r). It follows from Lemma 10.3.1 and (3.4) that there exm i i i ists θ = (θi )m i=1 ∈ R+ which satisﬁes (5.4) and such that Y = θ X , i = 1, . . . , m. This implies that θ = β(1, 1, . . . , 1) with some β ≥ 0. Therefore Y = βX. We set X(a, V ) = X, Z(a, V ) = Z, r(a, V ) = r. Clearly we have already established properties (a),(b),(d) for (a, V ). Property (c) for (a, V ) follows from relation (c) from Lemma 10.2.2 and property (ii) (see Section 10.1). Relations (c), (d) from Lemma 10.2.2 imply that property (e) holds for (a, V ). It is easy now to see that the functions (a, V ) → X(a, V ), Z(a, V ), (a, V ) ∈ U0 are continuous. This completes the proof of the lemma. n → R1 is called superlinear if it is superadditive A function φ : R+ + n → R1 is called strictly and positively homogeneous. A function φ : R+ + n , φ(x + y) = superlinear if it is superlinear and relations x ∈ Kn , y ∈ R+ φ(x) + φ(y) imply that y ∈ {λx : λ ∈ [0, ∞)}. n → R1 is called a CD-function if A function φ : R+ + 1

n

n φ(x) = γ(x1 )β . . . (xn )β , x ∈ R+

where γ > 0, β ∈ Kn , ni=1 β i = 1. n → R1 is strictly superlinear [75] and for each A CD-function φ : R+ + n → R1 such that p, θ ∈ Kn there exists a unique CD-function φ : R+ + n φ (θ) = p (here φ (θ) = (∂φ/∂xi (θ))i=1 ). (see [86]). For each number γ > 0 and each β ∈ Ks satisfying si=1 β i = 1 we deﬁne a function φ(γ, β) : Rs → R1 as s

φ(γ, β)(x) = γ(

β i (xi )2 )1/2 , x ∈ Rs .

(5.6)

i=1

The function φ(γ, β) is positively homogeneous and subadditive. Letp, θ ∈ Ks . It is easy to see that there exist γ > 0, β ∈ Ks such that si=1 β i = 1, φ(γ, β) (θ) = p. For s ∈ {1, . . . , m} we set Ls = {x ∈ Rn : xi = 0, i ∈ {1, . . . , n} \ Is }. Clearly Ls is isomorﬁc to Rq(s) where q(s) =Card(Is ). Proof of Theorem 10.1.1. Let (a0 .V0 ) ∈ M × L and let W0 be an open neighborhood of (a0 , V0 ) in M × L. In order to prove the theorem it is

A class of diﬀerential inclusions

313

suﬃcient to show that there exists a nonempty open set W ⊂ W0 such that each (a, V ) ∈ W satisﬁes the following conditions: There exists a generalized equilibrium state (α(a, V ), (X, α(a, V )X), p) with α(a, V ) > ||V ||, X, p ∈ (Kn )m ; the mapping Q(a, V ) has the turnpike property. Lemma 10.2.6 implies that there exists (a1 , V1 ) ∈ W0 such that 1, α(a1 , V1 ) > ||V1 ||

(5.7)

n )m \ {0} : α(a , V )x ∈ Q(a , V )(x)} ⊂ (K )m , and there and {x ∈ (R+ 1 1 1 1 n n )m such that Z i ∈ ai (Y i ), i = 1, . . . , m, exist Y ∈ (Kn )m , Z ∈ (R+ Z ij > 0 for each i ∈ {1, . . . , m}, j ∈ Ii and m

(α(a1 , V1 )Y i − V i X i − Z i ) = 0,

i=1 n ) \ {0} : p ∈ Q(a1 , V1 ) (α(a1 , V1 )p)} ⊂ (Kn )m . {p ∈ (R+

By these relations and Lemma 10.5.1 there is an open neighborhood W1 of (a1 , V1 ) in M×L such that W1 ⊂ W0 and that for each (a, V ) ∈ W1 properties (a)-(e) from Lemma 10.5.1 hold. For each (a, V ) ∈ W1 there exist n m X(a, V ) ∈ (Kn )m , Z(a, V ) ∈ (R+ ) , r(a, V ) ∈ Kn

guaranteed by properties (b),(c),(e) from Lemma 10.5.1. In view of Lemma 10.5.1 we may assume that the functions (a, V ) → α(a, V ), X(a, V ), Z(a, V ), (a, V ) ∈ W1 are continuous. By (5.7) and properties (b), (c), (e) from Lemma 10.5.1, we may assume without loss of generality that for each (a, V ) ∈ W1 , Z sj (a, V ) > 0, s ∈ {1, . . . , m}, j ∈ Is , ||V || < 1.

(5.8)

Put X1 = X(a1 , V1 ), Z1 = Z(a1 , V1 ), α1 = α(a1 , V1 ), r1 = r(a1 , V1 ). (5.9) ˆ 1 , V1 ∗ X ˆ 1 + Zˆ1 ) = Clearly (α1 , X1 , V1 , Z1 ) ∈ A. We will show that P (X 0. Let us assume the converse. Then there exist a sequence of numbers 1 ∞ n {λs }∞ s=1 ⊂ R and a sequence {θs }s=1 ⊂ R \ {0} such that λs > λs+1 > α1 , s = 1, 2, . . . , λs → α1 as s → ∞,

(5.10)

314

TURNPIKE PROPERTIES

ˆ 1 − Zˆ1 ) θs = 0, s = 1, 2, . . . . ˆ 1 − V1 ∗ X ||θs || = 1, (λs X In view of (5.9) and property (d) from Lemma 10.5.1, we may assume that θs → (1, 1, . . . , 1) as s → ∞. There exists an integer q ≥ 1 such that θq ∈ Km . Relations (5.9) and (5.10) imply that i i m λq ((θqi X1i )m i=1 ) ∈ Q(a1 , V1 )((θq X1 )i=1 ),

λq ≤ α(a1 , V1 ) = α1 < λq , a contradiction. The contradiction we have reached proves that ˆ 1 + Zˆ1 ) = 0. ˆ 1 , V1 ∗ X P (X

(5.11)

(5.8) implies that there is an open neighborhood U∞ of (α1 , X1 , V1 , Z1 ) in A such that for each (α, X, V, Z) ∈ U∞ , Z si > 0 (s = 1, . . . , m, i ∈ Is ), α > ||V ||.

(5.12)

Let ∈ (0, 1). Put U () = B((α1 , X1 , V1 , Z1 ), ) ∩ U∞ .

(5.13)

It follows from Lemma 10.4.6, properties (d), (e) from Lemma 10.5.1, (5.12), (5.11) and (5.8) that there exists a nonempty open set U1 () ⊂ U () such that each (α, X, V, Z) ∈ U1 () has properties (a),(b),(c) from Lemma 10.4.6 with r0 = r1 . Fix (α , X , V , Z ) ∈ U1 (). By property (a) from Lemma 10.4.6 there exists r ∈ Kn such that ˆ − Zˆ )r = 0. ˆ − V ∗ X ||r || = 1, ||r − r1 || ≤ , (α X

(5.14)

It follows from (5.12) and (5.13) that Zsi > 0, s = 1, . . . , m, j ∈ Is .

(5.15)

In view of (5.15) for i ∈ {1, . . . , m} there are diagonal matrices D1 (i, ) and D2 (i, ) of the dimension n × n such that D1 (i, )Xi = X1i , D2 (i, )Z1i = Zi , D2 (i, )y = y for each y ∈ Rn satisfying y j = 0, j ∈ Ii .

(5.16)

A class of diﬀerential inclusions

315

For each j ∈ {1, . . . , m} there exist a function φj : Lj → R1 of the n → R1 such that form (5.6) and a CD-function ψj : R+ (φj ) (Zj ) is a projection of r to Lj ,

(5.17)

(ψj ) (Xj ) =< r , Zj >−1 (α In − Vj )r . n deﬁne Let γ, Γ ∈ (0, 1). For i = 1, . . . , m, x ∈ R+

[ai ()](x) = {u ∈ D2 (i, )ai1 (D1 (i, )x) :< (α In − Vi )r , x >≥< r , u >}, (5.18) [ai (, γ)](x) = {u ∈ D2 (i, )ai1 (D1 (i, )x) : < (α In − Vi )r , x >≥ γ < r , u > +(1 − γ)φi (u)}, [ai (, γ, Γ)](x) = [Γai (, γ)](x) + (1 − Γ)ψi (x){u : 0 ≤ u ≤ Zi }. It is not diﬃcult to see that the mappings ai (), ai (, γ), ai (, γ, Γ), i = 1, . . . , m are normal convex processes and their graphs are closed. We will show that a(, γ, Γ) = (ai (, γ, Γ))m i=1 ∈ M. It is suﬃcient to show that for each s ∈ {1, . . . , m} the mapping as (, γ, Γ) has property (ii) (see Section 10.1). Suppose that x, g ∈ Kn and i ∈ {1, . . . , m}. It is suﬃcient to show that the problem < g, u >→ max, u ∈ [ai (, γ)](x) has a unique solution. Let (5.19) τ = sup{< g, u >: u ∈ [ai (, γ)](x)}, τ0 = sup{< g, u >: u ∈ D2 (i, )ai1 (D1 (i, )x)}, u0 ∈ D2 (i, )ai1 (D1 (i, )x), < g, u0 >= τ0 . We may assume that τ < τ0 . Let u1 , u2 ∈ [ai (, γ)](x), u1 = u2 , < g, us >= τ, s = 1, 2. Set u3 = 2−1 (u1 + u2 ). It is easy to see that u3 ∈ [ai (, γ)](x), < g, u3 >= τ , γ < r , u3 > +(1 − γ)φi (u3 ) << (α In − Vi )r , x > . For t ∈ (0, 1) we deﬁne ut = tu3 + (1 − t)u0 . It is easy to verify that if |t − 1| is small enough then ut ∈ [ai (, γ)](x), < g, ut >> τ . This contradicts (5.19). The contradiction we have reached proves that ai (, γ, Γ) has property (ii). Therefore a(, γ, Γ) ∈ M. For each , γ, Γ ∈ (0, 1) we have deﬁned a(, γ, Γ) ∈ M, an open nonempty set U1 () ⊂ B((α1 , X1 , V1 , Z1 ), )∩U∞ , (α , X , V , Z ) ∈ U1 () and r ∈ Kn . Let δ > 0. We will show that there exist ∈ (0, δ), γ, Γ ∈ (0, 1) such that d(ai (, γ, Γ), ai1 ) < δ, i = 1, . . . , m.

316

TURNPIKE PROPERTIES

Choose λ > 1 such that n , (λ2 − 1) sup{||y|| : y ∈ ai1 (x), x ∈ R+

(5.20)

||x|| ≤ 1, i = 1, . . . , m} < 3−1 δ. It follows from (5.12), (5.14), (5.16) and the deﬁnition of (α , X , V , Z ) that there is a positive number < δ such that λ−1 r1 ≤ r ≤ λr1 , λ−1 In ≤ D1 (i, ), D2 (i, ) ≤ λIn ,

(5.21)

λ−1 (α1 In − V1i )r1 ≤ (α In − Vi )r ≤ λ(α1 In − V1i )r1 , i = 1, . . . , m. We will show that d(ai (), ai1 ) < 3−1 δ, i = 1, . . . , m. n , i ∈ {1, . . . , m}, y ∈ ai (x). In view of (5.9), Lemma 10.2.3 Let x ∈ R+ 1 and property (e) from Lemma 10.5.1,

< (α1 In − V1i )r1 , x >≥< r1 , y > .

(5.22)

D2 (i, )ai1 (D1 (i, )x) ⊃ λ−2 ai1 (x) λ−2 y.

(5.23)

(5.21) implies that

The inequalities (5.21) and (5.22) imply that < (α In − Vi )r , x > ≥ λ−1 < (α1 In − V1i )r1 , x > ≥ λ−1 < r1 , y >≥ λ−2 < r , y > . Combined with (5.23) and (5.18) this implies that λ−2 y ∈ [ai ()](x), λ−2 ai1 (x) ⊂ [ai ()](x).

(5.24)

It follows from (5.18) and (5.21) that [ai ()](x) ⊂ D2 (i, )ai1 (D1 (i, )x) ⊂ λ2 ai1 (x). Combined with (5.20) and (5.24) this inclusion implies that d(ai (), ai1 ) < 3−1 δ, i = 1, . . . , m.

(5.25)

We will show that there exists γ ∈ (0, 1) for which d(ai (), ai (, γ)) < 3−1 δ, i = 1, . . . , m.

(5.26)

317

A class of diﬀerential inclusions

It follows from the deﬁnition of φi (see (5.17)), (5.18) that for i ∈ {1, . . . , m}, x ∈ Rn , numbers γ1 , γ2 satisfying 0 < γ1 < γ2 < 1 the following inclusion holds: [ai (, γ1 )](x) ⊂ ai (, γ2 )(x) ⊂ ai ()(x). On the other hand for each n , i ∈ {1, . . . , m}, y ∈ [ai ()](x), λ∗ ∈ (0, 1) x ∈ R+

there exists γ ∈ (0, 1) such that λ∗ y ∈ ai (, γ)(x). This implies the existence of γ ∈ (0, 1) for which (5.26) is valid. It is easy to see that there exists Γ ∈ (0, 1) such that d(ai (, γ, Γ), ai (, γ)) < 3−1 δ, i = 1, . . . , m. In view of this inequality, (5.25) and (5.26), d(ai1 , ai (, γ, Γ)) < δ, i = 1, . . . , m.

(5.27)

We have shown that for each δ ∈ (0, 1) there exist ∈ (0, δ), γ, Γ ∈ (0, 1) such that (5.27) holds. Therefore there exists 0 , Γ0 , γ0 ∈ (0, 1) such that (a(0 , γ0 , Γ0 ), V0 ) ∈ W1 .

(5.28)

We will show that α0 X0 ∈ Q(a(0 , γ0 , Γ0 ), V0 )X0 , α0 = α(a(0 , γ0 , Γ0 ), V0 ).

(5.29)

It follows from (5.17), (5.18) and Lemma 10.2.1 that (r0 , . . . , r0 ) ∈ Q(a(0 , γ0 , Γ0 ), V0 ) (α0 (r0 , . . . , r0 )).

(5.30)

By (5.16), property (c) from Lemma 10.5.1, (5.9), (5.17), (5.14) and (5.18) for i ∈ {1, . . . , m}, Zi0 ∈ D2 (i, 0 )ai1 (D1 (i, 0 )Xi0 ),

γ0 < r0 , Zi0 >

(5.31)

+(1 − γ0 )φi0 (Zi0 ) = γ0 < r0 , Zi0 > +(1 − γ0 ) << (φi0 ) (Zi0 ), Zi0 > =< r0 , Zi0 >=< (α0 In − Vi0 )r0 , Xi0 >, Zi0 ∈ ai (0 , γ0 )(Xi0 ). (5.31), (4.8) and (5.30) implies (5.29). Since (α0 , X0 , V0 , Z0 ) ∈ A it follows from (5.29), (5.28) and properties (b),(c) from Lemma 10.5.1 which holds for each (a, V ) ∈ W1 that ||X0 ||−1 X0 = X(a(0 , γ0 , Γ0 ), V0 ), ||X0 ||−1 Z0 = Z(a(0 , γ0 , Γ0 ), V0 ). (5.32)

318

TURNPIKE PROPERTIES

Note that the functions (a, V ) → X(a, V ), Z(a, V ), α(a, V ), (a, V ) ∈ W1 are continuous and properties (b),(c) from Lemma 10.5.1 hold for each (a, V ) ∈ W1 . Therefore there exists an open neighborhood W of (a(0 , γ0 , Γ0 ), V0 ) in M × L such that W ⊂ W1 and for each (a, V ) ∈ W , (α(a, V ), ||X0 ||X(a, V ), V, ||X0 ||Z(a, V )) ∈ U1 (0 ).

(5.33)

Let (a, V ) ∈ W . It follows from the deﬁnition of W, W1 that properties (a)-(e) from Lemma 10.5.1 hold for (a, V ) and (5.33) holds. By properties (a),(d) from Lemma 10.5.1, ˆ V ) − V ∗ X(a, ˆ V) α(a, V ) > ||V ||, {θ ∈ Rm : (α(a, V )X(a,

(5.34)

ˆ V )) θ = 0} = {λ(1, 1, . . . , 1) : λ ∈ R1 }. −Z(a, It follows from (5.33), the deﬁnition of U1 (0 ) and property (c) from Lemma 10.4.6 which holds with (α, X, V, Z) = (α(a, V ), ||X0 ||X(a, V ), V, ||X0 ||Z(a, V )) that

ˆ (a, V ), (V ∗ X(a, ˆ V ) + Z(a, ˆ V )) ) ∩ {z ∈ C : σ(X Rez = α(a, V )} = {α(a, V )}.

By this relation,(5.34), properties (a)-(e) from Lemma 10.5.1 which hold for (a, V ), Lemma 10.3.2, the mapping Q(a, V ) has the turnpike property. This completes the proof of the theorem.

10.6.

Example

In this section we will construct (a, V ) ∈ M × L such that there exists an equilibrium state (α(a, V ), (X, α(a, V )X), p) of Q(a, V ) satisfying α(a, V ) > ||V ||, X, p ∈ (Kn )m and such that Q(a, V ) does not have the turnpike property. We recall that Rez = x, Imz = y for z = x + iy ∈ C. Proposition 10.6.1 Let V ∈ L, a = (a1 , . . . , am ) ∈ M, Q = Q(a, V ), α = α(a, V ) > ||V ||, X ∈ (Kn )m , r ∈ Kn , p = (r, . . . , r) ∈ (Kn )m , αX ∈ Q(X), p ∈ Q (αp), Z i ∈ ai (X i ), i = 1, . . . , m,

(6.1)

A class of diﬀerential inclusions m

319

((αIn − V i )X i − Z i ) = 0.

i=1

Assume that there exist θ ∈ C, λ ∈ Cm \ {0} such that ˆ λ = (V ∗ X ˆ + Z) ˆ λ. Reθ = α, θ = α, θX

(6.2)

Then Q does not have the turnpike property. j m Proof. Set Reλ = (Reλj )m j=1 , Imλ = (Imλ )j=1 . For each γ ∈ (0, 1) n m j m we deﬁne yγ = (yγj )m j=1 : [0, ∞) → (R+ ) , uγ = (uγ )j=1 : [0, ∞) → n m (R+ ) as follows:

yγj (t) = [exp((α − 1)t) + γRe(exp((θ − 1)t)λj )]X j ,

(6.3)

ujγ (t) = [exp((α − 1)t) + γIm(exp((θ − 1)t)λj )]X j , t ∈ [0, ∞), j = 1, . . . , m. It is easy to verify that for each small enough γ > 0 the functions yγ , uγ are trajectories of Q which have the von Neumann growth rate. Assume that the mapping Q has the turnpike property. Then for small enough γ > 0 there exist numbers d1γ , d2γ > 0 such that limt→∞ exp((1 − α)t)yγ (t) = d1γ X, limt→∞ exp((1−α)t)uγ (t) = d2γ X. Together with (6.3) this implies that for j = 1, . . . , m, limt→∞ Re(exp((θ − 1 − α)t)λj ) = γ −1 (d1γ − 1), limt→∞ Im(exp((θ − 1 − α)t)λj ) = γ −1 (d2γ − 1), limt→∞ (exp((θ − 1 − α)t)λj ) = γ −1 (d1γ − 1) + iγ −1 (d2 γ − 1). This is contradictory to (6.2). The obtained contradiction proves that Q does not have the turnpike property. The proposition is proved. Assume that m, n = 3, P = ∅, Is = {s}, s = 1, . . . , n, e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1). Assume that a number β > 0 is large enough. We set 2 2 cij β = (3 + β )/(3(1 + β )),

(i, j) ∈ {(1, 1), (2, 2), (3, 3), (3, 2), (1, 2), (2, 3), (2, 1)}. It is easy to see that 11 2 2 −1 2 −1 2 − 3(c11 4−1 [2c11 β + ((cβ ) (β + 1)) β (1 + β )) ] 2 2 −1 > (c11 β ) − 3(1 + β ) .

(6.4)

320

TURNPIKE PROPERTIES

31 This implies that there exist c13 β , cβ > 0 which satisfy 13 11 2 2 −1 13 31 c31 β cβ = (cβ ) − 3(1 + β ) , cβ + cβ

(6.5)

2 11 2 −1 = 2c11 β + [(β + 1)(cβ ) ] 2 −1 −3[c11 β (1 + β )] .

Consider the matrix Cβ = (cij β ), i, j = 1, 2, 3. It follows from (6.4), (6.5) that the polynomial det(Cβ − θI3 ), θ ∈ C has roots 1, (1 − iβ)(1 + β 2 )−1 , (1 + iβ)(1 + β 2 )−1 . This implies that the polynomial det(θCβ − I3 ), θ ∈ C has roots 1, 1 + iβ, 1 − iβ. There exist Λ, r ∈ K3 such that

Set

Cβ Λ = Λ, Cβ r = r.

(6.6)

i2 i3 X i = Λi (ci1 β , cβ , cβ ), i = 1, 2, 3.

(6.7)

3 → R1 such that For i = 1, 2, 3 there exists a CD-function φi : R+ + i2 i3 i −1 (φi ) (ci1 β , cβ , cβ ) = (r ) r.

(6.8)

n . It is easy to verify Set V = 0, ai (x) = [0, φi (x)]ei , i = 1, 2, 3, x ∈ R+ that (a, V ) ∈ M × L, X ∈ Q(a, V )(X). (6.9)

By Lemma 10.2.1 and (6.8) (r, r, r) ∈ Q(a, V ) (r, r, r).

(6.10)

(6.10), (6.9) and (6.7) imply that α(a, V ) = 1. Since the polynomial det(θCβ − I3 ), θ ∈ C has a root 1 + iβ there exists λ = (λ1 , λ2 , λ3 ) ∈ C3 \ {0} for which (1 + iβ)Cβ λ = λ. Together with (6.7) this implies that ˆ (λi (Λi )−1 )3 = (1 + iβ) (1 + iβ)X i=1

3

X j (λj (Λj )−1 )

j=1

= (1 + iβ)

3 cjβ λj = (1 + iβ)Cβ λ = λ = Zˆ (λi (Λi )−1 )3i=1

j=1

Zi

φi (X i )ei

where = = Λi ei , i = 1, 2, 3. Together with (6.9), (6.10) and Proposition 10.6.1 this implies that Q(a, V ) does not have the turnpike property.

Chapter 11 CONVEX PROCESSES

In this chapter we study the dynamic properties of optimal trajectories n of convex processes G : K → 2R where K ⊂ Rn is a closed convex cone. We show that optimal trajectories of a convex process G spend most of the time in a small neighborhood of a von Neumann path. The turnpike theorem that we obtain generalizes the result of Rubinov [75] which n n → 2R+ n is the where R+ was established for a convex process G : R+ n cone of the elements of the Euclidean space R that have nonnegative coordinates. Also, we show that the turnpike phenomenon is stable under small perturbations of the convex process G.

11.1.

Preliminaries

In this chapter we investigate the turnpike property for optimal trajectories of convex processes. For each metric space Q denote by Π(Q) the collection of nonempty compact subsets of Q endowed with the Hausdorﬀ metric d(·, ·). Let K ⊂ Rn be a closed convex cone (K + K ⊂ K, λK ⊂ K, λ > 0) with nonempty interior in Rn which does not contain any nontrivial subspace of Rn . A set-valued mapping G : K → Π(Rn ) is a convex process if it satisﬁes λG(x) + µG(y) ⊂ G(λx + µy) for each x, y ∈ K and each λ, µ ≥ 0, or equivalently, if its graph is a convex cone. Convex processes are the set-valued analogs of linear mappings and certain properties of the latter can be also established for the former [48, 71, 75, 77].

322

TURNPIKE PROPERTIES

Denote by M the set of all convex processes G : K → Π(Rn ) such that the set graph(G) = {(x, v) ∈ Rn × Rn : x ∈ K, v ∈ G(x)} is closed and G(x) ∩ K = ∅ for each x ∈ K. Denote by (·, ·) the scalar product in Rn and by | · | the Euclidean norm in Rn . We say that x ≤ y (x, y ∈ Rn ) if and only if y − x ∈ K. Set K ∗ = {η ∈ Rn : (η, x) ≥ 0 for every x ∈ K}. It is a standard result (see [43]) that K ∗ has a nonempty interior. The interior of a set S is denoted by intS. Let G ∈ M. A real number λ0 is called an eigenvalue of G if there exists x0 ∈ K \{0} such that λ0 x0 ∈ G(x0 ). We then call x0 an eigenvector of G. A sequence {xt }Tt=0 ⊂ K where T ∈ {1, 2, . . .} ∪ {∞} is called a trajectory of G if xt+1 ∈ G(xt ) for all nonnegative integers t < T . In this chapter we study the dynamics properties of optimal trajectories of a convex process G ∈ M. We show that optimal trajectories of G spend most of the time in a small neighborhood of the set {λX : λ ∈ [0, ∞)} where X is a unique (up to scalar multiplication) eigenvector corresponding to the maximal eigenvalue of G. The turnpike theorem that we prove generalizes the result of Rubinov n → Π(Rn ) where [75] which was established for a convex process G : R+ + n R+ is the cone of the elements of the Euclidean space Rn that have nonnegative coordinates. The results of this chapter were obtained in [101]. The chapter is organized as follows. In Section 11.2 we discuss a suﬃcient condition for the asymptotic turnpike property established by Leizarowitz [43]. Section 11.3 contains statements of our turnpike results which are proved in Section 11.4. In Section 11.5 we discuss the stability of the turnpike phenomenon and state our stability results which are proved in Section 11.6.

11.2.

Asymptotic turnpike property

Let K be a convex closed cone with nonempty interior in Rn which does not contain any nontrivial subspace of Rn . Consider a convex process G : K → Π(Rn ) which is a continuous mapping. It may happen that there are no eigenvalues of G in K at all. However, it is easy to see that the set of eigenvalues of G in K is compact. Denote by αG the maximal eigenvalue of G. Set α0 = inf ∗ sup {(η, v)(η, x)−1 : v ∈ G(x)}. η∈K x∈K

(2.1)

323

Convex processes

Leizarowitz [43] studied the existence of eigenvectors of convex processes and established the following result. Proposition 11.2.1 Suppose that if x ∈ ∂K \ {0} and η ∈ ∂K ∗ \ {0} are such that (η, x) = 0, then (η, v) > 0 for some v ∈ G(x). Then the inﬁmum in (2.1) is attained inside intK ∗ and there is x0 ∈ K \ {0} for which α0 x0 ∈ G(x0 ) and α0 = αG . Assume that (X, αG X) ∈ G. We say that G has the asymptotic turnpike property if for any trajectory {xt }∞ t=1 of G there exists a number c0 ≥ 0 such that −t xt = c0 X. lim αG t→∞

The asymptotic turnpike property for convex processes was studied in [48, 75, 86]. The following result which is a discrete-time analog of Theorem 6.3 in [43] gives the necessary and suﬃcient condition for the asymptotic turnpike property. Theorem 11.2.1 Suppose that there is y ∈ K \{0} for which a(y)∩(K \ {0}) = ∅ and if x ∈ ∂K \{0}, η ∈ ∂K ∗ \{0} are such that (η, x) = 0, then (η, v) > 0 for some v ∈ G(x). Assume that λ0 , the maximal eigenvalue of G, corresponds to a unique (up to multiplication by positive numbers) eigenvector denoted X. Then for every trajectory {xt }∞ t=0 of G there −t exists a number c0 ≥ 0 such that limt→∞ λ0 xt = c0 X if and only if for each trajectory {zt }∞ t=0 of G of the form zt = λt0 (cos(ωt)a + sin(ωt)b + X), t = 0, 1, . . . for some a, b ∈ Rn , ω ∈ R1 the following relation holds: zt = c0 λ0 X, t = 0, 1, . . . with some constant c0 ≥ 0. The proof of Theorem 11.2.1 is analogous to the proof of Theorem 6.3 in [43]. One of the key ingredients in this proof is provided by the following result which is a discrete-time analog of Theorem 3.2 in [38]. Proposition 11.2.2 Let F ⊂ Rn × Rn be a convex compact set, domF = {x ∈ Rn : there exists y ∈ Rn such that (x, y) ∈ F }, F (x) = {y ∈ Rn : (x, y) ∈ F }, x ∈ domF . Assume that the following properties hold: (i) there is a unique Y ∈ domF such that Y ∈ F (Y ); (ii) if a, b ∈ Rn , α = 0 and a sequence zj = Y +cos(αj)a+sin(αj)b, j = 0, 1 . . . satisﬁes (zj , zj+1 ) ∈ F, j = 0, 1, . . . , then zj = Y, j = 0, 1, . . ..

324

TURNPIKE PROPERTIES

Then for each > 0 there exists an integer N () ≥ 1 such that for n each sequence {zj }∞ j=0 ⊂ R which satisﬁes (zj , zj+1 ) ∈ F, j = 0, 1, . . ., the relation |zj − Y | ≤ holds for all j ≥ N ().

11.3.

Turnpike theorems

Let G ∈ M. For each x ∈ K and each integer T ≥ 1 denote aT (x) = {y ∈ Rn : there exists a trajectory {xt }Tt=0 of G for which x0 = x, xT = y}. Evidently aT (x) is a convex compact set for each x ∈ K and each integer T ≥ 1. Let T be a natural number. A trajectory {xt }Tt=0 of G is optimal if there exists f ∈ K ∗ \ {0} such that (f, xT ) = sup{(f, y) : y ∈ aT (x)}. For x ∈ Rn , r > 0 set B(x, r) = {y ∈ Rn : |x − y| ≤ r}. Suppose that G ∈ M, αG > 0, X ∈ intK, p ∈ intK ∗ , |X| = 1, αG X ∈ G(X),

(3.1)

αG is a maximal eigenvalue of G and (αG p, x) ≥ (p, y) for each x ∈ K and each y ∈ G(x). We can easily prove the following result. Proposition 11.3.1 Assume that x0 ∈ intK and {xt }Tt=0 is a trajectory of G, T ≥ 1. Then this trajectory is optimal if and only if for each λ > 1 the relation λxT ∈ aT (x0 ) − K holds. Suppose that for each trajectory {xt }∞ t=0 ⊂ G there exists a number c0 ≥ 0 such that −t xt → c0 X as t → ∞. (3.2) αG For x ∈ K we deﬁne λ(x) = sup{β ∈ [0, ∞) : there exists a trajectory {xt }∞ t=0 of G (3.3) −t for which x0 = x, lim αG xt = βX}. t→∞

It is easy to verify that for each x ∈ K the number λ(x) is ﬁnite and there −t exists a trajectory {xt }∞ t=0 of G for which x0 = x and αG xt → λ(x)X as t → ∞. It is easy to see that λ(cx) = cλ(x) for each x ∈ K, c > 0, each x ∈ intK is a continuity point of the function λ : K → [0, ∞) and λ(x) ≤ λ(y) for each x, y ∈ K satisfying x ≤ y.

325

Convex processes

We will establish the following results which describe the structure of optimal trajectories of G. Theorem 11.3.1 Assume that a compact nonempty set Q ⊂ int K and > 0. Then there exists an integer L ≥ 1 such that for each integer T ≥ 2L and each optimal trajectory {xt }Tt=0 of G satisfying x0 ∈ Q the following relation holds: −t xt | ≤ , t = L, . . . , T − L. |λ(x0 )X − αG

Theorem 11.3.2 Let > 0. Then there exist an integer L ≥ 1 and a number δ > 0 such that for each integer T ≥ L and each optimal trajectory {xt }Tt=0 of G satisfying |x0 − X| ≤ δ the following relation holds: −t xt − X| ≤ , t = 0, . . . , T − L. |αG

11.4.

Proofs of Theorems 11.3.1 and 11.3.2

There exists r0 > 0 such that B(X, r0 ) ⊂ K, B(p, r0 ) ⊂ K ∗ .

(4.1)

The next auxiliary result shows that if {xt }Tt=0 is a trajectory of G and x0 , xT are close to X, then xt is close to X for all t = 0, . . . , T . Lemma 11.4.1 Let αG = 1 and be a positive number. Then there exists δ > 0 such that if an integer T ≥ 2 and if a trajectory {xt }Tt=0 of G satisﬁes x0 , xT ∈ B(X, δ), then xt ∈ B(X, ) for t = 1, . . . , T − 1. Proof. Let us assume the converse. Then for each δ ∈ (0, ) there T (δ) exists an integer T (δ) ≥ 2 and a trajectory {xt (δ)}t=0 of G such that xi (δ) ∈ B(X, δ), i = 0, T (δ), sup{|xt (δ) − X| : t = 1, . . . , T (δ) − 1} > . (4.2) ∞ ⊂ (0, 1) such that ⊂ (0, 1), {β } Choose sequences {γi }∞ i i=1 i=1 ∞ i=1

γi > 0, 2r0−1 (p, X)(1 −

∞ i=1

γi2 ) < 64−1 ,

∞

βi < (128(p, X))−1 r0 .

i=1

(4.3) There exists a monotone decreasing sequence of positive numbers {δi }∞ i=1 such that δ0 ≤ 32−1 , γi ≤ 1 − δi r0−1 , i = 1, 2, . . . ,

(4.4)

326

TURNPIKE PROPERTIES

(1 + δi r0−1 )2 (1 − δi r0−1 )−2 ≤ 1 + βi , i = 1, 2, . . . . (4.1) implies that for any natural number i, B(X, δi ) ⊂ {y ∈ Rn : (1 − δi r0−1 )X ≤ y ≤ (1 + δi r0−1 )X}. Put τj =

j

(4.5)

T (δi ), j = 1, 2, . . . .

(4.6)

i=1

Deﬁne a sequence {yt }∞ t=0 ⊂ K as follows: yt = xt (δ1 ), t = 0, . . . , T (δ1 ), yτj +t =

j i=1

(4.7)

((1 − δi r0−1 )(1 + δi r0−1 ))−1 xt (δj+1 ),

t = 1, . . . , T (δj+1 ), j = 1, 2, . . . Put ∆j =

j i=1

((1 − δi r0−1 )(1 + δi r0−1 )−1 ), j = 1, 2, . . . .

(4.8)

It follows from (4.7), (4.5), (4.6) and (4.2) that (1 − δ1 r0−1 )X ≤ y0 , yτ1 ≤ (1 + δ1 r0−1 )X, j i=1

≤

j i=1

(4.9)

[(1 − δi r0−1 )(1 + δi r0−1 )−1 ](1 − δj+1 r0−1 ) ≤ yτj+1

[(1 − δi r0−1 )(1 + δi r0−1 )−1 ](1 + δj+1 r0−1 )X, j = 1, 2, . . . , yτj ≥

j i=1

((1 − δi r0−1 )(1 + δi r0−1 )−1 )x0 (δj+1 ),

yτj ≤ (1 − δj r0−1 )−2 (1 + δj r0−1 )2 ∆j x0 (δj+1 ), j = 1, 2, . . . . For each natural number q there is a sequence {ut (q)}∞ t=0 ⊂ K such that ut (q) = 0, t = 0, . . . , τq − 1, uτq (q) = yτq − ∆q x0 (δq+1 ), ut+1 (q) ∈ G(ut (q)), t = τq , τq + 1, . . . .

(4.10)

327

Convex processes

Deﬁne a sequence {zt }∞ t=0 ⊂ K by zt = yt , t = 0, . . . , τ1 , zτq +t = yτq +t +

q j=1

uτq +t (j),

(4.11)

t = 1, . . . , T (δq+1 ), q = 1, 2, . . . . In view of (4.11), (4.7) and (4.10), zt+1 ∈ G(zt ), t = 0, 1, . . .. Therefore there exists a constant c0 ≥ 0 such that lim zt = c0 X.

(4.12)

t→∞

We will show that |zτ (q)+t − xt (δq+1 )| ≤ 16−1 , t = 1, . . . , T (δq+1 ), q = 1, 2, . . . .

(4.13)

(4.1) implies that −|x|r0−1 X ≤ x ≤ |x|r0−1 X for each x ∈ Rn \ {0},

(4.14)

|x| ≤ r0−1 (p, x) for each x ∈ K. By (4.7), (4.4), (4.14), (4.2) and (4.5), for each q ∈ {1, 2, . . .} and each t ∈ {1, . . . , T (δq+1 )}, |yτq +t − xt (δq+1 )| ≤ |xt (δq+1 )|(1 − ≤ |xt (δq+1 )|(1 −

∞ i=1

≤ r0−1 (1 −

∞ i=1

∞ i=1

(1 − δi r0−1 )2 )

γi2 ) ≤ r0−1 (p, xt (δq+1 ))(1 −

γi2 )(p, x0 (δq+1 )) ≤ r0−1 (1 −

∞ i=1

∞

(4.15)

γi2 )

i=1

γi2 )(1 + δq+1 r0−1 )(p, X).

It follows from (4.9), (4.10), (4.14), (4.2) and (4.5) that for each natural number q ≥ 1 and each nonnegative integer t, |ut (q)| ≤ r0−1 (p, ut (q)) ≤ r0−1 (p, uτq (q)) ≤ r0−1 (p, x0 (δq+1 ))∆q [(1 + δq r0−1 )2 (1 − δq r0−1 )−2 − 1] ≤ r0−1 ∆q [(1 + δq r0−1 )2 (1 − δq r0−1 )−2 − 1](1 + δq+1 r0−1 )(p, X). Combined with (4.8) and (4.4) this implies that |ut (q)| ≤ 2r0−1 (p, X)βq

328

TURNPIKE PROPERTIES

for each natural number q and each nonnegative integer t. It follows from this inequality and (4.11) that for each natural number q and each t ∈ {1, . . . , T (δq+1 )}, |yτq +t − zτq +t | ≤ 2r0−1 (p, X)

∞

βi .

i=1

It follows from this relation, (4.15), (4.4) and (4.3) that (4.13) holds for each natural number q and each t ∈ {1, . . . , T (δq+1 )}. Combined with (4.2), (4.6) and (4.4) this implies that for each integer q ≥ 2, |zτ (q) − X| ≤ 3 · 32−1 , sup{|zτ (q)+t − X| : t = 1, . . . , T (δq+1 ) − 1} ≥ (1 − 16−1 ). This is contradictory to (4.12). proves the lemma.

The contradiction we have reached

The next lemma shows that if an optimal trajectory {xt }Tt=0 satisﬁes x0 ∈ Q where Q is a given compact set and T is large enough, then xτ is close to the turnpike for some natural number τ . Lemma 11.4.2 Suppose that αG = 1, > 0 and Q ⊂ intK is a compact nonempty set. Then there exists a natural numberr T () such that the following property holds: If an integer T ≥ T () + 1 and if an optimal trajectory {xt }Tt=0 of G satisﬁes x0 ∈ Q, then there is τ ∈ {1, . . . , T ()} for which xτ = 0 and |xτ |−1 xτ ∈ B(X, ). Proof. Let us assume the converse. Then for each natural number i i of there exist an integer Ti ≥ i + 1 and an optimal trajectory {xt (i)}Tt=0 G such that (4.16) x0 (i) ∈ Q and ||xt (i)|−1 xt (i) − X| > for each t ∈ [1, . . . , i] satisfying xt (i) = 0. It follows from Proposition 11.3.1 that for each natural number i the Ti −1 is optimal. For each natural number i there is trajectory {xt (i)}t=0 pi ∈ K such that ||pi || = 1, (pi , xTi −1 (i)) = sup{(pi , y) : y ∈ aTi −1 (x0 (i))}.

(4.17)

It is easy to see that there exists a positive number γ0 such that x ≥ γ0 X for all x ∈ Q.

(4.18)

By (4.17), (4.18) and (4.1) for each natural number i ≥ 1, r0−1 (p, xTi −1 (i)) ≥ (pi , xTi −1 (i)) ≥ (pi , γ0 X) ≥ γ0 r0 ,

(4.19)

329

Convex processes

(p, xj (i)) ≥ γ0 r02 , j = 0, . . . , Ti − 1. We may assume without loss of generality that there exists a sequence n {yt }∞ t=0 ⊂ R such that xt (i) → yt as i → ∞ for each t ≥ 0.

(4.20)

Clearly {yt }∞ t=0 is a trajectory of G. Therefore there exists c0 ≥ 0 such that (4.21) lim yt = c0 X. t→∞

It follows from (4.16), (4.19) and (4.20) that (p, yt ) ≥ γ0 r02 , t = 0, 1, . . . , ||yt |−1 yt − X| ≥ , t = 1, 2, . . . . This is contradictory to (4.21). proves the lemma.

The contradiction we have reached

The following auxiliary result shows that if x0 is close to X, then an optimal trajectory {xt }Tt=0 is close to X for most t ∈ [0, T ]. Lemma 11.4.3 Suppose that αG = 1, > 0. Then there are δ ∈ (0, r0 ) and a natural number T () such that the following property holds: If a natural number T ≥ T () and if an optimal trajectory {xt }Tt=0 of G satisﬁes x0 ∈ B(X, δ), then xt ∈ B(X, ), t = 0, . . . , T − T ().

(4.22)

Proof. We may assume that < 4−1 r0 .

(4.23)

Lemma 10.4.1 implies that there exists 1 ∈ (0, 8−1 )

(4.24)

such that the following property holds: If an integer T ≥ 2 and if a trajectory {xt }Tt=0 of G satisﬁes x0 , xT ∈ B(X, 1 ), then (4.25) xt ∈ B(X, ), t = 1, . . . , T − 1. Choose numbers γ0 , 0 such that γ0 ∈ (0, 4−1 min{1 , 1}), 0 ∈ (0, 4−1 γ0 ),

(4.26)

(1 − γ0 )(1 + 0 r0−1 ) ≤ (1 − 2−1 γ0 ), (1 + γ0 )(1 − 0 r0−1 ) > (1 + 2−1 γ0 ). It follows from Lemma 11.4.2 and (4.1) that there exists a natural number L0 such that the following property holds:

330

TURNPIKE PROPERTIES

If an integer T ≥ L0 + 1 and if an optimal trajectory {xt }Tt=0 of G satisﬁes x0 ∈ B(X, 2−1 r0 ), then there exists an integer τ ∈ [1, L0 ] such that xτ = 0, |xτ |−1 xτ ∈ B(X, 0 ). (4.27) Choose an integer T () and a positive number δ such that T () > L0 + 1, δ < 2−1 0 , 4−1 r0 γ0 .

(4.28)

Assume that an integer T ≥ T () and an optimal trajectory {xt }Tt=0 of G satisﬁes x0 ∈ B(X, δ). (4.29) In view of the choice of L0 and (4.28) there exist a sequence of integers {Ti }qi=0 such that T0 = 0, Ti+1 − Ti ∈ [1, L0 ], T − Tq ∈ [0, L0 + 1],

(4.30)

xTi = 0, |xTi |−1 xTi ∈ B(X, 0 ), i = 1, . . . , q. We will show that xTi ∈ B(X, 1 ), i = 0, . . . , q.

(4.31)

In view of (4.30), (4.29) and (4.26) in order to prove the inequality (4.31) it is suﬃcient to show that ||xTi | − 1| ≤ γ0 , i = 1, . . . , q.

(4.32)

Let us assume the converse. Then there exists j ∈ {1, . . . , q} for which ||xTj | − 1| > γ0 .

(4.33)

There are two cases: (i) |xTj | > 1 + γ0 ; (ii) |xTj | < 1 − γ0 . Consider the case (i). We have |xTj | > 1 + γ0 . (4.34) Put

u = |xTj |−1 xTj − X.

(4.35)

|u| ≤ 0 .

(4.36)

(4.30) implies that By (4.35), (4.36), (4.34) and (4.1), u ≥ −0 r0−1 X, xTj = |xTj |X + |xTj |u ≥ (1 + γ0 )(X − 0 r0−1 X) = (1 + γ0 )(1 − 0 r0−1 )X.

(4.37)

331

Convex processes

It follows from (4.1) and (4.29) that −δr0−1 X ≤ x0 − X ≤ δr0−1 X. This inequality, (4.37) and (4.26) imply that (p, (1 + δr0−1 )X) ≥ (p, x0 ) ≥ (p, xTj ) ≥ (p, X)(1 + γ0 )(1 − 0 r0−1 ), (1 + δr0−1 ) ≥ (1 + γ0 )(1 − 0 r0−1 ) ≥ (1 + 2−1 γ0 ). This is contradictory to (4.28). Therefore in the case (i) we obtained a contradiction. Consider the case (ii). We have |xTj | < 1 − γ0 .

(4.38)

Deﬁne u by (4.35). Evidently (4.36) holds. By (4.35), (4.36), (4.38), (4.26) and (4.1), u ≤ 0 r0−1 X, xTj = |xTj |X + |xTj |u

(4.39)

≤ |xTj |(1 + 0 r0−1 )X ≤ (1 − γ0 )(1 + 0 r0−1 )X ≤ (1 − 2−1 γ0 )X. It follows from (4.1) and (4.29) that x0 ≥ (1 − δr0−1 )X.

(4.40)

The inequality (4.40) implies that there exists y0 ∈ aTj (x0 ) for which y0 ≥ (1 − δr0−1 )X. Combined with (4.39) and (4.28) this implies that y0 ≥ (1 − 4−1 γ0 )(1 − 2−1 γ0 )−1 xTj . In view of this inequality there exists y1 ∈ aT (x0 ) such that y1 ≥ (1 − 4−1 γ0 )(1 − 2−1 γ0 )−1 xT . On the other hand {xt }Tt=0 is an optimal trajectory of G. Therefore in the case (ii) we also obtained a contradiction which proves (4.32). Hence (4.31) holds. It follows from (4.31), (4.30) and the deﬁnition of 1 (see (4.24), (4.25)) that xt ∈ B(X, ), t = 0, . . . , T − L0 − 1. This completes the proof of the lemma. Proof of Theorem 11.3.1. We may assume without loss of generality that αG = 1. It is suﬃcient to show that for each y ∈ intK there exist a positive number γ and a natural number L such that B(y, γ) ⊂ intK and the following property holds: If T ≥ 2L and if an optimal trajectory {xt }Tt=0 of G satisﬁes x0 ∈ B(y, γ), then (4.41) xt ∈ B(λ(y)X, 2−1 ), t ∈ [L, T − L].

332

TURNPIKE PROPERTIES

Let y ∈ intK. We may assume that λ(y) = 1.

(4.42)

There exists r1 ∈ (0, r0 ) for which B(y, 2r1 ) ⊂ K.

(4.43)

Lemma 11.4.3 implies that there exist a natural number L1 and a positive number (4.44) δ1 < min{4−1 , r1 , 4−1 } such that the following property holds: If T ≥ L1 is an integer and if an optimal trajectory {xt }Tt=0 of G satisﬁes x0 ∈ B(X, δ1 ), then xt ∈ B(X, 4−1 ), t = 0, . . . , T − L1 .

(4.45)

Fix a positive number γ which satisﬁes γ < min{4−1 r1 , 32−1 δ1 r1 },

(4.46)

(1 − 32−1 δ1 )(1 − 64−1 δ1 )−1 ≤ 1 − γr1−1 . In view of Lemma 11.4.3 there exist a natural number L2 and a positive number (4.47) δ0 < 9−1 δ1 such that the following property holds: If T ≥ L2 is an integer and if an optimal trajectory {xt }Tt=0 of G satisﬁes x0 ∈ B(X, δ0 ), then xt ∈ B(X, 64−1 δ1 r0 ), t = 0, . . . , T − L2 .

(4.48)

We may assume without loss of generality that (1 + 8−1 δ1 )(1 − δ0 r0−1 ) > (1 + 2γr1−1 ),

(4.49)

(1 − 8−1 δ1 )(1 + δ0 r0−1 ) < (1 − 2γr1−1 ). It follows from Lemma 11.4.2 that there exists a natural number L3 such that the following property holds: If T ≥ L3 + 1 is an integer and if an optimal trajectory {xt }Tt=0 of G satisﬁes x0 ∈ B(y, r1 ), then there is τ ∈ {1, . . . , L3 } for which xτ = 0, |xτ |−1 xτ ∈ B(X, δ0 ).

(4.50)

333

Convex processes

By (4.42) there exists a trajectory {¯ zt }∞ ¯0 = y and t=0 of G such that z limt→∞ z¯t = X. Therefore there exists a natural number L4 such that z¯t ≥ (1 − 64−1 δ1 )X

(4.51)

L ≥ L1 + L2 + L3 + L4 + 2.

(4.52)

for all integers t ≥ L4 . Choose an integer

Assume that T ≥ 2L and {xt }Tt=0 is an optimal trajectory of G satisfying (4.53) x0 ∈ B(y, γ). It follows from the choice of L3 (see (4.50)), (4.46) and (4.52) that there exists a natural number τ ∈ [1, L3 ] which satisﬁes (4.50). We will show that (4.54) ||xτ | − 1| ≤ 8−1 δ1 . Let us assume the converse. There are two cases: (i) |xτ | > 1 + 8−1 δ1 ; (ii) |xτ | < 1 − 8−1 δ1 . (4.53), (4.43) and (4.46) imply that (1 − γr1−1 )y ≤ x0 ≤ (1 + γr1−1 )y.

(4.55)

It follows from (4.50) and (4.1) that (1 − δ0 r0−1 )X ≤ |xτ |−1 xτ ≤ (1 + δ0 r0−1 )X.

(4.56)

Consider the case (i). Then |xτ | > 1 + 8−1 δ1 . Combined with (4.56) and (4.49) this inequality implies that xτ ≥ (1 + 8−1 δ1 )(1 − δ0 r0−1 )X ≥ (1 + 2γr1−1 )X. In view of this inequality and (4.55) there exists u0 ∈ aτ (y) such that u0 ≥ (1 + 2γr1−1 )(1 + γr1−1 )−1 X. This is contradictory to (4.42). Consider the case (ii). Then |xτ | < 1 − 8−1 δ1 .

(4.57)

By (4.50), the choice of L2 , δ0 (see (4.47), (4.48)) and (4.52), |xτ |−1 xt ∈ B(X, 64−1 δ1 r0 ), t = τ, . . . , T − L2 . By this inclusion, (4.57) and (4.1) for each t ∈ {τ, . . . , T − L2 }, |xτ |−1 xt ≤ (1 + 64−1 δ1 )X,

(4.58)

334

TURNPIKE PROPERTIES

xt ≤ (1 − 8−1 δ1 )(1 + 64−1 δ1 )X ≤ (1 − 16−1 δ1 )X. In view of the choice of L4 (see (4.51)) and (4.58), τ + L4 ∈ [τ, T − L2 ], z¯τ +L4 ∈ aτ +L4 (y), z¯τ +L4 ≥ (1 − 64−1 δ1 )(1 − 16−1 δ1 )−1 xτ +L4 . By these relations, (4.45) and (4.46), there exists v ∈ aτ +L4 (x0 ) which satisﬁes v ≥ (1 − 64−1 δ1 )(1 − 16−1 δ1 )−1 (1 − γr1−1 )xτ +L4 ≥ (1 − 32−1 δ1 )(1 − 16−1 δ1 )−1 xτ +L4 . This is contradictory to the optimality of the trajectory {xt }Tt=0 . Since in both cases we obtained a contradiction we conclude that (4.54) holds. (4.50) and (4.54) imply that |xτ − X| ≤ |xτ − |xτ |X| + ||xτ | − 1| ≤ |xτ |δ0 + ||xτ | − 1| ≤ δ0 + (δ0 + 1)||xτ | − 1| ≤ δ0 + 4−1 δ1 ≤ 2−1 δ1 . It follows from this relation, the deﬁnition of L1 , δ1 and (4.52) that xt ∈ B(X, 4−1 ), t = τ, . . . , T − L1 . This completes the proof of the theorem. Theorem 11.3.2 follows from Lemma 11.4.3.

11.5.

Stability of the turnpike phenomenon

A sequence {Gt }Tt=0 ⊂ M where T ∈ {0, 1, . . .} is called a model. Let T ≥ 0 be an integer and let {Gt }Tt=0 ⊂ M be a model. A sequence +1 ⊂ Rn is a trajectory of the model {Gt }Tt=0 if xt ∈ K and {xt }Tt=0 +1 xt+1 ∈ Gt (xt ) for t = 0, . . . , T . A trajectory {xt }Tt=0 of the model {Gt }Tt=0 is called optimal if there exists f ∈ K ∗ \ {0} such that for +1 of the model satisfying y0 = x0 the relation each trajectory {yt }Tt=0 (f, xT +1 ) ≥ (f, yT +1 ) holds. We can easily prove the following result. Proposition 11.5.1 Assume that an integer T ≥ 1, −1 ⊂ M, Y ∈ intK, {Gt }Tt=0

β0 , . . . , βT −1 > 0, βt Y ∈ Gt (Y ) − K, t = 0, . . . , T − 1.

335

Convex processes

−1 Let {xt }Tt=0 be a trajectory of the model {Gt }Tt=0 with x0 ∈ intK. Then this trajectory is optimal if and only if for each λ > 1,

λxT + K ∩ {y ∈ Rn : there exists a trajectory {yt }Tt=0 −1 which satisﬁes y0 = x0 , yT = y} = ∅. of the model {Gt }Tt=0 −1 Let T ≥ 1 be an integer and {Gt }Tt=0 ⊂ M be a model. Assume that an integer q ≥ 1 and a number γ > 1. A trajectory {xt }Tt=0 of the model −1 is (q, γ)-optimal if for each integer τ satisfying 0 ≤ τ ≤ T − q {Gt }Tt=0 and each γ1 > γ,

(γ1 xτ +q + K) ∩ {y ∈ Rn : there exists a trajectory {yt }qt=0 of the model {Gt+τ }q−1 t=0 which satisﬁes y0 = xτ , yq = y} = ∅. We can easily prove the following result. −1 Proposition 11.5.2 Suppose that T, q are natural numbers, {Gt }Tt=0 T is a model, γ > 1 and {xt }t=0 is a (q, γ)-optimal trajectory of the model −1 . Then for each integer τ satisfying 0 ≤ τ ≤ T − q there exists {Gt }Tt=0 ∗ f ∈ K \ {0} such that the following property holds: q−1 If {yt }qt=0 is a trajectory of the model {Gτ +t }t=0 satisfying y0 = xτ , then γ(f, xτ +q ) ≥ (f, yq ).

For the set M we deﬁne a metric ρ(·, ·) as follows: ρ(G1 , G2 ) = d({(x, y) : x ∈ K, |x| ≤ 1, y ∈ G1 (x)}, {(x, y) : x ∈ K, |x| ≤ 1, y ∈ G2 (x)}), G1 , G2 ∈ M, where d(·, ·) is the Hausdorﬀ metric. Suppose that G ∈ M, αG > 0, X ∈ intK, p ∈ intK ∗ , |X| = 1,

(5.1)

αG X ∈ G(X), αG is the maximal eigenvalue of G, (αG p, x) ≥ (p, y), x ∈ K, y ∈ G(x) and that for each trajectory {xt }∞ t=0 of G there exists a number c0 ≥ 0 −t xt → c0 X as t → ∞. such that αG For each F ∈ M we deﬁne α(F ) = sup{β ∈ [0, ∞) : there exists x ∈ K \ {0} such that βx ∈ F (x) − K}.

(5.2)

336

TURNPIKE PROPERTIES

It is easy to see that α(G) = αG , there exists an open neighborhood V of G in M such that for each F ∈ V the number α(F ) is well deﬁned and G is a continuity point of the mapping F → α(F ), F ∈ V . We will establish the following results which show that the turnpike phenomenon is stable under small perturbations of G. Theorem 11.5.1 Suppose that x ∈ intK and is a positive number. Then there exist numbers δ > 0, γ > 1, integers q ≥ 1, Li ≥ 0, i = 1, 2 and an open neighborhood U of G in M such that B(x, δ) ⊂ intK and that the following property holds: −1 ⊂ U is a model and If T ≥ L1 + L2 is a natural number, {Gt }Tt=0 −1 T if {xt }t=0 is a (q, γ)-optimal trajectory of the model {Gt }Tt=0 satisfying x0 ∈ B(x, δ), then xt = 0 and ||xt |−1 xt − X| ≤ , t = L1 , . . . , T − L2 . Moreover if x = X, then L1 = 0. Theorem 11.5.2 Let x ∈ intK and > 0. Then there exist a number δ > 0, integers Li ≥ 0, i = 1, 2 and a neighborhood U of G in M such that B(x, δ) ⊂intK and for each F ∈ U , each natural number T ≥ L1 + L2 and each optimal trajectory {xt }Tt=0 of F satisfying |x0 − x| ≤ δ the following relation holds: |α(F )−t xt − λ(x)X| ≤ , t ∈ {L1 , . . . , T − L2 }. Moreover if x = X, then L1 = 0. Suppose that φt : K → [0, ∞), t = 0, 1, . . . are continuous functions, φt (ax) = aφt (x) for each a ∈ [0, ∞), x ∈ K and t = 0, 1, . . . ,

(5.3)

φt (x) ≤ φt (y) for each integer t ≥ 0 and each x, y ∈ K satisfying x ≤ y, 0 < c2 ≤ φt (x) ≤ c1 < ∞ for each t ∈ {0, 1, . . .}, each x ∈ K satisfying |x| ≤ 1 where 0 < c2 < c1 are constants. Consider an open neighborhood V of G in M such that for each F ∈ V the number α(F ) is well deﬁned and positive. Let z ∈ intK and µ be a positive number. For each pair (x, F ) belonging to a small neighborhood W of (z, G) in Rn × M and an integer T ≥ 0 we consider the following optimization problem: T t=0

α(F )−t φt (xt ) → max,

337

Convex processes

{xt }Tt=0 is a trajectory of F, x0 = x, α(F )−T xT ≥ µX. The problem (P) has a solution if the positive number µ and the neighborhood W are small enough. Theorem 11.5.3 Let z ∈ intK and let , µ be positive numbers. Then there exist a number δ > 0, integers L0 , L1 ≥ 0 and a neighborhood U of G in M such that B(z, δ) ⊂ intK, U ⊂ V and for each F ∈ U , each natural number T ≥ L1 + L0 , each x ∈ B(z, δ) and each optimal solution {xt }Tt=0 of the problem (P) satisfying x0 = x the following relations hold: xt = 0 and ||xt |−1 xt − X| ≤ , t ∈ {L1 , . . . , T − L0 }. Moreover if z = X, then L1 = 0.

11.6.

Proofs of Theorems 11.5.1, 11.5.2 and 11.5.3

Theorems 11.3.1 and 11.3.2 imply the following: Lemma 11.6.1 Let x ∈ intK and be a positive number. Then there exist δ > 0 and nonnegative integers L1 , L2 such that B(x, δ) ⊂ intK and the following property holds: If T ≥ L1 + L2 is a natural number and if {xt }Tt=0 is an optimal trajectory of G satisfying x0 ∈ B(x, δ), then xt = 0 and |xt |−1 xt ∈ B(X, ), t = L1 , . . . , T − L2 . Moreover if x = X, then L1 = 0. There exists a number r0 > 0 such that B(X, r0 ) ⊂ intK, B(p, r0 ) ⊂ intK ∗ .

(6.1)

The next auxiliary result shows that a (T, γ)-optimal trajectory of a −1 is close to an optimal trajectory of G if γ is close to 1 model {Gt }Tt=0 −1 and {Gt }Tt=0 is contained in a small neighborhood of G. Lemma 11.6.2 Assume that a nonempty compact set Q ⊂ intK, T is a natural number and is a positive number. Then there exist γ > 1, and a neighborhood U of G in M such that the following property holds: −1 ⊂ U and if {xt }Tt=0 is a (T, γ)-optimal trajectory of the If {Gt }Tt=0 T −1 model {Gt }t=0 satisfying x0 ∈ Q, then there exists an optimal trajectory {yt }Tt=0 of G for which y0 ∈ Q and |yt − xt | ≤ , t = 0, . . . , T .

338

TURNPIKE PROPERTIES

Proof. Let us assume the converse. Then for each integer s ≥ 1 there exist −1 ⊂ {F ∈ M : ρ(G, F ) ≤ s−1 } (6.2) {Gst }Tt=0 −1 and a (T, (1 + s−1 ))-optimal trajectory {xst }Tt=0 of the model {Gst }Tt=0 s such that x0 ∈ Q and

sup{|yt − xst | : t = 0, . . . , T } >

(6.3)

for each optimal trajectory {yt }Tt=0 of G satisfying y0 ∈ Q. Proposition 11.5.2 implies that for each natural number s there exists f s ∈ K ∗ such that ||f s || = 1 and the following property holds: −1 satisﬁes y0 = xs0 , then If a trajectory {yt }Tt=0 of the model {Gst }Tt=0 (1 + s−1 )(f s , xsT ) ≥ (f s , yT ).

(6.4)

We may assume without loss of generality that there are a sequence {xt }Tt=0 ⊂ Rn and f ∈ Rn for which f s → f, xst → xt as s → ∞, t = 0, . . . , T.

(6.5)

It is easy to see that x0 ∈ Q, {xt }Tt=0 is a trajectory of G. We will show that {xt }Tt=0 is an optimal trajectory of G. Assume the contrary. Then there exists a trajectory {zt }Tt=0 of G such that (6.6) z0 = x0 , (f, zT ) > sup{(f, xT ), 0} + 4∆ where ∆ is a positive number. We may assume without loss of generality that there exists c0 > 0 such that zt ≥ c0 X, t = 0, . . . , T.

(6.7)

There exist numbers M, Γ > 0 such that |xst | ≤ M, t = 0, . . . , T, s = 1, 2, . . . , |zt | ≤ M, t = 0, . . . , T,

(6.8)

Γ ∈ (0, 1), (1 − Γ2T +1 )(f, zT ) < ∆. It is not diﬃcult to see that there exists δ > 0 such that the following property holds: If F ∈ M satisﬁes ρ(F, G) ≤ δ and if u0 ∈ K, u1 ∈ G(u0 ) satisfy |u0 |, |u1 | ≤ M, u0 , u1 ≥ c0 X, then there are v0 ∈ K, v1 ∈ F (v0 ) for which v0 ≤ Γ−1 u0 , v1 ≥ Γu1 .

(6.9)

339

Convex processes

In view of (6.5) there exists a natural number s such that 4s−1 ≤ δ, s−1 < ∆(2M )−1 , |(f s − f, zT )| ≤ ∆,

(6.10)

|(f s , xsT ) − (f, xT )| ≤ ∆, xs0 ≥ Γx0 . It follows from (6.10), (6.7), (6.8), (6.2) and the choice of δ (see (6.9)) that for each t ∈ {0, . . . , T −1} there are ut ∈ K, ut+1 ∈ Gst (ut ) such that ut ≤ Γ−1 zt , ut+1 ≥ Γzt+1 . Combined with (6.6) and (6.10) this implies −1 such that that there exists a trajectory {vt }Tt=0 of the model {Gst }Tt=0 v0 = xs0 , vt ≥ Γ2t+1 zt , t = 1, . . . , T.

(6.11)

By (6.11), (6.8), (6.6), (6.10) and the deﬁnition of f s , (f s , vT ) ≥ Γ2T +1 (f s , zT ) ≥ Γ2T +1 (f, zT ) − ∆ ≥ (f, zT ) − 2∆ ≥ (f, xT ) + 2∆ ≥ (f s , xsT ) + ∆ ≥ 2−1 ∆ +(1 + (2M )−1 ∆)(f s , xsT ) ≥ 2−1 ∆ + (1 + s−1 )(f s , xsT ). In view of (6.11) this is contradictory to the deﬁnition of f s (see (6.4)). The obtained contradiction proves that {xt }Tt=0 is an optimal trajectory of G. This implies that (6.5) is contradictory to the deﬁnition of {xst }Tt=0 , s = 1, 2, . . . . The obtained contradiction proves the lemma. Corollary 11.6.1 Assume that a nonempty compact set Q ⊂ intK, T ≥ 1 is an integer, > 0. Then there exist γ > 1 and a neighborhood U −1 ⊂ U and each (T, γ)–optimal traof G in M such that for each {Gt }Tt=0 T −1 T jectory {xt }t=0 of the model {Gt }t=0 satisfying x0 ∈ Q, there is an optimal trajectory {yt }Tt=0 of G such that y0 ∈ Q and for each t ∈ {0, . . . , T }, xt , yt = 0, ||xt |−1 xt − |yt |−1 yt | ≤ . Proof of Theorem 11.5.1. We may assume that B(X, ) ⊂ intK. Lemma 11.6.1 implies that there exist δ1 ∈ (0, 2−1 ) and nonnegative integers N1 , N2 such that the following properties hold: (i) B(x, δ1 ) ⊂ intK and if T ≥ N1 + N2 is a natural number and if {xt }Tt=0 is an optimal trajectory of G satisfying x0 ∈ B(x, δ1 ), then xt = 0, |xt |−1 xt ∈ B(X, 2−1 ), t ∈ {N1 , . . . , T − N2 };

(6.12)

(ii) if T ≥ N2 is a natural number and if {xt }Tt=0 is an optimal trajectory of G satisfying x0 ∈ B(X, δ1 ), then relation (6.12) holds for each integer t ∈ {0, . . . , T − N2 }. Moreover N1 = 0 if x = X.

340

TURNPIKE PROPERTIES

It follows from Lemma 11.6.1 that there exist natural numbers N3 , N4 such that if T ≥ N3 + N4 is an integer and if {xt }Tt=0 is an optimal trajectory of G satisfying x0 ∈ B(X), then xt = 0, |xt |−1 xt ∈ B(X, 2−1 δ1 ), t ∈ {N3 , . . . , T − N4 }.

(6.13)

Put q = 2(N1 + N2 + N3 + N4 ).

(6.14)

In view of Corollary 11.6.1 there exist a number γ > 1 and a neighborhood U of G in M such that the following property holds: q (iii) If {Gt }q−1 t=0 ⊂ U is a model and if {xt }t=0 is a (q, γ)-optimal trajectory of the model {Gt }q−1 t=0 satisfying x0 ∈ B(X, ) (respectively x0 ∈ B(x, δ1 ), x0 ∈ B(X, δ1 )), then there exists an optimal trajectory {yt }qt=0 of G satisfying y0 ∈ B(X, ) (respectively y0 ∈ B(x, δ1 ), y0 ∈ B(X, δ1 )) such that xt , yt = 0, ||xt |−1 xt − |yt |−1 yt | ≤ 8−1 δ1 , t = 0, . . . , q.

(6.15)

Choose δ ∈ (0, 2−1 δ1 ) and put L1 = N1 , L2 = q.

(6.16)

−1 Assume that an integer T ≥ L1 + L2 , {Gt }Tt=0 ⊂ U and {xt }Tt=0 is a −1 (q, γ)-optimal trajectory of the model {Gt }Tt=0 satisfying x0 ∈ B(x, δ). We will show that for each t ∈ {L1 , . . . , T − L2 },

xt = 0, |xt |−1 xt ∈ B(X, ).

(6.17)

Assume the contrary. Then there exists an integer t1 ∈ {N1 , . . . , T − q} such that (6.17) does not hold with t = t1 . −1 . Consider a (q, γ)-optimal trajectory {xt }qt=0 of the model {Gt }Tt=0 q By property (iii) there exists an optimal trajectory {yt }t=0 of G such that y0 ∈ B(x, δ1 ) and (6.15) holds for t = 0, . . . , q. Combined with property (i) this implies that |yt |−1 yt ∈ B(X, 2−1 ), t ∈ {N1 , . . . , q − N2 },

(6.18)

|xt |−1 xt ∈ B(X, ), t ∈ {N1 , . . . , q − N2 }. Therefore t1 > q − N2 . Clearly a sequence {|xN1 |−1 xN1 +t }qt=0 is a (q, γ)optimal trajectory of the model {Gt+N1 }q−1 t=0 . By (6.18) and property (iii) there exists an optimal trajectory {zt }qt=0 of G such that z0 ∈ B(X, ) and for each t ∈ {0, . . . , q}, zt = 0, xt+N1 = 0, ||zt |−1 zt − |xt+N1 |−1 xt+N1 | ≤ 8−1 δ1 .

341

Convex processes

In view of these inequalities and the choice of N3 , N4 (see (6.13)), |zt |−1 zt ∈ B(X, 2−1 δ1 ), t ∈ {N3 , . . . , q − N4 },

(6.19)

|xt |−1 xt ∈ B(X, δ1 ), t ∈ {N1 + N3 , . . . , q + N1 − N4 }. Deﬁne E = {t ∈ {N1 , . . . , t1 } : xτ = 0, |xτ |−1 xτ ∈ B(X, ) for τ = N1 , . . . , t and |xt |−1 xt ∈ B(X, δ1 )}. (6.15), (6.18), (6.14) and (6.19) imply that N1 + N3 ∈ E. Denote by t2 the maximal element of E. Consider a (q, γ)-optimal trajectory {|xt2 |−1 xt+t2 }qt=0 of the model {Gt+t2 }q−1 t=0 . By the deﬁnition of E, t2 and property (iii) there exists an optimal trajectory {ht }qt=0 of G satisfying h0 ∈ B(X, δ1 ) such that xt+t2 = 0, ht = 0, ||xt+t2 |−1 xt+t2 − |ht |−1 ht | ≤ 8−1 δ1 , t = 0, . . . , q. (6.20) It follows from (6.20) and the property (ii) that for each t ∈ {0, . . . , q − N2 } |ht |−1 ht ∈ B(X, 2−1 ), |xt+t2 |−1 xt+t2 ∈ B(X, ).

(6.21)

In view if (6.14), (6.20) and the choice of N3 , N4 (see (6.13)) for each t ∈ {N3 , . . . , q − N4 }, |ht |−1 ht ∈ B(X, 2−1 δ1 ), |xt+t2 |−1 xt+t2 ∈ B(X, δ1 ). Combined with (6.21) and the deﬁnition of t1 this implies that t2 + N3 ∈ E. This is contradictory to the deﬁnition of t2 . The obtained contradiction proves the theorem. For each u, v ∈ Rn satisfying u ≤ v we put < u, v >= {z ∈ Rn : u ≤ z ≤ v}. Proof of Theorem 11.5.2. There exists a neighborhood U0 of G in M such that for each F ∈ U0 the number α(F ) is well deﬁned and positive. We may assume that B(λ(x)X, ) ⊂ intK. Put Λ = λ(x).

(6.22)

There exists a number θ > 1 for which Λ < θ−10 X, θ10 X > ⊂ B(ΛX, ).

(6.23)

342

TURNPIKE PROPERTIES

It follows from Theorem 11.5.1 and Proposition 11.3.1 that there exist a positive number δ1 < , a nonnegative integer N1 and a neighborhood U1 of G in M such that U1 ⊂ U0 , B(x, δ1 ) ⊂ intK, B(ΛX, δ1 ) ⊂ Λ < θ−1 X, θX >

(6.24)

and the following property holds: (i) If F ∈ U1 , T ≥ N1 is a natural number and if an optimal trajectory {xt }Tt=0 of F satisﬁes x0 ∈ B(ΛX, δ1 ), then xt ∈ |xt | < θ−1 X, θX > holds for t ∈ {0, . . . , T − N1 }. Since the function F → α(F ), F ∈ U0 is continuous at the point G there exists a neighborhood U2 ⊂ U1 of G in M such that for each F ∈ U2 , {x ∈ K : |x| = 1, α(F )x ∈ F (x) − K} ⊂ < θ−1 X, θX > .

(6.25)

It follows from Theorems 11.3.1 and 11.3.2 that there exist a positive number δ2 < 2−1 δ1 , a nonnegative integer N2 and a natural number N3 such that the following property holds: (ii) If T ≥ N2 + N3 is a natural number and if an optimal trajectory {xt }Tt=0 of G satisﬁes x0 ∈ B(x, δ2 ), then α(G)−t xt ∈ B(ΛX, 2−1 δ1 ), t ∈ {N2 , . . . , T − N3 }; moreover if x = X, then N2 = 0. In view of Lemma 11.6.2, the choice of N2 , N3 and the continuity of the function F → α(F ), F ∈ U0 at the point G there exist a positive number δ < δ2 and a neighborhood U ⊂ U2 of G in M such that the following property holds: 2 +N3 of F satisﬁes (iii) If F ∈ U and if an optimal trajectory {xt }N t=0 x0 ∈ B(x, δ), then α(F )−N2 xN2 ∈ B(ΛX, δ1 ). Put L1 = N2 , L2 = 2(N1 + N2 + N3 ).

(6.26)

Assume that F ∈ U , an integer T ≥ L1 +L2 and {xt }Tt=0 is an optimal trajectory of F such that x0 ∈ B(x, δ). There exists Y ∈ K such that |Y | = 1, α(F )Y ∈ F (Y ) − K.

(6.27)

In view of the choice of U2 (see (6.25)) Y ∈ < θ−1 X, θX > .

(6.28)

343

Convex processes

Let t0 ∈ {L1 , . . . , T − L2 }.

(6.29)

We will show that α(F )−t0 xt0 ∈ B(ΛX, ). Property (iii), (6.24) and (6.28) imply that α(F )−N2 xN2 ∈ B(ΛX, δ1 ) ⊂ Λ < θ−2 Y, θ2 Y > .

(6.30)

We may assume that t0 > N2 . −N2 of F . By propConsider an optimal trajectory {α(F )−N2 xN2 +t }Tt=0 erty (i) and (6.30), xt ∈ |xt | < θ−1 X, θX >, t ∈ {N2 , . . . , T − N1 }.

(6.31)

Combined with (6.30), (6.29) and (6.28) this implies that there exists h ∈ Rn such that h ∈ F t0 −N2 (θ2 ΛY ), h ≥ α(F )−N2 xt0 ≥ α(F )−N2 |xt0 |θ−2 Y,

(6.32)

F t0 −N2 (Y ) − K (Λθ4 α(F )N2 )−1 |xt0 |Y. There exists pF ∈ K ∗ \ {0} such that (α(F )pF , u) ≥ (pF , v) for each u ∈ K, v ∈ F (u). Combined with (6.32) this inequality implies that (Λθ4 α(F )N2 )−1 |xt0 | ≤ α(F )t0 −N2 , |xt0 | ≤ α(F )t0 Λθ4 .

(6.33)

We will show that |xt0 | > α(F )t0 Λθ−8 . Assume the contrary. Then by (6.31), (6.28), (6.29) and (6.26), xt0 ≤ |xt0 |θ2 Y ≤ θ−6 α(F )t0 ΛY.

(6.34)

The inclusion (6.30) implies that α(F )t0 −N2 ΛY ∈ F t0 −N2 (ΛY ) − K ⊂ F t0 −N2 (θ2 α(F )−N2 xN2 ) − K, θ−2 α(F )t0 ΛY ∈ F t0 −N2 (xN2 ) − K.

(6.35)

Since {xt }Tt=0 is an optimal trajectory of F relation (6.35) is contradictory to (6.34). The obtained contradiction proves that |xt0 | > α(F )t0 Λθ−8 . It follows from this relation, (6.31), (6.29), (6.33) and (6.23) that xt0 ∈ |xt0 | < θ−1 X, θX > ⊂ Λα(F )t0 < θ−9 X, θ9 X >,

344

TURNPIKE PROPERTIES

α(F )−t0 xt0 ∈ Λ < θ−9 X, θ9 X > ⊂ B(ΛX, ). This completes the proof of the theorem. Proof of Theorem 11.5.3. We preface the proof of Theorem 11.5.3 with two auxiliary results. Lemma 11.6.3 Suppose that z ∈ intK and µ > 0. Then there exist a pair of positive numbers δ, c3 and a neighborhood U of G in M such that B(z, δ) ⊂ intK, U ⊂ V , and the following property holds: If T ≥ 1 is an integer, F ∈ U , x ∈ B(z, δ) and if an optimal solution {xt }Tt=0 of the problem (P) satisﬁes x0 = x, then |α(F )−t xt | ≥ c3 , t = 0, . . . , T.

(6.36)

Proof. There exists a neighborhood U of G in M such that U ⊂ V and for each F ∈ U , {x ∈ K : |x| = 1, α(F )x ∈ F (x) − K} ⊂ < 2−1 X, 2X > .

(6.37)

Choose δ > 0 such that B(z, δ) ⊂ intK and choose c3 ∈ (0, 16−1 µr0 )

(6.38)

(recall r0 in (6.1)). Assume that F ∈ U, x ∈ B(z, δ), an integer T ≥ 1 and {xt }Tt=0 is an optimal solution of the problem (P) satisfying x0 = x. We will show that (6.36) holds. Let us assume the converse. Then there exists t ∈ {0, . . . , T } for which |α(F )−t xt | ≤ c3 .

(6.39)

There exists Y ∈ K such that |Y | = 1, α(F )Y ∈ F (Y ) − K.

(6.40)

In view of the choice of U (see (6.37)), Y ∈ < 2−1 X, 2X > .

(6.41)

There exists pF ∈ K ∗ \ {0} such that (α(F )pF , u) ≥ (pF , v) for each u ∈ K, v ∈ F (u). (6.39), (6.1), (6.41), (6.38) imply that α(F )−t xt ≤ c3 r0−1 X ≤ 2c3 r0−1 Y ≤ 8−1 µY,

(6.42)

345

Convex processes

2−1 α(F )T µY ≤ α(F )T µX ∈ F T −t (xt ) − K ⊂ F T −t (8−1 µα(F )t Y ) − K ⊂ 8−1 µα(F )t F T −t (Y ) − K, α(F )T −t Y ∈ 4−1 F T −t (Y ) − K. This is contradictory to (6.42). proves the lemma.

The contradiction we have reached

Lemma 11.6.4 Suppose that z ∈ intK. Then there exist a pair of positive numbers δ, c4 and a neighborhood U of G in M such that B(z, δ) ⊂ intK, U ⊂ V and the following property holds: If F ∈ U , T ≥ 1 is an integer and if a trajectory {xt }Tt=0 of F satisﬁes x0 ∈ B(z, δ), then (α(F )−t xt , p) ≤ c4 , t = 0, . . . , T. Proof. Choose a positive number < λ(z). By Theorem 11.5.2 there exist nonnegative integers L1 , L2 , a neighborhood U of G in M and a positive number δ such that U ⊂ V ∩ {F ∈ M : ρ(F, G) ≤ 4−1 min{1, δ}},

(6.43)

δ < |z|, sup{|u| : u ∈ G(v) for some v ∈ K satisfying |v| ≤ 1}, B(z, δ) ⊂ intK, {α(F ) : F ∈ U } ⊂ (2−1 α(G), 2α(G)) and that the following property holds: If F ∈ U , T ≥ L1 +L2 is a natural number and if an optimal trajectory {xt }Tt=0 of F satisﬁes x0 ∈ B(z, δ), then α(F )−t xt ∈ B(λ(z)X, ), |α(F )−t xt | ≤ 2λ(z)

(6.44)

for all t ∈ {L1 , . . . , T − L2 }. Choose a number c5 > sup{α(F )−t |y| : y ∈ F t (x), F ∈ U,

(6.45)

x ∈ B(z, δ), t ∈ {0, . . . , L1 + L2 }}(1 + |p|)(1 + 2λ(z)) and choose a number c4 > sup{c5 , 2λ(z), 2λ(z)(4α(G)−1 ) sup{|v| :

(6.46)

v ∈ G(u), u ∈ K, |u| ≤ 1}}(1 + |p|). Assume that F ∈ U , an integer T ≥ 1 and {xt }Tt=0 is a trajectory of F satisfying x0 ∈ B(z, δ). To prove the lemma it is suﬃcient to show that (6.47) (α(F )−T xT , p) ≤ c4 .

346

TURNPIKE PROPERTIES

We may assume that (xT , p) = sup{(p, y) : y ∈ F T (x0 )}. This implies that {xt }Tt=0 is an optimal trajectory of F . If T < L1 + L2 , then (6.47) follows from (6.45) and (6.46). Assume that T ≥ L1 + L2 . In view of the choice of U, δ, L1 , L2 (see (6.44)) |α(F )L2 −T xT −L2 | ≤ 2λ(z). Combined with (6.46), (6.43), (6.45) this implies (6.47). This completes the proof of the lemma. We will complete the proof of Theorem 11.5.3. Lemmas 11.6.3 and 11.6.4 imply that there exist positive numbers δ1 , c3 , c4 and a neighborhood U0 of G in M such that U0 ⊂ V, B(z, δ1 ) ⊂ intK and the following property holds: (a) If F ∈ U0 , T ≥ 1 is a natural number, x ∈ B(z, δ1 ), and if an optimal solution {xt }Tt=0 of the problem (P) satisﬁes x0 = x, then c3 ≤ |α(F )−t xt | ≤ c4 , t = 0, . . . , T. In view of Theorem 11.5.1 there exist δ ∈ (0, δ1 ), γ > 1, a natural number q ≥ 1, a pair of nonnegative integers Li , i = 1, 2 and a neighborhood U ⊂ U0 of G in M such that the following property holds: (b) If T ≥ L1 + L2 is a natural number, F ∈ U and if a (q, γ)-optimal trajectory {xt }Tt=0 of F satisﬁes x0 ∈ B(z, δ), then xt = 0, |xt |−1 xt ∈ B(X, ), t ∈ {L1 , . . . , T − L2 }; moreover if z = X, then L1 = 0. Choose a natural number L3 such that (γ − 1)L3 c3 c2 > qc1 c4 and put L0 = L2 + 2L3 + 2q. Assume that F ∈ U , a natural number T ≥ L1 + L0 , x ∈ B(z, δ) and {xt }Tt=0 is an optimal solution of the problem (P) satisfying x0 = x. We −L3 is a (q, γ)-optimal trajectory of F . will show that {xt }Tt=0 Let us assume the converse. Then there exists an integer τ ∈ [0, T − L3 − q] and a number γ1 > γ such that γ1 xτ +q ∈ F q (xτ ) − K. This implies that there exists a trajectory {yt }Tt=0 of F for which yt = xt , t = 0, . . . , τ, yt ≥ γ1 xt , t = τ + q, . . . , T . It follows from the optimality of {xt }Tt=0 , the deﬁnition of {yt }Tt=0 , L3 , (5.3) and property (a) that 0≥

T t=0

α(F )−t φt (yt ) −

T t=0

α(F )−t φt (xt ) ≥ (γ − 1)

T t=τ +q

φt (xt )α(F )−t

347

Convex processes

−

τ +q−1 t=τ

α(F )−t φt (xt ) ≥ (γ − 1)L3 c3 c2 − qc4 c1 > 0.

−L3 The obtained contradiction proves that {xt }Tt=0 is a (q, γ)-optimal trajectory of F . By property (b),

xt = 0, ||xt |−1 xt − X| ≤ , t = L1 , . . . , T − L2 − L3 . This completes the proof of the theorem.

Chapter 12 A DYNAMIC ZERO-SUM GAME

In this chapter we consider a class of dynamic discrete-time two-player zero-sum games. We show that for a generic cost function and each initial state there exists a pair of overtaking equilibria strategies over an inﬁnite horizon. We also establish that for a generic cost function f there exists a pair of stationary equilibria strategies (xf , yf ) such that each pair of “approximate” equilibria strategies spends almost all of its time in a small neighborhood of (xf , yf ).

12.1.

Preliminaries

Denote by | · | the Euclidean norm in Rm . Let X ⊂ Rm1 and Y ⊂ Rm2 be nonempty convex compact sets. Denote by M the set of all continuous functions f : X × X × Y × Y → R1 such that: for each (y1 , y2 ) ∈ Y × Y the function (x1 , x2 ) → f (x1 , x2 , y1 , y2 ), (x1 , x2 ) ∈ X × X is convex; for each (x1 , x2 ) ∈ X × X the function (y1 , y2 ) → f (x1 , x2 , y1 , y2 ), (y1 , y2 ) ∈ Y × Y is concave. For the set M we deﬁne a metric ρ : M × M → R1 by ρ(f, g) = sup{|f (x1 , x2 , y1 , y2 ) − g(x1 , x2 , y1 , y2 )| : x1 , x2 ∈ X,

y1 , y2 ∈ Y },

(1.1)

f, g ∈ M.

Clearly M is a complete metric space. Given f ∈ M and an integer n ≥ 1 we consider a discrete-time twoplayer zero-sum game over the interval [0, n]. For this game {{xi }ni=0 : xi ∈ X, i = 0, . . . , n} is the set of strategies for the ﬁrst player, {{yi }ni=0 :

350

TURNPIKE PROPERTIES

yi ∈ Y, i = 0, . . . , n} is the set of strategies for the second player, and the cost for the ﬁrst player associated with the strategies {xi }ni=0 , {yi }ni=0 is n−1 given by i=0 f (xi , xi+1 , yi , yi+1 ). Deﬁnition 1.1 Let f ∈ M, n ≥ 1 be an integer and let M ∈ [0, ∞). A yi }ni=0 ⊂ Y is called (f, M )-good if the pair of sequences {¯ xi }ni=0 ⊂ X, {¯ following properties hold: ¯0 , xn = x ¯n , for each sequence {xi }ni=0 ⊂ X satisfying x0 = x M+

n−1

f (xi , xi+1 , y¯i , y¯i+1 ) ≥

i=0

n−1

f (¯ xi , x ¯i+1 , y¯i , y¯i+1 );

(1.2)

i=0

for each sequence {yi }ni=0 ⊂ Y satisfying y0 = y¯0 , yn = y¯n , M+

n−1

f (¯ xi , x ¯i+1 , y¯i , y¯i+1 ) ≥

i=0

n−1

f (¯ xi , x ¯i+1 , yi , yi+1 ).

(1.3)

i=0

If a pair of sequences {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y is (f, 0)-good, it is called (f )-optimal. Our ﬁrst main result in this chapter deals with the so-called “turnpike property” of “good” pairs of sequences. Consider any f ∈ M. We say that the function f has the turnpike property if there exists a unique pair (xf , yf ) ∈ X × Y for which the following assertion holds: For each > 0 there exist an integer n0 ≥ 2 and a number δ > 0 such that for each integer n ≥ 2n0 and each (f, δ)-good pair of sequences {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y the relations |xi − xf |, |yi − yf | ≤ hold for all integers i ∈ [n0 , n − n0 ]. In this chapter our goal is to show that the turnpike property holds for a generic f ∈ M. We will prove the existence of a set F ⊂ M which is a countable intersection of open everywhere dense sets in M such that each f ∈ F has the turnpike property (see Theorem 12.2.1). We also study the existence of equilibria over an inﬁnite horizon for the class of zero-sum games considered in the chapter. Deﬁnition 1.2 Let f ∈ M. A pair of sequences yi }∞ {¯ xi }∞ i=0 ⊂ X, {¯ i=0 ⊂ Y is called (f )-overtaking optimal if the following properties hold: ¯0 , for each sequence {xi }∞ i=0 ⊂ X satisfying x0 = x T −1

lim sup[ T →∞

i=0

f (¯ xi , x ¯i+1 , y¯i , y¯i+1 ) −

T −1 i=0

f (xi , xi+1 , y¯i , y¯i+1 )] ≤ 0;

351

A dynamic zero-sum game

for each sequence {yi }∞ ¯0 , i=0 ⊂ Y satisfying y0 = y T −1

lim sup[ T →∞

f (¯ xi , x ¯i+1 , yi , yi+1 ) −

i=0

T −1

f (¯ xi , x ¯i+1 , y¯i , y¯i+1 )] ≤ 0.

i=0

Our second main result (see Theorem 12.2.2) shows that for a generic f ∈ M and each (x, y) ∈ X × Y there exists an (f )-overtaking optimal ∞ pair of sequences {xi }∞ i=0 ⊂ X, {yi }i=0 ⊂ Y such that x0 = x, y0 = y. The results of this chapter were obtained in [102].

12.2.

Main results

In this section we present the main results of the chapter. Theorem 12.2.1 There exists a set F ⊂ M which is a countable intersection of open everywhere dense sets in M such that for each f ∈ F the following assertions hold: 1. There exists a unique pair (xf , yf ) ∈ X × Y for which sup f (xf , xf , y, y) = f (xf , xf , yf , yf ) = inf f (x, x, yf , yf ).

y∈Y

x∈X

2. For each > 0 there exist a neighborhood U of f in M, an integer n0 ≥ 2 and a number δ > 0 such that for each g ∈ U , each integer n ≥ 2n0 and each (g, δ)-good pair of sequences {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y the relation (2.1) |xi − xf |, |yi − yf | ≤ holds for all integers i ∈ [n0 , n−n0 ]. Moreover, if |x0 −xf |, |y0 −yf | ≤ δ, then (2.1) holds for all integers i ∈ [0, n−n0 ], and if |xn −xf |, |yn −yf | ≤ δ, then (2.1) is valid for all integers i ∈ [n0 , n]. Theorem 12.2.2 There exists a set F ⊂ M which is a countable intersection of open everywhere dense sets in M such that for each f ∈ F the following assertion holds: For each x ∈ X and each y ∈ Y there exists an (f )-overtaking optimal ∞ pair of sequences {xi }∞ i=0 ⊂ X, {yi }i=0 ⊂ Y such that x0 = x, y0 = y. The chapter is organized as follows. Section 12.3 contains deﬁnitions and notation while Section 12.4 contains preliminary results. In Section 12.5 we prove the existence of a minimal pair of sequences. Section 12.6 contains auxiliary results for Theorem 12.2.1 while Section 12.7 contains auxiliary results for Theorem 12.2.2. Theorems 12.2.1 and 12.2.2 are proved in Section 12.8.

352

12.3.

TURNPIKE PROPERTIES

Deﬁnitions and notation

Let f ∈ M. Deﬁne a function f¯ : X × Y → R1 by f¯(x, y) = f (x, x, y, y),

x ∈ X, y ∈ Y.

(3.1)

Then there exists a saddle point (xf , yf ) ∈ X × Y for f¯. We have sup f¯(xf , y) = f¯(xf , yf ) = inf f¯(x, yf ).

(3.2)

µ(f ) = f¯(xf , yf ).

(3.3)

x∈X

y∈Y

Set Deﬁnition 3.1. Let f ∈ M. A pair of sequences ∞ {xi }∞ i=0 ⊂ X, {yi }i=0 ⊂ Y

is called (f )-minimal if for each integer n ≥ 2 the pair of sequences {xi }ni=0 , {yi }ni=0 is (f )-optimal. We will show in Section 12.5 (see Proposition 12.5.3) that for each f ∈ M, each x ∈ X and each y ∈ Y there exists an (f )-minimal pair of ∞ sequences {xi }∞ i=0 ⊂ X, {yi }i=0 ⊂ Y such that x0 = x, y0 = y. Let f ∈ M, n ≥ 1 be an integer, and let ξ = (ξ1 , ξ2 , ξ3 , ξ4 ) ∈ X × X × Y × Y . Deﬁne ΛX (ξ, n) = {{xi }ni=0 ⊂ X :

x0 = ξ1 , xn = ξ2 },

(3.4)

ΛY (ξ, n) = {{yi }ni=0 ⊂ Y :

y0 = ξ3 , yn = ξ4 },

(3.5)

f (ξ,n) ((x0 , . . . , xi , . . . , xn ), (y0 , . . . , yi , . . . yn )) =

n−1

f (xi , xi+1 , yi , yi+1 ),

i=0

(3.6) {xi }ni=0

∈ ΛX (ξ, n),

{yi }ni=0

∈ ΛY (ξ, n).

For x ∈ X, y ∈ Y , r > 0 put BX (x, r) = {z ∈ X : |x − z| ≤ r}, BY (y, r) = {z ∈ Y : |y − z| ≤ r}.

353

A dynamic zero-sum game

12.4.

Preliminary results

Let M, N be nonempty sets and let f : M × N → R1 . Set f a (x) = sup f (x, y), x ∈ M, y∈N

vfa = inf sup f (x, y), x∈M y∈N

Clearly

f b (y) = inf f (x, y), y ∈ N, x∈M

vfb = sup inf f (x, y). y∈N x∈M

vfb ≤ vfa .

(4.1) (4.2)

(4.3)

We have the following result (see Chapter 6, Section 2, Proposition 1 of [5]). Proposition 12.4.1 Let f : M × N → R1 , x ¯ ∈ M , y¯ ∈ N . Then sup f (¯ x, y) = f (¯ x, y¯) = inf f (x, y¯) x∈M

y∈N

(4.4)

if and only if vfa = vfb , sup f (¯ x, y) = vfa , inf f (x, y¯) = vfb . y∈N

x∈M

Let f : M × N → R1 . If (¯ x, y¯) ∈ M × N satisﬁes (4.4), it is called a saddle point (for f ). We have the following result (see Chapter 6, Section 2, Theorem 8 of [5]). Proposition 12.4.2 Let M ⊂ Rm , N ⊂ Rn be convex compact sets and let f : M × N → R1 be a continuous function. Assume that for each y ∈ N the function x → f (x, y), x ∈ M is convex and for each x ∈ M the function y → f (x, y), y ∈ N is concave. Then there exists a saddle point for f . Proposition 12.4.3 Let M, N be nonempty sets, f : M × N → R1 and −∞ < vfa = vfb < +∞, x0 ∈ M, y0 ∈ N, ∆1 , ∆2 ∈ [0, ∞), sup f (x0 , y) ≤ vfa + ∆1 ,

y∈N

inf f (x, y0 ) ≥ vfb − ∆2 .

x∈M

(4.5) (4.6)

Then sup f (x0 , y) − ∆1 − ∆2 ≤ f (x0 , y0 ) ≤ inf f (x, y0 ) + ∆1 + ∆2 .

y∈N

x∈M

(4.7)

Proof. By (4.6) and (4.5), sup f (x0 , y) − ∆1 − ∆2 ≤ vfa − ∆2 = vfb − ∆2 ≤ inf f (x, y0 ) ≤ f (x0 , y0 )

y∈N

x∈M

354

TURNPIKE PROPERTIES

≤ sup f (x0 , y) ≤ vfa + ∆1 = vfb + ∆1 ≤ inf f (x, y0 ) + ∆1 + ∆2 . x∈M

y∈N

The proposition is proved. Proposition 12.4.4 Let M, N be nonempty sets and let f : M × N → R1 . Assume that (4.5) is valid, x0 ∈ M , y0 ∈ N , ∆1 , ∆2 ∈ [0, ∞) and sup f (x0 , y) − ∆2 ≤ f (x0 , y0 ) ≤ inf f (x, y0 ) + ∆1 . x∈M

y∈N

(4.8)

Then sup f (x0 , y) ≤ vfa + ∆1 + ∆2 ,

y∈N

inf f (x, y0 ) ≥ vfb − ∆1 − ∆2 .

x∈M

(4.9)

Proof. It follows from (4.8), (4.2), (4.5) and (4.3) that vfb − ∆2 = vfa − ∆2 ≤ sup f (x0 , y) − ∆2 y∈N

≤ inf f (x, y0 ) + ∆1 ≤ vfb + ∆1 . x∈M

This implies (4.9). The proposition is proved.

12.5.

The existence of a minimal pair of sequences

Let f ∈ M, xf ∈ X, yf ∈ Y and sup f¯(xf , y) = f¯(xf , yf ) = inf f¯(x, yf ). x∈X

y∈Y

(5.1)

Proposition 12.5.1 Suppose that n ≥ 2 is a natural number and x ¯i = xf ,

y¯i = yf ,

i = 0, . . . , n.

(5.2)

yi }ni=0 is (f )-optimal. Then the pair of sequences {¯ xi }ni=0 , {¯ Proof. Assume that {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y and x0 , xn = xf ,

y0 , yn = yf .

The equalities (5.3), (5.2) and (5.1) imply that n−1 i=0

f (xi , xi+1 , y¯i , y¯i+1 ) =

n−1 i=0

f (xi , xi+1 , yf , yf )

(5.3)

355

A dynamic zero-sum game

≥ nf (n−1

n−1

xi , n−1

i=0

= nf (n−1

n−1

xi , n−1

i=0 n−1

n−1

xi+1 , yf , yf )

i=0

n−1

xi , yf , yf ) ≥ nf (xf , xf , yf , yf ),

i=0

f (¯ xi , x ¯i+1 , yi , yi+1 ) =

i=0

n−1

f (xf , xf , yi , yi+1 )

i=0

≤ nf (xf , xf , n−1

n−1

yi , n−1

i=0

= nf (xf , xf , n−1

n−1

yi , n−1

n−1

i=0

n−1

yi+1 )

i=0

yi ) ≤ nf (xf , xf , yf , yf ).

i=0

This completes the proof of the proposition. Proposition 12.5.2 Let n ≥ 2 be an integer and let (k)

(k)

({xi }ni=0 , {yi }ni=0 ) ⊂ X × Y,

k = 1, 2, . . .

be a sequence of (f )-optimal pairs. Assume that (k)

lim xi

k→∞

= xi ,

(k)

lim yi

k→∞

= yi ,

i = 0, 1, 2, . . . , n.

(5.4)

Then the pair of sequences ({xi }ni=0 , {yi }ni=0 ) is (f )-optimal. Proof. Let

{ui }ni=0 ⊂ X,

u0 = x0 , un = xn .

(5.5)

We will show that n−1

f (xi , xi+1 , yi , yi+1 ) ≤

i=0

n−1

f (ui , ui+1 , yi , yi+1 ).

(5.6)

i=0

Assume the contrary. Then there exists a positive number such that n−1 i=0

f (xi , xi+1 , yi , yi+1 ) >

n−1

f (ui , ui+1 , yi , yi+1 ) + 8.

(5.7)

i=0

There is a positive number δ < such that the following property holds: If z1 , z2 , z¯1 , z¯2 ∈ X, ξ1 , ξ2 , ξ¯1 , ξ¯2 ∈ Y satisfy |zi − z¯i |, |ξi − ξ¯i | ≤ δ,

i = 1, 2,

356

TURNPIKE PROPERTIES

then

z1 , z¯2 , ξ¯1 , ξ¯2 )| ≤ (8n)−1 . |f (z1 , z2 , ξ1 , ξ2 ) − f (¯

(5.8)

There is a natural number q such that (q)

(q)

|xi − xi |, |yi − yi | ≤ δ,

i = 0, . . . , n.

(5.9)

Set (q)

(q)

(q)

(q) u0 = x0 , u(q) n = xn ,

ui

= ui , i = 1, . . . , n − 1.

(5.10)

(q)

(q)

Since the pair of sequences ({xi }ni=0 , {yi }ni=0 ) is (f )-optimal it follows from (5.10) that n−1

(q)

(q)

(q)

(q)

f (xi , xi+1 , yi , yi+1 ) ≤

i=0

n−1

(q)

(q)

(q)

(q)

f (ui , ui+1 , yi , yi+1 ).

(5.11)

i=0

In view of the choice of δ (see (5.8)), (5.9), (5.10) and (5.5) for i = 0, . . . , n − 1, (q)

(q)

(q)

(q)

(q)

(q)

(q)

(q)

|f (xi , xi+1 , yi , yi+1 ) − f (xi , xi+1 , yi , yi+1 )| ≤ (8n)−1 , |f (ui , ui+1 , yi , yi+1 ) − f (ui , ui+1 , yi , yi+1 )| ≤ (8n)−1 . Combined with (5.7) these inequalities imply that n−1

(q) (q) (q) (q) f (xi , xi+1 , yi , yi+1 )

−

n−1

(q)

(q)

(q)

(q)

f (ui , ui+1 , yi , yi+1 ) > .

i=0

i=0

This is contradictory to (5.11). The obtained contradiction proves that (5.6) is valid. Analogously we can show that for each {ui }ni=0 ⊂ Y satisfying u0 = y0 , un = yn , the following relation holds: n−1 i=0

f (xi , xi+1 , yi , yi+1 ) ≥

n−1

f (xi , xi+1 , ui , ui+1 ).

i=0

This completes the proof of the proposition. The following proposition is the main result of this section. Proposition 12.5.3 Let f ∈ M and let x ∈ X, y ∈ Y . Then there ∞ exists an (f )-minimal pair of sequences {xi }∞ i=0 ⊂ X, {yi }i=0 ⊂ Y such that x0 = x, y0 = y. Proof. By Proposition 12.4.2 for each integer n ≥ 2 there exists an (f )(n) (n) (n) optimal pair of sequences {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y such that x0 =

357

A dynamic zero-sum game (n)

∞ x, y0 = y. There exist a pair of sequences {xi }∞ i=0 ⊂ X, {yi }i=0 ⊂ Y and a strictly increasing sequence of natural numbers {nk }∞ k=1 such that for each integer i ≥ 0, (nk )

xi

→ xi ,

(nk )

yi

→ yi as k → ∞.

It follows from Proposition 12.5.2 that the pair of sequences {xi }∞ i=0 , is (f )-minimal. The proposition is proved. {yi }∞ i=0

12.6.

Preliminary lemmas for Theorem 12.2.1

Let f ∈ M. There exist xf ∈ X, yf ∈ Y such that sup f (xf , xf , y, y) = f (xf , xf , yf , yf ) = inf f (x, x, yf , yf ). x∈X

y∈Y

(6.1)

For each r ∈ (0, 1) we deﬁne fr ∈ M which has the turnpike property such that fr → f as r → 0+ in M. Let r ∈ (0, 1). Deﬁne fr : X 2 × Y 2 → R1 by fr (x1 , x2 , y1 , y2 ) = f (x1 , x2 , y1 , y2 ) + r|x1 − xf | − r|y1 − yf |,

(6.2)

x1 , x2 ∈ X, y1 , y2 ∈ Y. Clearly fr ∈ M, sup fr (xf , xf , y, y) = fr (xf , xf , yf , yf ) = inf fr (x, x, yf , yf ). x∈X

y∈Y

(6.3)

We show that fr has the turnpike property. We begin with the following lemma which establishes that if a good pair of sequences {xi }ni=0 , {yi }ni=0 satisﬁes xn , x0 = xf and yn , y0 = yf , then {xi }ni=0 is contained in a small neighborhood of xf and {yi }ni=0 is contained in a small neighborhood of yf . Lemma 12.6.1 For each ∈ (0, 1) there exists a positive number δ < such that the following property holds: If n ≥ 2 is a natural number and if an (fr , δ)-good pair of sequences {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y satisﬁes xn , x0 = xf ,

yn , y0 = yf ,

(6.4)

then xi ∈ BX (xf , ), yi ∈ BY (yf , ),

i = 0, . . . , n.

(6.5)

358

TURNPIKE PROPERTIES

Proof. Let ∈ (0, 1). Choose a positive number δ < 8−1 r.

(6.6)

Assume that an integer n ≥ 2, {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y is an (fr , δ)good pair of sequences and (6.4) is valid. Set ξ1 , ξ2 = xf ,

ξ3 , ξ4 = yf ,

ξ = (ξ1 , ξ2 , ξ3 , ξ4 ).

(6.7)

Consider the sets ΛX (ξ, n), ΛY (ξ, n) and the functions (fr )(ξ,n) , f (ξ,n) (see (3.4)-(3.6)). Proposition 12.5.1 and (6.1) imply that n−1

sup{

f (xf , xf , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)} = nf (xf , xf , yf , yf )

i=0

(6.8) n−1

= inf{

f (pi , pi+1 , yf , yf ) : {pi }ni=0 ∈ ΛX (ξ, n)}.

i=0

In view of (6.8) and Proposition 12.4.1, n−1

sup{

f (xf , xf , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)}

(6.9)

i=0 n−1

= inf{sup{

f (pi , pi+1 , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)} :

i=0

{pi }ni=0 ∈ ΛX (ξ, n)}, n−1

inf{

f (pi , pi+1 , yf , yf ) : {pi }ni=0 ∈ ΛX (ξ, n)}

(6.10)

i=0 n−1

= sup{inf{

f (pi , pi+1 , ui , ui+1 ) : {pi }ni=0 ∈ ΛX (ξ, n)} :

i=0

{ui }ni=0 ∈ ΛY (ξ, n)}. Proposition 12.5.1 and (6.3) imply that n−1

sup{

fr (xf , xf , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)} = nfr (xf , xf , yf , yf )

i=0

(6.11) n−1

= inf{

i=0

fr (pi , pi+1 , yf , yf ) : {pi }ni=0 ∈ ΛX (ξ, n)}.

359

A dynamic zero-sum game

It follows from Proposition 12.4.1 and (6.11) that n−1

sup{

fr (xf , xf , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)}

(6.12)

i=0

= inf{sup{

n−1

fr (pi , pi+1 , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)} :

i=0

{pi }ni=0 ∈ ΛX (ξ, n)}, inf{

n−1

fr (pi , pi+1 , yf , yf ) : {pi }ni=0 ∈ ΛX (ξ, n)}

(6.13)

i=0

= sup{inf{

n−1

fr (pi , pi+1 , ui , ui+1 ) : {pi }ni=0 ∈ ΛX (ξ, n)} :

i=0

{ui }ni=0 ∈ ΛY (ξ, n)}. (6.7) and (6.4) imply that {xi }ni=0 ∈ ΛX (ξ, n),

{yi }ni=0 ∈ ΛY (ξ, n).

(6.14)

Since ({xi }ni=0 , {yi }ni=0 ) is an (fr , δ)-good pair of sequences we conclude that n−1

sup{

fr (xi , xi+1 , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)} − δ

(6.15)

i=0

≤

n−1

fr (xi , xi+1 , yi , yi+1 )

i=0 n−1

≤ inf{

fr (pi , pi+1 , yi , yi+1 ) : {pi }ni=0 ∈ ΛX (ξ, n)} + δ.

i=0

In view of Proposition 12.4.4, (6.15), (6.12) and (6.13), n−1

sup{

fr (xi , xi+1 , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)}

(6.16)

i=0 n−1

≤ sup{

fr (xf , xf , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)} + 2δ,

i=0 n−1

inf{

i=0

fr (pi , pi+1 , yi , yi+1 ) : {pi }ni=0 ∈ ΛX (ξ, n)}

(6.17)

360

TURNPIKE PROPERTIES n−1

≥ inf{

fr (pi , pi+1 , yf , yf ) : {pi }ni=0 ∈ ΛX (ξ, n)} − 2δ.

i=0

It follows from (6.2), (6.11), (6.16) and (6.8) that nf (xf , xf , yf , yf ) = nfr (xf , xf , yf , yf ) n−1

≥ sup{

(6.18)

fr (xi , xi+1 , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n)} − 2δ

i=0

≥ −2δ +

n−1

fr (xi , xi+1 , yf , yf ) = −2δ

i=0

+r

n−1

|xi − xf | +

i=0

n−1

f (xi , xi+1 , yf , yf )

i=0

≥ −2δ + r

n−1

|xi − xf | + nf (xf , xf , yf , yf ).

i=0

(6.2), (6.11), (6.17) and (6.8) imply that nf (xf , xf , yf , yf ) = nfr (xf , xf , yf , yf ) n−1

≤ inf{

(6.19)

fr (pi , pi+1 , yi , yi+1 ) : {pi }ni=0 ∈ ΛX (ξ, n)}

i=0

+2δ ≤ 2δ +

n−1

fr (xf , xf , yi , yi+1 )

i=0

= 2δ − r

n−1

|yi − yf | +

i=0

≤ 2δ − r

n−1

n−1

f (xf , xf , yi , yi+1 )

i=0

|yi − yf | + nf (xf , xf , yf , yf ).

i=0

(6.18), (6.19) and (6.6) imply that for i = 1, . . . , n − 1, |xi − xf | ≤ r−1 (2δ) < ,

|yi − yf | ≤ 2δr−1 < .

This completes the proof of the lemma. Choose a number D0 ≥ sup{|fr (x1 , x2 , y1 , y2 )| : x1 , x2 ∈ X, y1 , y2 ∈ Y }.

(6.20)

361

A dynamic zero-sum game

We can easily prove the following: Lemma 12.6.2 Suppose that n ≥ 2 is a natural number, M > 0 and {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y is an (fr , M )-good pair of sequences. Then the pair of sequences {¯ xi }ni=0 ⊂ X, {¯ yi }ni=0 ⊂ Y deﬁned by x ¯i = xi , y¯i = yi , i = 1, . . . , n − 1,

x ¯0 , x ¯n = xf ,

y¯0 , y¯n = yf

is (fr , M + 8D0 )-good. By using the uniform continuity of the function fr : X × X × Y × Y we can easily prove Lemma 12.6.3 For each > 0 there exists a positive number δ such that the following property holds: xi }ni=0 ⊂ If n ≥ 2 is a natural number and if sequences {xi }ni=0 , {¯ n n yi }i=0 ⊂ Y satisfy X, {yi }i=0 , {¯ yj − yj | ≤ δ, j = 0, n, |¯ xj − xj |, |¯

xj = x ¯j , yj = y¯j , j = 1, . . . , n − 1, (6.21)

then |

n−1

[fr (xi , xi+1 , yi , yi+1 ) − fr (¯ xi , x ¯i+1 , y¯i , y¯i+1 )]| ≤ .

(6.22)

i=0

Lemma 12.6.3 implies the following result. Lemma 12.6.4 For each positive number there exists a positive number δ such that the following property holds: If n ≥ 2 is a natural number, {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y is an (fr , )good pair of sequences, then each pair of sequences yi }ni=0 ⊂ Y {¯ xi }ni=0 ⊂ X, {¯ satisfying (6.21) is (fr , 2)-good. Lemmas 12.6.4 and 12.6.1 imply the following auxiliary result which shows that the property established in Lemma 12.6.1 also holds if x0 , xn belong to a small neighborhood of xf and y0 , yn belong to a small neighborhood of yf . Lemma 12.6.5 For each ∈ (0, 1) there exists a positive number δ < such that the following property holds: If n ≥ 2 is a natural number and if a (fr , δ)-good pair of sequences {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y satisﬁes |xj − xf |, |yj − yf | ≤ δ, j = 0, n, then |xi − xf |, |yi − yf | ≤ , i = 0, . . . , n.

362

TURNPIKE PROPERTIES

Denote by Card(E) the cardinality of a set E. The next auxiliary result shows that if an integer n is large enough and if a good pair of sequences {xi }ni=0 , {yi }ni=0 satisﬁes x0 , xn = xf and y0 , yn = yf , then (xj , yj ) belongs to a small neighborhood of (xf , yf ) for some j ∈ {1, . . . , n − 1}. Lemma 12.6.6 Suppose that M > 0 and let ∈ (0, 1). Then there exists a natural number n0 ≥ 4 such that the following property holds: 0 0 ⊂ X, {yi }ni=0 ⊂ Y satisIf an (fr , M )-good pair of sequences {xi }ni=0 ﬁes (6.23) x0 , xn0 = xf , y0 , yn0 = yf , then there exists j ∈ {1, . . . , n0 − 1} such that xj ∈ BX (xf , ), yj ∈ BY (yf , ).

(6.24)

Proof. Choose an integer n0 > 8 + 8(r)−1 M and put ξ1 , ξ2 = xf , 0 {xi }ni=0

ξ3 , ξ4 = yf ,

ξ = {ξi }4i=1 .

(6.25) (6.26)

0 {yi }ni=0

⊂ X, ⊂ Y is an (fr , M )-good pair of Assume that sequences and (6.23) holds. Proposition 12.4.4 implies that n 0 −1

sup{

0 fr (xi , xi+1 , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n0 )}

(6.27)

i=0 n 0 −1

≤ inf{sup{

0 ∈ ΛY (ξ, n0 )} : fr (pi , pi+1 , ui , ui+1 ) : {ui }ni=0

i=0 0 ∈ ΛX (ξ, n0 )} + 2M, {pi }ni=0

n 0 −1

inf{

0 fr (pi , pi+1 , yi , yi+1 ) : {pi }ni=0 ∈ ΛX (ξ, n0 )}

(6.28)

i=0 n 0 −1

≥ sup{inf{

0 fr (pi , pi+1 , ui , ui+1 ) : {pi }ni=0 ∈ ΛX (ξ, n0 )} :

i=0 0 ∈ ΛY (ξ, n0 )} − 2M. {ui }ni=0

It follows from Proposition 12.5.1, (6.3), Propositions 12.4.1 and 12.4.2 that n 0 −1

inf{sup{

i=0

0 fr (pi , pi+1 , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n0 )} :

(6.29)

363

A dynamic zero-sum game 0 {pi }ni=0 ∈ ΛX (ξ, n0 )}

= sup{inf{

n 0 −1

0 fr (pi , pi+1 , ui , ui+1 ) : {pi }ni=0 ∈ ΛX (ξ, n0 )} :

i=0 0 ∈ ΛY (ξ, n0 )} = n0 fr (xf , xf , yf , yf ). {ui }ni=0

In view of (6.29), (6.27), (6.2) and (6.28), n0 f (xf , xf , yf , yf ) = n0 fr (xf , xf , yf , yf ) ≥ −2M n 0 −1

+ sup{

(6.30)

0 fr (xi , xi+1 , ui , ui+1 ) : {ui }ni=0 ∈ ΛY (ξ, n0 )}

i=0

≥ −2M +

n 0 −1

fr (xi , xi+1 , yf , yf ) = −2M

i=0

+

n 0 −1

f (xi , xi+1 , yf , yf ) + r

i=0

n 0 −1

|xi − xf |,

i=0

n0 f (xf , xf , yf , yf ) = n0 fr (xf , xf , yf , yf ) ≤ 2M n 0 −1

+ inf{

(6.31)

0 fr (pi , pi+1 , yi , yi+1 ) : {pi }ni=0 ∈ ΛX (ξ, n0 )}

i=0

≤ 2M +

n 0 −1

fr (xf , xf , yi , yi+1 ) = 2M

i=0

+

n 0 −1

f (xf , xf , yi , yi+1 ) − r

i=0

n 0 −1

|yi − yf |.

i=0

Proposition 12.5.1 and (6.1) imply that n 0 −1 i=0

f (xi , xi+1 , yf , yf ) ≥ n0 f (xf , xf , yf , yf ) ≥

n 0 −1

f (xf , xf , yi , yi+1 ).

i=0

Combined with (6.30) and (6.31) this implies that n0 f (xf , xf , yf , yf ) ≥ −2M + n0 f (xf , xf , yf , yf ) + r

n 0 −1

|xi − xf |,

i=0

n0 f (xf , xf , yf , yf ) ≤ 2M + n0 f (xf , xf , yf , yf ) − r

n 0 −1 i=0

|yi − yf |,

364

TURNPIKE PROPERTIES

r

n 0 −1 i=0

|xi − xf | ≤ 2M,

r

n 0 −1

|yi − yf | ≤ 2M.

(6.32)

i=0

It follows from (6.23), (6.32) and (6.25) that Card{i ∈ {1, . . . , n0 − 1} : |xi − xf | ≥ } ≤ 2M r−1 , Card{i ∈ {1, . . . , n0 − 1} : |yi − yf | ≥ } ≤ 2M r−1 , Card{i ∈ {1, . . . , n0 − 1} : |xi − xf | < , |yi − yf | < } ≥ n0 − 1 − 4M (r)−1 > 6. This completes the proof of the lemma. Lemmas 12.6.6 and 12.6.2 imply the following. Lemma 12.6.7 For each ∈ (0, 1) and each M ∈ (0, ∞) there exists a natural number n0 ≥ 4 such that the following property holds: 0 0 ⊂ X, {yi }ni=0 ⊂ Y is an (fr , M )-good pair of sequences, If {xi }ni=0 then there is j ∈ {1, . . . , n0 − 1} for which xj ∈ BX (xf , ), yj ∈ B(yf , ). The next lemma shows that if an integer n is large enough, {xi }ni=0 , {yi }ni=0 is a good pair of sequences with respect to g belonging to a small neighborhood of fr , then (xj , yj ) belongs to a small neighborhood of (xf , yf ) for some j ∈ {1, . . . , n − 1}. Lemma 12.6.8 For each ∈ (0, 1) and each M ∈ (0, ∞) there exists a natural number n0 ≥ 4 and a neighborhood U of fr in M such that the following property holds: 0 0 ⊂ X, {yi }ni=0 ⊂ Y is a (g, M )-good pair of If g ∈ U and if {xi }ni=0 sequences, then there is j ∈ {1, . . . , n0 − 1} for which xj ∈ BX (xf , ), yj ∈ BY (yf , ).

(6.33)

Proof. Let ∈ (0, 1) and M ∈ (0, ∞). It follows from Lemma 12.6.7 that there exists a natural number n0 ≥ 4 such that for each 0 0 ⊂ X, {yi }ni=0 ⊂ Y there is (fr , M + 8)-good pair of sequences {xi }ni=0 j ∈ {1, . . . , n0 − 1} for which (6.33) is valid. Put U = {g ∈ M : ρ(fr , g) ≤ (16n0 )−1 }.

(6.34)

0 0 Assume that g ∈ U and {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y is a (g, M )-good 0 0 , {yi }ni=0 is pair of sequences. By (6.34) the pair of sequences {xi }ni=0

365

A dynamic zero-sum game

(fr , M + 8)-good. It follows from the deﬁnition of n0 that there exists j ∈ {1, . . . , n0 − 1} for which (6.33) is valid. The lemma is proved. The following auxiliary result is the main ingredient for the proof of Theorem 12.2.1. Lemma 12.6.9 For each ∈ (0, 1) there exist a neighborhood U of fr in M, a number δ ∈ (0, ) and a natural number n1 ≥ 4 such that the following property holds: If g ∈ U , n ≥ 2n1 is a natural number and if {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y is a (g, δ)-good pair of sequences, then xi ∈ BX (xf , ), yi ∈ BY (yf , ),

(6.35)

for all i ∈ [n1 , n − n1 ]. Moreover if x0 ∈ BX (xf , δ), y0 ∈ BY (yf , δ), then (6.35) holds for all i ∈ [0, n − n1 ], and if xn ∈ BX (xf , δ), yn ∈ BY (yf , δ), then (6.35) is valid for all i ∈ [n1 , n]. Proof. Let ∈ (0, 1). Lemma 12.6.5 implies that there exists δ0 ∈ (0, ) such that for each natural number n ≥ 2 and each (fr , δ0 )-good pair of sequences {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y satisfying xj ∈ BX (xf , δ0 ), yj ∈ B(yf , δ0 ), j = 0, n,

(6.36)

relation (6.35) is valid for i = 0, . . . , n. By Lemma 12.6.8 there exists an integer n0 ≥ 4 and a neighborhood U0 of fr in M such that for each 0 0 ⊂ X, {yi }ni=0 ⊂Y g ∈ U0 and each (g, 8)-good pair of sequences {xi }ni=0 there is j ∈ {1, . . . , n0 − 1} for which xj ∈ BX (xf , δ0 ), yj ∈ BY (yf , δ0 ).

(6.37)

n1 ≥ 4n0

(6.38)

δ ∈ (0, 4−1 δ0 ).

(6.39)

Fix an integer and a number Deﬁne U = U0 ∩ {g ∈ M :

ρ(g, fr ) ≤ 16−1 δn−1 1 }.

(6.40)

Assume that g ∈ U , an integer n ≥ 2n1 and {xi }ni=0 ⊂ X, {yi }ni=0 ⊂ Y is a (g, δ)-good pair of sequences. It follows from (6.39), (6.38) and the

366

TURNPIKE PROPERTIES

deﬁnition of n0 , U0 that there exists a sequence of integers {ti }ki=1 ⊂ [0, n] such that t1 ≤ n0 , ti+1 − ti ∈ [n0 , 3n0 ], i = 1, . . . , k − 1, n − tk ≤ n 0 ,

(6.41)

|xti − xf |, |yti − yf | ≤ δ0 , i = 1, . . . , k

and, moreover, if |x0 − xf |, |y0 − yf | ≤ δ, then t1 = 0, and if |xn − xf |, |yn − yf | ≤ δ, then tk = n. Clearly k ≥ 2. Fix q ∈ {1, . . . , k − 1}. To complete the proof of the lemma it is suﬃcient to show that for each integer i ∈ [tq , tq+1 ] the relation (6.35) holds. (q) tq+1 −tq (q) tq+1 −tq Deﬁne sequences {xi }i=0 ⊂ X, {yi }i=0 ⊂ Y by (q)

xi

(q)

= xi+tq , yi (q) t

= yi+tq , −t

i ∈ [0, tq+1 − tq ].

(q) t

(6.42)

−t

q q q+1 q+1 It is easy to see that {xi }i=0 , {yi }i=0 is a (g, δ)-good pair of sequences. Combined with (6.40), (6.39) and (6.41) this implies that the (q) tq+1 −tq (q) tq+1 −tq pair of sequences {xi }i=0 , {yi }i=0 is (fr , δ0 )-good. It follows from (6.42), (6.41), (6.39) and the deﬁnition of δ0 (see (6.36)) that

(q)

xi

(q)

∈ BX (xf , ), yi

∈ BY (yf , ),

i = 0, . . . , tq+1 − tq .

Together with (6.42) this implies that xi ∈ BX (xf , ), yi ∈ BY (yf , ), i = tq , . . . , tq+1 . This completes the proof of the lemma.

12.7.

Preliminary lemmas for Theorem 12.2.2

For each metric space K denote by C(K) the space of all continuous functions on K with the topology of uniform convergence (||φ|| = sup{|φ(z)| : z ∈ K}, φ ∈ C(K)). Let f ∈ M. There exist xf ∈ X, yf ∈ Y such that sup f (xf , xf , y, y) = f (xf , xf , yf , yf )

y∈Y

(7.1)

= inf f (x, x, yf , yf ) x∈X

(see (6.1)). Let r ∈ (0, 1). Deﬁne fr : X × X × Y × Y → R1 by fr (x1 , x2 , y1 , y2 ) = f (x1 , x2 , y1 , y2 ) + r|x1 − xf |

(7.2)

367

A dynamic zero-sum game

−r|y1 − yf |,

x1 , x2 ∈ X, y1 , y2 ∈ Y (X)

(see (6.2)). Clearly fr ∈ M. Deﬁne functions fr (Y ) fr : Y × Y → R1 by

: X × X → R1 ,

fr(X) (x1 , x2 ) = fr (x1 , x2 , yf , yf ),

x1 , x2 ∈ X,

(7.3)

fr(Y ) (y1 , y2 ) = fr (xf , xf , y1 , y2 ),

y1 , y2 ∈ Y.

(7.4)

Lemma 12.7.1 For each ∈ (0, 1) there exists a positive number δ < for which the following condition holds: If n ≥ 2 is a natural number, {xi }ni=0 ⊂ X,

x0 , xn = xf

(7.5)

zn = xn

(7.6)

and if for each {zi }ni=0 ⊂ X satisfying z0 = x0 , the inequality n−1

fr(X) (xi , xi+1 ) ≤

i=0

n−1

fr(X) (zi , zi+1 ) + δ

(7.7)

i=0

holds, then |xi − xf | ≤ ,

i = 0, . . . , n.

(7.8)

Proof. Let ∈ (0, 1). Choose a number δ ∈ (0, 8−1 r).

(7.9)

Assume that an integer n ≥ 2, {xi }ni=0 ⊂ X, (7.5) is valid and for each sequence {zi }ni=0 ⊂ X satisfying (7.6), relation (7.7) holds. This implies that n−1

fr (xi , xi+1 , yf , yf ) ≤ nfr (xf , xf , yf , yf ) + δ

i=0

= nf (xf , xf , yf , yf ) + δ. By (7.2), (7.5) and (7.1), n−1

fr (xi , xi+1 , yf , yf ) = r

i=0

n−1

|xi − xf |

i=0

+

n−1 i=0

f (xi , xi+1 , yf , yf )

(7.10)

368

TURNPIKE PROPERTIES

≥r

n−1

|xi − xf | + nf (n−1

i=0

n−1

xi , n−1

i=0

≥r

n−1

n−1

xi , yf , yf )

i=0

|xi − xf | + nf (xf , xf , yf , yf ).

i=0

It follows from this relation, (7.10) and (7.9) that for each i ∈ {0, . . . , n− 1} the inequality |xi − xf | ≤ r−1 δ < is true. This completes the proof of the lemma. Deﬁnition 7.1. Let g ∈ C(X × X), n be a natural number, and let M be a nonnegative number. A sequence {¯ xi }ni=0 ⊂ X is called (g, X, M )good if M+ for each sequence

n−1

g(xi , xi+1 ) ≥

i=0 n {xi }i=0 ⊂

n−1

g(¯ xi , x ¯i+1 )

i=0

X satisfying x0 = x ¯0 , xn = x ¯n .

Deﬁnition 7.2. Let g ∈ C(Y ×Y ), n be a natural number and let M be a nonnegative number. A sequence {¯ yi }ni=0 ⊂ Y is called (g, Y, M )-good if n−1

g(yi , yi+1 ) ≤ M +

i=0

n−1

g(¯ yi , y¯i+1 )

i=0

for each sequence {yi }ni=0 ⊂ Y satisfying y0 = y¯0 , yn = y¯n . Deﬁnition 7.3. Let n1 ≥ 0, n2 > n1 be a pair of integers, M be a nonnegative number, and let 2 −1 ⊂ C(X × X). {gi }ni=n 1

n2 −1 2 A sequence {¯ xi }ni=n ⊂ X is called ({gi }i=n , X, M )-good if 1 1

M+

n 2 −1 i=n1

gi (xi , xi+1 ) ≥

n 2 −1

gi (¯ xi , x ¯i+1 )

i=n1

2 ⊂ X satisfying xn1 = x ¯n1 , xn2 = x ¯n2 . for each sequence {xi }ni=n 1

Deﬁnition 7.4. Let n1 ≥ 0, n2 > n1 be integers, and let 2 −1 ⊂ C(Y × Y ), M ∈ [0, ∞). {gi }ni=n 1 2 2 −1 A sequence {¯ yi }ni=n ⊂ Y is called ({gi }ni=n , Y, M )-good if for each se1 1 n2 quence {yi }i=n1 ⊂ Y satisfying

yn1 = y¯n1 , yn2 = y¯n2

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A dynamic zero-sum game

the following inequality holds: n 2 −1 i=n1

gi (yi , yi+1 ) ≤

n 2 −1

gi (¯ yi , y¯i+1 ) + M.

i=n1

Analogously to Lemma 12.7.1 we can establish the following Lemma 12.7.2 For each ∈ (0, 1) there exists a positive number δ < such that the following property holds: (Y ) If n ≥ 2 is a natural number and if an (fr , Y, δ)-good sequence {yi }ni=0 ⊂ Y satisﬁes y0 , yn = yf , then yi ∈ BY (yf , ), i = 0, . . . , n. By using Lemmas 12.7.1 and 12.6.3 we can easily deduce Lemma 12.7.3 For each ∈ (0, 1) there exists a positive number δ such that the following property holds: (X) If n ≥ 2 is a natural number and if an (fr , X, δ)-good sequence n {xi }i=0 ⊂ X satisﬁes x0 , xn ∈ BX (xf , δ), then xi ∈ BX (xf , ), i = 0, . . . , n. By using Lemmas 12.7.2 and 12.6.3 we can easily deduce Lemma 12.7.4 For each ∈ (0, 1) there exists a positive number δ such that the following property holds: (Y ) If n ≥ 2 is a natural number and if an (fr , Y, δ)-good sequence {yi }ni=0 ⊂ Y satisﬁes y0 , yn ∈ BY (yf , δ), then yi ∈ BY (yf , ), i = 0, . . . , n. Lemma 12.7.5 For each ∈ (0, 1) and each M > 0 there exists a natural number n0 ≥ 4 such that the following property holds: (X) 0 If an (fr , X, M )-good sequence {xi }ni=0 ⊂ X satisﬁes x0 = xf , xn0 = xf ,

(7.11)

then there is j ∈ {1, . . . , n0 − 1} such that xj ∈ BX (xf , ).

(7.12)

Proof. Let ∈ (0, 1) and M > 0. Choose an integer n0 > 8 + 8M (r)−1 . (X)

(7.13)

0 Assume that {xi }ni=0 ⊂ X is an (fr , X, M )-good sequence and (7.11) is valid. It is not diﬃcult to see that

M + n0 f (xf , xf , yf , yf ) = n0 fr (xf , xf , yf , yf ) + M

370

TURNPIKE PROPERTIES n 0 −1

≥

fr (xi , xi+1 , yf , yf ) = r

n 0 −1

i=0

+

n 0 −1

f (xi , xi+1 , yf , yf ) ≥ r

n 0 −1

i=0

|xi − xf |

i=0

+n0 f (n−1 0 ≥r

|xi − xf |

i=0

n 0 −1

n 0 −1 i=0

xi , n−1 0

n 0 −1

xi , yf , yf )

i=0

|xi − xf | + n0 f (xf , xf , yf , yf ).

i=0

Combined with (7.13) this implies that there is j ∈ {1, . . . , n0 − 1} for which (7.12) is valid. This completes the proof of the lemma. Analogously to Lemma 12.7.5 we can establish the following Lemma 12.7.6 For each ∈ (0, 1) and each M > 0 there exists a natural number n0 ≥ 4 such that the following property holds: (Y ) 0 If an (fr , Y, M )-good sequence {yi }ni=0 ⊂ Y satisﬁes y0 = yf , yn0 = yf , then there exists j ∈ {1, . . . , n0 − 1} such that yj ∈ BY (yf , ). Choose a number D0 ≥ sup{|fr (x1 , x2 , y1 , y2 )| : x1 , x2 ∈ X, y1 , y2 ∈ Y }. We can easily prove the following lemma. Lemma 12.7.7 1. Assume that n ≥ 2 is an integer, M is a posi(X) ¯0 = tive number, a sequence {xi }ni=0 ⊂ X is (fr , X, M )-good and x ¯n = xf , x ¯i = xi , i = 1, . . . , n − 1. Then the sequence {¯ xi }ni=0 is xf , x (X) (fr , X, M + 8D0 )-good. 2. Assume that n ≥ 2 is an integer, M is a positive number, a se(Y ) quence {yi }ni=0 ⊂ Y is (fr , Y, M )-good and y¯0 = yf , y¯n = yf , y¯i = (Y ) yi , i = 1, . . . , n − 1. Then the sequence {¯ yi }ni=0 is (fr , Y, M + 8D0 )good. Lemmas 12.7.5, 12.7.6 and 12.7.7 imply the following two results. Lemma 12.7.8 For each ∈ (0, 1) and each M > 0 there exists a natural number n0 ≥ 4 such that the following condition holds: (X) 0 For each (fr , X, M )-good sequence {xi }ni=0 ⊂ X, min{|xj − xf | : j ∈ {1, . . . , n0 − 1}} ≤ .

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A dynamic zero-sum game

Lemma 12.7.9 For each ∈ (0, 1) and each M > 0 there exists a natural number n0 ≥ 4 such that the following condition holds: (Y ) 0 For each (fr , Y, M )-good sequence {yi }ni=0 ⊂Y, min{|yj − yf | : j ∈ {1, . . . , n0 − 1}} ≤ . By using Lemmas 12.7.8 and 12.7.9 analogously to the proof of Lemma 12.6.8, we can establish the following two results. Lemma 12.7.10 For each ∈ (0, 1) and each M > 0 there exists a (X) natural number n0 ≥ 4 and a neighborhood U of fr in C(X × X) such that the following property holds: 0 0 −1 0 −1 ⊂ X is a ({gi }ni=0 , X, M )-good se⊂ U and if {xi }ni=0 If {gi }ni=0 quence, then there is j ∈ {1, . . . , n0 − 1} for which xj ∈ BX (xf , ). Lemma 12.7.11 For each ∈ (0, 1) and each M > 0 there exists a (Y ) natural number n0 ≥ 4 and a neighborhood U of fr in C(Y × Y ) such that the following property holds: 0 −1 0 0 −1 ⊂ U and if {yi }ni=0 ⊂ Y is a ({gi }ni=0 , Y, M )-good seIf {gi }ni=0 quence, then there is j ∈ {1, . . . , n0 − 1} for which yj ∈ BY (yf , ). (X)

Lemma 12.7.12 For each ∈ (0, 1) there exist a neighborhood U of fr in C(X × X), a positive number δ < and a natural number n1 ≥ 4 such that the following property holds: n If n ≥ 2n1 is a natural number, {gi }n−1 i=0 ⊂ U and if {xi }i=0 ⊂ X is a n−1 ({gi }i=0 , X, δ)-good sequence, then xi ∈ BX (xf , )

(7.14)

for all integers i ∈ [n1 , n − n1 ]. Moreover if x0 ∈ BX (xf , δ), then (7.14) holds for all integers i ∈ [0, n − n1 ], and if xn ∈ BX (xf , δ), then (7.14) is valid for all integers i ∈ [n1 , n]. Proof. Let ∈ (0, 1). Lemma 12.7.3 implies that there exists a positive number δ0 < such that the following property holds: (X) If n ≥ 2 is a natural number and if an (fr , X, δ0 )-good sequence {xi }ni=0 ⊂ X satisﬁes x0 , xn ∈ BX (xf , δ0 ), then the relation (7.14) is valid for i = 0, . . . , n. In view of Lemma 12.7.10 there exist a natural number n0 ≥ 4 and a (X) neighborhood U0 of fr in C(X × X) such that the following property holds: 0 −1 0 −1 ⊂ U0 and if {xi }ni=0 ⊂ X is a ({gi }ni=0 , X, 8)-good seIf {gi }ni=0 quence, then there is j ∈ {1, . . . , n0 − 1} for which xj ∈ BX (xf , δ0 ).

372

TURNPIKE PROPERTIES

Choose a natural number n1 ≥ 4n0 and a positive number δ < 4−1 δ0 and set U = U0 ∩ {g ∈ C(X × X) : ||g − fr(X) || ≤ (16n1 )−1 δ}. n Assume that an integer n ≥ 2n1 , {gi }n−1 i=0 ⊂ U and a sequence {xi }i=0 n−1 ⊂ X is ({gi }i=0 , X, δ)-good. Arguing as in the proof of Lemma 12.6.9 we can show that (7.14) is valid for all integers i ∈ [n1 , n−n1 ] and, moreover, if x0 ∈ BX (xf , δ), then (7.14) holds for all integers i ∈ [0, n − n1 ], and if xn ∈ BX (xf , δ), then (7.14) is valid for all integers i ∈ [n1 , n]. The lemma is proved.

Analogously to Lemma 12.7.12 we can prove the following (Y )

Lemma 12.7.13 For each ∈ (0, 1) there exist a neighborhood U of fr in C(Y × Y ), a positive number δ < and a natural number n1 ≥ 4 such that the following property holds: n If n ≥ 2n1 is a natural number, {gi }n−1 i=0 ⊂ U and if {yi }i=0 ⊂ Y is a n−1 ({gi }i=0 , Y, δ)-good sequence, then yi ∈ BY (yf , )

(7.15)

for all integers i ∈ [n1 , n − n1 ]. Moreover if y0 ∈ BY (yf , δ), then (7.15) holds for all integers i ∈ [0, n − n1 ], and if yn ∈ BY (yf , δ), then (7.15) is valid for all integers i ∈ [n1 , n].

12.8.

Proofs of Theorems 12.2.1 and 12.2.2

We will use the notation from sections 12.1-12.7. Let f ∈ M. There exists a pair (xf , yf ) ∈ X × Y such that (6.1) holds. Let r ∈ (0, 1) and let i be a natural number. Consider the function fr : X × X × Y × Y deﬁned by (6.2). Clearly all lemmas from sections 12.6 and 12.7 are valid for fr . By Lemma 12.7.12 there exist a number

a number

γ1 (f, r, i) ∈ (0, 2−i ),

(8.1)

δ1 (f, r, i) ∈ (0, 2−i )

(8.2)

and an integer n1 (f, r, i) ≥ 4 such that the following property holds: (a) If n ≥ 2n1 (f, r, i) is a natural number, {gj }n−1 j=0 ⊂ C(X × X) satisﬁes ||gj − fr(X) || ≤ γ1 (f, r, i), j = 0, . . . , n − 1,

373

A dynamic zero-sum game

and if {xj }nj=0 ⊂ X is a ({gj }n−1 j=0 , X, δ1 (f, r, i))-good sequence, then xj ∈ BX (xf , 2−i ),

j ∈ [n1 (f, r, i), n − n1 (f, r, i)].

(8.3)

Lemma 12.7.13 implies that there exist numbers δ2 (f, r, i), γ2 (f, r, i) ∈ (0, 2−i )

(8.4)

and a natural number n2 (f, r, i) ≥ 4 such that the following property holds: (b) If n ≥ 2n2 (f, r, i) is a natural number, {gj }n−1 j=0 ⊂ C(Y × Y ) satisﬁes ||gj − fr(Y ) || ≤ γ2 (f, r, i), j = 0, . . . , n − 1 and if {yj }nj=0 ⊂ Y is a ({gj }n−1 j=0 , Y, δ2 (f, r, i))-good sequence, then yj ∈ BY (yf , 2−i ),

j ∈ [n2 (f, r, i), n − n2 (f, r, i)].

(8.5)

Put n3 (f, r, i) = n1 (f, r, i) + n2 (f, r, i),

(8.6)

δ3 (f, r, i) = min{δ1 (f, r, i), δ2 (f, r, i)}, γ3 (f, r, i) = min{γ1 (f, r, i), γ2 (f, r, i)}. In view of the uniform continuity of the function fr there exists a number (8.7) δ4 (f, r, i) ∈ (0, δ3 (f, r, i)) such that if x1 , x2 , x ¯1 , x ¯2 ∈ X, y1 , y2 , y¯1 , y¯2 ∈ Y satisfy |xj − x ¯j |, |yj − y¯j | ≤ δ4 (f, r, i),

j = 1, 2,

then |fr (x1 , x2 , y1 , y2 ) − fr (¯ x1 , x ¯2 , y¯1 , y¯2 )| ≤ 16−1 γ3 (f, r, i).

(8.8)

It follows from Lemma 12.6.9 that there exist numbers γ4 (f, r, i) ∈ (0, 16−1 γ3 (f, r, i)), δ5 (f, r, i) ∈ (0, 8−1 δ4 (f, r, i))

(8.9)

and a natural number n4 (f, r, i) ≥ 4 such that the following property holds: (c) If g ∈ M satisﬁes ρ(g, fr ) ≤ γ4 (f, r, i), n ≥ 2n4 (f, r, i) is a natural number and if {xj }nj=0 ⊂ X, {yj }nj=0 ⊂ Y is a (g, δ5 (f, r, i))-good pair of sequences, then xj ∈ BX (xf , 8−1 δ4 (f, r, i)), yj ∈ BY (yf , 8−1 δ4 (f, r, i))

(8.10)

374

TURNPIKE PROPERTIES

for all j ∈ [n4 (f, r, i), n − n4 (f, r, i)]; moreover if x0 ∈ BX (xf , δ5 (f, r, i)), y0 ∈ BY (yf , δ5 (f, r, i)), then (8.10) holds for all integers j ∈ [0, n − n4 (f, r, i)], and if xn ∈ BX (xf , δ5 (f, r, i)), yn ∈ BY (yf , δ5 (f, r, i)), then (8.10) is valid for all integers j ∈ [n4 (f, r, i), n]. Lemma 12.6.9 implies that there exist numbers γ(f, r, i) ∈ (0, 8−1 γ4 (f, r, i)), δ(f, r, i) ∈ (0, 8−1 δ5 (f, r, i))

(8.11)

and a natural number n5 (f, r, i) ≥ 4 such that the following property holds: (d) If g ∈ M satisﬁes ρ(g, fr ) ≤ γ(f, r, i), n ≥ 2n5 (f, r, i) is a natural number and if {xj }nj=0 ⊂ X, {yj }nj=0 ⊂ Y is a (g, δ(f, r, i))-good pair of sequences, then xj ∈ BX (xf , 8−1 δ5 (f, r, i)), yj ∈ BY (yf , 8−1 δ5 (f, r, i))

(8.12)

for all j ∈ [n5 (f, r, i), n − n5 (f, r, i)]. Set U (f, r, i) = {g ∈ M : ρ(g, fr ) < γ(f, r, i)}.

(8.13)

Deﬁne F = ∩∞ k=1 ∪ {U (f, r, i) : f ∈ M, r ∈ (0, 1), i = k, k + 1, . . .}.

(8.14)

It is easy to see that F is a countable intersection of open everywhere dense sets in M. Proof of Theorem 12.2.1. Let h ∈ F. There exists a pair (x1 , y1 ) ∈ X × Y such that sup h(x1 , x1 , y, y) = h(x1 , x1 , y1 , y1 ) = inf h(x, x, y1 , y1 ) x∈X

y∈Y

(8.15)

(see (3.1) and (3.2)). Assume that (x2 , y2 ) ∈ X × Y and sup h(x2 , x2 , y, y) = h(x2 , x2 , y2 , y2 ) = inf h(x, x, y2 , y2 ). x∈X

y∈Y

(8.16)

We will show that x2 = x1 ,

y2 = y1 .

(8.17)

375

A dynamic zero-sum game (1)

(2)

(1)

(2)

∞ ∞ ∞ Deﬁne sequences {xj }∞ j=0 , {xj }j=0 ⊂ X, {yj }j=0 , {yj }j=0 ⊂ Y by (1)

xj

(2)

= x1 , xj

= x2 ,

(1)

yj

(2)

= y1 , yj

= y2 ,

j = 0, 1, . . . .

(8.18)

It follows from (8.15), (8.18) and Proposition 12.5.1 that the pairs of sequences (1)

(1)

∞ ({xj }∞ j=0 , {yj }j=0 ),

(2)

(2)

∞ ({xj }∞ j=0 , {yj }j=0 )

are (h)-minimal. Let ∈ (0, 1). Choose a natural number k such that 2−k < 64−1 .

(8.19)

There exist f ∈ M, r ∈ (0, 1) and an integer i ≥ k such that h ∈ U (f, r, i). (1)

(8.20)

(1)

(2)

(2)

∞ ∞ ∞ Since the pairs of sequences ({xj }∞ j=0 , {yj }j=0 ), ({xj }j=0 , {yj }j=0 ) are (h)-minimal it follows from (8.19), (8.20), (8.18), property (d) and (8.13) that

|x1 − xf |, |x2 − xf |, |y1 − yf |, |y2 − yf | ≤ 8−1 δ5 (f, r, i) < 2−i < , |x1 − x2 |, |y1 − y2 | ≤ 2. Since is an arbitrary number in the interval (0, 1) we conclude that (8.17) is valid. Therefore we have shown that there exists a unique pair (xh , yh ) ∈ X × Y such that sup h(xh , xh , y, y) = h(xh , xh , yh , yh ) = inf h(x, x, yh , yh ).

(8.21)

x∈X

y∈Y

Let > 0. Choose a natural number k for which (8.19) holds. There exist f ∈ M, r ∈ (0, 1) and an integer i ≥ k for which (8.20) is valid. (h) (h) ∞ Consider the sequences {xj }∞ j=0 ⊂ X, {yj }j=0 ⊂ Y deﬁned by (h)

xj

= xh ,

(h)

yj

= yh ,

j = 0, 1, . . . . (h)

(8.22) (h)

∞ It was shown above that the pair of sequences {xj }∞ j=0 , {yj }j=0 is (h)-minimal. It follows from (8.19), (8.22), (8.13) and property (d) that

xh ∈ BX (xf , 8−1 δ5 (f, r, i)), yh ∈ BY (yf , 8−1 δ5 (f, r, i)).

(8.23)

Assume that g ∈ U (f, r, i), an integer n ≥ 2n4 (f, r, i) and {xj }nj=0 ⊂ X, {yj }nj=0 ⊂ Y is a (g, δ5 (f, r, i))-good pair of sequences. It follows

376

TURNPIKE PROPERTIES

from property (c), (8.11), (8.13) and (8.23) that the following properties hold: xj ∈ BX (xf , 8−1 δ4 (f, r, i)), yj ∈ BY (yf , 8−1 δ4 (f, r, i)),

(8.24)

and |xj − xh |, |yj − yh | ≤ for all integers j ∈ [n4 (f, r, i), n − n4 (f, r, i)]; if |x0 − xf |, |y0 − yf | ≤ δ5 (f, r, i), then (8.24) holds for all integers j ∈ [0, n − n4 (f, r, i)]; if |xn − xf |, |yn − yf | ≤ δ5 (f, r, i), then (8.24) holds for all integers j ∈ [n4 (f, r, i), n]. Together with (8.23) this implies that the following properties hold: if x0 ∈ BX (xh , 2−1 δ5 (f, r, i)), y0 ∈ BY (yf , 2−1 δ5 (f, r, i)), then (8.24) hold for all integers j ∈ [0, n − n4 (f, r, i)]; if xn ∈ BX (xh , 2−1 δ5 (f, r, i)), yn ∈ BY (yf , 2−1 δ5 (f, r, i)), then (8.24) is valid for all integers j ∈ [n4 (f, r, i), n]. This completes the proof of the theorem. Proof of Theorem 12.2.2. Let h ∈ F, z ∈ X, ξ ∈ Y . By Theorem 12.2.1 there exists a unique pair (xh , yh ) ∈ X × Y such that sup h(xh , xh , y, y) = h(xh , xh , yh , yh )

y∈Y

= inf h(x, x, yh , yh ). x∈X

(8.25)

By Proposition 12.5.3 there is an (h)-minimal pair of sequences {¯ xj }∞ yj }∞ j=0 ⊂ X, {¯ j=0 ⊂ Y for which x ¯0 = z,

y¯0 = ξ.

(8.26)

We show that the pair of sequences ({¯ xj }∞ yj }∞ j=0 , {¯ j=0 ) is (h)-overtaking optimal. Theorem 12.2.1 implies that x ¯j → xh ,

y¯j → yh as j → ∞.

(8.27)

Let {xi }∞ i=0 ⊂ X and x0 = z. We will show that T −1

lim sup[ T →∞

j=0

h(¯ xj , x ¯j+1 , y¯j , y¯j+1 ) −

T −1 j=0

h(xj , xj+1 , y¯j , y¯j+1 )] ≤ 0. (8.28)

377

A dynamic zero-sum game

Assume the contrary. Then there exists a number Γ0 > 0 and a strictly increasing sequence of natural numbers {Tk }∞ k=1 such that for all integers k ≥ 1, T k −1

h(¯ xj , x ¯j+1 , y¯j , y¯j+1 ) −

j=0

T k −1

h(xj , xj+1 , y¯j , y¯j+1 ) ≥ Γ0 .

(8.29)

j=0

We will show that xj → xh as j → ∞.

(8.30)

For j = 0, 1, . . . deﬁne a function gj : X × X → R1 by gj (u1 , u2 ) = h(u1 , u2 , y¯j , y¯j+1 ),

u1 , u1 ∈ X.

(8.31)

Clearly gj ∈ C(X × X), j = 0, 1, . . . . Let > 0. Choose a natural number q such that (8.32) 2−q < 64−1 . There exist f ∈ M, r ∈ (0, 1) and an integer p ≥ q such that h ∈ U (f, r, p).

(8.33)

Since the pair of sequences ({¯ xj }∞ yj }∞ j=0 , {¯ j=0 ) is (h)-minimal it follows from the deﬁnition of U (f, r, p) (see (8.13)), (8.33) and property (d) that for all integers j ≥ n5 (f, r, p), x ¯j ∈ BX (xf , 8−1 δ5 (f, r, p)), y¯j ∈ BY (yf , 8−1 δ5 (f, r, p)).

(8.34)

By (8.25), Proposition 12.5.1, (8.33) and property (d), xh ∈ BX (xf , 8−1 δ5 (f, r, p)), yh ∈ BY (yf , 8−1 δ5 (f, r, p)).

(8.35)

Since the pair of sequences ({¯ xj }∞ yj }∞ j=0 , {¯ j=0 ) is (h)-minimal there exists a constant c0 > 0 such that for each integer T ≥ 1, T −1

T −1

h(¯ xj , x ¯j+1 , y¯j , y¯j+1 ) ≤ inf{

j=0

h(uj , uj+1 , y¯j , y¯j+1 ) :

(8.36)

j=0

{uj }Tj=0 ⊂ X,

u0 = z} + c0 .

(8.36), (8.31) and (8.29) imply that the following property holds: (e) For each ∆ > 0 there exists an integer j(∆) ≥ 1 such that for 2 is each pair of integers n1 ≥ j(∆), n2 > n1 the sequence {xj }nj=n 1 n2 −1 ({gj }j=n1 , X, ∆)-good.

378

TURNPIKE PROPERTIES (X)

Consider the function fr : X × X → R1 deﬁned by (7.3). For j = 0, 1, . . . deﬁne a function g¯j : X × X → R1 by g¯j (u1 , u2 ) = fr (u1 , u2 , y¯j , y¯j+1 ),

u1 , u2 ∈ X.

(8.37)

It follows from (7.3), (8.37), (8.34), (8.9) and the deﬁnition of δ4 (f, r, p) (see (8.7), (8.8)) that for all integers j ≥ n5 (f, r, p), ||¯ gj − fr(X) || ≤ 16−1 γ3 (f, r, p).

(8.38)

By (8.38), (8.37), (8.31), (8.33), (8.13), (8.11), (8.9) for all integers j ≥ n5 (f, r, p), ||gj − fr(X) || ≤ 16−1 γ3 (f, r, p) + γ(f, r, p) < γ3 (f, r, p).

(8.39)

It follows from (8.39), properties (e) and (a) and (8.6) that there exists an integer m0 ≥ 1 such that |xj − xf | ≤ 2−p for all integers j ≥ m0 . Together with (8.32) and (8.35) this implies that for all integers j ≥ m0 the relation |xj − xh | ≤ 2−p + 2−p < is true. Since is an arbitrary positive number we conclude that lim xj = xh .

(8.40)

j→∞

There exists a number 0 > 0 such that for each z1 , z2 , z¯1 , z¯2 ∈ X and each ξ1 , ξ2 , ξ¯1 , ξ¯2 ∈ Y which satisfy |zj − z¯j |, |ξj − ξ¯j | ≤ 20 ,

j = 1, 2

(8.41)

the following relation holds: z1 , z¯2 , ξ¯1 , ξ¯2 )| ≤ 8−1 Γ0 . |h(z1 , z2 , ξ1 , ξ2 ) − h(¯

(8.42)

By (8.40) and (8.27) there exists an integer j0 ≥ 8 such that for all integers j ≥ j0 , |xj − xh | ≤ 2−1 0 ,

|¯ xj − xh | ≤ 2−1 0 .

(8.43)

There exists an integer s ≥ 1 such that Ts > j0 .

(8.44)

Deﬁne a sequence {x∗j }sj=0 ⊂ X by x∗j = xj , j = 0, . . . , Ts − 1,

x∗Ts = x ¯Ts .

(8.45)

379

A dynamic zero-sum game

Since the pair of sequences ({¯ xj }∞ yj }∞ j=0 , {¯ j=0 ) is (h)-minimal we conclude that, by (8.45), T s −1

h(¯ xj , x ¯j+1 , y¯j , y¯j+1 ) −

j=0

T s −1

h(x∗j , x∗j+1 , y¯j , y¯j+1 ) ≤ 0.

(8.46)

j=0

On the other hand it follows from (8.45), (8.29), (8.43), (8.44) and the deﬁnition of 0 (see (8.41),(8.42)) that T s −1

h(¯ xj , x ¯j+1 , y¯j , y¯j+1 ) −

j=0

=

T s −1

T s −1

h(x∗j , x∗j+1 , y¯j , y¯j+1 )

j=0

h(¯ xj , x ¯j+1 , y¯j , y¯j+1 ) −

T s −1

j=0

h(xj , xj+1 , y¯j , y¯j+1 )

j=0

+h(xTs −1 , xTs , y¯Ts −1 , y¯Ts ) − h(x∗Ts −1 , x∗Ts , y¯Ts −1 , y¯Ts ) ≥ Γ0 + h(xTs −1 , xTs , y¯Ts −1 , y¯Ts ) − h(xTs −1 , x ¯Ts , y¯Ts −1 , y¯Ts ) ≥ Γ0 − 8−1 Γ0 . This is contradictory to (8.46). The obtained contradiction proves that (8.28) holds. Analogously we can show that for each sequence {yj }∞ j=0 ⊂ Y satisfying y0 = ξ, T −1

lim sup[ T →∞

−

T −1

h(¯ xj , x ¯j+1 , yj , yj+1 )

j=0

h(¯ xj , x ¯j+1 , y¯j , y¯j+1 )] ≤ 0.

j=0

This implies that the pair of sequences ({¯ xj }∞ yj }∞ j=0 , {¯ j=0 ) is (h)-overtaking optimal. This completes the proof of the theorem.

Comments

Chapter 1. In this chapter we discuss three notions of optimality for inﬁnite horizon problems. The notion of (f )-minimal solutions was introduced by Aubry and Le Daeron [6] in their study of the discrete Frenkel–Kontorova model related to dislocations in one-dimensional crystals. In [6] Aubry and Le Daeron established the existence of (f )-minimal solutions and obtained a full description of their structure. A minimal solution was called in [6] a minimal energy conﬁguration. The theory developed in [6] is of great interest from the point of view of inﬁnite horizon optimal control as well as from the point of view of the theory of dynamical systems. Leizarowitz and Mizel [44] used the notion of (f )-minimal solutions in their study of a class of variational problems arising in continuum mechanics. They established the existence of periodic (f )-minimal solutions under a certain technical assumption which was removed in [90]. The notions of overtaking optimal solutions and good solutions were introduced in the economics literature [4, 33, 81]. Usually the existence of overtaking optimal solutions is a diﬃcult problem and we solve it for nonconvex variational problems only in a generic setting. But good solutions exist for all inﬁnite horizon variational problems considered in the book. Note that for practical needs it is enough to obtain approximate solutions which are good functions. The results which we obtained for good solutions are an important tool in our existence theory of overtaking optimal solutions. In order to establish the existence of overtaking optimal solutions we usualy verify that all good functions have the same asymptotic behavior and then show that a good minimal function is overtaking optimal. Note that many results on inﬁnite horizon optimal control problems are collected in [16, 26, 48, 60, 61, 67].

382

TURNPIKE PROPERTIES

Chapter 2. This chapter contains the turnpike results for nonautonomous nonconvex variational problems, the strongest and the most general results of this book. These results allow us to consider the turnpike property as a general phenomenon which holds for large classes of problems. We consider the complete metric space of integrands M and show that the turnpike property holds for most integrands of M in the sense of the Baire category. Such an approach is common in many areas of Mathematical Analysis [19, 21-23, 25, 35, 70, 106, 107]. The example of an integrand belonging to the space M which does not have the turnpike property given in Section 2.6 shows that the main results of the chapter cannot be improved. In [109] we obtained results which may help us to verify if a given integrand has the turnpike property. The turnpike property is very important for applications. Suppose that our integrand has the turnpike property and we know a ﬁnite number of “approximate” solutions of the variational problems with this integrand. Then we know the turnpike X, or at least its approximation, and the constant τ which is an estimate for the time period required to reach the turnpike. This information can be useful if we need to ﬁnd an “approximate” solution of a new variational problem with a new time interval [T1 , T2 ] and the new values y, z at the end points T1 and T2 . Namely, instead of solving this new problem on the “large” interval [T1 , T2 ] we can ﬁnd an “approximate” solution of the variational problem on the “small” interval [T1 , T1 + τ ] with the values y, X(T1 + τ ) at the end points and an “approximate” solution of the variational problem on the “small” interval [T2 − τ, T2 ] with the values X(T2 − τ ), z at the end points. Then the concatenation of the ﬁrst solution, the function X(t), t ∈ [T1 + τ, T2 − τ ] and the second solution is an “approximate” solution of the variational problem on the interval [T1 , T2 ] with the values y, z at the end points. Numerical applications of the turnpike theory are discussed in [62, 63, 67]. Chapter 3. The notion of a weakly optimal function was introduced in the economic literature by Brock [12]. The results of this chapter are obtained for the subspace A of the space of integrands M considered in Chapters 1 and 2. This subspace A consists of all time-independent integrands belonging to M. Since the subspace A is a small set in M and the results of Chapter 2 are of generic nature, they cannot be applied for the subspace A. The results which we obtain in this chapter are weaker

COMMENTS

383

than their analogs in Chapter 2. In Chapter 2 we establish a generic existence of an overtaking optimal function, while in this chapter we obtain a generic existence of a weakly optimal function. Also the convergence in the turnpike results for the autonomous case are weaker than the convergence in their analogs for the nonautonomous case. This fact is natural since in a large space we have more possibilities for perturbations. In Chapter 2 for a given integrand we use a perturbation which depends on t and obtain a new integrand which has the strong turnpike property. In Chapter 3 for a given integrand we may only use a perturbation which does not depend on t. As a result, for the space A we obtain a weaker turnpike property than for the space M. Some nongeneric turnpike results for the space A were obtained in [110]. In this chapter we show that the turnpike property holds for approximate solutions on ﬁnite intervals if the integrand has the so-called asympotic turnpike property, which means that all good functions on the inﬁnite interval [0, ∞) have the same asymptotic behavior. An analogous result for one-dimensional second order variational problems arising in continuum mechanics was obtained in [55]. Chapter 4. Chapter 4 is a continuation of Chapter 3. It contains a detailed analysis of the structure of optimal solutions on inﬁnite horizon problems for an integrand which has the asymptotic turnpike property. In Chapter 3 we associate with any integrand the so-called long run average cost growth rate and a certain continuous function. In this chapter we show that this long run average cost growth rate and the continuous function depend on the integrand continuously if the integrand has the asymptotic turnpike property. This result may be useful if we try to apply some numerical procedures in order to calculate weakly optimal solutions. It implies stability of such procedures. In this chapter we improve some turnpike results for optimal solutions of inﬁnite horizon problems. For example, we show that all optimal solutions with initial points belonging to a given bounded set converge to the turnpike uniformly.

384

TURNPIKE PROPERTIES

Chapter 5. In Chapter 5 our goal is to improve the turnpike results obtained in Chapter 2 for approximate solutions of autonomous variational problems on ﬁnite intervals. In order to obtain this improvement we need to suppose additional assumptions on integrands. Namely, we assume that the integrands are smooth and their partial derivatives satisfy certain growth conditions. These assumptions imply, in particular, that minimizers of variational problems are solutions of the corresponding Euler–Lagrange equations. We establish the turnpike property for integrands which have the asymptotic turnpike property. Chapter 6. In this chapter we study the turnpike properties of a class of linear control problems arising in engineering. This class includes an inﬁnite horizon problem of tracking of the periodic trajectory studied in [3]. An integrand is assumed to be periodic with respect to the time variable and strictly convex as the function of the state variable and the control variable. This assumption implies that the turnpike is a periodic trajectory. The strict convexity assumption implies the uniqueness of overtaking optimal solutions. As in the previous chapters we associate with our linear control problem a related discrete time optimal control problem which is in this case autonomous and strictly convex. Chapter 7. This chapter is devoted to the study of the turnpike properties for a class of linear control problems arising in engineering. This class includes the class of linear control problems discussed in Chapter 6 and, in particular, the inﬁnite horizon problem of tracking of the periodic trajectory studied in [3]. An integrand is assumed to be strictly convex as the function of the state variable and the control variable, but we do not assume that it is periodic with respect to the time variable. The strict convexity assumption implies the uniqueness of overtaking optimal solutions which are not periodic in general. Chapter 8. Discrete-time control systems considered in this chapter appear in many areas of applied mathematics: in mathematical economics [47, 48, 56-61], continuum mechanics [20, 44] and in the theory of dynamical systems [6]. As we have already seen in the previous chapters these

COMMENTS

385

systems are also useful tools in the study of continuous-time optimal control problems. With any continuous-time control problem we associate a related discrete-time control problem. It turns out that there is a simple correspondence between solutions of the continuous-time problem and solutions of the related discrete-time problem. In Chapter 8 we study an autonomous problem with convex cost function on a bounded closed convex subset of a Banach space and a nonautonomous nonconvex problem on a complete metric space. Some results on inﬁnity horizon autonomous nonconvex problems on a complete metric space were established in [108]. Chapter 9. The primary area of appications of inﬁnite-dimensional optimal continuous-time control problems concerns models of regional economic growth discussed in [36], cattle ranching models proposed in [24], and systems with distributed parameters and boundary controls related to engineering [8, 31] and to water resources problems [62, 63]. In this chapter we obtain the convergence to the turnpike in the weak topology. In order to establish the convergence to the turnpike in the strong topology we need to assume that the integrand is strictly convex. Chapter 10. This chapter is devoted to applications of the turnpike theory to mathematical economics. We consider a large class of nonlinear Leontjev type models of multisector economics. It should be mentioned that we obtain a suﬃcient condition for the turnpike property which can be veriﬁed if we know a generalized equilibrium state of the model. A version of this model which takes into account lag was considered in [87]. Chapter 11. The general Neumann–Gale model has been studied in many publications. Many results on this model are collected in [48, 75, 77]. A stochastic version of the Neumann–Gale model was studied in [2]. Chapter 12. This chapter is devoted to applications of the turnpike theory to game theory. It is based on the paper [102]. For other applications see [15, 17]. In [17] overtaking equilibria was studied for switching regulator and tracking games. Turnpike properties for inﬁnite horizon open-loop competitive processes were discussed in [15]. It is not clear if it is possible to obtain turnpike results in game theory without convexity assumptions.

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Index

Absolutely continuous function, 2 Attractor, 75 Banach space, 223 Base of a uniformity, 4 Borel measurable function, 173 Cardinality, 74 Cauchy sequence, 7 CES-function, 289 Compact metric space, 183 Compact set, 71 Complete metric space, 1 Complete uniform space, 4 Cone, 283 Convex process, 321 Convex programming problem, 259 Determinant, 299 Diﬀerential inclusion, 283 Dual mapping, 284 Eigenvalue, 322 Eigenvector, 322 Epigraph, 181 Equiabsolutely integrable sequence of functions, 179 Euclidean norm, 3 Fatou’s lemma, 180 Frenkel–Kontorova model, 2 Generalized equilibrium state, 284 Generator, 257 Good function, 5 Good trajectory-control pair, 175 Greatest common divizor, 299 Hausdorﬀ metric, 74 Hausdorﬀ space, 4 Hilbert space, 257

Identity operator, 283 Increasing function, 3 Inﬁnite horizon problem, 1 Lebesgue measure, 91 Linear control system, 173 Metrizable space, 4 Mild solution, 257 Minimal element, 95 Minimal long-run average cost growth rate, 73 Minimal pair of sequences, 352 Minimal sequence, 238 Minimal solution, 2 Normal mapping, 284 Optimal steady-state, 259 Overtaking optimal function, 2 Overtaking optimal trajectory-control pair, 175 Polynomial, 299 Positive deﬁnite symmetric matrix, 174 Positive semideﬁnite symmetric matrix, 174 Relative topology, 63 Representation formula, 120 Saddle point, 352 Strictly convex function, 174 Strictly superlinear function, 312 Strongly continuous semigroup, 257 Subgradient vector, 181 Superlinear function, 312 Topological subspace, 63 Trajectory-control pair, 174 Uniform space, 4 Uniform strict convexity, 197 Von Neumann growth rate, 285 Weakly optimal function, 73 Weakly overtaking trajectory-control pair, 259