Optimality Conditions: Abnormal and Degenerate Problems

304 Pages·2000·7.36 MB·English

by  Aram V. Arutyunov (auth.)

Optimality Conditions:
Abnormal and Degenerate Problems
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 526
Optimality Conditions:
Abnormal and Degenerate
Problems
by
Aram V. Arutyunov
Peoples' Friendship University,
Moscow, Russia
and
Moscow State University,
Department of Computational Mathematics and Cybernetics,
Moscow, Vorb 'ery Gory, Russia
Springer-Science+Business Media, B.V.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5596-5 ISBN 978-94-015-9438-7 (eBook)
DOI 10.1007/978-94-015-9438-7
This is a completely revised and updated version of "Extremum conditions: Abnormal and
degenerate problems". Published by Factorial in 1997, Moscow.
Translated by S.A. Vakhrameev.
Printed on acid-free paper
All Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000.
Softcover reprint of the hardcover 1s t edition 2000
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
PREFACE vii
1 EXTREMAL PROBLEMS WITH CONSTRAINTS 1
1.1 Extremal Problems with Constraints. Normal and Abnormal
Points . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Elementary Statements (Finite-Dimensional Case) .. 8
1.3 Certain Notation and Concepts . . . . . . . . . . . . . 11
1.4 Statement of the First- and Second-Order Conditions. 15
1.5 Lower Estimates for Upper Topological Limits of Sequences
of Subspaces ............. 22
1.6 Proof of Theorems 4.1, 4.2, and 4.3 . 23
1.7 Proof of Theorem 5.1 . . . . . . . . . 31
1.8 Sufficient Second-Order Conditions . 36
1.9 Interconnection of Necessary and Sufficient Second-Order
Conditions. 2-Normal Mappings. . 41
1.10 Properties of 2-Normal Mappings. . . 44
1.11 Lagrange-Avakov Function
and Necessary Extremality Conditions 53
1.12 Theorem on the Tangent Cone. Tuples 60
1.13 Proof of Theorem 11.1 . . . . . . . . . 69
1.14 Higher-Order Necessary Conditions. . 71
1.15 Sufficient Conditions for Abnormal Problems. Higher-Order
Sufficient Conditions . . . . . . . 74
1.16 Proof of Theorems 15.1 and 15.2 . . . . . . . . . . . . . .. 78
2 OPTIMAL CONTROL PROBLEM.
PONTRYAGIN MAXIMUM PRINCIPLE 89
2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . 89
2.2 Basic Assumptions and Notation . . . . . . . . . . . . . . 92
2.3 Pontryagin Maximum Principle for the Simplest Problem 98
2.4 Statement of the Pontryagin Maximum Principle.
State Constraints and the Degeneration Phenomenon. 107
2.5 Linear-Convex Problems. . . . . . . . . . . . . . . 115
2.6 Proof of the Weakened Maximum Principle for a
Linear-Convex Problem Without State Constraints 123
2.7 Proof of the Maximum Principle in a Linear-Convex Problem
with State Constraints . . . . . . . . . . . . . . . . . . . .. 133
2.8 Proof of the Pontryagin Maximum Principle.
Finite-Dimensional Approximation Method 141
VI
2.9 Penalty Method. Necessary Conditions in the f,l-Problem .. 147
2.10 Completing the Proof of the Weakened Maximum Principle 155
2.11 v-Problem and Completing the Proof
of the Maximum Principle . . . . . . . . . . . . . . . . . .. 164
2.12 A Little More About the Nondegeneracy of the Maximum
Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167
2.13 Relaxations and Perturbations of Optimal Control Problems 174
3 DEGENERATE QUADRATIC FORMS OF
THE CALCULUS OF VARIATIONS 181
3.1 Statement of the Problem . . . . . . . . . . . . . . . . . .. 181
3.2 Constructions and the Notation. The spaces Wl B[Ol, 02J and
Wn[01,02J . . . . . . . . . . . . . . . . ..' 183
3.3 Statement of the Main Results ...... 188
3.4 Discussion of the Main Results. Examples 196
3.5 Proof of Theorem 3.1 . 198
3.6 Proof of Theorem 3.2. . . . . . . . . . . . 208
3.7 Proof of Theorem 3.4. . . . . . . . . . . . 221
3.8 Necessary and Sufficient Conditions for a Local Minimum in
Degenerate Problems of the Calculus of Variations. . . . .. 242
4 STUDY OF MAPPINGS IN A NEIGHBORHOOD
OF AN ABNORMAL POINT 245
4.1 Implicit Function Theorem and Abnormal Points . . 245
4.2 Discussion and Auxiliary Results . . . . . . . . . . . 249
4.3 Proof of the Inverse and Implicit Function Theorems 258
4.4 On the Existence of Regular Zeros for a Quadratic Mapping 265
4.5 Level Set of a Smooth Mapping in a Neighborhood of an
Abnormal Point. . . . . . . . . . . . . . . . . . . . . . . .. 268
4.6 Criterion for the Strong 2-Regularity of Quadratic Mappings 279
REFERENCES 287
INDEX 293
LIST OF NOTATION 297
PREFACE
This book is devoted to one of the main questions of the theory of extremal prob
lems, namely, to necessary and sufficient extremality conditions. It is intended
mostly for mathematicians and also for all those who are interested in optimiza
tion problems. The book may be useful for advanced students, post-graduated
students, and researchers.
The book consists of four chapters. In Chap. 1 we study the abstract
minimization problem with constraints, which is often called the mathemati
cal programming problem. Chapter 2 is devoted to one of the most important
classes of extremal problems, the optimal control problem. In the third chapter
we study one of the main objects of the calculus of variations, the integral
quadratic form. In the concluding, fourth, chapter we study local properties of
smooth nonlinear mappings in a neighborhood of an abnormal point.
The problems which are studied in this book (of course, in addition to their
extremal nature) are united by our main interest being in the study of the
so called abnormal or degenerate problems. This is the main distinction of
the present book from a large number of books devoted to theory of extremal
problems, among which there are many excellent textbooks, and books such as,
e.g., [13, 38, 59, 78, 82, 86, 101, 112, 119], to mention a few.
What does 'abnormal' mean? We explain this by examining the following
example of the problem studied in Chap. 1:
f(x) -t min /Fi(X) = 0, i = 1, k, (0.1)
where j and Fi are given smooth functions on the n-dimensional arithmetical
=
space X Rn.
Let a point Xo satisfy the constraints of problem (0.1). This point is said to
be normal if the gradients Fl(xo), i = 1,k, are linearly independent; otherwise
it is said to be abnormal. It turns out that if the point Xo is abnormal then,
for any function j, the Lagrange multiplier rule holds automatically at this
point (with AO = 0); this rule yields first-order necessary extremality conditions
independently of the fact whether or not Xo is a minimum point. Therefore at
an abnormal point the Lagrange multiplier rule gives no additional information
(degenerates), and hence, generally speaking, this rule is not instrumental in
studying the extremality of abnormal points. Moreover, the classical second
order necessary conditions do not hold at an abnormal point, in general.
The study of abnormal extremal problems has been attracting the attention
of mathematicians for a long time. For the first time the concept of an abnormal
point for a constrained extremal problem was introduced by C. Caratheodory
and G.A. Bliss; they also have recognized the importance and difficulty of study
ing abnormal points. They did so by examining a finite-dimensional problem
and the Bolza problem of the classical calculus of variations (see (78) and ref
erences given there). The following words are from G.A. Bliss: " ... it seems
to be not quite possible to create a complete theory in the immediate future
without certain normality assumptions because of a large number of singular
vii
Vlll
cases which has appeared" [78], p. 2231.
Recently, the interest in abnormal problems was again stimulated. This is
owed to the study of geodesics on sub-Riemannian manifolds. After several
unsuccessfully attempts it became clear that a geodesic on sub-Riemannian
manifolds can be abnormal, and this is not a pathology; after that an active
study of abnormal geodesics was started (see, e.g., [11, 12, 71, 109, 110, 117]).
The shortest geodesics are solutions of nonholonomic variational problems.
"The problems of such a type arise in a natural way in various fields of math
ematics, mechanics, and physics in those cases where the Riemannian structure
degenerates in one sense or another. In general relativity theory light propagates
along geodesics of a certain Riemannian structure in accordance with Fermat's
principle. In the case where this structure 'blows up' to a certain distribution
(i.e., motion in a direction which is orthogonal to this distribution is retarded),
it is natural to consider the 'limit structure'. Under certain additional assump
tions this limit structure turns out to be Riemannian. In differential geometry
the works by M. Gromov and his students are devoted to various blow-up phe
nomena of Riemannian structure". (See [71], p. 788.)
Chapter 1 is devoted to the study of extremality conditions without a priori
assumptions of normality. In this chapter, on the basis of perturbation method
and the elaborated auxiliary technique for problem (0.1) (along with equality
type of constraints, it also contains inequality type of constraints), we obtain
new necessary extremality conditions of the first, second, and higher orders.
These results are also substantive at abnormal points, and, at normal points
they transforms into the known ones. We study the 'gap' between the obtained
necesssary and sufficient local minimum condition. We isolate a class of con
straints (the mappings defining these constraints are called 2-normal mappings)
for which the mentioned 'gap' is the minimally possible one, and it is proved
that this class is generic. These results are also generalized to the case where X
is an infinite-dimensional vector space. However, we note (and this is a princi
pal point) that all results indicated above are new and substantive in the case
X = Rn, and their generalization to the infinite-dimensional case is carried out
in a sufficiently standard way. Also, we note that in the first chapter we suggest
two different approaches to the study of extremal problems; Secs. 2-10 and
11-16, respectively, are devoted to these approaches.
In Chap. 2 we study the optimal control problem with endpoint, mixed,
and state constraints. Using the elaborated finite-dimensional approximati
on methods, under very general smoothness assumptions we prove the maxi
mum principle for it. Also, for linear-in-control problems we describe a simple
method for proving the Pontryagin maximum principle, which is based on the
perturbation method.
In various optimal control problems we again encounter the degeneration
phenomenon; namely, under state constraints, in many interesting (in particular,
from the point of view of applications) problems, the versions of the maximum
principle which were known earlier are automatically degenerating, i.e., any
ITranslated into English from the Russian translation of [78].
ix
admissible control satisfies them, and this makes these versions inapplicable for
the study. In Chap. 2 we pay special attention to the study of this phenomenon
and to obtaining a nondegenerate version of the maximum principle.
In Chap. 3 we study the positive semi-definitness of the quadratic form
=
U(x) fOl(A(t);i;(t),;i;(t)) +(B(t)x(t),x(t))
+2(C(t);i;(t), x(t)) dt
+(0 (x(O), x(l)) , (x(O), x(l))) .
Moreover, principal attention is paid to the case where the strengthened Legen
dre condition is violated at certain points t, i.e., the matrix A(t) is not positive
definite. Such forms are said to be degenerate. It is known that the classical
Jacobi theory of conjugate points is not applicable for degenerate forms. In
Chap. 3 we present necessary and sufficient conditions for definitness of de
generate quadratic forms and also obtain a general formula for computing their
index.
Although the fourth chapter is not directly devoted to the theory of extremal
problems, it is, however, directly related to this theory, especially to the theory
developed in the first chapter. The first four sections of Chap. 4 are devoted to
the implicit and inverse function theorems at abnormal points. Using the neces
sary extremality conditions proved in Chap. 1 we obtain sufficient conditions for
solvability of nonlinear equations, which lead to the implicit function theorem.
We note that under the normality assumption these results transform into the
classical implicit function theorem, and are new at an abnormal point even for
the finite-dimensional case. Moreover, in the fourth section it is proved that in
a neighborhood of an abnormal point the set of zeros of a nonlinear 2-regular
mapping is locally diffeomorphic to the set of zeros of its second differential.
Also we present the criterion for the strong 2-regularity of a quadratic mapping.
The general theory of abnormal and degenerate problems is still very far
from being complete, and the present book is the first attempt at elaborating
the methods for studying abnormal problems (we also note that the books [58,
67] which have appeared in Russia recently contain certain results related to
abnormal problems; in [58] the creation of numerical methods is initiated). Our
book contains mainly new results and also results that were only published in
journals.
In many respects my own vision of abnormal problems has been formed in
many discussions with Professors E.R. Avakov and A.A. Agrachev, and I am
very gratefull to them. I express my gratitude to Professor V.M. Tikhomirov
whose views have played a very important role in the formation of my point of
view on extremal problems in general. Also, I express my gratitude to Professors
A.A. Agrachev, F.P. Vasil'ev, A.F. Izmailov, N.M. Novikova, M.F. Sukhinin,
and A.A. Shananin who have read separate parts of the manuscripts, and made
a number of important remarks which have improved the presentation. In par
ticular, it was F.P. Vasil'ev, who advised me to write Sec. 2 of Chap. 1. I
x
express my gratitude to Professors R.V. Gamkrelidze, H.J. Jongen. and A.B.
Zhishchenko for support. I thank my wife Natalya Chernikova for her help and
her constant support; without them this book would not have been written.
I would like here to say a few words about my teacher, Professor N.T.
Tynyanskii, who died at the peak of his creative powers.
The English edition of the book that is proffered to the reader is essentially
different from the Russian edition which appeared in 1997 2. Namely, Chap. 4
is new (only its last section appeared in the Appendix of the Russian edition).
In the first chapter I have made certain proofs more detailed and have added
new results and examples which are intended to provide better understanding.
The scheme of the proofs in Chap. 2 is slightly changed and, moreover, its
last section is new. The bibliography has been completed, and there are other
less significant distinctions. Moreover, some misprints and gaps of the Russian
editions have been removed. Some of them were pointed to me by my students,
D.Yu. Karamzin and V. Jachimovic, and I express my gratitude to them.
Also, I express my gratitude to Professor S. A. Vakhrameev who has kindly
agreed to translate this book into English. His careful translation and editing
of the book have improved the presentation.
A.V. Arutyunov
2 A.V. Arutyunov, Extremality Conditions. Abnormal and Degenerate Problems, (Usloviya
extremuma. Anormal'nye i vyrozhdennye zadachiJ, Faktorial, Moscow (1997).