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Graduate Texts in Mathematics
264
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Graduate Texts in Mathematics
Series Editors:
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California, Berkeley, CA, USA
Advisory Board:
Colin Adams, Williams College, Williamstown, MA, USA
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Roger E. Howe, Yale University, New Haven, CT, USA
David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA
Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USA
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Amie Wilkinson, University of Chicago, Chicago, IL, USA
Graduate Texts in Mathematics bridge the gap between passive study and creative
understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks
in graduate courses, they are also suitable for individual study.
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Francis Clarke
Functional Analysis,
Calculus of
Variations and
Optimal Control
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Francis Clarke
Institut Camille Jordan
Université Claude Bernard Lyon 1
Villeurbanne, France
ISSN 0072-5285 Graduate Texts in Mathematics
ISBN 978-1-4471-4820-3 (eBook)
ISBN 978-1-4471-4819-7
DOI 10.1007/978-1-4471-4820-3
Springer London Heidelberg New York Dordrecht
Library of Congress Control Number: 2013931980
Mathematics Subject Classification: 46-01, 49-01, 49K15, 49J52, 90C30
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To France. To its country lanes and market
towns, its cities and caf´es, its mountains and
Roman ruins. To its culture and history, its
wine and food, its mathematicians and its fast
trains. To French society. To France.
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Preface
The famous old road from V´ezelay in Burgundy to Compostela in Spain is a long
one, and very few pilgrims walk the entire route. Yet every year there are those
who follow some part of it. We do not expect that many readers of this book will
accompany us step by step from the definition of a norm on page 3, all the way to an
advanced form of the Pontryagin maximum principle in the final chapter, though we
would welcome their company. In describing the itinerary, therefore, we shall make
some suggestions for shorter excursions.
The book consists of four parts. The first of these is on functional analysis, the last
on optimal control. It may appear that these are rather disparate topics. Yet they
share the same lineage: functional analysis (Part I) was born to serve the calculus
of variations (Part III), which in turn is the parent of optimal control (Part IV). Add
to this observation the need for additional elements from optimization and nonsmooth analysis (Part II), and the logic of the four parts becomes clear. We proceed
to comment on them in turn.
Part I: Functional analysis. The prerequisites are the standard first courses in real
analysis, measure and integration, and general topology. It seems likely to us, then,
that the reader’s backpack already contains some functional analysis: Hilbert spaces,
at least; perhaps more. But we must set off from somewhere, and we do not, strictly
speaking, assume this. Thus, Part I serves as an introduction to functional analysis.
It includes the essential milestones: operators, convex sets, separation, dual spaces,
uniform boundedness, open mappings, weak topologies, reflexivity. . .
Our course on functional analysis leads to a destination, however, as does every
worthwhile journey. For this reason, there is an emphasis on those elements which
will be important later for optimization, for the calculus of variations, and for control
(that is, for the rest of the book).
Thus, compactness, lower semicontinuity, and minimization are stressed. Convex
functions are introduced early, together with directional derivatives, tangents, and
normals. Minimization principles are emphasized. The relevance of the smoothness
of the norm of a Banach space, and of subdifferentials, is explained. Integral functionals are studied in detail, as are measurable selections. Greater use of optimization is made, even in proving classical results. These topics manage to walk hand in
hand quite amiably with the standard ones.
The reader to whom functional analysis is largely familiar territory will nonetheless
find Part I useful as a guide to certain areas.
vii
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Preface
Part II: Optimization and nonsmooth analysis. The themes that we examine in
optimization are strictly mathematical: existence, necessary conditions, sufficient
conditions. The goal is certainly not to make the reader an expert in the field, in
which modeling and numerical analysis have such an important place. So we are
threading our way on a fairly narrow (if scenic) path. But some knowledge of the
subject and its terminology, some familiarity with the multiplier rule, a good understanding of the deductive (versus the inductive) method, an appreciation of the role
of convexity, are all important things to acquire for future purposes. Some students
will not have this background, which is why it is supplied here.
Part II also contains a short course on nonsmooth analysis and geometry, together
with closely related results on invariance of trajectories. These subjects are certainly
worth a detour in their own right, and the exposition here is streamlined and innovative in some respects. But their inclusion in the text is also based on the fact that
they provide essential infrastructure for later chapters.
Part III: Calculus of variations. This is meant to be a rather thorough look at the
subject, from its inception to the present. In writing it, we have tried to show the
reader not only the landmarks, but also the contours of this beautiful and ongoing
chapter in mathematics. This is done in part by advancing in stages, along a path
that reveals its history.
A notable feature in this landscape is the presence of recent advanced results on regularity and necessary conditions. In particular, we encounter a refined multiplier rule
that is completely proved. We know of no textbook which has such a thing; certainly
not with the improvements to be found here. Another important theme is the existence question, where we stress the need to master the direct method. This is made
possible by the earlier groundwork in functional analysis. There are also substantial examples and exercises, involving such topics as viscosity solutions, nonsmooth
Lagrangians, the logarithmic Sobolev inequality, and periodic trajectories.
Part IV: Optimal control. Control theory is a very active subject that regularly
produces new kinds of mathematical challenges. We focus here upon optimality, a
topic in which the central result is the Pontryagin maximum principle. This important theorem is viewed from several different angles, both classical and modern, so
as to fully appreciate its scope. We demonstrate in particular that its extension to
nonsmooth data not only unifies a variety of special cases mathematically, but is
itself of intrinsic interest.
Our survey of optimal control does not neglect existence theory, without which the
deductive approach cannot be applied. We also discuss Hamilton-Jacobi methods,
relaxation, and regularity of optimal controls. The exercises stem in part from several fields of application: economics, finance, systems engineering, and resources.
The final chapter contains general results on necessary conditions for differential
inclusions. These theorems, which provide a direct route to the maximum princi-
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Preface
ix
ple and the multiplier rule, appear here for the first time in a text; they have been
polished and refined for the occasion.
A full proof of a general maximum principle, or of a multiplier rule, has never been
an easy thing; indeed, it has been famously hard. One may say that it has become
more streamlined; it has certainly become more general, and more unified; but it has
not become easy. Thus, there is a difficult stretch of road towards the end of Part IV;
however, it leads to a fully self-contained text.
Intended users. The author has himself used the material of this book many times,
for various courses at the first-year or second-year graduate level. Accordingly,
the text has been planned with potential instructors in mind. The main question is
whether to do in detail (most of) Part I, or just refer to it as needed for background
material. The answer must depend on the experience and the walking speed of the
audience, of course.
The author has given one-semester courses that did not stray from Part I. For some
audiences, this could be viewed as a second course on functional analysis, since,
as we have said, the text adopts a somewhat novel emphasis and choice of material
relative to other introductions. The instructor must also decide on how much of
Chapter 8 to cover (it’s nothing but problems). If the circumstances of time and
audience permit, one could tread much of Part I and still explore some chapters
from Part II (as an introduction to optimization and nonsmooth analysis) or Part
III (the calculus of variations). As an aid in doing so, Part I has been organized in
such a way that certain material can be bypassed without losing one’s way. We refer
especially to Sections 4.3–4.4, 5.4, 6.2–6.4, and Sections 7.2–7.4 (or all of Chapter
7, if in fact the audience is familiar with Hilbert spaces).
Here is a specific example. To give a course on functional analysis and calculus
of variations, one could choose to travel lightly and drop from Part I the material
just mentioned. Then, Chapter 9 might be done (minus the last section, perhaps).
Following this, one could skip ahead to the first three or four chapters of Part III; they
constitute in themselves a viable introduction to the calculus of variations. (True, the
proof of Tonelli’s theorem in Chapter 16 uses unseen elements of Chapter 6, but we
indicate a shortcut to a special case that circumvents this.)
For advanced students who are already competent in functional analysis, an entirely different path can be taken, in which one focuses entirely on the latter half of
the book. Then Part I (and possibly Part II) can play the role of a convenient and
strangely relevant appendix (one that happens to be at the front), to be consulted as
needed. As regards the teaching of optimal control, we strongly recommend that it
be preceded by the first three or four chapters of Part III, as well as Section 19.1 on
verification functions.
In addition to whatever merits it may have as a text, we believe that the book has considerable value as a reference. This is particularly true in the calculus of variations
and optimal control; its advanced results make it stand out in this respect. But its ac-
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Preface
cessible presentation of certain other topics may perhaps be appreciated too (convex
analysis, integral semicontinuity, measurable selections, nonsmooth analysis, and
metric regularity, for example). We dare to hope that it will be of interest to both the
mathematics and the control engineering communities for all of these reasons, as
well as to certain related ones (such as operations research and economics).
A word about the exercises: there are hundreds. Some of them stand side by side
with the text, for the reader to meet at an early stage. But additional exercises (many
of them original in nature, and more difficult) lie waiting at the end of each part, in
a separate chapter. Solutions (full, partial, or just hints) are given for quite a few of
them, in the endnotes. A list of notation is given at the beginning of the index.
Acknowledgements.
I shall end this preface on a more personal note. My mathematical education has
been founded on the excellent books and the outstanding teachers that I have met
along the way. Among the latter, I have long wanted to give thanks to Thomas
Higgins of the order of Christian Brothers; to Norbert Schlomiuk of l’Universit´e de
Montr´eal; to Basil Rattray and Michael Herschorn of McGill University; to Terry
Rockafellar, Victor Klee, and Ernest Michael at the University of Washington. In the
endnotes, I mention the books that have had the greatest influence on me.
It is a pleasure to acknowledge the unfailing support of l’Institut Camille Jordan and
l’Universit´e de Lyon over the years, and that of le Centre national de recherche scientifique (CNRS). I have also benefited from regularly teaching at l’Ecole normale
sup´erieure de Lyon.
The book was written in large part during the ten years in which I held a chair at
l’Institut universitaire de France; this was a major contribution to my work. Thanks
are also due to Pierre Bousquet for his many insightful comments. And to Carlo
Mariconda, who read through the entire manuscript (V´ezelay to Compostela!) and
made countless helpful suggestions, I say: mille grazie.
On quite a different plane, I avow my heartfelt gratitude to my wife, Gail Hart, for
her constant love and companionship on the journey.
Francis Clarke
Burgundy, France
July 2012
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Contents
Part I Functional Analysis
1
Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Linear mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 The dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Derivatives, tangents, and normals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2
Convex sets and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Properties of convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Extended-valued functions, semicontinuity . . . . . . . . . . . . . . . . . . . . .
2.3 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Separation of convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
27
30
32
41
3
Weak topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Induced topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The weak topology of a normed space . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The weak∗ topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
47
51
53
56
4
Convex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Subdifferential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Conjugate functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 The minimax theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
59
67
71
73
5
Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Completeness of normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Perturbed minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Open mappings and surjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Metric regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Reflexive spaces and weak compactness . . . . . . . . . . . . . . . . . . . . . . . .
75
75
82
87
90
96
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Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1 Uniform convexity and duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Measurable multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 Integral functionals and semicontinuity . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Weak sequential closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7
Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2 A smooth minimization principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3 The proximal subdifferential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4 Consequences of proximal density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8
Additional exercises for Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Part II Optimization and Nonsmooth Analysis
9
Optimization and multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.1 The multiplier rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.2 The convex case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.3 Convex duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
10 Generalized gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.2 Calculus of generalized gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.3 Tangents and normals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.4 A nonsmooth multiplier rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
11 Proximal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
11.1 Proximal calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
11.2 Proximal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
11.3 A proximal multiplier rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
11.4 Dini and viscosity subdifferentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
12 Invariance and monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
12.1 Weak invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
12.2 Weakly decreasing systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
12.3 Strong invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
13 Additional exercises for Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Part III Calculus of Variations
14 The classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
14.1 Necessary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
14.2 Conjugate points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
14.3 Two variants of the basic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
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15 Nonsmooth extremals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
15.1 The integral Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
15.2 Regularity of Lipschitz solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
15.3 Sufficiency by convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
15.4 The Weierstrass necessary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
16 Absolutely continuous solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
16.1 Tonelli’s theorem and the direct method . . . . . . . . . . . . . . . . . . . . . . . . 321
16.2 Regularity via growth conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
16.3 Autonomous Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
17 The multiplier rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
17.1 A classic multiplier rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
17.2 A modern multiplier rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
17.3 The isoperimetric problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
18 Nonsmooth Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
18.1 The Lipschitz problem of Bolza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
18.2 Proof of Theorem 18.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
18.3 Sufficient conditions by convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
18.4 Generalized Tonelli-Morrey conditions . . . . . . . . . . . . . . . . . . . . . . . . 363
19 Hamilton-Jacobi methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
19.1 Verification functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
19.2 The logarithmic Sobolev inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
19.3 The Hamilton-Jacobi equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
19.4 Proof of Theorem 19.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
20 Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
20.1 The classical context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
20.2 Lipschitz solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
20.3 Hilbert-Haar theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
20.4 Solutions in Sobolev space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
21 Additional exercises for Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Part IV Optimal Control
22 Necessary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
22.1 The maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
22.2 A problem affine in the control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
22.3 Problems with variable time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
22.4 Unbounded control sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
22.5 A hybrid maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
22.6 The extended maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
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Contents
23 Existence and regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
23.1 Relaxed trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
23.2 Three existence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
23.3 Regularity of optimal controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
24 Inductive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
24.1 Sufficiency by the maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . 491
24.2 Verification functions in control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
24.3 Use of the Hamilton-Jacobi equation . . . . . . . . . . . . . . . . . . . . . . . . . . 500
25 Differential inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
25.1 A theorem for Lipschitz multifunctions . . . . . . . . . . . . . . . . . . . . . . . . 504
25.2 Proof of the extended maximum principle . . . . . . . . . . . . . . . . . . . . . . 514
25.3 Stratified necessary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
25.4 The multiplier rule and mixed constraints . . . . . . . . . . . . . . . . . . . . . . 535
26 Additional exercises for Part IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
Notes, solutions, and hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
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Part I
Functional Analysis
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Chapter 1
Normed Spaces
There are only two kinds of math books: those you cannot read
beyond the first sentence, and those you cannot read beyond the
first page.
C. N. Yang
What we hope ever to do with ease, we must learn first to do
with diligence.
Samuel Johnson
We now set off on an expedition through the vast subject of functional analysis. No
doubt the reader has some familiarity with this place, and will recognize some of
the early landmarks of the journey. Our starting point is the study of normed spaces,
which are situated at the confluence of two far-reaching mathematical abstractions:
vector spaces, and topology.
1.1 Basic definitions
The setting is that of a vector space over the real numbers R. There are a dozen or
so axioms that define a vector space (the number depends on how they are phrased),
bearing upon the existence and the properties of certain operations called addition
and scalar multiplication. It is more than probable that the reader is fully aware of
these, and we shall say no more on the matter. We turn instead to the central idea of
this chapter.
A norm on the vector space X corresponds to a reasonable way to measure the size
of an element, one that is consistent with the vector operations. Given a point x ∈ X,
the norm of x is a nonnegative number, designated x . We also write x X at
times, if there is a need to distinguish this norm from others. In order to be a norm,
the mapping x → x must possess the following properties:
0 ∀ x ∈ X ; x = 0 if and only if x = 0 (positive definiteness);
•
x
•
x+y
•
t x = |t | x
x + y
∀ x, y ∈ X (the triangle inequality);
∀t ∈ R , x ∈ X (positive homogeneity).
Once it has been equipped with a norm, the vector space X, or, more precisely
perhaps, the pair ( X, · ), is referred to as a normed space.
F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control,
Graduate Texts in Mathematics 264, DOI 10.1007/978-1-4471-4820-3 1,
© Springer-Verlag London 2013
3
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4
1 Normed Spaces
We have implied that vector spaces and topology are to meet in this chapter; where,
then, is the topology? The answer lies in the fact that a norm induces a metric on
X in a natural way: the distance d between x and y is d(x, y) = x − y . Thus, a
normed space is endowed with a metric topology, one (and this is a crucial point)
which is compatible with the vector space operations.
Some notation. The closed and open balls in X are (respectively) the sets of the
form
B(x, r) =
y ∈ X : y−x
r , B ◦(x, r) =
y ∈ X : y−x < r},
where the radius r is a positive number. We sometimes write B or B X for the closed
unit ball B(0,1), and B ◦ for the open unit ball B ◦(0,1). A subset of X is bounded
if there is a ball that contains it.
If A and C are subsets of X and t is a scalar (that is, an element of R), the sets A +C
and t A are given by
A +C =
a+c : a ∈ A, c ∈ C ,
tA =
ta : a ∈ A .
(Warning: A+A is different from 2 A in general.) Thus, we have B x, r = x +r B.
We may even ask the reader to tolerate the notation B(x, r) = x + r B. The closure
of A is denoted cl A or A, while its interior is written int A or A◦.
Given two points x and y in X, the closed interval (or segment) [ x, y ] is defined as
follows:
[ x, y ] = z = (1 − t) x + t y : t ∈ [ 0,1] .
When t is restricted to (0,1) in the definition, we obtain the open interval (x, y). The
half-open intervals [ x, y) and (x, y ] are defined in the evident way, by allowing t to
vary in [ 0,1) and (0,1] respectively.
The compatibility between the vector space and its norm topology manifests itself
by the fact that if U is an open subset of X, then so is its translate x + U and its
scalar multiple tU (if t = 0). This follows from the fact that balls, which generate
the underlying metric topology, cooperate most courteously with the operations of
translation and dilation:
B(x, r) = x + B(0, r) , B(0, t r) = t B(0, r) (t > 0).
It follows from this, for example, that we have int (t A) = t int A when t = 0, and
that a sequence x i converges to a limit x if and only if the difference x i − x converges
to 0. There are topologies on X that do not respect the vector operations in this way,
but they are of no interest to us. We shall have good reasons later on to introduce
certain topologies on X that differ from that of the norm; they too, however, will be
compatible with the vector operations.
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1.1 Basic definitions
5
A vector space always admits a norm. To see why this is so, recall the well-known
consequence of Zorn’s lemma which asserts that any vector space has a basis {eα },
in the sense of linear algebra. This means that any x has a unique representation
x = ∑ α x α eα , where all but a finite number of the coefficients x α are 0. The reader
will verify without difficulty that x := ∑ α | x α | defines a norm on X. In practice,
however, there arises the matter of choosing a good norm when a space presents
multiple possibilities.
Sometimes a good norm (or any norm) is hard to find. An example of this: the space
of all continuous functions f from R to R. Finding an explicit norm on this space is
problematic, and the space itself has found use only when endowed with a topology
that is not that of a norm. In other cases, several norms may come to mind.
1.1 Example. The vector space of continuous functions f from [ 0,1] to R admits,
among others, the two following norms:
f
∞
1
= max | f (t)| ,
f
t ∈ [ 0,1]
1
=
0
| f (t)| dt ,
both of which are well defined. One of these (the first, it turns out) is a better choice
than the other, for reasons that will become completely clear later.
Two norms · 1 and ·
constants c, d such that
x
1
2
on X are said to be equivalent if there exist positive
c x
2,
x
d x
2
1
∀ x ∈ X.
As the reader may verify, this amounts to saying that each ball around 0 of one
type contains a ball around 0 of the other type. This property, in turn, is easily seen
to characterize the equality of the topologies induced by the two metrics. Thus we
may say: two norms on X are equivalent if and only if they induce the same metric
topology.
1.2 Exercise. Are the two norms of Example 1.1 equivalent?
When we restrict attention to a linear subspace of a normed space, the subspace
is itself a normed space, since the restriction of the norm is a norm. This is an
internal mechanism for creating smaller normed spaces. Another way to create new
(larger) normed spaces is external, via Cartesian products. In this case, there is some
flexibility in how to define the norm on the product.
Let X and Y be normed spaces, with norms · X and ·
product Z = X ×Y may be equipped with any of the norms
x
X
+ y
Y,
( x
X)
2
+( y
Y)
2
1/2
,
max
Y.
x
Then the Cartesian
X
, y
Y
,
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6
1 Normed Spaces
among others. That these are all norms on X×Y is simple to check, and it is not much
more work to go on to verify that all these product norms are equivalent.
1.3 Example. We denote Euclidean n-space by R n ; this is the vector space consisting of n-tuples x = (x 1 , x 2 , . . . , x n ) of real numbers. By default, we always consider
that it is equipped with the Euclidean norm
x 12 + x 22 + · · · + x n2
|x| =
1/2
, x ∈ Rn.
No other norm is awarded the honor of being written with single bars.
1.4 Example. (Continuous functions on a compact set) Let K be a compact metric space. We denote by C(K), or C(K, R) if more precision is desired, the vector
space of continuous functions f : K → R , equipped with the norm
=
f
f
C(K)
= max | f (x)|.
x∈K
Notation: When K is an interval [ a, b ] in R , we write C[ a, b ] for C(K).
1.5 Exercise. Prove that C[ 0,1] is an infinite dimensional vector space, by exhibiting a linearly independent set with infinitely many elements.
1.6 Example. (Spaces of sequences) Some useful examples of normed spaces are
obtained by considering sequences with certain properties. For fixed p ∈ [ 1, ∞), we
define
x
=
p
∑i
1
1/p
| xi|p
,
where x = (x 1 , x 2 , . . .) is any sequence of real numbers. As we show below, the set
of all sequences x for which x p < ∞ is a vector space. It is equipped with the
norm · p , and designated p . The vector space of all bounded sequences, denoted
∞, is turned into a normed space with the norm x
∞ := sup i 1 | x i |.
We shall require Hăolders inequality, which, in the present context, asserts that
∑i
1
| xi yi|
x
p
y
p∗
for any two sequences x = (x 1 , x 2 , . . . ) and y = (y 1 , y 2 , . . . ). Here, p lies in [ 1, ∞ ],
and p∗ signifies the conjugate exponent of p, the unique number in [ 1, ∞ ] such that
1/p + 1/p∗ = 1. (Thus, p∗ = ∞ when p = 1, and vice versa.)
Other members of the cast include the following spaces of convergent sequences:
c=
x = (x 1 , x 2 , . . .) : lim x i exists , c 0 =
i→∞
x = (x 1 , x 2 , . . .) : lim x i = 0 .
i→∞
Another vector space of interest consists of those sequences having finitely many
nonzero terms; it is denoted c∞ . (The letter c in this context stands for compact
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1.1 Basic definitions
7
support.) We equip c, c 0 , and c∞ with the same norm as ∞, of which they are
evidently linear subspaces. It is easy to see that all these normed spaces are infinite
dimensional; that is, an algebraic basis for the underlying vector space must have
infinitely many elements.
1.7 Proposition. For 1
∞,
p
p
is a vector space and
·
p
is a norm on
p.
Proof. The cases p = 1, p = ∞ are simple, and are left to the reader as exercises.
For 1 < p < ∞, the inequality 1
(a + b) p
2 p a p + b p , a, b
0
p
implies that is stable under addition. It is clearly stable under scalar multiplication
as well; thus, p is a vector space. We need only verify that the putative norm · p
satisfies the triangle inequality. To this end, we observe first the following:
| x i + yi | p
| x i + y i | p−1 | x i | + | x i + y i | p−1 | y i | .
Then, we take the sum over i, and we proceed to invoke Hăolders inequality for the
two terms on the right; the triangle inequality results.
1.8 Exercise.
(a) Let 1
p
∞. Show that p ⊂ q , and that the injection (that is, the
identity map Λ x = x from p to q ) is continuous.
(b) It is clear that 1 is a subspace of c 0 , and that c 0 is a subspace of
case can we say closed subspace?
∞.
In which
1.9 Example. (Lebesgue spaces) We proceed to revisit some familiar facts and establish some notation. Let Ω be a nonempty open subset of R n, and let f : Ω → R
be a Lebesgue measurable function. We write dx for Lebesgue measure on R n . The
(Lebesgue) integral
Ω
| f (x)| dx
is then well defined, possibly as +∞. When it is finite, we say that f is summable
(on Ω ). The class of summable functions f is denoted by L1 (Ω ). More generally,
for any p ∈ [ 1, ∞), we denote by L p (Ω ), or by L p(Ω , R), the set of all functions f
such that | f | p is summable, and we write
f
p
=
f
L p (Ω )
:=
Ω
| f (x)| p dx
1/p
.
There remains the case p = +∞. The function f is essentially bounded when, for
some number M, we have | f (x)|
M a.e. The abbreviation “a.e.” stands for “almost everywhere,” which in this context means that the inequality holds except for
This inequality will be evident to us quite soon, once we learn that the function t → t p is convex
on the interval (0, ∞); see page 36.
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8
1 Normed Spaces
the points x in a null set (that is, a subset of Ω of measure zero). We define L∞ (Ω ) to
be the class of measurable functions f : Ω → R that are essentially bounded, with
norm
f ∞ = f L∞ (Ω ) = inf M : | f (x)| M, x ∈ Ω a.e. .
Since the infimum over the empty set is +∞, we see that f ∞ = +∞ precisely
when f fails to be essentially bounded. Thus, for any p in [ 1, ∞ ], we may say that a
measurable function f belongs to L p(Ω ) if and only if f p < ∞.
1.10 Exercise. Let f ∈ L∞ (Ω ). Prove that | f (x)|
f
∞,
x ∈ Ω a.e.
Notation: When Ω is an interval (a, b) in R, we write L p(a, b) for L p(Ω ).
We shall affirm below that L p(Ω ) is a vector space and · p is a norm, but let
us first recall a familiar convention. We identify two elements f and g of L p(Ω )
when f (x) = g(x) for almost every x ∈ Ω . Thus, the elements of L p(Ω ) are really
equivalence classes of functions { f }, where g ∈ { f } if and only if f (x) = g(x) a.e.
This distinction is not reflected in our notation,2 but it explains why it is absurd to
speak of (for example) the set of functions f ∈ L p(0,1) satisfying f (1/2) = 0: this
does not correspond to a property of an equivalence class. On the other hand, it
makes sense to speak of those f ∈ L p(Ω ) which satisfy | f (x)| 1 a.e. in Ω : this
property is stable with respect to elements in an equivalence class.
Let us once again recall Hăolders inequality, which, in the current context, affirms
that if f ∈ L p(Ω ) and g ∈ L p∗ (Ω ), where 1 p ∞, then the function f g belongs
to L1(Ω ), and we have
fg 1
f p g p∗ .
We can use this to adapt the proof of Prop. 1.7, and we obtain
1.11 Proposition. For each p ∈ [ 1, ∞ ] , the class L p(Ω ) is a vector space, and ·
is a norm on L p(Ω ).
1.12 Exercise. Show that L q (0,1) is a strict subspace of L p(0,1) if 1 p < q
(This is true generally of L p(Ω ), when Ω is bounded; compare Exer. 1.8.)
p
∞.
1.13 Example. (Absolutely continuous functions) Let [ a, b ] be an interval in R.
A function x : [ a, b ] → R is said to be absolutely continuous if x is continuous in
the following sense: for every ε > 0, there exists δ > 0 such that, for every finite
collection { [ a i , b i ] } of disjoint subintervals of [ a, b ], we have
∑ (bi − ai ) < δ
i
=⇒
∑ | x(bi ) − x(ai )| < ε .
i
It is shown in elementary courses in integration that a continuous function x possesses this property if and only if it is an indefinite integral; that is, there exists a
function v ∈ L1(a, b) such that
2
Rudin: “We relegate this distinction to the status of a tacit understanding.”
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1.2 Linear mappings
9
t
x(t) = x(a) +
a
v(τ ) d τ ,
t ∈ [ a, b ].
(∗)
In this case, the theory of integration tells us that x is differentiable at almost every
point in (a, b), with x (t) = (d/dt) x(t) = v(t), t ∈ (a, b) a.e. Thus, absolutely
continuous functions are well behaved, in that they coincide with the integral of
their derivative. For this reason, they constitute the customary class in which the
theory of ordinary differential equations is developed, and they will play a central
role later when we study the calculus of variations.
The vector space of absolutely continuous functions on [ a, b ] is denoted AC[ a, b ],
and we endow it with the norm
x
AC
= | x(a)| +
b
a
| x (t)| dt .
More generally, for 1
p
∞, we denote by AC p [ a, b ] the class of continuous
functions x which admit a representation of the form (∗) with v ∈ L p(a, b). The
norm on AC p [ a, b ] is given by
x
AC p
= | x(a)| + x
L p (a,b) .
The function x on [ a, b ] is called Lipschitz if there exists M such that
| x(s) − x(t)|
M | s − t | ∀ s, t ∈ [ a, b ].
Such a function x is easily seen to be absolutely continuous, with a derivative x (almost everywhere) that satisfies | x (t)| M a.e. Thus, a Lipschitz function belongs
to AC ∞ [ a, b ]. Conversely, one shows that an element x of AC ∞ [ a, b ] satisfies the
Lipschitz condition above, the minimal M for this being x L∞ (a,b) .
1.14 Exercise. Show that the function x(t) =
interval [ 0,1], but is not Lipschitz.
√
t is absolutely continuous on the
1.2 Linear mappings
A linear map (or application, or transformation) Λ between two vector spaces X and
Y is one that exhibits a healthy respect for the underlying vector space structure; it
preserves linear combinations:
Λ (t 1 x 1 + t 2 x 2 ) = t 1 Λ (x 1 ) + t 2 Λ (x 2 ),
x1, x 2 ∈ X , t1, t 2 ∈ R .
Such maps turn out to be of central importance in the theory of normed spaces. Note
that they constitute in themselves a vector space, since a linear combination of two
linear maps is another linear map.
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1 Normed Spaces
Notation: When Λ is linear, Λ (x) is also written as Λ x or Λ , x .
The vector space of linear applications from X to Y is denoted by L(X,Y ). If Λ
belongs to L(X,Y ), then Λ (0) = 0 necessarily. If Λ is continuous at 0, then Λ −1 (BY )
contains a neighborhood of 0, and therefore a ball rB X in X. Thus
x
r =⇒ Λ x ∈ B Y ,
X
or, in a formulation that is easily seen to be equivalent to this,
Λx
(1/r) x
Y
∀ x ∈ X.
X
It also follows readily that the continuity of Λ at 0 is equivalent to its continuity
everywhere, and to its being bounded above on a neighborhood of 0. In summary,
and without further proof, we may say:
1.15 Proposition. Let X and Y be normed spaces, and let Λ ∈ L(X,Y ). Then the
following are equivalent :
(a) Λ is continuous ;
(b) Λ is bounded above on a neighborhood of 0 ;
(c) There exists M such that Λ x
Y
M ∀ x ∈ B(0,1) ;
(d) There exists M such that Λ x
Y
M x
X
∀x ∈ X.
1.16 Exercise. Let y i (i = 1, 2, . . . , n) be elements in a normed space Y , and let
Γ : R n → Y be defined by
Γ λ = Γ (λ 1, λ 2 , . . . , λ n ) =
∑ i = 1 λ i yi .
n
Prove that Γ is continuous.
The elements of L(X,Y ) are often referred to as operators. We reserve the term linear functional for the elements of L(X, R); that is, the real-valued linear applications
on X. For any element Λ of L(X,Y ) we write
Λ
= Λ
L(X,Y )
= sup
Λx
Y
: x ∈ X, x
X
1
.
1.17 Exercise. Let Λ ∈ L(X,Y ), where X, Y are normed spaces. Then
Λ
=
sup
x ∈ X, x = 1
Λx
Y
=
sup
x ∈ X, x < 1
Λx
Y
=
sup
x ∈ X, x = 0
Λx Y
.
x X
The reader will notice that two of the expressions displayed in this exercise are
inappropriate if X happens to be the trivial vector space {0}. Thus, it is implicitly
assumed that the abstract space X under consideration is nontrivial; that is, that X
contains nonzero elements. If the exclusion of the trivial case occasionally goes
