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Functional Analysis, Calculus of Variations and Optimal Control
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Francis Clarke
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Springer Science & Business Media
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2013
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Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This self-contained textbook gives a complete course on all these topics. It is written by a leading specialist who is also a noted expositor. This book provides a thorough introduction to functional analysis and includes many novel elements as well as the standard topics. A short course on nonsmooth analysis and geometry completes the first half of the book whilst the second half concerns the calculus of variations and optimal control. The author provides a comprehensive course on these subjects, from their inception through to the present. A notable feature is the inclusion of recent, unifying developments on regularity, multiplier rules, and the Pontryagin maximum principle, which appear here for the first time in a textbook. Other major themes include existence and Hamilton-Jacobi methods. The many substantial examples, and the more than three hundred exercises, treat such topics as viscosity solutions, nonsmooth Lagrangians, the logarithmic Sobolev inequality, periodic trajectories, and systems theory. They also touch lightly upon several fields of application: mechanics, economics, resources, finance, control engineering. Functional Analysis, Calculus of Variations and Optimal Control is intended to support several different courses at the first-year or second-year graduate level, on functional analysis, on the calculus of variations and optimal control, or on some combination. For this reason, it has been organized with customization in mind. The text also has considerable value as a reference. Besides its advanced results in the calculus of variations and optimal control, its polished presentation of certain other topics (for example convex analysis, measurable selections, metric regularity, and nonsmooth analysis) will be appreciated by researchers in these and related fields.

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Part 1
Functional Analysis

Francis Clarke Graduate Texts in Mathematics Functional Analysis, Calculus of Variations and Optimal Control 2013 10.1007/978-1-4471-4820-3_1

Springer-Verlag London 2013

1. Normed Spaces

Francis Clarke 1

(1)

Institut Camille Jordan, Universit Claude Bernard Lyon 1, Villeurbanne, France

Abstract

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. The setting is that of a vector space over the real numbers

. The central idea of this chapter is that of a norm . It is studied in detail, along with the attendant concept of linear operators. The dual space is introduced. The chapter ends with a presentation of derivates, directional derivatives, tangent and normal vectors. These constructs, which allow one to reduce nonlinear situations to linear ones, will play a central role in later developments.

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

1.1 Basic definitions

The setting is that of a vector space over the real numbers

. 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:

x 0 x X ; x =0 if and only if x =0 ( positive definiteness );
x + y x + y x , y X ( the triangle inequality );
( 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 .

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

where the radius r is a positive number We sometimes write B or B X for the - photo 4

where the radius r is a positive number. We sometimes write B or B X for the closed unit ball B (0,1), and

for the open unit ball

. 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 Functional Analysis Calculus of Variations and Optimal Control - image 7

), the sets A + C and tA are given by

Functional Analysis Calculus of Variations and Optimal Control - image 8

(Warning: A + A is different from 2 A in general.) Thus, we have Functional Analysis Calculus of Variations and Optimal Control - image 9

. 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

, while its interior is written int A or Given two points x and y in X the closed interval or segment x y is - photo 11

Given two points x and y in X , the closed interval (or segment) [ x , y ] is defined as follows:

When t is restricted to 01 in the definition we obtain the open interval - photo 12

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:

It follows from this for example that we have int tA t int A when t 0 - photo 13

It follows from this, for example, that we have int( tA )= 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.

A vector space always admits a norm. To see why this is so, recall the well-known consequence of Zorns lemma which asserts that any vector space has a basis { e }, in the sense of linear algebra. This means that any x has a unique representation x = x e , where all but a finite number of the coefficients x are 0. The reader will verify without difficulty that x := | x | 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

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