The Calculus of Variations and Functional Analysis With Optimal Control and Applications in Mechanics (Series on Stability, Vibration and Control of Systems, Series A - Vol. 12)

435 Pages·2003·15.33 MB·English

by  Michael J. Cloud

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
Series A Volume 12
The Calculus of Variations
and Functional Analysis
With Optimal Control and Applications in Mechanics
Leonid P. Lebedev & Michael J. Cloud
^
World Scientific
The Calculus of Variations and
Functional Analysis
With Optimal Control and Applications in Mechanics
SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
Founder and Editor: Ardeshir Guran
Co-Editors: C. Christov, M. Cloud, F. Pichler & W. B. Zimmerman
About the Series
Rapid developments in system dynamics and control, areas related to many other
topics in applied mathematics, call for comprehensive presentations of current
topics. This series contains textbooks, monographs, treatises, conference proceed
ings and a collection of thematically organized research or pedagogical articles
addressing dynamical systems and control.
The material is ideal for a general scientific and engineering readership, and is
also mathematically precise enough to be a useful reference for research specialists
in mechanics and control, nonlinear dynamics, and in applied mathematics and
physics.
Selected Volumes in Series B
Proceedings of the First International Congress on Dynamics and Control of Systems,
Chateau Laurier, Ottawa, Canada, 5-7 August 1999
Editors: A. Guran, S. Biswas, L. Cacetta, C. Robach, K. Teo, and T. Vincent
Selected Volumes in Series A
Vol. 2 Stability of Gyroscopic Systems
Authors: A. Guran, A. Bajaj, Y. Ishida, G. D'Eleuterio, N. Perkins,
and C. Pierre
Vol. 3 Vibration Analysis of Plates by the Superposition Method
Author: Daniel J. Gorman
Vol. 4 Asymptotic Methods in Buckling Theory of Elastic Shells
Authors: P. E. Tovstik and A. L Smirinov
Vol. 5 Generalized Point Models in Structural Mechanics
Author: I. V. Andronov
Vol. 6 Mathematical Problems of Control Theory: An Introduction
Author: G. A. Leonov
Vol. 7 Analytical and Numerical Methods for Wave Propagation in
Fluid Media
Author: K. Murawski
Vol. 8 Wave Processes in Solids with Microstructure
Author: V. I. Erofeyev
Vol. 9 Amplification of Nonlinear Strain Waves in Solids
Author: A. V. Porubov
Vol. 10 Spatial Control of Vibration: Theory and Experiments
Authors: S. O. Reza Moheimani, D. Halim, and A. J. Fleming
Vol. 11 Selected Topics in Vibrational Mechanics
Editor: I. Blekhman
SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
<Hfe> Series A Volume 12
Founder and Editor: Ardeshir Guran
Co-Editors: C. Christov, M. Cloud,
F. Pichler & W. B. Zimmennan
The Calculus of Variations
and Functional Analysis
With Optimal Control and Applications in Mechanics
Leonid P. Lebedev
National University of Colombia, Colombia &
Rostov State University, Russia
Michael J. Cloud
Lawrence Technological University, USA
\jJ5 World Scientific
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THE CALCULUS OF VARIATIONS AND FUNCTIONAL ANALYSIS:
WITH OPTIMAL CONTROL AND APPLICATIONS IN MECHANICS
Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd.
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Foreword
A foreword is essentially an introductory note penned by an invited writer,
scholar, or public figure. As a new textbook does represent a pedagogical
experiment, a foreword can serve to illuminate the author's intentions and
provide a bit of insight regarding the potential impact of the book.
Alfred James Lotka — the famous chemist, demographer, ecologist, and
mathematician — once stated that "The preface is that part of a book
which is written last, placed first, and read least." Although the follow
ing paragraphs do satisfy Lotka's first two conditions, I hope they will not
satisfy the third. For here we have a legitimate chance to adopt the sort
of philosophical viewpoint so often avoided in modern scientific treatises.
This is partly because the present authors, Lebedev and Cloud, have ac
cepted the challenge of unifying three fundamental subjects that were all
rooted in a philosophically-oriented century, and partly because the varia
tional method itself has been the focus of controversy over its philosophical
interpretation. The mathematical and philosophical value of the method
is anchored in its coordinate-free formulation and easy transformation of
parameters. In mechanics it greatly facilitates both the formulation and
solution of the differential equations of motion. It also serves as a rigor
ous foundation for modern numerical approaches such as the finite element
method. Through some portion of its history, the calculus of variations
was regarded as a simple collection of recipes capable of yielding neces
sary conditions of minimum for interesting yet very particular functionals.
But simple application of such formulas will not suffice for reliable solu
tion of modern engineering problems — we must also understand various
convergence-related issues for the popular numerical methods used, say, in
elasticity. The basis for this understanding is functional analysis: a rel
atively young branch of mathematics pioneered by Hilbert, Wiener, von
v
VI Calculus of Variations and Functional Analysis
Neumann, Riesz, and many others. It is worth noting that Stefan Banach,
who introduced what we might regard as the core of modern functional
analysis, lectured extensively on theoretical mechanics; it is therefore not
surprising that he knew exactly what sort of mathematics was most needed
by engineers.
For a number of years I have delivered lecture courses on system dynam
ics and control to students and researchers interested in Mechatronics at
Johannes Kepler University of Linz, the Technical University of Vienna, and
the Technical University of Graz. Mechatronics is an emerging discipline,
frequently described as a mixture of mechanics, electronics, and comput
ing; its principal applications are to controlled mechanical devices. Some
engineers hold the mistaken view that mechatronics contains nothing new,
since both automatic control and computing have existed for a long time.
But I believe that mechatronics is a philosophy which happens to overlap
portions of the above-mentioned fields without belonging to any of them
exclusively. Mechanics, of course, rests heavily on the calculus of variations,
and has a long history dating from the works of Bernoulli, Leibniz, Euler,
Lagrange, Fermat, Gauss, Hamilton, Routh, and the other pioneers. The
remaining disciplines — electronics and computing — are relatively young.
Optimal control theory has become involved in mechatronics for obvious
reasons: it extends the idea of optimization embodied in the calculus of
variations. This involves a significant extension of the class of problems to
which optimization can be applied. It also involves an extension of tradi
tional "smooth" analysis tools to the kinds of "non-smooth" tools needed
for high-powered computer applications. So again we see how the tools of
modern mathematics come into contact with those of computing, and are
therefore of concern to mechatronics.
Teaching a combination of the calculus of variations and functional anal
ysis to students in engineering and applied mathematics is a real challenge.
These subjects require time, dedication, and creativity from an instructor.
They also take special care if the audience wishes to understand the rigor
ous mathematics used at the frontier of contemporary research. A principal
hindrance has been the lack of a suitable textbook covering all necessary
topics in a unified and sensible fashion. The present book by Professors
Lebedev and Cloud is therefore a welcome addition to the literature. It is
lucid, well-connected, and concise. The material has been carefully cho
sen. Throughout the book, the authors lay stress on central ideas as they
present one powerful mathematical tool after another. The reader is thus
prepared not only to apply the material to his or her own work, but also
Foreword vn
to delve further into the literature if desired.
An interesting feature of the book is that optimal control theory arises as
a natural extension of the calculus of variations, having a more extensive set
of problems and different methods for their solution. Functional analysis,
of course, is the basis for justifying the methods of both the calculus of
variations and optimal control theory; it also permits us to qualitatively
describe the properties of complete physical problems. Optimization and
extreme principles run through the entire book as a unifying thread.
The book could function as both (i) an attractive textbook for a course
on engineering mathematics at the graduate level, and (ii) a useful refer
ence for researchers in mechanics, electrical engineering, computer science,
mechatronics, or related fields such as mechanical, civil, or aerospace engi
neering, physics, etc. It may also appeal to those mathematicians who lean
toward applications in their work. The presence of homework problems at
the end of each chapter will facilitate its use as a textbook.
As Poincare once said, mathematicians do not destroy the obstacles
with which their science is spiked, but simply push them toward its bound
ary. I hope that some particular obstacles in the unification of these three
branches of science (the calculus of variations, optimal control, and func
tional analysis) and technology (mechanics, control, and computing) will
continue to be pushed out as far as possible. Professors Lebedev and Cloud
have made a significant contribution to this process by writing the present
book.
Ardeshir Guran
Wien, Austria
March, 2003
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Preface
The successful preparation of engineering students, regardless of specialty,
depends heavily upon the basics taught in the junior year. The general
mathematical ability of students at this level, however, often forces instruc
tors to simplify the presentation. Requiring mathematical content higher
than simple calculus, engineering lecturers must present this content in a
rapid, often cursory fashion. A student may see several different lecturers
present essentially the same material but in very different guises. As a re
sult "engineering mathematics" often comes to be perceived as a succession
of procedures and conventions, or worse, as a mere bag of tricks. A student
having this preparation is easily confounded at the slightest twist of a prob
lem. Next, the introduction of computers has brought various approximate
methods into engineering practice. As a result the standard mathematical
background of a modern engineer should contain tools that belonged to
the repertoire of a scientific researcher 30-40 years ago. Computers have
taken on many functions that were once considered necessary skills for the
engineer; no longer is it essential for the practitioner to be able to carry
out extensive calculations manually. Instead, it has become important to
understand the background behind the various methods in use: how they
arrive at approximations, in what situations they are applicable, and how
much accuracy they can provide. In large part, for solving the boundary
value problems of mathematical physics, the answers to such questions re
quire knowledge of the calculus of variations and functional analysis. The
calculus of variations is the background for the widely applicable method of
finite elements; in addition, it can be considered as the first part of the the
ory of optimal control. Functional analysis allows us to deal with solutions
of problems in more or less the same way we deal with vectors in space. A
unified treatment of these portions of mathematics, together with examples
IX