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Lectu re Notes in
Operations Research and
Mathematical Economics
Edited by M. Beckmann, Providence and H. P. Künzi, Zürich
3
Aaron Strauss
University of Maryland
An Introduction to
Optimal Control Theory
1968
Springer-Verlag Berlin . Heidelberg . New York
This research was supported in part by the National Science Foundation under Grant NSF -GP 6167.
ISBN 978-3-540-04252-5 ISBN 978-3-642-51001-4 (eBook)
DO! 10.1007/978-3-642-51001-4
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from
Springer Verlag. © by Springer-Verlag Berlin . Heidelberg 1968.
Library of Congress Catalog Card Number 68-24470.
PREFACE
This paper is intended for the beginner. It is not a state
of-the-art paper for research workers in the field of control theory.
Its purpose is to introduce the reader to some of the problems and
results in control theory, to illustrate the application of these re
sults, and to provide a guide for his further reading on this subject.
I have tried to motivate the results with examples, especial
ly with one canonical, simple example described in §3. Many results,
such as the maximum principle, have long and difficult proofs. I have
omitted these proofs. In general I have included only the proofs which
are either (1) not too difficult or (2) fairly enlightening as to the
nature of the result. I have, however, usually attempted to draw the
strongest conclusion from a given proof. For example, many existing
proofs in control theory for compact targets and uniqueness of solutions
also hold for closed targets and non-uniqueness.
Finally, at the end of each section I have given references
to generalizations and origins of the results discussed in that section.
I make no claim of completeness in the references, however, as I have
often been content merely to refer the reader either to an exposition
or to a paper which has an extensive bibliography.
IV
These 1ecture notes are revisions of notes I used for aseries
of nine 1ectures on contro1 theory at the International Summer Schoo1 on
Mathematica1 Systems and Economics held in Varenna, Ita1y, June 1967.
Those notes in turn were condensed from a one semester course on contro1
theory which I gave at the University of Mary1and in 1965. I would guess
that these 1ecture notes are, in their present form, most suitab1e for a
5 - 10 week introductory course on contro1 theory. I wou1d say that some
know1edge of ordinary differential equations (CODDINGTON and LEVINSON
[1, Chapters 1-3]) and measure theory is essential for a good understand-
ing of these notes.
This manuscript was written part1y whi1e I held aNational
Science Foundation postdoctora1 fellowship to the University of F10rence
and part1y whi1e I was supported by National Science Foundation Grant
NSF - GP 6167 at the University of Mary1and.
Fina11y, I am deep1y gratefu1 to Professors A. Ha1anay and
J. A. Yorke for many he1pfu1 and stimu1ating discussions, and to my
students at Mary1and and Varenna whose penetrating questions and sug-
gestions have stimu1ated my own interest in contro1 theory.
Aaron Strauss
College Park, Mary1and
January, 1968
CONTENTS
1. INTRODUCTION 1
2. CONTROL PROBLEM 3
Notation and definitions, motivation of later results
3. RAILROAD TRAIN EXAMPLE 17
Motivation of a "bang-bang principle," the time optimal
and minimum fuel problems
4. CONTROLLABILITY 31
Restricted and unrestricted control, the controllability
matrix
s.
CONTROLLABILITY USING SPECIAL CONTROLS . 47
Piecewise constant controls and bang-bang controls
6. LINEAR TIME OPTIMAL SYSTEMS 59
Existence, necessary conditions, extremaI control,
reachable cone, normal systems and bang-bang controls,
application to synthesis of railroad train example
7. GENERAL CONTROL SYSTEMS: EXISTENCE 81
Motivation behind existence results, two existence
theorems and extensions, examples
8. GENERAL CONTROL SYSTEMS: NECESSARY CONDITIONS. 107
Principle of Optimality, Pontryagin maximum principle,
transversality conditions, application to synthesis
of railroad train example
9. FURTHER TOPICS • 141
REFERENCES 143
LIST OF SYMBOLS 149
INDEX FOR DEFINITIONS 153
1
1. INTRODUCTION
The modern theory of contro1 not only offers many intriguing
problems to delight the mathematicians but also, perhaps to an extent
unparalleled in mathematics, has attracted attention throughout our
soeiety. Philosophically, this is understandable, for from a eompletely
natural point of view the entirety of human enterprise may be thought
of as an effort to contro1 or influence processes of one type or another.
It is, of course, true that the objectives and criteria for performance
in many situations are diffuse and defy tractable analysis. Nevertheless,
the basic concepts are clear and establish a procedure firmly based in
logie and practicality.
In control theory, in a general sense, attention is first of
all centered on a process, that is, some action or motion taking place
or existing in time. Along with the notion of process one considers
controls for influencing the particular process in question. If analysis
is to be performed, it is necessary to formulate a structure called the
dynamics of the process or a law which governs change in state. When
a policy of influence has b€en specified, our dynamics by definition
provides me ans whereby on the basis of knowledge of the state x(t) of
the proeess for times t t ,t some arbitrary point in our time set,
~
o 0
one can determine the evolutions of x(t) for t > t
o
The next element required in the general formulation of a control
problem is an objective. That is, we set some goal to be achieved by our
process through a properly applied control policy. An objective is usual1y
specified as the acquisition of some desired state target for the process.
2
One question which arises natura11y is whether or not me ans for inf1uencing
the process are sufficient1y strong to allow the achievement of a specified
objective. If such means exist, then we have a properly formu1ated contro1
structure. Starting from some arbitrary initial state for a process, one
may consider the set of all states which can be acquired through inf1uence
policies at our disposal. Such a set is ca1led the reachab1e set for the
process defined relative to the specified initial state. A proper1y formu-
1ated control structure exists if an objective state for the process lies
in the reachable set relative to the present state.
In contro1 problems there are, in general, a number of ways in
which the objective for a process may be accomp1ished. Within the set of
possibilities, taking into account imposed constraints, one may desire to
systematical1y choose the "best" approach with respect to some performance
criterion. If with respect to some performance criterion one seeks in the
set of all policies for achieving an objective the one that is "best," then
the formulation is an optimal control problem.
In this paper we shall assume that the dynamics is in the form
of a vector ordinary differential eguation and the control, whose range is
contained in some pre-assigned control region, is a function belonging to
a certain admissible -cl-as-s. The target will be a continuously varying
c10sed set, and the performance criterion will be the integral. of ~
real valued function. We shall discuss such questions as whether a given
initial point can be "steered to the target" using some control, whether
controls required to belong to some special class of functions would also
steer this initial point to the target, whether an optimal control exists
md, if so, how it can be found.
3
2. CONTROL PROBLEM
Let Rn denote Euclidean n-space and let rl C Rm • For
i .th
x denote the 1. component of x and let
<
)= 1 1 n n
Let x, y xy+ ... +xy Let
and
1 n m n .
Let f R x R x R + R be contlnuous. Then
(E) x = f (t, x, u)
n
has solutions in the following sense: for each and each x ~ R ,
o
there exists at least one absolutely continuous function
x(t) = x(t t , x, u('»
o 0
such that
4
,
=
(2.1) x(t) f (t, x(t} , u(t) )
almost everywhere on [to' t2] ,where t2 is some real number
=
and such that x(t) x • 1f at
o 0
least one solution x(·) of (2.1) exists on all of [to' tl ],
then we call u(') asontrol and x(·) a response. A control
can have many responses, although we shall often assume that
(E) has uniqueness (i.e., (2.1) has only one solution through
x for each x and u IE U) in which case a control has exact-
o 0
ly one response.
Let
p = {p eRn P is closed} •
Let and
G ['0' '1] -+ P •
Then a control problem consists of the following five items:
an ordinary differential eguation (E)
a control region n
an admissible control class U
an initial point x
o
'1]
and a target G( • ) on [ '0'
