Optimal Control Theory: Applications to Management Science and Economics

577 Pages·2019·9.15 MB·English

by  Suresh P. Sethi

Suresh P. Sethi
Optimal
Control Theory
Applications to Management Science
and Economics
Third Edition
Optimal Control Theory
Suresh P. Sethi
Optimal Control Theory
Applications to Management Science
and Economics
Third Edition
123
SureshP.Sethi
JindalSchoolofManagement,SM30
UniversityofTexasatDallas
Richardson,TX,USA
ISBN978-3-319-98236-6 ISBN978-3-319-98237-3 (eBook)
https://doi.org/10.1007/978-3-319-98237-3
LibraryofCongressControlNumber:2018955904
2ndedition:©Springer-VerlagUS2000
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This book is dedicated to the memory of
my parents
Manak Bai and Gulab Chand Sethi
Preface to Third Edition
The third edition of this book will not see my co-author Gerald L.
Thompson, who very sadly passed away on November 9, 2009. Gerry
and I wrote the first edition of the 1981 book sitting practically side by
side, and I learned a great deal about book writing in the process. He
was also my PhD supervisor and mentor and he is greatly missed.
After having used the second edition of the book in the classroom
for many years, the third edition arrives with new material and many
improvements. Examples and exercises related to the interpretation of
the adjoint variables and Lagrange multipliers are inserted in Chaps.2–
4. Direct maximum principle is now discussed in detail in Chap.4 along
with the existing indirect maximum principle from the second edition.
Chattering or relaxed controls leading to pulsing advertising policies are
introduced in Chap.7. An application to information systems involving
chattering controls is added as an exercise.
The objective function in Sect.11.1.3 is changed to the more popular
objectiveofmaximizingthetotaldiscountedsociety’sutilityofconsump-
tion. Further discussion leading to obtaining a saddle-point path on the
phasediagramleadingtothelong-runstationaryequilibriumisprovided
in Sect.11.2. For this purpose, a global saddle-point theorem is stated
in Appendix D.7. Also inserted in Appendix D.8 is a discussion of the
Sethi-Skiba points which lead to nonunique stable equilibria. Finally,
a new Sect.11.4 contains an adverse selection model with continuum of
the agent types in a principal-agent framework, which requires an appli-
cation of the maximum principle.
Chapter 12 of the second edition is removed except for the material
on differential games and the distributed parameter maximum principle.
The differential game material joins new topics of stochastic Nash differ-
ential games and Stackelberg differential games via their applications to
marketing to form a new Chap.13 titled Differential Games. As a result,
Chap.13 of the second edition becomes Chap.12. The material on the
distributed parameter maximum principle is now Appendix D.9.
The exposition is revised in some places for better reading. New
exercises are added and the list of references is updated. Needless to say,
the errors in the second edition are corrected, and the notation is made
consistent.
vii
viii Preface to Third Edition
Thanks are due to Huseyin Cavusoglu, Andrei Dmitruk, Gustav Fe-
ichtinger, Richard Hartl, Yonghua Ji, Subodha Kumar, Sirong Lao, Hel-
mut Maurer, Ernst Presman, Anyan Qi, Andrea Seidl, Atle Seierstad,
Xi Shan, Lingling Shi, Xiahong Yue, and the students in my Optimal
Control Theory and Applications course over the years for their sug-
gestions for improvement. Special thanks go to Qi (Annabelle) Feng
for her dedication in updating and correcting the forthcoming solution
manual that went with the first edition. I cannot thank Barbara Gordon
and Lindsay Wilson enough for their assistance in the preparation of
the text, solution manual, and presentation materials. In addition, the
meticulous copy editing of the entire book by Lindsay Wilson is much
appreciated. Anshuman Chutani, Pooja Kamble, and Shivani Thakkar
are also thanked for their assistance in drawing some of the figures in
the book.
Richardson, TX, USA Suresh P. Sethi
June 2018
Preface to Second Edition
The first edition of this book, which provided an introduction to op-
timalcontroltheoryanditsapplicationstomanagementsciencetomany
students in management, industrial engineering, operations research and
economics, went out of print a number of years ago. Over the years we
have received feedback concerning its contents from a number of instruc-
tors who taught it, and students who studied from it. We have also kept
up with new results in the area as they were published in the literature.
For this reason we felt that now was a good time to come out with a
new edition. While some of the basic material remains, we have made
several big changes and many small changes which we feel will make the
use of the book easier.
The most visible change is that the book is written in Latex and the
figures are drawn in CorelDRAW, in contrast to the typewritten text
and hand-drawn figures of the first edition. We have also included some
problems along with their numerical solutions obtained using Excel.
The most important change is the division of the material in the
old Chap.3, into Chaps.3 and 4 in the new edition. Chapter 3 now
contains models having mixed (control and state) constraints, current
value formulations, terminal conditions and model types, while Chap.4
covers the more difficult topic of pure state constraints, together with
mixed constraints. Each of these chapters contain new results that were
not available when the first edition was published.
Thesecondmostimportantchangeistheexpansionofthematerialin
the old Sect.12.4 on stochastic optimal control theory and its becoming
the new Chap.13. The new Chap.12 now contains the following ad-
vanced topics on optimal control theory: differential games, distributed
parameter systems, and impulse control. The new Chap.13 provides a
brief introduction to stochastic optimal control problems. It contains
formulations of simple stochastic models in production, marketing and
finance, and their solutions. We deleted the old Chap.11 of the first
edition on computational methods, since there are a number of excellent
referencesnowavailableonthistopic. Someofthesereferencesarelisted
in Sect.4.2 of Chap.4 and Sect.8.3 of Chap.8.
ix
x Preface to Second Edition
The emphasis of this book is not on mathematical rigor, but rather
on developing models of realistic situations faced in business and man-
agement. For that reason we have given, in Chaps.2 and 8, proofs of the
continuous anddiscretemaximumprinciplesbyusingdynamicprogram-
ming and Kuhn-Tucker theory, respectively. More general maximum
principles are stated without proofs in Chaps.3, 4 and 12.
One of the fascinating features of optimal control theory is its ex-
traordinarily wide range of possible applications. We have covered some
of these as follows: Chap.5 covers finance; Chap.6 considers production
and inventory problems; Chap.7 covers marketing problems; Chap.9
treats machine maintenance and replacement; Chap.10 deals with prob-
lems of optimal consumption of natural resources (renewable or ex-
haustible); and Chap.11 discusses a number of applications of control
theory to economics. The contents of Chaps.12 and 13 have been de-
scribed earlier.
Finally, four appendices cover either elementary material, such as
the theory of differential equations, or very advanced material, whose
inclusion in the main text would interrupt its continuity. At the end
of the book is an extensive but not exhaustive bibliography of relevant
material on optimal control theory including surveys of material devoted
to specific applications.
We are deeply indebted to many people for their part in making this
edition possible. Onur Arugaslan, Gustav Feichtinger, Neil Geismar,
RichardHartl,SteffenJørgensen,SubodhaKumar,HelmutMaurer,Ger-
hard Sorger, and Denny Yeh made helpful comments and suggestions
about the first edition or preliminary chapters of this revision. Many
students who used the first edition, or preliminary chapters of this revi-
sion, also made suggestions for improvements. We would like to express
our gratitude to all of them for their help. In addition we express our
appreciation to Eleanor Balocik, Frank (Youhua) Chen, Feng Cheng,
Howard Chow, Barbara Gordon, Jiong Jiang, Kuntal Kotecha, Ming
Tam, and Srinivasa Yarrakonda for their typing of the various drafts of
the manuscript. They were advised by Dirk Beyer, Feng Cheng, Sub-
odha Kumar, Young Ryu, Chelliah Sriskandarajah, Wulin Suo, Houmin
Yan, Hanqin Zhang, and Qing Zhang on the technical problems of using
LATEX.
We also thank our wives and children—Andrea, Chantal, Anjuli,
Dorothea, Allison, Emily, and Abigail—for their encouragement and un-
derstanding during the time-consuming task of preparing this revision.
Preface to Second Edition xi
Finally, while we regret that lack of time and pressure of other du-
ties prevented us from bringing out a second edition soon after the first
editionwentoutofprint,wesincerelyhopethatthewaithasbeenworth-
while. In spite of the numerous applications of optimal control theory
which already have been made to areas of management science and eco-
nomics, we continue to believe there is much more that remains to be
done. We hope the present revision will rekindle interest in furthering
such applications, and will enhance the continued development in the
field.
Richardson, TX, USA Suresh P. Sethi
Pittsburgh, PA, USA Gerald L. Thompson
January 2000