Table Of ContentIntegral Equations and
Stability of Feedback Systems
CONSTANTIN CORDUNEANU
UniversityofJassy
Jassy,Romania
ACADEMIC PRESS New York and London 1973
A Subsidiary ofHarcourt BraceJovanovich, Publishers
COPYRIGHTCO1973,BYACADEMICPRESS,INC.
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ACADEMIC PRESS, INC.
111FifthAvenue.NewYork,New York10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LID.
24/28Oval Road. LondonNWI
Library ofCongressCataloging inPublicationData
Corduneanu,C
Integralequationsand stabilityoffeedback systems.
Bibliography: p.
I. Controltheory. 2. Feedback controlsystems.
3. Integralequations. I. Title.
QA402.3.C66 629.8'312 72-88341
ISBN0-12-188350-7
AMS (MOS) 1970 Subject Classifications: 45M10, 45D05,
45M05,45EIO
PRINTEDIN THE UNITEDSTATES OFAMERICA
Contents
PREFACE vii
Preliminaries
1.1 Some FunctionSpaces 1
1.2 AFixed-PointTheorem 7
1.3 Fourierand Laplace Transforms 12
1.4 A FactorizationProblem 18
Exercises 22
2 Admissibility and Hammerstein Equations
2.1 Admissibility and Continuity 24
2.2 Admissibility of the Pair (C,; CG) with Respect to the VolterraOperator 25
2.3 Admissibility ofthe Pair u(C;g",",Ct) 28
2.4 Admissibility of the Pair CG) 34
2.5 Admissibility of the Pair(P, CG), 1< P< co 37
2.6 Admissibility of the Pair(Lg"",LG"") 39
2.7 Admissibility with Respect to the ConvolutionOperator 42
2.8 ReviewofOtherAdmissibility Results 46
2.9 Admissibility for DifferentialSystems 49
2.10 IntegralEquationsin C 51
G
v
vi Contents
2.11 PerturbedIntegralEquationsin C 56
G
2.12 Existence ofConvergentSolutions 59
2.13 FurtherExistenceResultsin C 64
G
2.14 ConvolutionEquations 66
2.15 Existence of MeasurableSolutions 70
Exercises 74
BibliographicalNotes 77
3 Frequency Techniques and Stability
3.1 General RemarksandStatement ofSome Problems 83
3.2 Equation(1.1) with an IntegrableKernel 85
3.3 Equation(1.1) with k(t)+PIntegrable, P>0 90
3.4 FurtherInvestigation ofthe PrecedingCase 95
3.5 AnotherCase of a NonintegrableKernel 101
3.6 AStabilityResult in the Vector Case 105
3.7 An Integra-DifferentialEquation 109
3.8 Existence ofL' Solutions 112
3.9 ControlSystems with Time Lag 116
3.10 A Resultof Yakubovitch 119
3.11 EnergeticStability 124
3.12 ACriterionfor LPStability 131
Exercises 133
BibliographicalNotes 137
4 Wiener-HopfEquations
4.1 ASpecial Class of Kernels 144
4.2 Equation(WH)with Index Zero 146
4.3 Equation(WH)with Positive Index 151
4.4 Equation(WH)with NegativeIndex 156
4.5 Some Examples 162
4.6 PerturbedEquations 167
Exercises 171
BibliographicalNotes 173
5 Further Methods and Topics
5.1 Some Results of Levin 177
5.2 ASystem Occurringin the Dynamics ofNuclear Reactors 185
5.3 APositivityConditionfor the Kernel 194
5.4 Linearization of VolterraIntegral Equations 198
5.5 Volterra-StieltjesIntegralEquations 201
5.6 TauberianResults 213
Exercises 216
BibliographicalNotes 217
References 221
AUTHOR INDEX 233
SUBJECT INDEX 236
Preface
The mathematical description ofthe dynamical processes encountered in
physical, biological, and applied sciencesnowadays requires the investigation
of(ordinary and partial) differential equations, equations with delay, integral
or integro-differential equations, and other kinds of functional equations.
The first integral equation mentionedin the mathematicalliterature isdue
to Abel and can be found in almost any book on this subject (see, for in-
stance, the author's book [12]). Abel found this equation in 18I2, starting
from a problem in mechanics. He gave a very elegant solution that was
published in 1826in Crelle's Journal.
Starting in 1896, Vito Volterra built up a theory of integral equations,
viewingtheir solution as a problem of finding the inversesof certain integral
operators.
In 1900, Ivar Fredholm made his famous contribution that led to a
fascinating period in the development of mathematical analysis. Poincare,
Frechet, Hilbert, Schmidt, Hardy, and Riesz were involved in this new area
of research. The impactof Fredholm'stheory onthe foundation and develop-
ment of functional analysis has also been outstanding. These facts could
explain why Volterra's equations, whose role in the investigation of some
dynamical processes (mainly in biology) had been emphasized by Volterra
himself,tookaplace ofsecondaryimportance.Actually, the integral equations
vii
viii Preface
of Volterra type are present any time we deal with a differential equation.
According to J. Dieudonne, "... differential equations constitute a swindling.
In fact, there exist no differential equations. The only interesting equations
are the integral ones" (Nico, 1969,No.2).
Duringthe last 15-20 years, Volterra equations have emerged vigorously
in such applied fields as automatic control theory, network theory, and the
dynamics of nuclear reactors. The fact that causal operators (i.e., Volterra
operators) alone are adequate to describe dynamical systems has led to an
increased interest in Volterra equations. Satisfactory solutions have been
found for nonlinear problems, which very often arise in applications. Many
results are now available, and several books on this subject have appeared in
the last fewyears (see Kudrewicz [5],Miller [7],and J. C. Willems [3]).
Except for the introductory chapter and a fewresults inChapters2and 4,
this book contains concepts and results related to Volterra equations.
Some topics are not necessarily related only to Volterra equations and,
therefore, a broaderframework has been considered in the exposition.
Chapter 2 is concerned with linear integral operators (admissibility re-
sults) and Hammerstein equations. The first part of the chapter contains
necessary and sufficient conditions for the continuity of various classes of
linear integral operators. The reader interested in applications willrecognize
here various criteria of stability for linear input-output systems. The second
part of the chapter is devoted to the existence theory for Hammerstein
equations, using fixed-point theorems as the main tool.
Chapter 3 contains results pertaining to the frequency method. Besides
Popov's original stability theorem, there are some generalizations and related
topics. Applications are also included.
Chapter 4 is dedicated to the theory of the Wiener-Hopf equation, as
it appears in M. G. Krein's work [I]. The underlying space here is richer
than in Krein's original theory. This theory has significant applications (the
transfer of radiant energy, for instance) that have not been included in this
book.
The last chapterisheterogeneous incharacterand containsvarious results
related to the useof Liapunov functionals (energy method), positive kernels,
the linearization problem, and Volterra-Stieltjes equations, as well as an
application to the dynamics of nuclear reactors (due to Levin and Nohel
[6]).
Theapplied mathematicianand the research engineer willfindinthis book
some tools of current use in systems theory. The pure mathematician can
see how concepts and results such as Banach algebras, the closed-graph
theorem, or fixed-point theorems are used to reach very profane purposes.
Many results included in this book have been discussed at the weekly
meetings of the seminar on qualitative problems with the Faculty of Mathe-
matics, University of Iasi, during the academic year 1966/1967 and since
Preface ix
1968. A seminar on integral equations was held inthe Department of Mathe-
maticsat the University of Rhode Island duringthe academic year 1967(1968,
whilethe author held a visiting position there.
The author is indebted to Professor D. Petrovanu, Professor R. K.
Miller, Dr. V. Barbu, and Dr. G. Bantas for helpful discussions. In particu-
lar, Dr. Barbu cooperated in writing Section 4.4. Wetake this opportunityto
express our sincere thanks.
We also wish to thank Miss Roza Oxenberg for her cooperation in
preparing the manuscript.
Finally, our thanks and appreciation go to Professor R. Bellman, and
to allpersons at Aca lemic Press involved in carrying out this project.
Preliminaries
This chapter provides some auxiliary results we shall need in subsequent
chapters. Becausemany ofthese results can beeasilyfound inthe mathemati-
cal literature, wedo not include their proofs. Only a sketch of the proofand
adequate references are given. There are some results for which we givethe
complete proofs. We intend to spare the reader's effort in better understand-
ing some theorems that play an important role in our exposition and whose
interest overreaches the framework of this book.
1.1 SomeFunction Spaces
We shall deal first with some spaces of continuous functions on the posi-
tive half-axis R+ ={t: t ;;::: O}. Following Bourbaki [1], let us denote by
CcCR+, Rn) the linear space of all continuous mappings from R+.into R",
the topology being that of uniform convergence on any compact subset
of R+.
1
2 1 Preliminaries
For any x E CcCR+,R") and natural number n ~ I, denote
Ix], = sup{lIx(t)II:0 ~ t ~ n}, (1.1)
where 11·11 stands for the Euclidean norm in R". The mapping x -+ IxI"
is a seminorm on Cc(R+, R") and the topology of uniform convergence
on any compact subset of R+ isthat generated by the family of seminorms
{Ixl":n = 1,2, ...}.A distance function can be defined on Cc(R+, R") by
means of the formula
1 Ix-YI"
p(x, y) = ~00 2"1+ I - I (1.2)
"-1 X Y"
The topology induced by the distance function p is the same topology of
uniform convergence on any compact subset ofR+ . It can be easily shown
that Cc(R+, R") iscomplete.
Finally, let us mention that a family of functions Me Cc(R+, R") is
relativelycompactifand only ifthe followingpropertyholds: let T> 0 bean
arbitrary numberand considerthe set ofrestrictions ofthe functions belong-
ing to M on [0,T], say MT; then MT is relatively compact in C([O,T], R"),
i.e., it is uniformly bounded and equicontinuous on [0,T].
Consider now a continuous positive functiong(t), t E R+ . By Cg(R+, R")
we will denote the linear space ofall continuous mappings from R+ into R"
such that
sup{llx(t)ll/g(t):tE R+} < +00. (1.3)
Ifwedefine
IxIC = sup{llx(t)ll/g(t) :tE R+} (1.4)
g
for all x E Cg(R+, R"), then we can see easily that x -+ IxICgis a norm.
Moreover, Cg(R+, R") isa Banachspace. To provethis statementweshall
consider a Cauchy sequence {x"'} c Cg(R+, R"), and show that there exists
x E Cg(R+, R") with the property
lim Ix" - xIC = O. (1.5)
m....oo g
Indeed, to any e> 0 there corresponds N = N(e) > 0 such that
1x"'-xPlcg<e (1.6)
whenever m, p ~ N(e). But (1.6) implies
11x"'(t) - xP(t)II< eg(t) (1.7)
for tE R+ and m, p ~ N(e). From (1.7) we easily derive the existence of a
mapping x(t) such that x(t) =lim x"'(t) as m-+ 00, for any fixed tE R+ .
