Stability of Neutral Functional Differential Equations

311 Pages·2014·2.919 MB·English

by  Michael Gil'

Atlantis Studies in Differential Equations
Series Editor: Michel Chipot
Michael I. Gil’
Stability of Neutral
Functional
Differential
Equations
Atlantis Studies in Differential Equations
Volume 3
Series editor
Michel Chipot, Zürich, Switzerland
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’
Michael I. Gil
Stability of Neutral
Functional Differential
Equations
Michael I.Gil’
Department of Mathematics
Ben-Gurion Universityof theNegev
BeerSheva
Israel
ISSN 2214-6253 ISSN 2214-6261 (electronic)
ISBN 978-94-6239-090-4 ISBN 978-94-6239-091-1 (eBook)
DOI 10.2991/978-94-6239-091-1
LibraryofCongressControlNumber:2014950675
©AtlantisPressandtheauthor2014
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Preface
1. Thesuggestedbookdealswiththestabilityoflinearandnonlinearvectorneutral
type functional differential equations. Equations with neutral type linear parts
and nonlinear causal mappings are also considered. Explicit conditions for the
exponential, absolute and input-to-state stabilities are derived. Moreover, solu-
tion estimates for the considered equations are established. These estimates
provide the bounds for regions of attraction of steady states. The main meth-
odologypresentedinthebookisbasedonacombinedusageoftherecentnorm
estimates for matrix-valued functions with the following methods and results:
the generalized Bohl-Perron principle, the integral version of the generalized
Bohl-Perronprincipleandthepositivityconditionsforfundamentalsolutionsto
scalar neutral equations. We also apply the so-called generalized norm. A sig-
nificant part of the book is devoted to the generalized Aizerman problem.
2. Neutral type functional differential equations (NDEs) naturally arise in various
applications, such as control systems, mechanics, nuclear reactors, distributed
networks, heat flows, neural networks, combustion, interaction of species,
microbiology, learning models, epidemiology, physiology, and many others.
The theory offunctional differential equations has been developed in the works
of V. Volterra, A.D. Myshkis, N.N. Krasovskii, B. Razumikhin, N. Minorsky,
R. Bellman, A. Halanay, J. Hale and other mathematicians.
The problem of stability analysis of various neutral type equations continues to
attract the attention of many specialists despite its long history. It is still one of
themostburningproblemsbecauseoftheabsenceofitscompletesolution.The
basic method for the stability analysis is the method of the Lyapunov type
functionals. By that method many very strong results are obtained. We do not
consider the Lyapunov functionals method because several excellent books
cover this topic. It should be noted that finding the Lyapunov type functionals
for vector neutral type equations is often connected with serious mathematical
difficulties, especially in regard to nonautonomous and nonlinear equations. On
thecontrary,thestabilityconditionspresentedinthesuggestedbookaremainly
formulated in terms of the determinants and eigenvalues of auxiliary matrices
v
vi Preface
dependentonaparameter.Thisfactallowsustoapplythewell-knownresultsof
the theory of matrices to the stability analysis.
3. Theaimofthebookistoprovidenewtoolsforspecialistsinthestabilitytheory
offunctional differential equations, control system theory and mechanics.
This is the first book that:
(i) givesasystematicexpositionoftheapproachtostabilityanalysisofvector
neutral type functional differential equations which is based on estimates
for matrix-valued functions allowing us to investigate various classes of
equations from the unified viewpoint;
(ii) contains a solution of the generalized Aizerman problem for NDEs;
(iii) presentsthegeneralizedBohl-Perronprincipleforneutraltypesystemsand
its integral version;
(iv) suggests explicit stability conditions for semilinear equations with linear
neutral type parts and nonlinear causal mappings.
The book is intended not only for specialists in the theory of functional
differentialequations,butforanyoneinterestedinvariousapplicationswho
has had at least a first year graduate level course in analysis.
I was very fortunate to have fruitful discussions with the late Professors
M.A. Aizerman, M.A. Krasnosel’skii, A.D. Myshkis, A. Pokrovskii and
A.A. Voronov, to whom I am very grateful for their interest in my
investigations.
4. The book consists of 9 chapters.
Chapter 1 is of preliminary character. In that chapter we present standard facts,
mainly from the theory of Banach and ordered spaces, required in further
consideration. In addition, we establish norm estimates for operators of the
forms
Zη
f ! dR ðsÞfðt(cid:2)sÞ
0
0
and
Zη
f ! dsRðt;sÞfðt(cid:2)sÞ ðt(cid:3)0Þ;
0
where R (s) and R(t, s) arereal matrix valued functions having bounded variations
0
in s. These estimates play an essential role in the book.
In Chap. 2 we accumulate norm estimates for matrix-valued functions and bounds
for the eigenvalues of matrices. In particular, we suggest estimates for the resol-
vents, powers of matrices as well as for matrix exponentials. We also present
bounds for the spectral radius and investigate perturbations of eigenvalues. The
material of this paper is systematically applied in the rest of the book chapters.
Chapter 3 deals with linear autonomous and time-variant vector difference-delay
Preface vii
equationswithcontinuoustime.Undervariousassumptions,normestimatesforthe
Cauchy(resolvent)operatorsoftheconsideredequationsarederived.Inthesequel
chapters these estimates give us stability conditions for the neutral type functional
differential equations.
InChap.4weconsidervectorlineardifferential-delayequations(DDEs).Wederive
estimatesfortheLp-andC-normsoftheCauchyoperatorsofautonomousandtime-
variantdifferential-delayequations.Theseestimatesenableustoinvestigateneutral
type equations.
Chapter 5 is devoted to vector linear autonomous NDEs. We derive estimates for
the Lp- and C-norms of the characteristic matrix functions and fundamental solu-
tions of the considered equations, as well as establish stability conditions for
autonomous NDEs.
InChap.6weinvestigatelinearvectortime-variant(non-autonomous)neutraltype
functionaldifferentialequations.Inparticular,weextendtheBohl-Perronprinciple
toaclassofneutraltypefunctionaldifferentialequations.Namely,itisprovedthat
thehomogeneousequationisexponentiallystable,providedthecorrespondingnon-
homogeneousequationwiththezeroinitialconditionandanarbitraryboundedfree
term has a bounded solution. We also establish the integral version of the gen-
eralized Bohl-Perron principle for NDEs, i.e. it is shown that the homogeneous
equation is exponentially stable, if the corresponding non-homogeneous equation
with the zero initial condition and an arbitrary free term from
Lpð0;1Þ¼Lpð½0;1Þ;CnÞ, has a solution belonging to Lpð0;1Þ. As applications
of these principles, the stability conditions for time-variant NDEs close to auton-
omous systems are derived. In addition, we investigate time-variant systems with
smallprincipaloperatorsandobtainstabilityconditionsindependentofdelayinthe
non-autonomous case.
Chapter 7 deals with nonlinear vector equations having linear autonomous neutral
type parts and nonlinear causal mappings. Explicit conditions for the exponential,
absolute and input-to-state stabilities are derived.
Chapter 8 is concerned with scalar nonlinear neutral type functional differential
equations with autonomous linear parts. We derive explicit absolute L2-stability
conditionsintermsofthenormsoftheCauchyoperatorstothelinearpartsandthe
Lipschitz constants of the nonlinearities. In addition, we consider the generalized
Aizerman problem.
InChap.9wediscusscertainpropertiesofthecharacteristicvaluesofautonomous
vector NDEs. In particular, bounds for characteristic values and perturbations
results are derived.
Contents
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Banach Spaces and Linear Operators. . . . . . . . . . . . . . . . . . . 1
1.2 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Some Theorems From Functional Analysis. . . . . . . . . . . . . . . 7
1.4 Ordered Banach Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 The Abstract Gronwall Lemma. . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Integral Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 The Kantorovich Generalized Norm . . . . . . . . . . . . . . . . . . . 13
1.8 Autonomous Difference Operators . . . . . . . . . . . . . . . . . . . . 15
1.9 Time-Variant Difference Operators . . . . . . . . . . . . . . . . . . . . 24
1.10 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.11 Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Eigenvalues and Functions of Matrices . . . . . . . . . . . . . . . . . . . . 33
2.1 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Representations of Matrix Functions . . . . . . . . . . . . . . . . . . . 35
2.3 Norm Estimates for Resolvents. . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Spectral Variations of Matrices. . . . . . . . . . . . . . . . . . . . . . . 39
2.5 Norm Estimates for Matrix Functions . . . . . . . . . . . . . . . . . . 41
2.5.1 Estimates via the Resolvent . . . . . . . . . . . . . . . . . . . 41
2.5.2 Functions Regular on the Convex Hull
of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.3 Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . 44
2.6 Absolute Values of Elements of Matrix Functions. . . . . . . . . . 46
2.6.1 Statement of the Result . . . . . . . . . . . . . . . . . . . . . . 46
2.6.2 Proof of Theorem 2.6 . . . . . . . . . . . . . . . . . . . . . . . 48
2.7 Diagonalizable Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.8 Perturbations of Matrix Exponentials . . . . . . . . . . . . . . . . . . 52
2.9 Functions of Matrices with Nonnegative Off-Diagonals. . . . . . 56
2.10 Perturbations of Determinants. . . . . . . . . . . . . . . . . . . . . . . . 59
2.11 Bounds for the Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 63
ix
x Contents
2.11.1 Gerschgorin’s Circle Theorem . . . . . . . . . . . . . . . . . 63
2.11.2 Cassini Ovals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.11.3 The Perron Theorems . . . . . . . . . . . . . . . . . . . . . . . 65
2.11.4 Bounds for the Eigenvalues of Matrices “Close”
to Triangular Ones . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.12 Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Difference Equations with Continuous Time. . . . . . . . . . . . . . . . . 71
3.1 Autonomous Difference-Delay Equations. . . . . . . . . . . . . . . . 71
3.2 Application of the Laplace Transform . . . . . . . . . . . . . . . . . . 76
3.3 Lp-Norms of Solutions to Autonomous Equations. . . . . . . . . . 77
3.3.1 Equations with One Delay in Lp; p(cid:3)1 . . . . . . . . . . . 77
3.3.2 Autonomous Difference-Delay Equations
with Several Delays . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 L2-Norms of Solutions to Autonomous Equations. . . . . . . . . . 82
3.5 Solution Estimates Via Determinants. . . . . . . . . . . . . . . . . . . 84
3.6 Time-Variant Equations: The General Case . . . . . . . . . . . . . . 87
3.7 Time-Variant Difference Equations with One Delay . . . . . . . . 90
3.8 Time-Variant Equations with Several Delays . . . . . . . . . . . . . 92
3.9 Difference Equations with Commensurable Delays . . . . . . . . . 94
3.10 Perturbations of Characteristic Values . . . . . . . . . . . . . . . . . . 97
3.11 Perturbations of Characteristic Determinants. . . . . . . . . . . . . . 101
3.12 Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4 Linear Differential Delay Equations. . . . . . . . . . . . . . . . . . . . . . . 105
4.1 Homogeneous Autonomous Equations. . . . . . . . . . . . . . . . . . 105
4.2 Non-homogeneous Autonomous Equations. . . . . . . . . . . . . . . 107
4.3 Estimates for Characteristic Matrices. . . . . . . . . . . . . . . . . . . 110
4.4 The Cauchy Operator of an Autonomous Equation . . . . . . . . . 113
4.4.1 L2-norm Estimates for the Cauchy Operator. . . . . . . . 113
4.4.2 Integrals of Characteristic Functions . . . . . . . . . . . . . 114
4.4.3 Integrals of Fundamental Solutions. . . . . . . . . . . . . . 118
4.4.4 An Estimate for the C-norm of the Fundamental
Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4.5 C- and Lp-norms of the Cauchy Operator. . . . . . . . . . 121
4.5 Systems with Several Distributed Delays. . . . . . . . . . . . . . . . 122
4.6 Scalar Autonomous Differential Delay Equations . . . . . . . . . . 123
4.6.1 The General Case. . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.6.2 Equations with Positive Fundamental Solutions . . . . . 127
4.6.3 Additional Lower Bounds for Quasipolynomials. . . . . 132
4.7 Autonomous Systems with One Distributed Delay . . . . . . . . . 134
4.7.1 The General Case. . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.7.2 Application of Lemma 4.9. . . . . . . . . . . . . . . . . . . . 135