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QUALITATIVE METHODSIN
NONLINEAR DYNAMICS
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PURE AND APPLIED
MATHEMATICS
A Programof Monographs,Textbooks, and Lecture Notes
EXECUTIVE EDITORS
EarlJ. Taft
Rutgers Univers#y
NewBrunswick, NewJersey
Zuhair Nashed
University of Delaware
Newark, Delaware
EDITORIAL BOARD
M. S. Baouendi
Universityof California,
San Diego
Jane Cronin
RutgersUniversity
Jack K. Hale
GeorgiaInstitute of Technology
Anil Nerode
Cornell University
Donald Passman
Universityof Wisconsin,
Madison
Fred S. Roberts
RutgersUniversity
S. Kobayashi
UniversityofCalifornia,
Berkeley
DavidL. Russell
VirginiaPolytechnicInstitute
andState University
Marvin Marcus
Universityof California,
Santa Barbara
Walter Schempp
Universitiit Siegen
W. S. Massey
Yale University
Mark Teply
Universityof Wisconsin,
Milwaukee
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MONOGRAPHS AND TEXTBOOKS IN
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19. N.R. Wallach, HarmonicAnalysis on Homogeneous
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22. B.-Y. Chen,Geometry
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33. K.R. Goodearl,RingTheory(1976)
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35. N.J. Pullman,MatrixTheoryandIts Applications(1976)
36. B.R. McDonald,
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39. C. O. Chdstenson
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41. R. L. Long,AlgebraicNumber
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42. W.F.Pfeffer, Integrals andMeasures
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43. R.L. Wheeden
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44. J.H. Curtiss, Introductionto Functions
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45. K. Hrbacek
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46. W.S. Massey,Homology
47. M. Marcus,Introductionto Modem
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48. E. C. Young,VectorandTensorAnalysis(1978)
49. S.B. Nad/er,Jr., Hyperspaces
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50. S.K. Segal,Topicsin GroupKings(1978)
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51. A. C. M.van Rooij, Non-Archimedean
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53. C. Sadosky,
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67. J. K. Beem
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71. T. W.Wieting, TheMathematical
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73. R.L. Faber,Foundations
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163.A. Charlieretal., Tensors
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166. S. Guerre-Delabddre,
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168. S.H. KulkamiandB.
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169. W.C.Brown,MatdcesOverCommutative
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172. E. C. Young,VectorandTensorAnalysis:Second
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174. M. Pavel, Fundamentals
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175.S. A. Albeverioet al., Noncommutative
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176. W. Fulks, Complex
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182. X. Mao,Exponential
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183. B.S. Thomson,
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184. J. E. Rubio,OptimizationandNonstandard
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186. A. N. MichelandK. Wang,Qualitative Theoryof Dynamical
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187. M.R.Dame/,Theoryof Lattice-OrderedGroups(1995)
188. Z. Naniewiczand P. D. Panagiotopoulos,MathematicalTheoryof Hemivadational
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193. P.A. Gdllet, Semigroups:
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195. V.F. Krotov, GlobalMethods
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196. K.I. Beidaretal.,Ringswith Generalized
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198. G. Sierksma,Linear andInteger Programming
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199. R. Lasser,Introductionto FouderSedes
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200. V. Sima,Algorithmsfor Linear-Quadratic
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201. D. Redmond,
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202. J. K. Beem
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203. M. Fontanaet al., Pr0fer Domains
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204. H. Tanabe,
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205. C. Q. Zhang,Integer FlowsandCycleCoversof Graphs(1997)
206. E. SpiegelandC. J. O’Donnell,IncidenceAlgebras(1997)
207. B. JakubczykandW.Respondek,
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Additional Volumes
in Preparation
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QUALITATIVE METHODSIN
NONLINEAR DYNAMICS
Novel Approachesto
Liapunov’sMatrix Functions
A. A. Martynyuk
Institute of Mechanics
National Academyof Sciences of Ukraine
Kiev, Ukraine
MARCEL
MARCEL DEKKER,
DEKKER
INC.
NEWYO~K- BASEL
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ISBN:0-8247-0735-4
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PREFACE
An important place among modern qualitative
methods in nonlinear
dynamics of systems is occupied by those associated with the development
of Poincar~’s and Liapunov’s ideas for investigating nonlinear systems of
differential equations.
Liapunov divides into two categories all methodsfor the solution of the
problem of stability of motion. He includes in the first category those methods that reduce the consideration of the disturbed motion to the determination of the general or particular solution of the equation of perturbed
motion. It is usually necessary to search for these solutions in a variety
of forms, of which the simplest are those that reduce to the usual method
of successive approximations. Liapunovcalls the totality of all methods of
this first category the "first method".
It is possible, however,to indicate other methodsof solution of the problem of stability which do not necessitate the calculation of a particular or
the general solution of the equations of perturbed motion, but which reduce to the search for certain functions possessing special properties. Liapunov calls the totality of all methodsof this second category the "second
method".
During the post-Liapunov period both the first and second Liapunov’s
methods have been developed considerably. The second method, or the
direct Liapunov method, based first on scalar auxiliary function, w~s replenished with new ideas and new classes of auxiliary functions. This allowed one to apply this fruitful technique in the solution of manyapplied
problems. The ideas of the direct Liapunov method are the source of new
modern techniques of qualitative analysis in nonlinear systems dynamics.
A considerable number of publications appearing annually in this direction
provide a modern tool for qualitative analysis of processes and phenomena
in the real world.
The aim of this monographis to introduce the reader to a new direction
in nonlinear dynamicsof systems. This direction is closely connected with a
iii
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iv
PREFACE
new class of matrix-valued function of particular importance in construction
of an appropriate Liapunov function for the system under consideration.
It is knownthat the problem of stability is important not only for the
continuous systems modeled by ordinary differeatial equations. Therefore,
in this monographthe methods of qualitative analysis are presented for
discrete-time and impulsive systems. Further, in view of the importance
of the problem of estimating the domains of asymptotic stability,
a new
methodfor its solution is set out in a separate chapter.
The monographcontains five chapter and is arranged as follows.
The first chapter contains all necessary results associated with the method of matrix-valued Liapunov functions. It also provides general information on scalar and vector functions including the cone-valued ones. General
theorems on various types of stability of the equilibrium state of the systems cited in this chapter are basic for establishing the sufficient stability
tests in subsequent chapters.
The second chapter deals with the construction of matrix-valued functions and corresponding scalar auxiliary Liapunov functions. Here new
methods of the initial system decomposition are discussed, including those
of hierarchical decomposition. The corresponding sufficient tests for various types of stability and illustrative examples are presented for every
case under consideration. Along with the classical notion of stability major attention is paid to new types of motion stability, in particular, to the
exponential polystability of separable motions as well as the integral and
Lipschitz stability.
The third chapter addresses the methods of stability analysis of discrete-time systems. Our attention is focussed mostly on the development
of the methodof matrix-valued functions in stability theory of discrete-time
systems.
In the fourth chapter the problems of dynamics of nonlinear systems
in the presence of impulsive perturbations are discussed. The method of
matrix-valued Liapunov functions is adapted here for the class of impulsive
systems that were studied before via the scalar Liapunov function. The
proposed development of the direct Liapunov method for the given class of
systems enables us to makean algorithm constructing the appropriate Liaptmov
functions and to increase efficiency of this method.
In the final chapter the problem of estimating the domains of
asymptotic stability is discussed in terms of the method of matrix-valued
Liapunov functions. By means of numerous examples considered earlier by
Abdullin, Anapolskii, et al. [1], Michel, Sarabudla, et al. [1], and ~iljak [1] it
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PREFACE
v
is shownthat the application of matrix-valued functions involves an essential
extension of the domains of asymptotic stability constructed previously.
I wish to acknowledgethe essential technical assistance provided by my
colleagues in the Stability of Processes Department of S.P.Timoshenko Institute of Mechanics, National Academyof Sciences of Ukraine.
The bibliographical
information used in the monograph was checked
by CD-ROMCompact MATH,which was kindly provided by Professor,
Dr. Bernd Wegner and Mrs. Barbara Strazzabosco from the Zentralblatt
MATH.
I express my sincere gratitude to all persons mentioned above. I am
also grateful to the staff of MarcelDekker,Inc., for their initiative and kind
assistance.
A. A. Martynyuk
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CONTENTS
Preface
111
1 Preliminaries
1
1.1 Introduction
1
1.2 Nonlinear Continuous Systems
1.2.1 General equations of nonlinear dynamics
1.2.2 Perturbed motion equations
1.3 Definitions of Stability
1.4 Scalar, Vector and Matrix-Valued Liapunov Functions
1.4.1 Auxiliary scalar functions
1.4.2 Comparison functions
1.4.3 Vector Liapunov functions
1.4.4 Matrix-valued metafunction
14
1.5 Comparison Principle
1.6 Liapunov-Like Theorems
1.6.1 Matrix-valued function and its properties
1.6.2 A version of the original theorems of Liapunov
23
23
1.7 Advantages of Cone-Valued Liapunov Functions
1.7.1 Stability with respect to two measures
1.7.2 Stability analysis of large scale systems
~4
1.8 Liapunov’s Theorems for Large Scale Systems in General
1.8.1 Whyare matrix-valued Liapunov functions needed?
1.8.2 Stability and instability of large scale systems
41
41
42
1.9 Notes
47
2 Qualitative
Analysis
of Continuous
2.1 Introduction
vii
Systems
49
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viii
CONTENTS
2.2 Nonlinear Systems with Mixed Hierarchy of Subsystems
2.2.1 Mixed hierarchical structures
2.2.2 Hierarchical matrix function structure
2.2.3 Structure of hierarchical matrix function derivative
2.2.4 Stability and instability conditions
2.2.5 Linear autonomous system,
2.2.6 Examples of third order systems
50
50
52
56
59
60
63
2.3 Dynamics of the Systems with Regular Hierarchy Subsystems
2.3.1 Ikeda-~iljak hierarchical decomposition
2.3.2 Hierarchical Liapunov’s matrix-valued functions
2.3.3 Stability and instability conditions
2.3.4 Linear nonautonomous systems
68
68
69
74
79
2.4 Stability Analysis of Large Scale Systems
2.4.1 A class of large scale systems
2.4.2 Construction of nondiagonal elements of
matrix-valued function
2.4.3 Test for stability analysis
2.4.4 Linear large scale system
2.4.5 Discussion and numerical example
90
90
91
94
94
97
2.5 Overlapping Decomposition and Matrix-Valued Function
Construction
2.5.1 Dynamical system extension
2.5.2 Liapunov matrix-valued function construction
2.5.3 Test for stability of system (2.5.1)
2.5.4 Numerical example
100
100
105
105
106
2.6 Exponential Polystability Analysis of Separable Motions
2.6.1 Statement of the Problem
2.6.2 A method for the solution of the problem
2.6.3 Autonomous system
2.6.4 Polystability by the first order approximations
108
108
110
118
122
2.7 Integral and Lipschitz Stability
Definitions
Sufficient conditions for integral and asymptotic
integral stability
2.7.3 Uniform Lipschitz stability
127
127
2.8 Notes
135
128
133
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CONTENTS
3 Qualitative
4
Analysis
ix
of Discrete-Time
Systems
139
3.1 Introduction
3.2 Systems Described by Difference Equations
3.3 Matrix-Valued Liapunov Functions Method
3.3.1 Auxiliary results
3.3.2 Comparison principle application
3.3.3 General theorems on stability
3.4 Large Scale System Decomposition
139
140
143
3.5 Stability and Instability of Large Scale Systems
3.5.1 Auxiliary estimates
3.5.2 Stability and instability conditions
151
151
157
3.6 Autonomous Large Scale Systems
3.7 Hierarchical Analysis of Stability
3.7.1 Hierarchical decomposition and stability
3.7.2 Novel tests for connective stability
3.8 Controlled Systems
159
166
166
172
179
3.9 Notes
181
Nonlinear
Dynamics of Impulsive
143
144
147
149
conditions
Systems
4.1 Introduction
183
183
4.2 Large Scale Impulsive Systems in General
4.2.1 Notations and definitions
4.2.2 Auxiliary results
4.2.3 Sufficient stability conditions
4.2.4 Instability conditions
Hierarchical
Impulsive Systems
4.3
184
184
186
195
197
4.4 Analytical Construction of Liapunov Function
4.4.1 Structure of hierarchical matrix-valued Liapunov
function
4.4.2 Structure of the total derivative of hierarchical
matrix-valued function
204
4.5 Uniqueness and Continuability
215
222
of Solutions
4.6 On Boundedness of the Solutions
4.7 Novel Methodologyfor Stability
4.7.1 Stability conditions
4.8 Notes
201
204
207
228
228
238
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x
CONTENTS
5 Applications
5.1 Introduction
5.2 Estimations of Asymptotic Stability Domainsin General
5.2.1 A fundamental Zubov’s result
5.2.2 Someestimates for quadratic
matrix-valued functions
5.2.3 Algorithm of constructing a point network covering
boundary of domain E
5.2.4 Numerical realization and discussion of the algorithm
5.2.5 Illustrative examples
5.3 Construction of Estimate for the Domain E of Power
System
5.4 Oscillations and Stability of SomeMechanical Systems
5.4.1 Three-mass systems
5.4.2 Nonautonomousoscillator
5.5 Absolute Stability of Discrete Systems
5.6 Notes
239
239
239
239
241
245
250
254
263
267
267
269
270
274
References
Subject
Index
295
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1
PRELIMINARIES
1.1 Introduction
This chapter contains an extensive overview of the qualitative methods in
nonlinear dynamics and is arranged as follows.
Section 1.2 is short and gives information about continuous nonlinear systems that is important for applications in investigation of the mechanical,
electrical and electromechanical systems. Also discussed are the equations
of perturbed motion of nonlinear systems which are the object of investigation in this monograph.
For the reader’s convenience, in Section 1.3 the definitions we use of
motion stability of various types are formulated. These formulations result from an adequate description of stability properties of nonlinear and
nonautonomous systems.
Section 1.4 deals with three classes of Liapunovfunctions: scalar, vector and matrix-valued ones, as well as the possibilities of their application
in motion stability theory. Along with the well-known results, some new
notions are introduced, for example, the notion of the "Liapunov metafunction".
Basic theorems of the comparison principle for SL-class and VL-class of
the Liapunov functions are set out in Section 1.5. Also, some important
corollaries of the comparison principle related to the results of Zubovare
presented here.
Section 1.6 deals with generalization of the main Liapunov and Barbashin-Krasovskii theorems established by the author in terms of matrixvalued functions. Somecorollaries of general theorems contain new sufficient stability (instability) tests for the equilibrium state of the system
under consideration.
In Section 1.7 the vector and cone-valued functions are applied in the
problem of stability with respect to two measures and in stability theory of
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2
1. PRELIMINARIES
large scale systems. Detailed discussion of possibilities of these approaches
may prove to be useful for manybeginners in the field.
In the final Section 1.8, the formulations of theorems of the direct Liapunov method are set out based on matrix-valued functions and intended
for application in stability investigation of large scale systems.
Generally, the results of this chapter are necessary to get a clear idea
of the results presented in Chapters 2-5. Throughout Chapters 2-5 references to one or an other section of Chapter 1 are made.
1.2 Nonlinear
1.2.1
General equations
Continuous
of nonlinear
Systems
dynamics
The systems without nonintegrability differential constraints represent a
wide class of mechanical systems with a finite numberof degrees of freedom.
Let the state of such system in the phase space Rn, n = 2k, be determined
by the vectors
q = (ql,...,
qk) w and
~ = (~1,...,
w.
~k)
It is knownthat the general motion equations of such a mechanical system
are
(1.2.1)
d (or’~
OT _ Us, s = 1,2,...,k.
Here T is the kinetic energy of the mechanical system and Us are the
generalized forces.
The system of equations (1.2.1) is simplified, if for the forces affecting
the system a force function U = U(t,q~,... ,qk) exists such that
OU
us= , s=l,2,...,k.
The simplified system obtained so far,
d(O(T+V)~
dt \ 00,
]
O(T+U)_o,
Oqs
s
= 1,2,...,k,
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1.2 NONLINEARCONTINUOUS
SYSTEMS
3
can be presented in the canonical form
dqs
dt
OR
Op~
dps
dt
OR
Oqs
s = 1,2,...,k,
OT and R = T2-To-U. Here To is the totality
of the
where p~ = 04--7
velocity-independent terms in the expression of the kinetic energy, and T2
is the totality of the second order terms with respect to velocities.
The qualitative analysis of equations (1.2.1) and its particular cases
the principle point of the investigations in nonlinear dynamicsof continuous
systems.
1.2.2
Perturbed
motion equations
Under certain assumptions the equations (1.2.1) can be represented in the
scalar form
dy_~i = Y~(t, Yl,...
,Y2k), i = 1,
2k,
or in the equivalent vector form
dy = Y(t, y),
dt
(1.2.2)
where* y = (Yl,Y2,... ,Y2k)T E 2k and Y= (Y1,Y~,... ,Y2k) T, Y:7-×
Rek -+ R~k. A motionof (1.2.2) is denoted by y(t; to, Y0), ~(to; to, Y0)
and the reference motion r/r(t; to, Yro). Fromthe physical point of view the
reference motion should be realizable by the system. From the mathematical point of view this means that the reference motion is a solution
of (1.2.2),
(1.2.3)
&?r(t; t0, Yro)_=Y[t, ~/~(t; to, Y~0)].
dt
Let the Liapunov transformation of coordinates be used,
x = y - Yr,
(1.2.4)
where yr(t) -- ~lr(t;to,Yro).
(1.2.5)
Let f: T x R~k -~ R2k be defined by
f(t, x) = Y[t, y~(t) + x] - Y[t, Yr].
! !
~T
*In Liapunov’s
notationy ---- (ql,q2,... ,qk, ql,q2,’’’
,qkJ
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4
1. PRELIMINARIES
It is evident that
(1.2.6)
f(t,O) =_
Now(1.2.2)- (1.2.5) yield
dx
d~- = f(t, x).
(1.2.7)
In this way, the behavior of perturbed motions related to the reference
motion (in total coordinates) is represented by the behavior of the state
deviation x with respect to the zero state deviation. The reference motion
in the total coordinates Yi is represented by the zero deviation x = 0 in
state deviation coordinates xi. With this in mind, the following result emphasizes complete generality of both Liapunov’s second method and results
represented by Liapunov [1] for the system (1.2.7). Let Q: R2k ’~,
-r R
n = 2k is admissible but not required.
In the monographGrujid, et al. [1] the following assertion is proved.
Proposition 1.2.1. Stability of x = 0 of systena (1.2.7) with respect
to Q = x is necessary and sufficient for stability of the reference motion
of system (1.2.2) with respect to every vector function Q that is continuous
in y.
This theorem reduced the problem of the stability of the reference motion
of (1.2.2) with respect to Q to the stability problem of x = 0 of (1.2.7)
with respect to x.
For the sake of clarity we state
Definition 1.2.1. State x* of the system (1.2.7) is its equilibrium state
over 7~ iff
(1.2.8)
x(t;to,x*)
= x*, for all
t E To, and to
The expression "over 7~" is omitted iff 7~ = R.
Proposition 1.2.2. For x* ~ Rn to be an equilibrium state
system (1.2. 7) over Ti it is necessary and sufficient that both
of the
(i) for every to q T/ there is the unique solution x(t; to, x*) of (1.2.7),
which is defined for all to ~ To
and
(ii) f(t,x*) = 0, for a/l t e To, and to e 7~.
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1.3 DEFINITIONSOF STABILITY
5
The conditions for existence and uniqueness of the solutions of system
(1.2.7) can be found in manywell-knownbooks by Dieudonne[1], Hale [1],
Hirsch and Smale [1], Simmons[1], Yoshizawa[1], etc.
The next result provides a set of sufficient conditions for the uniqueness
of solutions for initial value problem
(1.2.9)
d-¥ = f(t,
x), X(to) =
Proposition 1.2.3. Let :D C Rn+l be an open and connected set.
Assume f ¯ C(:D, Rn) and for every compact K C ~), f satisfies
the
Lipschitz condition
[]f(t,x) f( t,y)[[ <_L[[x - y[[
for all (t,x), (t,y) 6 K, where L is a constant depending only on
Then (1.2.9) has at most one solution on any interval [to, to + c), c > 0.
Definition 1.2.2. A solution x(t;to,Xo) of (1.2.7) defined on the interval (a, b) is said to be boundedif there exists /~ > 0 such that [[x(t; to, x0)[]
< fl for all t ¯ (a, b), where/~ maydepend on each solution.
For the system (1.2.7) the following result can be easily demonstrated.
Proposition 1.2.4. Assume f ¯ C(J x Rn,Rn), where J = (a,b)
a finite or infinite interval. Let every solution of (1.2.7) is bounded. Then
every solution of (1.2.7) can be continued on the entire interval (a, b).
1.3 Definitions
of Stability
Consider the differential system (1.2.7), where f ¯ C(%n, Rn). Suppose that the function f is smooth enough to guarantee existence, uniqueness and continuous dependenceof solutions x(t; to, x0) of (1.2.7). We
present various definitions of stability (see Grujid [1] and Grujid, et al. [1]).
Definition 1.3.1. The state x = 0 of the system (1.2.7) is:
(i) stable with respect to 7~ iff for every to ¯ T~ and every e > 0 there
exists 5(to,e) > 0, such that [[Xo[[ < 5(to,e) implies
all t ¯ %;
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1. PRELIMINARIES
(ii) uniformly stable with respect to To iff both (i) holds and for every
¢ > 0 the corresponding maximal ~M obeying (i) satisfies
inf[t~M(t,~): t ¯ T/] >
(iii)
stable in the whole with respect to Ti iff both (i) holds and
5M(t,e)--~+oo
as e-~+oo,
for
all
t¯T/;
(iv) uniformly stable in the wholewith respect to T, iff both (ii) and (iii)
hold;
(v) unstable with respect to 7~ iff there are to ¯ T/, e ¯ (0, +oo) and
T ¯ To, V > tO, such that for every 5 ¯ (0,+oo) there is Xo,
Ilxoll< 5, forwhich
IIx(T;to, xo)ll> ~.
The expression "with respect to 7~" is omitted from (i)- (v) iff 7~
These stability properties hold as t -~ +oo but not for t = +oo.
Further the definitions on solution attraction are cited. The examples
by Hahn [2], Krasovskii [1], and Vinograd [1] showed that the attraction
property does not ensure stability.
Definition 1.3.2. The state x = 0 of the system (1.2.7) is:
(i) attractive with respect to Ti ifffor every to ¯ 7~ there exists A(to)
0 and for every ~ > 0 there exists ~’(to;zo,~) ¯ [0,+oo) such
that Ilzoll < A(to) implies IIx(t;to,Xo)ll
< ¢, for all t ¯ (to
r(to; xo, ¢), +oo);
(ii) Xo-uniformly attractive with respect to 7~ iff both (i) is true and for
every to ¯ T/ there exists A(to) > 0 and for every ~ ¯ (0, +oc)
there exists r~,[to, A(to), ~] ¯ [0, +oo) such that
sup [T,~(t0; X0,¢): X0¯ T/] = T=(7~,X0,
(iii) to-uniformly attractive with respect to 7~ iff (i) is true, there is A >
0 and for every (x0, ~) Ba ì (0, +oÂ) there exists ru(Ti, Xo, ~)
[0, +o¢) such that
sup [rm(to); xo, (): to ¯ Ti] = ~’u(7~,x0,
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1.3 DEFINITIONSOF STABILITY
7
(iv) uniformlyattractive with respect to Ti iff both (ii) and (iii) hold,
is, that (i) is true, there exists A > 0 and for every ~ E (0, +c~)
there is T~(T/, A, ¢) E [0, +~) such that
sup [~-m(to; x0, ~): (to, x0) ~ T/x Ba] = r(T/, A,
(v) The properties (i)- (iv) hold "in the whole" iff (i) true for every
A(t0) ~ (0, +oo) and every to ~
The expression "with respect to Ti" is omitted iff T/= R.
Definitions 1.3.1 and 1.3.2 enable us to define various types of asymptotic
stability as follows.
Definition 1.3.3. The state x = 0 of the system (1.2.7) is:
(i) asymptotically stable with respect to Ti iff it is both stable with
respect to T/and attractive with respect to 7~;
(ii) equi-asymptoticallystable with respect to Ti iff it is both stable with
respect to
(iii) quasi-uniformlyasymptotically stable with respect to Ti iff it is both
uniformly stable with respect to 7~ and t0-uniformly attractive with
respect to 7~;
(iv) uniformly asymptotically stable with respect to 7~ iff it is both uniformly stable with respect to 7~ and uniformly attractive with respect to
(v) the properties (i)- (iv) "in t he whole" iff b oth the c orresponding stability of x = 0 and the corresponding attraction of x = 0
hold in the whole;
(vi) exponentially stable with respect to Ti iff there are A > 0 and real
numbers c~ _> 1 and fl > 0 such that HXoll < A implies
IlX(t;to,xo)]] <_~llXoll exp[-fl(t- to)],
for all
teTo, and for all
to
This holds "in the whole" iff it is true for A = +oo.
The expression "with respect to 7~" is omitted iff 7~ = R.
