Methods of A. M. Lyapunov and their Application

✍ Scribed by V. I. Zubov

• Publisher: P. NOORDHOFF LTD
• Year: 1964
• Tongue: English
• Leaves: 281
• Category: Library

✦ Synopsis

The purpose of the present edition is to acquaint the reader with
new results obtained in the theory of stability of motion, and also
to summarize certain researches by the author in this field of
mathematics. It is known that the problem of stability reduces not
only to an investigation of systems of ordinary differential equations
but also to an investigation of systems of partial differential
equations. The theory is therefore developed in this book in such
a manner as to make it applicable to the solution of stability problems
in the case of systems of ordinary differential equations as
well as in the case of systems of partial differential equations.
For the reader's benefit, we shall now list briefly the contents of
the present monograph.
This book consists of five chapters.
In Sections 1-5 of Chapter I we give the principal information
connected with the concept of metric space, and also explain the
meaning of the terms which will be used below. Sections 6 and 7 are
preparatory and contain examples of dynamical systems in various
spaces. In Section 8 we define the concept of dynamical systems
in metric space, and also give the principal theorems from the
book [5] of Nemytsky and Stepanov. In Sections 9-10 we give
the principal definitions, connected with the concept of stability
in the sense of Lyapunov of invariant sets of a dynamical system,
and also investigate the properties of certain stable invariant sets.
In Section 11 we solve the problem of a qualitative construction
of a neighborhood of a stable (asymptotically stable) invariant set. In
particular, it is established that for stability in the sense of Lyapunov
of an invariant set M of a dynamical system f(p, t) it is necessary,
and in the case of the presence of a sufficiently small compact neighborhood of the set M it is also sufficient, that there exist no
motions· f(p, t), P eM, having ex-limit points in M. The results
obtained here are new even to the theory of ordinary differential
equations. In Sections 12-13 we give criteria for stability and
instability of invariant sets with the aid of certain functionals.
These functionals are the analogue of the Lyapunov function and
therefore the method developed here can be considered as a certain
extension of Lyapunov's second method. All the results of these
sections are local in character. We cite, for example, one of these.
In order for an invariant set M to be uniformly asymptotically
stable, it is necessary and sufficient that in a certain neighborhood
S(M, r) of M there exists a functional V having the following
properties:
1. Given a number c1 > 0, it is possible to find c2 > 0 such
that V(P) > c2 for p(p, M) > c1.
2. V(p) ~ 0 as p(p, M) ~ 0.
3. The function V(f(p, t)) does not increase for f(p, t) e S(M, r)
and V(f(p, t)) ~ 0 as t ~ + oo uniformly relative to p e S(M, <h),
where d1 is a certain number such that 0 < d1 < r. In Section 14
we describe one of the central theorems that has a non-local
character. It is shown there that in order for an invariant open
set A, containing a sufficiently small neighborhood of the set M, to be
the region of asymptotic stability of the uniformly asymptotically
stable and uniformly attracting set M, it is necessary and sufficient
that there exist two functionals V(p) and fP(p) such that:
l. V(p) is defined and continuous in A, fP(p) is defined and
continuous in R, fP(p) = 0, p eM, and - 1 < V(p) < 0 for
p e A; fP(p) > 0 for p(p, M) =I= 0.
2. For /'2 > 0 it is possible to find /'1 and cx1 such that
V(p) < -y1, fP(p) > cx1 for p(p, M) > /'2·
3. V and (/) ~ 0 as p(p, M) ~ 0.
4. dVfdt = fP(1 + V).
5. V(p) ~ -1 as p(p, q) ~ 0, peA, q E A"-.A, and q eM.
Here, as above, p and q are elements of tl;te space R, and p(p, M)
is the metric distance from the point p to the set M. Section 15 contains a method that makes it possible to estimate the distance
from the motion to the investigated invariant set. The theorems
obtained in this section can be considered as supplements to
Sections 12-14. Sections 1-15 cover the contents of the first chapter,
devoted to an investigation of invariant sets of dynamical systems.
In the second chapter we give a developed application of the
ideas and methods of the first chapter to the theory of ordinary
differential equations. In Section 1 of Chapter 2 we develop the
theorem of Section 14 for stationary systems of differential equations,
and it is shown thereby that the Lyapunov function V can
be selected differentiable to the same order as the right members
of the system. In the same section we give a representation of
this function as a curvilinear integral and solve the problem of
the analytic structure of the right members of the system, which
right members have a region of asymptotic stability that is prescribed
beforehand. In Section 2 of Chapter II we consider the
case of holomorphic right members. The function V, the existence
of which is established in Section 1 of this chapter, is represented
in this case in the form of convergent series, the analytic continuation
of which makes it possible to obtain the function in the entire
region of asymptotic stability. The method of construction of such
series can be used for an approximate solution of certain non-local
problems together with the construction of bounded solutions in
the form of series, that converge either for t > 0 or for t e (- oo,
+ oo). These series are obtained from the fact that any bounded
solution is described by functions that are analytic with respect
to t in a certain strip or half strip, containing the real half-axis.
In Section 3 of Chapter II we develop the theory of equations with
homogeneous right members. It is shown in particular that in
order for the zero solution of the system to be asymptotically
stable, it is necessary and sufficient that there exist two homogeneous
functions: one positive definite W of order m, and one
negative definite V of order (m + 1 - #). such that dVfdt = W,
where # is the index of homogeneity of the right members of the
system. If the right members of the system are differentiable, then
these functions satisfy a system of partial differential equations,
the solution of which can be found in closed form. This circumstance
makes it possible to give a necessary and sufficient condition for asymptotic stability in the case when the right members
are forms of degree p., directly on the coeffilients of these forms.
In Sections 4 and 5 of Chapter II we consider several doubtful
cases: k zero roots and 2k pure imaginary roots. We obtain here
many results on the stability, and also on the existence of integrals
of the system and of the family of bounded solutions. In Section 6
of Chapter II the theory developed in Chapter I is applied to the
theory of non-stationary systems of equations. In it are formulated
theorems that follow from the results of Section 14, and a method
is also proposed for the investigation of periodic solutions.
In Section 1 of Chapter III we solve the problem of the analytic
representation of solutions of partial differential equations in the
case when the conditions of the theorem of S. Kovalevskaya are
not satisfied. The theorems obtained here are applied in Section 2
of Chapter III to systems of ordinary differential equations. This
supplements the investigations of Briot and Bouquet, H. Poincare,
Picard, Horn, and others, and makes it possible to develop in
Section 3 of Chapter III a method of constructing series, describing
a family of 0-curves for a system of equations, the expansions of
the right members of which do not contain terms which are linear
in the functions sought. The method of construction of such series
has made it possible to give another approach to the solution of the
problem of stability in the case of systems considered in Sections 3-5
of Chapter II and to formulate theorems of stability, based on the
properties of solutions of certain systems of nonlinear algebraic
equations. Thus, the third chapter represents an attempt at
solving the problem of stability with the aid of Lyapunov's first
method.
In Chapter IV we again consider metric spaces and families of
transformations in them. In Section I of Chapter IV we introduce
the concept of a general system in metric space.
A general system is a two-parameter family of operators from
R into R, having properties similar to those found in solutions of
the Cauchy problem and the mixed problem for partial differential
equations. Thus, the general systems are an abstract model of
these problems. We also develop here the concept of stability of
invariant sets of general systems. In Section 2 of Chapter IV,
Lyapunov's second method is extended to include the solution of problems of stability of invariant sets of general systems. The
theorems obtained here yield necessary and sufficient conditions.
They are based on the method of investigating two-parameter
families of operators with the aid of one-parameter families of
functionals. We also propose here a general method for estimating
the distance from the motion to the invariant set. In Section 3 of
Chapter IV are given several applications of the developed theory
to the Cauchy problem for systems of ordinary differential equations.
Results are obtained here that are not found in the known literature.
The fifth chapter is devoted to certain applications of the developed
theory to the investigation of the problem of stability of the
zero solution of systems of partial differential equations in the case
of the Cauchy problem or the mixed problem. In Section I of
Chapter V are developed general theorems, which contain a method of
solving the stability problem and which are orientative in character.
In Sections 2-3 of Chapter V are given specific systems of partial
differential equations, for which criteria for asymptotic stability are
found. In Section 3 the investigation of the stability of a solution
of the Cauchy problem for linear systems of equations is carried
out with the aid of a one-parameter family of quadratic functionals,
defined in W~N>. Stability criteria normalized to W~NJ are
obtained here. However, the imbedding theorems make it possible
to isolate those cases when the stability will be normalized in C.
In the same section are given several examples of investigation
of stability in the case of the mixed problem.
For a successful understanding of the entire material discussed
here, it is necessary to have a knowledge of mathematics equivalent
to the scope of three university courses. However, in some places
more specialized knowledge is also necessary.

✦ Table of Contents

FOREWORD*......Page 7
FOREWORD......Page 8
FROM THE AUTHOR......Page 11
TABLE OF CONTENTS......Page 17
INTRODlJCTION......Page 19
l. Metric space......Page 27
2. Operators and functionals in R......Page 28
3. Neighborhood of a set......Page 29
5. Normed linear spaces......Page 30
6. Dynamical systems inn-dimensional Euclidean space En......Page 32
7. Examples of dynamical systems in functional spaces......Page 35
8. Dynamical systems in metric space......Page 38
9. Formulation of the problem of stability of invariant setsPrincipal definitions......Page 40
10. Uniformly asymptotically stable and uniformly attracting invariantsets of a dynamical system f(p, t)......Page 45
II. Qualitative description, from the point of view of the Lyapunovstability, of a neighborhood of an invariant set......Page 49
12. Necessary and sufficient conditions for stability......Page 59
13. Necessary and sulficient conditions for instability......Page 66
14. Necessary and sufficient conditions for the existence of uniformlyasymptotically stable and uniformly attracting invariant sets......Page 70
IS. Method of estimating the distance from the motion f(p, t) to acertain invariant set M with the aid of functionals......Page 79
I. Stationary systems of differential equations......Page 85
2. Case of analytic right members of the system (2.1)......Page 105
3. Systems of differential equations with homogeneous right members......Page 122
4. Case of k zero roots......Page 148
5. Case of several pairs of pure imaginary roots......Page 162
6. System of non-stationary differential equations......Page 170
1. Auxiliary theorems from the theory of partial differential equations......Page 184
2. Representation of solutions of systems of ordinary dilferentialequatiort,s in a neighborhood of a singular point......Page 191
3. Analytic representation of the 0-curves of a special system ofdilferential equations......Page 197
I. General systems. Basic definitions......Page 214
2. Conditions for stability and instability of an invariant set M of ageneral system in metric space......Page 220
3. Certain applications to non-stationary systems of ordinarydifferential equations......Page 242
I. Some general propositions......Page 253
2. Stability of solutions of quasi-linear systems of special type......Page 258
3. Stability of the trivial solution of a system of partial ditferentialequations......Page 266
BIBLIOGRAPHY......Page 277
INDEX......Page 280