Methods of A. M. Lyapunov and their Application

281 Pages·1964·11.908 MB·English

by  V. I. Zubov

METHODS OF A. M. L Y APUNOV
AND THEIR APPLICATION
©Copyright I964 by P. Noordholf, Ltd., Groningen, The Netherlands.
This book or parts thereof may not be reproduced in any form without writtc
permission of the publishers.
Printed in The Netherlands
METHODSOFA.M.LYAPUNOV
AND THEIR APPLICATION
by
V.I. ZUBOV
Translation prepared under the auspices of the
UNITED STATES ATOMIC ENERGY COMMISSION
Edited for this English edition by
LEO F. BORON
THE PENNSYLVANIA STATE UNIVERSITY
1964
P. NOORDHOFF LTD · GRONINGEN · THE NETHERLANDS
nPE)U1CJIOBHE
Co3.LI.aTeJieM Teop1111 ycTOHlii1BOCTI1 .li.811JKeHI1H H8JIHeTCH
Bhi.LI.810lll.HHCH MaTeMaTI1K, aKa.LI.eMI1K A. M. JIHnyHos (1857-
1918 r.r:)~Ero TPY.ll.hi HBI1JIHCh oTnpa8HOH TOliKOH .LI.JIH W11-
poKoro Kpyra HCCJie,li.088HI1H no np06JieMe ycTOHlii180CTI1.
3TO B nOJIHOH Mepe OTHOCI1TCSI 11 K H8CTOHlll.eH MOHOrpaqml1,
pyCCKOe H3,li.8HI1e KOTOpOH 6b!JIO np11ypolleHO K 100-JieTHIO
co AHH poJK.LI.eHI1H A. M. JIHnyHosa 11 OTKpbiBaJIOCh nopTpe
TOM :nora 3aMellaTeJihHoro ylleHoro.
ro,ll.bl, npowe,li.WI1e CO BpeMeHI1 8biXOii.a 8 C8eT pyCCKOrO
113,li.8HI1SI, 03H8MeH088JII1Cb ,li.8JibHeHWI1M pa3811TI1eM MeTO,li.08
JIHnyHo8a H pacwHpeHI1eM ccpepbi 11x npHMeHeHI1H. CJie.LI.yeT
OTMeTHTb nJIO,li.OT80pHOe COe,li.I1HeHI1e H,ll.eH H3 TeOpH11 YCTOH
li1180CTI1 ,li.811JKeHI1SI 11 113 · TeOp1111 38TOM3TI1lleCKOrO ynpa8-
JieHI1H, KOTOpoe HaMeTI1JIOCb 8 nOCJie.LI.Hee speMH, liTO np11-
8eJIO K napaJIJieJibHOMY 6biCTpOMY pa3811TI110 B 06JiaCTI1
TeOpHH OnTHM8JibHOrO ynpaBJieHI1SI H B 06JiaCTI1 TeOpl111
yCTOHliH80CTH .li.811JKeH11H. 0.LI.H11M 113 np11Mepo8 TaKara
pO.LI.a MOJKeT CJIYJKHTb rJiy6oK8SI no C80HM pe3yJibT8T8M
pa6oTa KaJihMaHa.
5I Ha,ll.eiOCb', liTO H8CT0Hlll.ee H3,li.8HI1e nOCJIYJKI1T .LI.8Jib
HeflweMy pa3811TI110 M8TeM8TI1lleCKHX MeTO,li.OB HCCJie,li.OBa
HHSI HeJIHHeHHbiX CHCTeM.
JleHHHrpa.LI., cpeopaJib 1964 r. B. H. 3 y6 o 8
A. M. LY APUNOV
1857-1918

FOREWORD*
The creator of the theory of the stability of motion was the dis
tinguished mathematician Academician A. M. Lyapunov (1857-
1918}. His works were the point of departure for extensive in
vestigations of the problem of stability. This, in full measure,
pertains also to the present monograph, the Russian edition of
which was timed to coincide with the hundredth anniversary of the
birth of A. M. Lyapunov and opened with a portrait of this re
markable scholar.
The years whkh have passed since the time of the appearance
of the Russian edition are marked with a further development of
the methods of Lyapunov and the extension of the spheres of their
application. We must note the fruitful combination of ideas from
the theory of the stability of mqtjqn and from the theory of auto
matic regulation which has accumulated recently and which has led
to a rapid development in the area of the theory of optimal regu
lation and in the area of the theory of stability of motion. One
example of this sort is the work of Kalman which is characterized
by profound results.
I hope that the present edition will serve to further develop the
mathematical methods of investigating non-linear systems.
Leningrad, V. I. ZUBOV
February 1964
* Written expressly for this English edition.
FOREWORD
This monograph by V.I. Zubov is directly allied in its contents to
the celebrated work by A. M. Lyapunov, "General Problem of
Stability of Motion." Let us start with a brief mention of the
principal results of this work. A. M. Lyapunov reduces the problem
of stability of motion to an investigation of the stability of the
trivial solution x1 = x2 = . . . = x11 = 0 of the system of ordinary
differential equations
dx
dt8 = la(XI. ... , Xn, t), (s = 1, ••• , n), (1)
where the right members are power series in the variables x1, •.. , Xn
without the free term, converging in a neighborhood of
X1 = X2 = o • , = Xn = 0.
If the right members, i.e., the coefficients of the above-mentioned
series, do not contain t, then A.M. Lyapunov speaks of a steady-state
motion (autonomous system). In a general investigation of the
stability of the trivial solution of (1), he employed two methods
developed by him: expansion of the solutions of the system (1)
in series of a special type, and the use of the functions V(x1, ... , x t)
11,
which together with their derivatives with respect to t, calculated
by virtue of (1):
dV oV n oV
W(x1, ..• , Xn, t) = -d =-+~--;-fa,
t ot s=1 uXa
have certain properties, as functions of (x1. •.. , Xn, t). Such
functions, when constructed, give sufficient conditions for stability
or instability. In the construction of functions of this kind, Lyapu
nov used the general theorem that he proved concerning the solution
FOREWORD VII
of systems of partial differential equations that do not satisfy the con
ditionsofthewellknown theorem ofS. V. Kovalevskaya. The functions
V, about which we spoke before, are called Lyapunov functions
and the theorem just mentioned will be called the auxiliary theorem.
In his works Lyapunov has established the conditions under
which the linear terms of Eq. ( 1) alone solve the problem of stability,
and has considered many important particular cases, when this
problem can be solved only by an examination of higher-order
terms (so-called doubtful cases). In one of the doubtful cases of an
autonomous system (1}, a method is given for constructing periodic
solutions. Upon further investigations systems of differential equa
tions were dispensed with, and only the points P were considered
in ann-dimensional space that moved with varying tin accordance
with definite laws - so-called dynamical systems. Such systems
were considered further not in n-dimensional Euclidean space
(x1, ... , Xn) but in an arbitrary metric space.
In the first chapter of this monograph there are considered dyna
mical systems in metric space and the problem of stability (in different
meanings) of invariant sets, i.e. sets consisting of trajectories. In
the application of Lyapunov's second method, the functions V are
replaced by functionals. An investigation is made of the behavior
of the trajectory in a neighborhood of an invariant set in the
presence of stability and under certain conditions the entire region
of asymptotic stability is characterized with the aid of a corre
sponding functional. This chapter is purely theoretical in nature
and contains no constructions of the corresponding functionals.
In the second chapter, the results of the first chapter are applied
to systems of differential equations of type (1). In particular, the
case is considered when the right members are homogeneous
functions of (x1. ... , Xn)· Necessary and sufficient conditions for
asymptotic stability are found for this case.
Next, for the case of analytic right members of Equations (1} and
of autonomous systems, the doubtful case is considered, when the
characteristic equation of first approximation has k zero and k pairs
of identical pure imaginary roots with simple elementary divisors,
and the remaining roots have a negative real part. The condition
for the existence of k-holomorphic integrals is given, which leads
to a sufficiency condition for the stability of the zero solution. Also
VIII FOREWORD
given is a sufficiency condition for asymptotic stability. In the
absence of zero roots, a family of bounded solutions is constructed
(analogous to the family of periodic solutions of A. M. Lyapunov,
which were mentioned above). The author considers also the case
when the right members of Eqs. (I) contain t.
The third chapter is connected essentially with a generalization
of Lyapunov's auxiliary theorem concerning systems of partial
differential equations and the application of his first method. The
result mentioned leads, in a natural manner, to a generalization
of the well known results of Briot and Bouquet and also those of
Poincare concerning the construction of solutions of differential
equations in a neighborhood of singular points. Autonomous
systems (I) are considered, the right members of which are holo
morphic and contain no linear terms. Under certain additional
assumptions, integral curves are constructed, which tend to the
+
origin as t ~ oo. If the origin is a point of asymptotic stability,
all the integral curves are obtained in this manner. In certain cases,
necessary and sufficient conditions for asymptotic stability are given.
In the fourth chapter the concept of a dynamical system in metric
space is generalized to obtain a scheme that serves to investigate
the stability, in various meanings, for problems connected with
partial differential equations. As in the first chapter, an analogue
of the Lyapunov second method is developed. Moreover, estimates
are given for the distance from a moving point to the investigated
invariant set. In applications to systems of ordinary equations this
yields new results.
In the fifth chapter, the results of the preceding chapter are
applied to partial differential equations.
In his foreword to "General Problem of Stability of Motion"
Lyapunov wrote: "My only purpose in this work is to explain what
I succeeded to do towards solving the problem I formulated and what
may serve as a starting point for further research of similar character."
These words were justified to a great extent. Hundreds of papers
directly connected with Lyapunov's research have been published.
In the present monograph by V. I. Zubov these investigations
receive a substantial generalization and development.
Academician V. I. SMIRNOV