Ebook: Introduction to the theory of stability

The first chapter gives an account of the method of Lyapunov functions
originally expounded in a book by A. M. Lyapunov with the title The
general problem of stability of motion which went out of print in 1892.
Since then a number of monographs devoted to the further development
of the method of Lyapunov functions has been published: in the USSR,
those by A. I. Lurie (22], N. G. Chetaev (26], I. G. Malkin [8], A. M.
Letov [23], N. N. Krasovskii [7], V. I. Zubov [138]; and abroad, J. La
Salle and S. Lefshets [11], W. Hahn [137].
Our book certainly does not pretend to give an exhaustive account of these
methods; it does not even cover all the theorems given in the monograph
by Lyapunov. Only autonomous systems are discussed and, in the linear
case, we confine ourselves to a survey of Lyapunov functions in the form
of quadratic forms only. In the non-linear case we do not consider the
question of the invertibility of the stability and instability theorems
On the other hand, Chapter 1 gives a detailed account of problems pertaining
to stability in the presence of any initial perturbation, the theory
of which was first propounded during the period 1950-1955. The first
important work in this field was that of N. P. Erugin [133-135, 16] and
the credit for applying Lyapunov functions to these problems belongs to
L'!lrie and Malkin. Theorems of the type 5.2, 6.3, 12.2 presented in Chapter
1 played a significant role in the development of the theory of stability
on the whole. In these theorems the property of stability is explained by the
presence of a Lyapunov function of constant signs and not one of fixed
sign differentiated with respect to time as is required in certain of Lyapunov's
theorems. The fundamental role played by these theorems is
explained by the fact that almost any attempt to construct simple
Lyapunov functions for non-linear systems leads to functions with the
above property.
In presenting the material of Chapter 1, the method of constructing the
Lyapunov functions is indicated where possible. Examples are given at
the end of the Chapter, each of which brings out a particular point of
interest.
Chapter 2 is devoted to problems pertaining to systems with variable
structure. From a mathematical point of view such systems represent a
very narrow class of systems of differential equations with discontinuous
right-hand sides, a fact that has enabled the author and his collaborators
to construct a more or less complete and rigorous theory for this class of
systems. Special note should be taken of the importance of studying the
stability of systems with variable structure since such systems are capable
of stabilising objects whose parameters are varying over wide limits.
Some of the results of Chapter 2 were obtained jointly with the engineers
who not only elaborated the theory along independent lines but also constructed
analogues of the systems being studied.
The method of Lyapunov function finds an application here also but the
reader interested in Chapter 2 can acquaint himself with the contents
independently of the material of the preceding Chapter.
In Chapter 3 the stability of the solutions of differential equations in
Banach space is discussed. The reasons for including this chapter are the
following. First, at the time work commenced on this chapter, no monograph
or even basic work existed on this subject apart from the articles
by L. Massera and Schaffer [94, 95, 139, 140]. The author also wished
to demonstrate the part played by the methods of functional analysis in
the theory of stability. The first contribution to this subject was that of
M. G. Krein [99]. Later, basing their work in particular on Krein's
method, Massera and Schaffer developed the theory of stability in functional
spaces considerably further. By the time work on Chapter 3 had
been completed, Krein's book [75] had gone out of print. However, the
divergence of scientific interests of Krein and the present author were such
that the results obtained overlap only when rather general problems are
being discussed.
One feature of the presentation of the material in Chapter 3 deserves
particular mention. We treat the problem of perturbation build-up as a
problem in which one is seeking a norm of the operator which will transform
the input signal into the output signal. Considerable importance is
given to the theorems of Massera and Schaffer, these theorems again
being discussed from the point of view of perturbation build-up but this
time over semi-infinite intervals of time.
It has become fashionable to discuss stability in the context of stability
with respect to a perturbation of the input signal. If we suppose that a
particular unit in an automatic control system transforms a.Ii input signal
into some other signal then the law of transformation of these signals is
given by an operator. In this case, stability represents the situation in
which a small perturbation of the input signal causes a small perturbation
of the output signal. From a mathematical point of view this property
corresponds tC? the property of continuity of the operator in question. It is
interesting to give the internal characteristic of such operators. As a rule
this characteristic reduces to a description of the asymptotic behaviour
of a Cauchy matrix (of the transfer functions). The results of Sections 5 and
6 will be discussed within this framework.
We should note that the asymptotic behaviour of the Cauchy matrix of
the system is completely characterised by the response behaviour of the
unit to an impulse. Thus the theorems given in Section 5 and 6 may be
regarded as theorems which describe the response of a system to an
impulse as a function of the response of the system when acted upon by
other types of perturbation. For this reason problems relating to the
transformation of impulse actions are of particular importance. Here,
the elementary theory of stability with respect to impulse actions is based
on the concept of functions of limited variations and on the notion of a
Stieltjes integral. This approach permits one to investigate from one and
the same point of view both stability in the Lyapunov sense (i.e. stability
with respect to initial perturbations) and stability with respect to continuously
acting perturbations.
The last paragraph of Chapter 3 is devoted to the problem of programmed
control. The material of Sections 6 and 7 has been presented in such a way
that no difficulty will be found in applying it for the purpose of solving
the problem of realising a motion along a specified trajectory. To develop
this theory, all that was necessary was to bring in the methods and results
of the theory of mean square approximations.
It should be noted that Chapter 3 demands of the reader a rather more
extensive mathematical groundwork than is required for the earlier
Chapters. In that Chapter we make use of the basic ideas of functional
analysis which the reader can acquaint himself with by reading, for
example, the book by Kantorovich and Akilov [71]. However, for the
convenience of the reader, all the basic definitions and statements of
functional analysis which we use in Chapter 3 are presented in Section 1
of that Chapter.
At the end of the book there is a detailed bibliography relating to the
problems discussed.