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The first chapter gives an account of the method of Lyapunov functionsoriginally expounded in a book by A. M. Lyapunov with the title Thegeneral problem of stability of motion which went out of print in 1892.Since then a number of monographs devoted to the further developmentof the method of Lyapunov functions has been published: in the USSR,those by A. I. Lurie (22], N. G. Chetaev (26], I. G. Malkin , A. M.Letov , N. N. Krasovskii , V. I. Zubov ; and abroad, J. LaSalle and S. Lefshets , W. Hahn .Our book certainly does not pretend to give an exhaustive account of thesemethods; it does not even cover all the theorems given in the monographby Lyapunov. Only autonomous systems are discussed and, in the linearcase, we confine ourselves to a survey of Lyapunov functions in the formof quadratic forms only. In the non-linear case we do not consider thequestion of the invertibility of the stability and instability theoremsOn the other hand, Chapter 1 gives a detailed account of problems pertainingto stability in the presence of any initial perturbation, the theoryof which was first propounded during the period 1950-1955. The firstimportant work in this field was that of N. P. Erugin [133-135, 16] andthe credit for applying Lyapunov functions to these problems belongs toL'!lrie and Malkin. Theorems of the type 5.2, 6.3, 12.2 presented in Chapter1 played a significant role in the development of the theory of stabilityon the whole. In these theorems the property of stability is explained by thepresence of a Lyapunov function of constant signs and not one of fixedsign differentiated with respect to time as is required in certain of Lyapunov'stheorems. The fundamental role played by these theorems isexplained by the fact that almost any attempt to construct simpleLyapunov functions for non-linear systems leads to functions with theabove property.In presenting the material of Chapter 1, the method of constructing theLyapunov functions is indicated where possible. Examples are given atthe end of the Chapter, each of which brings out a particular point ofinterest.Chapter 2 is devoted to problems pertaining to systems with variablestructure. From a mathematical point of view such systems represent avery narrow class of systems of differential equations with discontinuousright-hand sides, a fact that has enabled the author and his collaboratorsto construct a more or less complete and rigorous theory for this class ofsystems. Special note should be taken of the importance of studying thestability of systems with variable structure since such systems are capableof stabilising objects whose parameters are varying over wide limits.Some of the results of Chapter 2 were obtained jointly with the engineerswho not only elaborated the theory along independent lines but also constructedanalogues of the systems being studied.The method of Lyapunov function finds an application here also but thereader interested in Chapter 2 can acquaint himself with the contentsindependently of the material of the preceding Chapter.In Chapter 3 the stability of the solutions of differential equations inBanach space is discussed. The reasons for including this chapter are thefollowing. First, at the time work commenced on this chapter, no monographor even basic work existed on this subject apart from the articlesby L. Massera and Schaffer [94, 95, 139, 140]. The author also wishedto demonstrate the part played by the methods of functional analysis inthe theory of stability. The first contribution to this subject was that ofM. G. Krein . Later, basing their work in particular on Krein'smethod, Massera and Schaffer developed the theory of stability in functionalspaces considerably further. By the time work on Chapter 3 hadbeen completed, Krein's book  had gone out of print. However, thedivergence of scientific interests of Krein and the present author were suchthat the results obtained overlap only when rather general problems arebeing discussed.One feature of the presentation of the material in Chapter 3 deservesparticular mention. We treat the problem of perturbation build-up as aproblem in which one is seeking a norm of the operator which will transformthe input signal into the output signal. Considerable importance isgiven to the theorems of Massera and Schaffer, these theorems againbeing discussed from the point of view of perturbation build-up but thistime over semi-infinite intervals of time.It has become fashionable to discuss stability in the context of stabilitywith respect to a perturbation of the input signal. If we suppose that aparticular unit in an automatic control system transforms a.Ii input signalinto some other signal then the law of transformation of these signals isgiven by an operator. In this case, stability represents the situation inwhich a small perturbation of the input signal causes a small perturbationof the output signal. From a mathematical point of view this propertycorresponds tC? the property of continuity of the operator in question. It isinteresting to give the internal characteristic of such operators. As a rulethis characteristic reduces to a description of the asymptotic behaviourof a Cauchy matrix (of the transfer functions). The results of Sections 5 and6 will be discussed within this framework.We should note that the asymptotic behaviour of the Cauchy matrix ofthe system is completely characterised by the response behaviour of theunit to an impulse. Thus the theorems given in Section 5 and 6 may beregarded as theorems which describe the response of a system to animpulse as a function of the response of the system when acted upon byother types of perturbation. For this reason problems relating to thetransformation of impulse actions are of particular importance. Here,the elementary theory of stability with respect to impulse actions is basedon the concept of functions of limited variations and on the notion of aStieltjes integral. This approach permits one to investigate from one andthe same point of view both stability in the Lyapunov sense (i.e. stabilitywith respect to initial perturbations) and stability with respect to continuouslyacting perturbations.The last paragraph of Chapter 3 is devoted to the problem of programmedcontrol. The material of Sections 6 and 7 has been presented in such a waythat no difficulty will be found in applying it for the purpose of solvingthe problem of realising a motion along a specified trajectory. To developthis theory, all that was necessary was to bring in the methods and resultsof the theory of mean square approximations.It should be noted that Chapter 3 demands of the reader a rather moreextensive mathematical groundwork than is required for the earlierChapters. In that Chapter we make use of the basic ideas of functionalanalysis which the reader can acquaint himself with by reading, forexample, the book by Kantorovich and Akilov . However, for theconvenience of the reader, all the basic definitions and statements offunctional analysis which we use in Chapter 3 are presented in Section 1of that Chapter.At the end of the book there is a detailed bibliography relating to theproblems discussed.
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