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.