دانلود کتاب Methods of A. M. Lyapunov and their Application

The purpose of the present edition is to acquaint the reader withnew results obtained in the theory of stability of motion, and alsoto summarize certain researches by the author in this field ofmathematics. It is known that the problem of stability reduces notonly to an investigation of systems of ordinary differential equationsbut also to an investigation of systems of partial differentialequations. The theory is therefore developed in this book in sucha manner as to make it applicable to the solution of stability problemsin the case of systems of ordinary differential equations aswell as in the case of systems of partial differential equations.For the reader's benefit, we shall now list briefly the contents ofthe present monograph.This book consists of five chapters.In Sections 1-5 of Chapter I we give the principal informationconnected with the concept of metric space, and also explain themeaning of the terms which will be used below. Sections 6 and 7 arepreparatory and contain examples of dynamical systems in variousspaces. In Section 8 we define the concept of dynamical systemsin metric space, and also give the principal theorems from thebook [5] of Nemytsky and Stepanov. In Sections 9-10 we givethe principal definitions, connected with the concept of stabilityin 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 constructionof a neighborhood of a stable (asymptotically stable) invariant set. Inparticular, it is established that for stability in the sense of Lyapunovof 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 nomotions· f(p, t), P eM, having ex-limit points in M. The resultsobtained here are new even to the theory of ordinary differentialequations. In Sections 12-13 we give criteria for stability andinstability of invariant sets with the aid of certain functionals.These functionals are the analogue of the Lyapunov function andtherefore the method developed here can be considered as a certainextension of Lyapunov's second method. All the results of thesesections are local in character. We cite, for example, one of these.In order for an invariant set M to be uniformly asymptoticallystable, it is necessary and sufficient that in a certain neighborhoodS(M, r) of M there exists a functional V having the followingproperties:1. Given a number c1 > 0, it is possible to find c2 > 0 suchthat 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, 0 for p(p, M) =I= 0.2. For /'2 > 0 it is possible to find /'1 and cx1 such thatV(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 distancefrom the motion to the investigated invariant set. The theoremsobtained in this section can be considered as supplements toSections 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 theideas and methods of the first chapter to the theory of ordinarydifferential equations. In Section 1 of Chapter 2 we develop thetheorem of Section 14 for stationary systems of differential equations,and it is shown thereby that the Lyapunov function V canbe selected differentiable to the same order as the right membersof the system. In the same section we give a representation ofthis function as a curvilinear integral and solve the problem ofthe analytic structure of the right members of the system, whichright members have a region of asymptotic stability that is prescribedbeforehand. In Section 2 of Chapter II we consider thecase of holomorphic right members. The function V, the existenceof which is established in Section 1 of this chapter, is representedin this case in the form of convergent series, the analytic continuationof which makes it possible to obtain the function in the entireregion of asymptotic stability. The method of construction of suchseries can be used for an approximate solution of certain non-localproblems together with the construction of bounded solutions inthe form of series, that converge either for t > 0 or for t e (- oo,+ oo). These series are obtained from the fact that any boundedsolution is described by functions that are analytic with respectto 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 withhomogeneous right members. It is shown in particular that inorder for the zero solution of the system to be asymptoticallystable, it is necessary and sufficient that there exist two homogeneousfunctions: one positive definite W of order m, and onenegative definite V of order (m + 1 - #). such that dVfdt = W,where # is the index of homogeneity of the right members of thesystem. If the right members of the system are differentiable, thenthese functions satisfy a system of partial differential equations,the solution of which can be found in closed form. This circumstancemakes it possible to give a necessary and sufficient condition for asymptotic stability in the case when the right membersare forms of degree p., directly on the coeffilients of these forms.In Sections 4 and 5 of Chapter II we consider several doubtfulcases: k zero roots and 2k pure imaginary roots. We obtain heremany results on the stability, and also on the existence of integralsof the system and of the family of bounded solutions. In Section 6of Chapter II the theory developed in Chapter I is applied to thetheory of non-stationary systems of equations. In it are formulatedtheorems that follow from the results of Section 14, and a methodis also proposed for the investigation of periodic solutions.In Section 1 of Chapter III we solve the problem of the analyticrepresentation of solutions of partial differential equations in thecase when the conditions of the theorem of S. Kovalevskaya arenot satisfied. The theorems obtained here are applied in Section 2of Chapter III to systems of ordinary differential equations. Thissupplements the investigations of Briot and Bouquet, H. Poincare,Picard, Horn, and others, and makes it possible to develop inSection 3 of Chapter III a method of constructing series, describinga family of 0-curves for a system of equations, the expansions ofthe right members of which do not contain terms which are linearin the functions sought. The method of construction of such serieshas made it possible to give another approach to the solution of theproblem of stability in the case of systems considered in Sections 3-5of Chapter II and to formulate theorems of stability, based on theproperties of solutions of certain systems of nonlinear algebraicequations. Thus, the third chapter represents an attempt atsolving the problem of stability with the aid of Lyapunov's firstmethod.In Chapter IV we again consider metric spaces and families oftransformations in them. In Section I of Chapter IV we introducethe concept of a general system in metric space.A general system is a two-parameter family of operators fromR into R, having properties similar to those found in solutions ofthe Cauchy problem and the mixed problem for partial differentialequations. Thus, the general systems are an abstract model ofthese problems. We also develop here the concept of stability ofinvariant 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. Thetheorems obtained here yield necessary and sufficient conditions.They are based on the method of investigating two-parameterfamilies of operators with the aid of one-parameter families offunctionals. We also propose here a general method for estimatingthe distance from the motion to the invariant set. In Section 3 ofChapter IV are given several applications of the developed theoryto 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 developedtheory to the investigation of the problem of stability of thezero solution of systems of partial differential equations in the caseof the Cauchy problem or the mixed problem. In Section I ofChapter V are developed general theorems, which contain a method ofsolving the stability problem and which are orientative in character.In Sections 2-3 of Chapter V are given specific systems of partialdifferential equations, for which criteria for asymptotic stability arefound. In Section 3 the investigation of the stability of a solutionof the Cauchy problem for linear systems of equations is carriedout with the aid of a one-parameter family of quadratic functionals,defined in W~N>. Stability criteria normalized to W~NJ areobtained here. However, the imbedding theorems make it possibleto isolate those cases when the stability will be normalized in C.In the same section are given several examples of investigationof stability in the case of the mixed problem.For a successful understanding of the entire material discussedhere, it is necessary to have a knowledge of mathematics equivalentto the scope of three university courses. However, in some placesmore specialized knowledge is also necessary.