Theory and Application of Liapunov's Direct Method

THEORY & APPLICATION

OF

LIAPUNOV’S

DIRECT METHOD

WOLFGANG HAHN

English Edition Prepared by

Siegfried H. Lehnigk

Translation by

Hans H. Hosenthien and

Siegfried H. Lehnigk

DOVER PUBLICATIONS, INC.

Mineola, New York

Bibliographical Note

This Dover edition, first published in 2019, is an unabridged republication of the work originally published in the Prentice-Hall International Series in Applied Mathematics by Prentice-Hall, Inc., Englewood Cliffs, New Jersey, in 1963.

Library of Congress Cataloging-in-Publication Data

Names: Hahn, Wolfgang, 1911-1998, author. | Lehnigk, Siegfried H.

Title: Theory and application of Liapunov’s direct method / Wolfgang Hahn : English edition prepared by Siegfried H. Lehnigk ; translation by Hans H. Hosenthien and Siegfried H. Lehnigk.

Other titles: Theorie und Anwendung der direkten Methode von Ljapunov. English

Description: Dover edition. | Mineola, New York : Dover Publications, Inc., 2019. | Series: Dover books on mathematics | Text in English, translated from German. | English edition originally published: Englewood Cliffs, N.J. : Prentice-Hall, 1963. | Includes bibliographical references and indexes.

Identifiers: LCCN 2018045874 | ISBN 9780486833606 | ISBN 0486833607

Subjects: LCSH: Lyapunov functions. | Differential equations.

Classification: LCC QA871 .H183 2019 | DDC 515/.39—dc23

LC record available at https://lccn.loc.gov/2018045874

Manufactured in the United States by LSC Communications 83360701    2019

www.doverpublications.com

TABLE OF CONTENTS

1FUNDAMENTAL CONCEPTS

1. Notations

2. The concept of stability according to Liapunov

3. The idea of the direct method of Liapunov

2SUFFICIENT CONDITIONS FOR STABILITY OR INSTABILITY OF THE EQUILIBRIUM

4. The main theorems on stability

5. Theorems on instability

3APPLICATIONS OF THE STABILITY THEOREMS TO CONCRETE PROBLEMS

6. Fundamental remarks on the applications

7. Equations with definite first integrals

8. Construction of a Liapunov function for a linear equation with constant coefficients

9. Simple stability considerations for nonautonomous linear differential equations

10. Equations with linear principal parts

11. Bounds for the initial values

12. Estimates for the stability domain of the parameters

13. The problem of Aizerman and its modifications

14. The problem of Lur’e and its generalizations

15. Estimates for the solutions

4THE CONVERSE OF THE MAIN THEOREMS

16. Statement of the problem

17. Uniform stability

18. The inversion of the stability theorems

19. The inversion of the instability theorems

20. On the stability theory of dynamical systems

21. Zubov’s method of construction

5LIAPUNOV FUNCTIONS WITH CERTAIN PROPERTIES OF RATE OF CHANGE

22. Order number and exponential stability

23. Differential equations with homogeneous right hand sides

24. The stability behavior of linear differential equations

25. The order numbers of a linear differential equation

6THE SENSITIVITY OF THE STABILITY BEHAVIOR TO PERTURBATIONS

26. Stability according to the first approximation

27. The theorem of Liapunov on regular differential equations

28. Total stability

7THE CRITICAL CASES

29. General remarks on the critical cases

30. The two simplest critical cases

31. Malkin’s comparison theorems

32. Special investigations of critical cases

33. The boundary of the stability domain in the parameter space

8GENERALIZATIONS OF THE CONCEPT OF STABILITY

34. Stability in a finite interval

35. Differential equations with bounded solutions

36. The application of the direct method in general metric spaces

37. Stability in the case of partial differential equations

38. Application of the direct method to differential-difference equations

39. Application of the direct method to difference equations

BIBLIOGRAPHY

AUTHOR INDEX

SUBJECT INDEX

PREFACE TO THE ENGLISH EDITION

The English edition of the Theorie und Anwendung der direkten Methode von Ljapunov is, apart from the new Section 35, a translation of the German edition of 1959. I took the opportunity to make a few corrections and, at least, to quote the most recent publications. A closer evaluation of the results, however, was not possible.

Liapunov’s method has been recognized in several textbooks, monographs, and introductory presentations, since the German edition of this book was published. In particular, the reader is referred to the following English language publications: Antosiewicz [2], Cesari [1], Cunningham [1], Kalman and Bertram [1], LaSalle and Lefschetz [1], and the summarizing report AIEE Workshop on Liapunov’s Second Method, published by the University of Michigan, Industry Program of the College of Engineering, Ann Arbor, Michigan, 1960.

My thanks are due to Professor Siegfried H. Lehnigk, Huntsville, Alabama, to Mr. Hans H. Hosenthien, Huntsville, Alabama, and to their co-workers, for editing this translation and for valuable comments concerning the text.

w. HAHN

PREFACE TO THE GERMAN EDITION

The fundamental work of A. M. Liapunov (1857-1918) on the stability of motion published in Russian in 1892 and in a French translation in 1907 (Liapunov [1] in the bibliography), originally received only little attention and for a long time was nearly forgotten. Just 25 years ago, these investigations were resumed by some Soviet mathematicians. It was noticed that Liapunov’s methods are applicable to concrete problems in Physics and Engineering. Henceforth, as can be seen from the increasing number of publications, mathematicians deal more and more with the stability theory founded by Liapunov. This is particularly true for the so-called second or direct method which Liapunov actually used only to establish stability theorems in Theoretical Mechanics. Nowadays, this method is applied to practical problems in the realm of mechanical and electrical oscillations and, particularly, in Control Engineering. On the other hand, it has been recognized that the direct method can serve as a fundamental principle of a general theory of stability comprising considerably more problems than the one ensuing from ordinary differential equations.

The theory of the direct method has been considerably advanced within recent years, and it has reached a certain state of completion. Therefore, it can now be presented in a summarizing progress report. A review offering this subject to a larger circle appears the more indicated since nearly all of the literature was published in Russian, and parts of this literature are difficult to obtain. Among books published outside the USSR, the handbook of Sansone and Conti [1] as well as the textbook of Lefschetz [1] devote only one chapter to Liapunov’s method. The excellent textbook of Malkin [19] on the theory of stability is available in German and English translations.

In the subsequent report I have included papers dealing with the direct method which either extend the theory, or use the method as a tool. Hereby, I have endeavored to include the pertinent publications through 1957 as completely as possible. In regard to the fact that the overwhelming majority of these publications is devoted to the derivation and application of stability criteria for ordinary differential equations, I have emphasized the stability theory of ordinary differential equations in the Euclidian phase space, and I have developed it as far as the direct method permits.

With the exception of occasional references, I have omitted the topological methods (cf., e.g., Elsgolts [1], Nemyckii and Stepanoff [1]), termed qualitative methods by the Soviet authors, as well as other methods of stability investigation, such as those developed by Perron [1,4] and others.

The material is divided in the following manner. The first two chapters contain the elementary part of the theory, the knowledge of which is necessary and practically also sufficient for the applications. A knowledge of the fundamentals of the theory of differential equations and of matrix calculus are the only prerequisites. In these chapters, the primary facts have been fully substantiated; secondary results and extensions have been referred to the Remarks. Applications in the narrower sense, especially with respect to technical problems, are treated as a whole in Chapter 3. In this manner, I believe, the importance of the problem and of the individual papers is more emphasized than it would be by arranging the results according to strictly systematic points of view. In Chapters 4 to 7, the theory is extended further. Some sections (26, 28, 32, 33) of these chapters, however, are as well of interest for applications.

The concluding Chapter 8 shows that the direct method is not restricted to differential equations, but that with its help an essentially more general stability theory can be established. Chapter 8 requires a more profound knowledge of topology. However, the necessary generalizations are prepared by the formulation of the fundamental definitions and theorems of the previous chapters. In the results, and occasionally also in the proofs, I have mentioned the place of the first publication. Evidently, this cannot be done with full certainty in each case. Particularly important definitions and theorems have been emphasized by putting them in italics. These formulations, however, are not always those of utmost generality.

I have not dealt in large with second-hand presentations. They are included in the references. The recently published book of Zubov [6], however, the only monograph on the direct method published as yet, deserves particular mention. The author, starting from the concept of the dynamical system (cf. Sec. 20) and applying his own method of construction (cf. Sec. 21), arrives at a very elegant derivation of the principal results of the theory. His presentation aims at the utmost generality and, in spite of the title, not at applications in the sense of Chapter 3 of my report.

I would like to express my sincere gratitude to Prof. Dr. F. K. Schmidt, who suggested the writing of this report.

I also offer thanks to Dr. André, Dr. Hornfeck, and Dr. Tietz for their assistance in proof reading as well as for their valuable suggestions and, last, not least, to the publisher for his cooperation and prompt completion of the publication.

W. HAHN

Braunschweig, April 27, 1958

1FUNDAMENTAL CONCEPTS

1. NOTATIONS

(a) The concepts stability and instability originated in mechanics as characterizations of the equilibrium of a rigid body. The equilibrium is said to be stable if the body resumes its original position after every sufficiently small displacement. Similarly, a motion is called stable if it is insensitive to small perturbations and to changes in the initial values and in the parameters. Here, in the most simple case, motion means the variation of a point with respect to time, but more generally, we shall understand by a motion the quantities which determine the state of a physical system as a function of time (such as the Lagrange coordinates). A motion can also be interpreted as the trajectory of a point in a space of sufficiently high dimension, thereby permitting the concept of motion to be explained without reference to physical interpretations.

An exact definition of the concept of stability of a motion (which must, of course, comprise the stability of an equilibrium as a special case) can be given in various ways (cf. Moisseev [5]). The definition as formulated by Liapunov [1] has been found to be particularly convenient (cf. Sec. 2). The present work will be based primarily on this definition.

(b) A point of the real, n-dimensional Euclidean space shall be denoted by the coordinates x1 . . . , xn. The system of values {x1 . . . , xn}, denoted by x, is to be treated as a column vector of n components. The row vector which corresponds to x shall be denoted by xT. Analogously, symbols such as y, z, a, . . . , will be used for column vectors with n components. In the case of several vectors x1, x2, . . . , the components x1j, . . . , xnj refer to the vector xj. Vectors will be denoted by lower case sans serif letters.

The absolute value of the vector x is, as usual, the quantity

In addition to the n-dimensional x-space, which is also called phase space, we shall refer to the (n + 1)-dimensional space of the quantities x1, . . . xn, t, which will be called motion space.

(c) The notation x = x(t) indicates that the components xi of x are functions of t. If these functions are continuous, then the point (x(t), t) of the motion space moves along a segment of a curve as t runs from t1 to t2, t1 t t2. This segment forms a portion of the motion of x(t) in the motion space. The t-axis represents the particular motion belonging to the null vector x(t) ≡ 0.

The projection of a motion upon the phase space is called the phase curve, or trajectory, of the motion. In this case the quantity t plays the role of a curve parameter. If x(t) is defined for all t t0, or t t0 there corresponds to this right or left infinite parameter-interval a branch of the motion, sometimes called half-trajectory.

(d) Scalar functions will be denoted by lower case italic, or, sometimes, Greek letters. Functions of the form f(x1, . . . , xn), or f(x1, . . . , xn, t), which are defined in the phase space or motion space, shall be abbreviated by the notation f(x), or f(x, t), respectively. Occasionally, several such functions f1(x, t), f(x, t), ... , may be combined to form a column vector f(x, t).

Capital sans serif letters denote matrices; If denotes a matrix, will denote its transpose and its inverse. will always denote the unit matrix. Capital German letters will be used to denote point sets. The concept domain will be used in the general sense, i.e., for open as well as for closed point sets.

(e) The function f(x, t) is said to belong in to the class C0(f C0) if it satisfies a Lipschitz condition at each point of a domain , i.e., if in a certain neighborhood of the point (x, t) an inequality of the form

is satisfied, where the constant m is independent of t and xi. The function f is said to belong to the class Cr if f has continuous partial derivatives with respect to the xij up to the order r. If there exists for the whole domain a uniform Lipschitz constant m, or if the partial derivatives are uniformly bounded, then or respectively. Cω denotes the class of the real analytic functions.

The statement φ(r) belongs to the class K means that φ(r) is a continuous real function defined in the closed interval 0 r h, and that φ(r) vanishes at r = 0 and increases strictly monotonically with r.

(f) The vector differential equation

stands for a system of n scalar differential equations of the first order and it might sometimes be equivalent to a single scalar differential equation of the order n. In the following, no distinction will be made in general between vector and scalar differential equations.

If the right hand side of (1.2) is continuous and of such a nature that the existence and uniqueness of the solutions, as well as their continuous dependence on the initial values is assured, f shall be said to belong to the class E, f E. The initial values represent the initial instant to (henceforth, we assume t0 0) and the initial point x0. If f ∈ E, then p(t, x0, t0) will denote that well defined solution which takes on the value x0 at t = t0, i.e., p(t0, x0, t0) = x0. A constant solution, p(t, x0, t0) ≡ x0 is said to be an equilibrium, or a singular point, of the differential equation. If x0 is the only constant solution in a neighborhood of x0, it is called an isolated equilibrium.

If the right hand side of (1.2) does not depend on t, or if it is periodic in t, the equation is called autonomous, or periodic, respectively. Linear differential equations are written in matrix notation as

(g) Among the scalar functions, the so-called Liapunov functions (cf. Definition 4.1) play a particular role. For these functions we shall reserve the letters u, v, and ω. For the most part, they are defined in the spherical neighborhood

of the origin of the phase space, or in a half-cylindrical neighborhood

of the t-axis of the motion space. In all of the following it is assumed that these functions are continuous and that they have continuous partial derivatives with respect to all arguments. The symbols and shall always have the meaning explained by (1.4) and (1.5). The number t0 shall mean a fixed, nonnegative and in certain cases sufficiently large constant. To give an exact characterization of a Liapunov function, it is necessary to introduce some definitions.

DEFINITION 1.1: A function v(x) is called positive (negative) semi-definite if v(0) = 0 and if in a neighborhood of the origin v(x) 0 ( 0). The case that v vanishes identically is included. If v(0) = 0 and v(x) > 0 (< 0) for x ≠ 0, the function is called positive (negative) definite.

DEFINITION 1.2: A function v(x,t), defined in a domain of the motion space, is called positive (negative) semi-definite if v(0, t) = 0, (t (t0), and if, with suitable hi1 h, v(x, t) 0 ( 0) in The function v(x, t) is called positive (negative) definite if v(0, t) = 0 and if in a domain of the phase space, a positive definite function w(x) exists, such that the relation

holds.

Example: is positive semi-definite, but not positive definite; however,

is postitive definite.

DEFINITION 1.3: A function v(x, t), defined in a half-space t t0, is called radially unbounded if for each α > 0 there is a β > 0 such that v(x, t) < α whenever |x| > β and t t0.

The concept of radial unboundedness has been introduced by Barbašin and Krasovskii [1] with the terminology "v becomes infinitely large."

DEFINITION 1.4: A function v(x, t) is called decrescent if the relation

holds uniformly in t. This is equivalent to the existence of a positive definite function u(x) which is independent of t and which satisfies the inequality

Example: is not decrescent, but is decrescent.

In papers of Liapunov and others who followed him, the terminology "v admits an infinitesimal small upper bound is used instead of v is decrescent."

(h) Using the fact that the functions v(x, t)