Theory and Application of Liapunov's Direct Method

by Wolfgang Hahn

Overview

The groundbreaking work of Russian mathematician A. M. Liapunov (1857–1918) on the stability of dynamical systems was overlooked for decades because of political turmoil. During the Cold War, when it was discovered that his method was applicable to the stability of aerospace guidance systems, interest in his research was rekindled. It has remained high ever since.
This monograph on both the theory and applications of Liapunov's direct method reflects the work of a period when the theory had been studied seriously for some time and reached a degree of completeness and sophistication. It remains of interest to applied mathematicians in many areas. Topics include applications of the stability theorems to concrete problems, the converse of the main theorems, Liapunov functions with certain properties of rate of change, the sensitivity of the stability behavior to perturbations, the critical cases, and generalizations of the concept of stability.

Product Details

ISBN-13:	9780486839868
Publisher:	Dover Publications
Publication date:	04/17/2019
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	192
File size:	9 MB

About the Author

Wolfgang Hahn (1911–98) was on the faculty of Braunschweig University of Technology in Germany.

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PREFACE

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 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 mathematician s. 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 as well as the textbook of Lefschetz devote only one chapter to Liapunov's method. The excellent textbook of Malkin 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, Nemyckii and Stepanoff), termed "qualitative" methods by the Soviet authors, as well as other methods of stability investigation, such as those developed by Perron 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, 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. Andre, 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

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Table of Contents

Preface to the English Edition
Preface to the German Edition
Fundamental Concepts
Sufficient Conditions for Stability or Instability of the Equilibrium
Applications of the Stability Theorems to Concrete Problems
The Converse of the Main Theorems
Liapunov Functions with Certain Properties of Rate of Change
The Sensitivity of the Stability Behavior to Perturbations
The Critical Cases
Generalizations of the Concept of Stability
Bibliography
Author Index
Subject Index