Differential equations stability oscillations time lags

Differential Equations STABILITY, OSCILLATIONS, T I M E LAGS M A T H E MATICS I N SCIENCE AND ENGINEERING A SERIES OF MONOGRAPHS AND T E X T B O O K S Edited by Richard Bellman 10 11 12 13 14 15 TRACY Y THOMAS Concepts from Tensor Analysis and Differential Geometry Second Edition 1965 TRACY Y THOMAS Plastic Flow and Fracture in Solids 1961 RUTHERFORD ARIS The Optimal Design of Chemical Reactors: A Study in Dynamic Programming 1961 JOSEPHLA SALLEand SOLOMON LEFSCHETZ.Stability by Liapunov’s Direct Method with Applications 1961 GEORGE LEITMANN (ed ) Optimization Techniques: With Applications to Aerospace Systems 1962 RICHARDBELLMANand KENNETHL COOKE.Differential-Difference Equations 1963 FRANKA HAIGHT.Mathematical Theories of Traffic Flow 1963 F V ATKINSON Discrete and Continuous Boundary Problems 1964 A JEFFREY and T TANIUTI Non-Linear Wave Propagation: With Applications to Physics and Magnetohydrodynamics 1964 JULIUS T Tou Optimum Design of Digital Control Systems 1963 HARLEY FLANDERS Differential Forms : With Applications to the Physical Sciences 1963 SANFORD M ROBERTS.Dynamic Programming in Chemical Engineering and Process Control 1964 SOLOMON LEFSCHETZ Stability of Nonlinear Control Systems 1965 DIMITRISN CHORAFAS Systems and Simulation 1965 A A PERVOZVANSKII Random Processes in Nonlinear Control Systems 1965 16 17 18 MARSHALL C PEASE,111 Methods of Matrix Algebra 1965 V E BENEE.Mathematical Theory of Connecting Networks and Telephone Traffic 1965 WILLIAM F AMES.Nonlinear Partial DifferentialEquations in Engineering 1965 MATHEMATICS IN SCIENCE A N D ENGINEERING 19 20 21 22 23 J A C Z ~ LLectures on Functional Equations and Their Applications 1965 R E MURPHY Adaptive Processes in Economic Systems 1965 S E DREYFUS Dynamic Programming and the Calculus of Variations 1965 Optimal Control Systems 1965 A A FEL’DBAUM A HALANAY Differential Equations: Stability, Oscillations, Time Lags 1965 In preparczfion DIMITRISN CHORAFAS Control Systems Functions and Programming Approaches DAVIDSWORDER Optimal Adaptive Control Systems NAMIKO&JZT&ELI Time Lag Control Processes MILTONASH Optimal Shutdown Control in Nuclear Reactors This page intentionally left blank Differential Equations STABILITY, OSCILLATIONS, TIME LAGS A Halanay BUCHAREST UNIVERSITY INSTITUTE OF MATHEMATICS ACADEMY OF THE SOCIALIST REPUBLIC OF RUMANIA 1966 ACADEMIC PRESS New York and London COPYRIGHT 1966, BY ACADEMIC PRESSINC ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS ACADEMIC PRESS INC 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS INC (LONDON) LTD Berkeley Square House, London W.l LIBRARY OF CONGRESS CATALOG CARDNUMBER: 65-25005 PRINTED I N THE UNITED STATES OF AMERICA DIFFERENTIAL EQUATIONS: STABILITY, OSCILLATIONS, TIME LAGS THIS BOOK WAS ORIGINALLY PUBLISHED AS: TEORIA CALITATIVA A ECUATIILOR DIFERENTIALE STABILITATEA OSCILATII SISTEME CU ARGUMENT INTIRZIAT EDITURA ACADEMIEI REPUBLIC11 POPULARE ROMINE, BUCHAREST, 1963 DUPA LIAPUNOV Preface to the English Edition When it was proposed that an English version of my book on stability and oscillation in differential and differential-difference equations be published, my first intention was to modify it substantially Indeed, although the Rumanian original appeared in 1963, most of it was written in 1961 In the meantime, progress which deserved to be reported was achieved in all the fields covered in the book Many remarkable works appeared in the United States However, the desire not to delay the appearance of this book too much finally prompted me to renounce the initial plan and to content myself with making extensive changes only in the last chapter on systems with time lag In this chapter, only Sections 1, 2, 5, 13-16 (Sections 1, 2, 4, 11-15 in the Rumanian edition) remain unchanged; the other paragraphs are either newly introduced or completely transformed Also, Section 11 of Chapter 111 was completely rewritten I n the new -jersion of Chapter IV the results concerning stability of linear systems with constant coefficients are deduced from the general theory of linear systems with periodic coefficients Since the book of N N Krasovskii on stability theory is now available in English, -4ppendix I11 of the Rumanian edition was not included Throughout the text a number of misprints and oversights in the Rumanian edition were corrected Some new titles (89-log), considerably fewer than would have been necessary had my original plan been carried out, were added to the Bibliography The notations are not always those: currently used in the American literature, but this will not create difficulties in reading because all the notation is explained, and the sense will be clear from the context I wish to express here my gratitude and warmest thanks to Dr R Bellman for his keen interest in this book and for the tiring work he kindly undertook to check the entire translation and to improve it It is for me a great privilege and pleasure to have this book published in his very interesting and useful series, which in the short time since its inception has won unanimous acceptance A HALANAY December, 1965 Bucharest vii This page intentionally left blank Preface to the Rumanian Edition T h e qualitative theory of differential equations is in a process of continuous development, reflected in the great number of books and papers dedicated to it I t is well known that the beginning of the qualitative theory of differential equations is directly connected with the classical works of PoincarC, Lyapunov, and Birkhoff on problems of ordinary and celestial mechanics From these origins, stability theory, the mathematical theory of oscillations with small parameters, and the general theory of dynamic systems have been developed T h e great upsurge, circa 1930, of the qualitative theory of differential equations in the USSR began, on the one hand, with the study, at the Aviation Institute in Kazan, of problems of stability theory with applications in aircraft stability research, and on the other hand, i n Moscow, as a consequence of the observations of A A Andronov of the useful role played by the theory of periodic solutions of nonlinear equations in explaining some phenomena of radiotechniques It is the period marked, for example, by the renowned book on the theory of oscillations by A A Andronov, Haikin, and Witt, and the well-known book on nonlinear mechanics by Krylov and Bogoliubov From 1930 on at Moscow University, a seminar was held on the qualitative theory of differential equations with particular emphasis on fundamental theoretical problems T h e results of the first seminar appeared in the two editions of V V Nemytzki and V V Stepanov’s monograph, “ T he Qualitative Theory of Differential Equations.’’ Wide circulation of this book led to the growth of research in this area I n the last ten to fifteen years, stability theory and the theory of periodic solutions (to which is added the problem of almost periodic solutions) have been given a new impulse because they represent essential parts of the mathematical apparatus of modern control theory A characteristic feature of the development of the qualitative theory of differential equations is the fact that, in solving its problems, a most varied mathematical apparatus is used: topology and functional analysis, linear algebra, and the theory of functions of a complex variable ix 516 APPENDIX We write +(a) = j m e-i"ty(t)dt -02 On the basis of properties and we have ;a+(.) = m co c i a t y ' ( t ) d t= c i s m e-%ty(t)dt = -2~i+'(a) OD It follows that a +'(a) = E +(a)9 +(a) - a 2E ' hence It follows that E b We apply Theorem to the functions y and h It follows that For E - + O we obtain jmI +(a)Iz da = -a 24; jmh(0)e-y' dy = 24&(0) m I m e-v2 dy = ~ ) a? However, J -w J m The theorem is thus proved Theorem Then Let f , , f E L1 nL and+, , +2 be the Fourier transforms A.1 THEORY OF THE FOURIER TRANSFORM 517 On the basis of the preceding theorem, Proof 2a -m Consequently, J" -m ifi 1' dt f jm m if2 1' dt 4- jm( f i f z ffifi)dt m Taking into account the preceding theorem, we obtain the formula in the statement Consequence If fl and fi are real, the formula becomes j" fit2dt =& jm(2 Re +l+e)dt; m -m hence However, hence +,