Mathematics 305 PURE AND APPLIED MATHEMATICS A SERIES OF MONOGRAPHS AND TEXTBOOKS Weakly Connected Nonlinear Systems: Boundedness and Stability of Motion provides a systematic study on the boundedness and stability of weakly connected nonlinear systems, covering theory Weakly Connected and applications previously unavailable in book form. It contains many essential results needed for carrying out research on nonlinear systems of weakly connected equations. Nonlinear Systems After supplying the necessary mathematical foundation, the book illustrates recent approaches to studying the boundedness of motion of weakly connected nonlinear systems. The authors consider Boundedness and Stability of Motion conditions for asymptotic and uniform stability using the auxiliary vector Lyapunov functions and explore the polystability of the motion of a nonlinear system with a small parameter. Using the generalization of the direct Lyapunov method with the asymptotic method of nonlinear mechanics, they then study the stability of solutions for nonlinear systems with small perturbing forces. They also present fundamental results on the boundedness and stability of systems in Banach spaces with weakly connected subsystems through the generalization of the direct Lyapunov method, using both vector and matrix-valued auxiliary functions. Designed for researchers and graduate students working on systems Anatoly Martynyuk with a small parameter, this book will help readers get up to date on the knowledge required to start research in this area. Larisa Chernetskaya Vladislav Martynyuk K16508 K16508_Cover.indd 1 10/16/12 12:44 PM Martynyuk, Chernetskaya, Weakly Connected Nonlinear Systems and Martynyuk Weakly Connected Nonlinear Systems Boundedness and Stability of Motion K16508_Printer_PDF.indd 1 9/27/12 2:14 PM PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair Nashed Rutgers University University of Central Florida Piscataway, New Jersey Orlando, Florida EDITORIAL BOARD Jane Cronin Freddy van Oystaeyen Rutgers University University of Antwerp, Belgium S. Kobayashi University of California, Donald Passman Berkeley University of Wisconsin, Madison Marvin Marcus University of California, Fred S. Roberts Santa Barbara Rutgers University W. S. Massey David L. Russell Yale University Virginia Polytechnic Institute and State University Anil Nerode Cornell University Walter Schempp Universität Siegen K16508_Printer_PDF.indd 2 9/27/12 2:14 PM MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS Recent Titles Kevin J. Hastings, Introduction to the Mathematics of Operations Research with Mathematica®, Second Edition (2006) Robert Carlson, A Concrete Introduction to Real Analysis (2006) John Dauns and Yiqiang Zhou, Classes of Modules (2006) N. K. Govil, H. N. Mhaskar, Ram N. Mohapatra, Zuhair Nashed, and J. Szabados, Frontiers in Interpolation and Approximation (2006) Luca Lorenzi and Marcello Bertoldi, Analytical Methods for Markov Semigroups (2006) M. A. Al-Gwaiz and S. A. Elsanousi, Elements of Real Analysis (2006) Theodore G. Faticoni, Direct Sum Decompositions of Torsion-Free Finite Rank Groups (2007) R. 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Pfeffer, The Divergence Theorem and Sets of Finite Perimeter (2012) Willi Freeden and Christian Gerhards, Geomathematically Oriented Potential Theory (2013) Anatoly Martynyuk, Larisa Chernetskaya, and Vladislav Martynyuk, Weakly Connected Nonlinear Systems: Boundedness and Stability of Motion (2013) K16508_Printer_PDF.indd 3 9/27/12 2:14 PM This page intentionally left blank Weakly Connected Nonlinear Systems Boundedness and Stability of Motion Anatoly Martynyuk Larisa Chernetskaya Vladislav Martynyuk K16508_Printer_PDF.indd 5 9/27/12 2:14 PM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20121023 International Standard Book Number-13: 978-1-4665-7087-0 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface xi Acknowledgments xv 1 Preliminaries 1 1.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fundamental Inequalities . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Gronwall type inequalities . . . . . . . . . . . . . . . . 2 1.2.2 Bihari type inequalities . . . . . . . . . . . . . . . . . 7 1.2.3 Differential inequalities . . . . . . . . . . . . . . . . . 12 1.2.4 Integral inequalities . . . . . . . . . . . . . . . . . . . 16 1.3 Stability in the Sense of Lyapunov . . . . . . . . . . . . . . . 17 1.3.1 Lyapunov functions . . . . . . . . . . . . . . . . . . . 17 1.3.2 Stability theorems . . . . . . . . . . . . . . . . . . . . 21 1.4 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . 29 1.5 Stability of Systems with a Small Parameter . . . . . . . . . 32 1.5.1 States of equilibrium . . . . . . . . . . . . . . . . . . . 33 1.5.2 Definitions of stability . . . . . . . . . . . . . . . . . . 34 1.6 Comments and References . . . . . . . . . . . . . . . . . . . 35 2 Analysis of the Boundedness of Motion 37 2.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 38 2.3 µ-Boundedness with Respect to Two Measures . . . . . . . . 40 2.4 Boundedness and the Comparison Technique . . . . . . . . . 47 2.4.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . 47 2.4.2 Conditions for the boundedness of motion . . . . . . . 48 2.5 Boundedness with Respect to a Part of Variables . . . . . . . 56 2.6 Algebraic Conditions of µ-Boundedness . . . . . . . . . . . . 61 2.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.7.1 Lienard oscillator . . . . . . . . . . . . . . . . . . . . . 66 2.7.2 Connected systems of Lurie–Postnikov equations . . . 67 2.7.3 A nonlinear system with weak linear connections . . . 69 2.8 Comments and References . . . . . . . . . . . . . . . . . . . 71 vii viii 3 Analysis of the Stability of Motion 73 3.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 73 3.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 74 3.3 Stability with Respect to Two Measures . . . . . . . . . . . . 76 3.4 Equistability Via Scalar Comparison Equations . . . . . . . . 86 3.5 Dynamic Behavior of an Individual Subsystem . . . . . . . . 90 3.6 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . 95 3.6.1 Uniform asymptotic stability . . . . . . . . . . . . . . 95 3.6.2 The global uniform asymptotic stability . . . . . . . . 98 3.6.3 Exponential stability . . . . . . . . . . . . . . . . . . . 99 3.6.4 Instability and full instability . . . . . . . . . . . . . . 103 3.7 Polystability of Motion . . . . . . . . . . . . . . . . . . . . . 105 3.7.1 General problem of polystability . . . . . . . . . . . . 105 3.7.2 Polystability of the system with two subsystems . . . 106 3.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.8.1 Analysis of longitudinal motion of an aeroplane . . . . 109 3.8.2 Indirect control of systems . . . . . . . . . . . . . . . . 112 3.8.3 Control system with an unstable free subsystem . . . 114 3.9 Comments and References . . . . . . . . . . . . . . . . . . . 116 4 Stability of Weakly Perturbed Systems 119 4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 119 4.2 Averaging and Stability . . . . . . . . . . . . . . . . . . . . . 120 4.2.1 Problem and auxiliary results . . . . . . . . . . . . . . 120 4.2.2 Conditions for stability . . . . . . . . . . . . . . . . . 122 4.2.3 Conditions of instability . . . . . . . . . . . . . . . . . 126 4.2.4 Conditions for asymptotic stability . . . . . . . . . . . 131 4.3 Stability on a Finite Time Interval . . . . . . . . . . . . . . . 135 4.4 Methods of Application of Auxiliary Systems . . . . . . . . . 141 4.4.1 Development of limiting system method . . . . . . . . 141 4.4.2 Stability on time-dependent sets . . . . . . . . . . . . 146 4.5 Systems with Nonasymptotically Stable Subsystems . . . . . 151 4.6 Stability with Respect to a Part of Variables . . . . . . . . . 163 4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.7.1 Analysis of two weakly connected oscillators . . . . . . 166 4.7.2 System of n oscillators . . . . . . . . . . . . . . . . . . 171 4.8 Comments and References . . . . . . . . . . . . . . . . . . . 177 5 Stability of Systems in Banach Spaces 179 5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 179 5.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . 179 5.3 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 181 5.4 Generalized Direct Lyapunov Method . . . . . . . . . . . . . 182 5.5 µ-Stability of Motion of Weakly Connected Systems . . . . . 185 5.6 Stability Analysis of a Two-Component System . . . . . . . 196 ix 5.7 Comments and References . . . . . . . . . . . . . . . . . . . 200 Bibliography 203 Index 211