Weakly connected nonlinear systems : boundedness and stability of motion

228 Pages·2010·1.58 MB·English

by  A A Martyni︠u︡k, L N Chernet︠s︡kai︠a︡, Vladislav Martynyuk

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
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Martynyuk, Chernetskaya,
Weakly Connected Nonlinear Systems and Martynyuk
Weakly Connected
Nonlinear Systems
Boundedness and Stability of Motion
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Weakly Connected
Nonlinear Systems
Boundedness and Stability of Motion
Anatoly Martynyuk
Larisa Chernetskaya
Vladislav Martynyuk
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CRC Press
Taylor & Francis Group
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Version Date: 20121023
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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