This monograph is a collective work. The names appear ing on the front cover are those of the people who worked on every chapter. But the contributions of others were also very important: C. Risito for Chapters I, II and IV, K. Peiffer for III, IV, VI, IX R. J. Ballieu for I and IX, Dang Chau Phien for VI and IX, J. L. Corne for VII and VIII. The idea of writing this book originated in a seminar held at the University of Louvain during the academic year 1971-72. Two years later, a first draft was completed. However, it was unsatisfactory mainly because it was ex ce~sively abstract and lacked examples. It was then decided to write it again, taking advantage of -some remarks of the students to whom it had been partly addressed. The actual text is this second version. The subject matter is stability theory in the general setting of ordinary differential equations using what is known as Liapunov's direct or second method. We concentrate our efforts on this method, not because we underrate those which appear more powerful in some circumstances, but because it is important enough, along with its modern developments, to justify the writing of an up-to-date monograph. Also excellent books exist concerning the other methods, as for example R. Bellman [1953] and W. A. Coppel [1965].
Erscheinungsdatum: 09.07.1977

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zlib/Mathematics/N. Rouche, P. Habets, M. Laloy (auth.)/Stability Theory by Liapunov’s Direct Method_1161135.pdf

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大手笔是怎样炼成的(套装共4册)

cN. Rouche, P. Habets, M. LaLoy.,Springer-Verlag

Nicolas Rouche; Patrick Habets; Michel Laloy

Rouche, Nicolas, Habets, P., Laloy, M.

closed connection without sending a

谢亦森

Springer Spektrum. in Springer-Verlag GmbH

Springer-Verlag, Lightning Source UK Ltd

Steinkopff. in Springer-Verlag GmbH

Springer Science & Business Media

Springer London, Limited

Copernicus

长江文艺出版社

Telos

Applied mathematical sciences ;, 22, Applied mathematical sciences (Springer-Verlag New York Inc.) ;, v. 22., New York, New York State, 1977

Applied mathematical sciences (Springer-Verlag New York Inc.), v. 22, New York, Milton Keynes UK, 2012?], c1977

Softcover reprint of the original 1st ed. 1977, FR, 1977

United States, United States of America

Springer Nature, New York, NY, 2012

Germany, Germany

2011 12 30

sm35895066

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Bibliography: p. [370]-385.
Includes indexes.

PREFACE 4
SOME NOTATIONS AND DEFINITIONS 6
TABLE OF CONTENTS 10
CHAPTER I. ELEMENTS OF STABILITY THEORY 14
1. A First Glance at Stability Concepts 14
2. VARIOUS Definitions of STABILITY and Attractivity 19
3. Auxiliary Functions 24
4. Stability and Partial Stability 26
5.Instability 32
6. Asymptotic Stability 38
7. Converse Theorems 57
8. Bibliographical Note 60
CHAPTER II. SIMPLE TOPICS IN STABILITY THEORY 62
1. Theorems of E.A. Barbashin and N.N. Krasovski for Autonomous and Periodic Systems 63
2. A Theorem of V.M. Matrosov on Asymptotic Stability 73
3.Introduction to the Comparison Method 86
4.Total Stability 93
5. The Frequency Method for Stability of Control Systems 97
6. Non-Differentiable Liapunov Functions 102
7. Bibliographical Note 108
CHAPTER III. STABILITY OF A MECHANICAL EQUILIBRIUM 110
1. Introduction 110
2. The Lagrange-Dirichlet Theorem and Its variants 111
3. Inversion of the Lagrange-Dirichlet Theorem Using Auxiliary Functions 118
4. Inversion of the Lagrange-Dirichlet Theorem Using the First Approximation 121
5. Mechanical Equilibrium in the Presence of Dissipative Forces 126
6. Mechanical Equilibrium in the Presence of Gyroscopic Forces 128
7. Bibliographical Note 139
CHAPTER IV. STABILITY IN THE PRESENCE OF FIRST INTEGRALS 141
1. Introduction 141
2. General Hypotheses 142
3. How to Construct Liapunov Functions 143
4. Eliminating Part of the Variables 147
5. Stability of Stationary Motions 152
6. Stability of a Betatron 158
7. Construction of Positive Definite Functions 164
8. Bibliographical Note 178
CHAPTER V. INSTABILITY 181
1. Introduction 181
2. Definitions and General Hypotheses 183
3. Fundamental Proposition 185
4. Sectors 186
5. Expellers 193
6. Example of an Equation of Nth Order 197
7. Instability of the Betatron 201
8. Example of an Equation of Third Order 204
9. Exercises 208
10. Bibliographical Notes 211
CHAPTER VI. A SURVEY OF QUALITATIVE CONCEPTS 214
1. Introduction 214
2. A View of Stability and Attractivity Concepts 217
3. Qualitative Concepts in General 220
4. Equivalence Theorems for Qualitative Concepts 228
5. A Tentative Classification of concepts 235
6. Weak Attractivity -1
7. Asymptotic Stability 248
8. Bibliographical Note 251
CHAPTER VII. ATTRACTIVITY FOR AUTONOMOUS EQUATIONS 254
1. Introduction 254
2. General Hypotheses 255
3. The Invariance Principle 255
4. An Attractivity and a Weak Attractivity Theorem 259
5. Attraction of a Particle by a Fixed Center 262
6. A Class of Nonlinear Electrical Networks 267
7. The Ecological Problem of Interacting Populations 273
8. Bibliographical Note 282
CHAPTER VIII. ATTRACTIVITY FOR NON AUTONOMOUS EQUATIONS 283
1 Introduction -1
2 The Families of Auxiliary Functions 284
3 Another Asymptotic Stability Theorem 293
4 Extensions of the Invariance Principle and Related Questions 298
5 The Invariance Principle for Asymptotically Autonomous and Related Equations 307
6 Dissipative Periodic systems 318
7 Bibliographical Note 323
CHAPTER IX. THE COMPARISON METHOD 326
1. Introduction 326
2. Differential Inequalities 327
3. A Vectorial Comparison Equation in Stability Theory 333
4. Stability of Composite Systems 340
5. An Example from Economics 345
6. A General Comparison Principle 349
7. Bibliographical Note 355
APPENDIX I. DIN! DERIVATIVES AND MONOTONIC FUNCTIONS 358
1. The Dini Derivatives 358
2. Continuous Monotonic Functions 360
3. The Derivative of a Monotonic Function 363
4. Dini Derivative of a Function along the Solutions of a Differential Equation 365
APPENDIX II. THE EQUATIONS OF MECHANICAL SYSTEMS 368
APPENDIX III. LIMIT SETS 376
LIST OF EXAMPLES 381
BIBLIOGRAPHY 383
AUTHOR INDEX 399
SUBJECT INDEX 406
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This monograph is a collective work. The names appearƯ ing on the front cover are those of the people who worked on every chapter. But the contributions of others were also very important: C. Risito for Chapters I, II and IV, K. Peiffer for III, IV, VI, IX R.J. Ballieu for I and IX, Dang Chau Phien for VI and IX, J.L. Corne for VII and VIII. The idea of writing this book originated in a seminar held at the University of Louvain during the academic year 1971-72. Two years later, a first draft was completed. However, it was unsatisfactory mainly because it was exƯ ce~sively abstract and lacked examples. It was then decided to write it again, taking advantage of -some remarks of the students to whom it had been partly addressed. The actual text is this second version. The subject matter is stability theory in the general setting of ordinary differential equations using what is known as Liapunov's direct or second method. We concentrate our efforts on this method, not because we underrate those which appear more powerful in some circumstances, but because it is important enough, along with its modern developments, to justify the writing of an up-to-date monograph. Also excellent books exist concerning the other methods, as for example R. Bellman [1953] and W.A. Coppel [1965]

Front Matter....Pages i-xii
Elements of Stability Theory....Pages 1-48
Simple Topics in Stability Theory....Pages 49-96
Stability of a Mechanical Equilibrium....Pages 97-127
Stability in the Presence of First Integrals....Pages 128-167
Instability....Pages 168-200
A Survey of Qualitative Concepts....Pages 201-240
Attractivity for Autonomous Equations....Pages 241-269
Attractivity for Non Autonomous Equations....Pages 270-312
The Comparison Method....Pages 313-344
Back Matter....Pages 345-396

2012-02-04