Stability Theory by Liapunov's Direct Method

ISBN-13: 9780387902586 / Angielski / Miękka / 1977 / 396 str.

Nicolas Rouche

Opis

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].

Spis treści

I. Elements of Stability Theory.- 1. A First Glance at Stability Concepts.- 2. Various Definitions of Stability and Attractivity.- 3. Auxiliary Functions.- 4. Stability and Partial Stability.- 5. Instability.- 6. Asymptotic Stability.- 7. Converse Theorems.- 8. Bibliographical Note.- II. Simple Topics in Stability Theory.- 1. Theorems of E.A. Barbashin and N.N. Krasovski for Autonomous and Periodic Systems.- 2. A Theorem of V.M. Matrosov on Asymptotic Stability.- 3. Introduction to the Comparison Method.- 4. Total Stability.- 5. The Frequency Method for Stability of Control Systems.- 6. Non-Differentiable Liapunov Functions.- 7. Bibliographical Note.- III. Stability of a Mechanical Equilibrium.- 1. Introduction.- 2. The Lagrange-Dirichlet Theorem and Its Variants.- 3. Inversion of the Lagrange-Dirichlet Theorem Using Auxiliary Functions.- 4. Inversion of the Lagrange-Dirichlet Theorem Using the First Approximation.- 5. Mechanical Equilibrium in the Presence of Dissipative Forces.- 6. Mechanical Equilibrium in the Presence of Gyroscopic Forces.- 7. Bibliographical Note.- IV. Stability in the Presence of First Integrals.- 1. Introduction.- 2. General Hypotheses.- 3. How to Construct Liapunov Functions.- 4. Eliminating Part of the Variables.- 5. Stability of Stationary Motions.- 6. Stability of a Betatron.- 7. Construction of Positive Definite Functions.- 8. Bibliographical Note.- V. Instability.- 1. Introduction.- 2. Definitions and General Hypotheses.- 3. Fundamental Proposition.- 4. Sectors.- 5. Expellers.- 6. Example of an Equation of N Order.- 7. Instability of the Betatron.- 8. Example of an Equation of Third Order.- 9. Exercises.- 10. Bibliographical Notes.- VI. A Survey of Qualitative Concepts.- 1. Introduction.- 2. A View of Stability and Attractivity Concepts.- 3. Qualitative Concepts in General.- 4. Equivalence Theorems for Qualitative Concepts.- 5. A Tentative Classification of Concepts.- 6. Weak Attractivity, Boundedness, Ultimate Boundedness.- 7. Asymptotic Stability.- 8. Bibliographical Note.- VII. Attractivity for Autonomous Equations.- 1. Introduction.- 2. General Hypotheses.- 3. The Invariance Principle.- 4. An Attractivity and a Weak Attractivity Theorem.- 5. Attraction of a Particle by a Fixed Center.- 6. A Class of Nonlinear Electrical Networks.- 7. The Ecological Problem of Interacting Populations.- 8. Bibliographical Note.- VIII. Attractivity for Non Autonomous Equations.- 1. Introduction, General Hypotheses.- 2. The Families of Auxiliary Functions.- 3. Another Asymptotic Stability Theorem.- 4. Extensions of the Invariance Principle and Related Questions.- 5. The Invariance Principle for Asymptotically Autonomous and Related Equations.- 6. Dissipative Periodic Systems.- 7. Bibliographical Note.- IX. The Comparison Method.- 1. Introduction.- 2. Differential Inequalities.- 3. A Vectorial Comparison Equation in Stability Theory.- 4. Stability of Composite Systems.- 5. An Example from Economics.- 6. A General Comparison Principle.- 7. Bibliographical Note.- Appendix I. DINI Derivatives and Monotonic Functions.- 1. The Dini Derivatives.- 2. Continuous Monotonic Functions.- 3. The Derivative of a Monotonic Function.- 4. Dini Derivative of a Function along the Solutions of a Differential Equation.- Appendix II. The Equations of Mechanical Systems.- Appendix III. Limit Sets.- List of Examples.- Author Index.