Author | Rouche, N. author |
---|---|
Title | Stability Theory by Liapunov's Direct Method [electronic resource] / by N. Rouche, P. Habets, M. Laloy |
Imprint | New York, NY : Springer New York, 1977 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9362-7 |
Descript | XII, 396 p. online resource |
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