Stability Theory by Liapunov's Direct Method

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.