The theory of the stability of motion has gained increasing signifi cance in the last decades as is apparent from the large number of publi cations on the subject. A considerable part of this work is concerned with practical problems, especially problems from the area of controls and servo-mechanisms, and concrete problems from engineering were the ones which first gave the decisin' impetus for the expansion and modern development of stability theory. In comparison with the many single publications, which are num bered in the thousands, the number of books on stability theory, and especially books not \\Titten in Russian, is extraordinarily small. Books which giw the student a complete introduction into the topic and which simultaneously familiarize him with the newer results of the theory and their applications to practical questions are completely lacking. I hope that the book which I hereby present will to some extent do justice to this double task. I haw endeavored to treat stability theory as a mathe matical discipline, to characterize its methods, and to prove its theorems rigorollsly and completely as mathematical theorems. Still I always strove to make reference to applications, to illustrate the arguments with examples, and to stress the interaction between theory and practice. The mathematical preparation of the reader should consist of about two to three years of university mathematics.
Table of ContentsI. Generalities.- § 1. The Stability Concept in Mechanics.- § 2. Stability in the Sense of Liapunov.- II. Linear Functional Equations with Constant Coefficients.- § 3. Transfer Units.- § 4. Linear Differential Equations with Constant Coefficients.- § 5. Geometrical Criteria for Stability.- § 6. Algebraic Criteria for Stability.- § 7. Orlando’s Formula.- § 8. Linear Transfer Systems.- § 9. An Example.- § 10. The Nyquist Criterion.- § 11. The Boundary of Stability.- § 12. Linear Differential Difference Equations.- § 13. Stability for Linear Differential Difference Equations with Constant Coefficients.- § 14. Linear Difference Equations with Constant Coefficients.- § 15. Linear Operators.- III. The Equilibrium of Autonomous Differential Equations.- § 16. Fundamental Concepts, Definitions and Notations.- § 17. Homogeneous Right Side.- § 18. General Systems of the Second Order.- § 19. Second Order Systems with Homogeneous Right Sides.- § 20. Second Order Linear Systems.- § 21. Perturbed Second Order Linear Systems.- § 22. Conservative Second Order Systems.- IV. The Direct Method of Liapunov.- § 23. Geometric Interpretation.- § 24. Some Subsidiary Considerations.- § 25. The Principal Theorems of the Direct Method for Autonomous Differential Equations.- § 26. Supplements to the Principal Theorems.- § 27. Construction of a Liapunov Function for a Linear Equation.- § 28. Liapunov Functions for Perturbed Linear Equations.- § 29. The Problem of Aizerman.- § 30. Further Applications of the Direct Method.- § 31. Absolute Stability.- § 32. Popov’s Criterion.- § 33. The Domain of Attraction.- § 34. Zubov’s Theorem.- V. The Direct Method for General Motions.- § 35. The General Stability Concept.- § 36. Extensions and Modifications of the Basic Definitions.- § 37. Instability and Non-Uniform Stability.- § 38. Relationships between the Stability Types.- § 39. Realizing Some Stability Types.- § 40. An Example for Instability.- § 41. Liapunov Functions.- § 42. Tests for Stability.- § 43. Applications and Examples. I. Differential and Difference Equations.- § 44. Applications and Examples. II. Functional and Partial Differential Equations.- § 45. System Stability and Stability of Invariant Sets.- § 46. Boundedness Criteria. The Parallel Theorems.- VI. The Converse of the Stability Theorems.- § 47. Formulation of the Problem.- § 48. The Converse of the Theorems on Non-Asymptotic Stability.- § 49. The Converse of Theorems on Asymptotic Stability.- § 50. Examples for the Converse Theorems.- § 51. Refinements of the Converse Theorems for Ordinary Differential Equations.- § 52. The Converse of the Instability Theorems.- VII. Stability Properties of Ordinary Differential Equations.- § 53. The Meaning of the Decrescence of Liapunov Functions.- § 54. Existence of a Liapunov Function in Case of Non-Uniform Asymptotic Stability.- § 55. Modified Stability Criteria.- § 56. Perturbed Equations.- § 57. Equations with Homogeneous Right Side.- VIII. Linear Differential Equations.- § 58. The General Solution of a Linear Homogeneous Differential Equation.- § 59. The Nonhomogeneous Linear Equation.- § 60. Linear Equations with Periodic Coefficients.- § 61. The Liapunov Reducibility Theorem.- § 62. Stability Criteria for Special Linear Differential Equations.- § 63. The Order Numbers of a Differential Equation.- § 64. Regular Differential Equations.- § 65. Stability in the First Approximation.- IX. The Liapunov Expansion Theorem.- § 66. Families of Solutions Depending on a Parameter.- § 67. The Liapunov Expansion Theorem.- X. The Critical Cases for Differential Equations.- § 68. General Remarks Concerning Critical Cases; Subsidiary Results.- § 69. The Principal Theorem of Malkin.- § 70. Simple Critical Cases for Autonomous Equations.- XI. Periodic and Almost Periodic Motions.- § 71. General Remarks on periodic Motions.- § 72. Nonhomogeneous Linear Equations with Periodic External Force.- § 73. Forced Almost Periodic Oscillations.- § 74. Piecewise Linear Equations.- § 75. A System with Several Discontinuity Types.- § 76. Perturbed Linear Equations.- § 77. Perturbed Linear Equations for the Resonance Case.- § 78. Periodic Solutions of Autonomous Equations.- § 79. Critical Cases of Second Order Autonomous Systems.- § 80. The Associated Coordinate System of a Periodic Solution.- § 81. Stability Properties of a Periodic Solution.- § 82. Examples: Testing for Stability.- Author Index.