The material is divided in the following manner. The first two chapters contain the elementary part of the theory, the knowledge of which is necessary and practically also sufficient for the applications. A knowledge of the fundamentals of the theory of differential equations and of matrix calculus are the only prerequisites. In these chapters, the primary facts have been fully substantiated; secondary results and extensions have been referred to the “Remarks.” Applications in the narrower sense, especially with respect to technical problems, are treated as a whole in Chapter 3. In this manner, I believe, the importance of the problem and of the individual papers is more emphasized than it would be by arranging the results according to strictly systematic points of view. In Chapters 4 to 7, the theory is extended further. Some sections (26, 28, 32, 33) of these chapters, however, are as well of interest for applications.
Author(s): Wolfgang Hahn
Series: Prentice-Hall International series in applied mathematics
Publisher: Prentice-Hall International
Year: 1963
Language: English
Pages: 194
City: Englewood Cliffs, N.J.
Hahn, W. Theory and application of Liapunov's direct method (P-H, 1963)(ISBN 9780060690274)(600dpi)(194p) ......Page 4
Copyright ......Page 5
PREFACE TO THE ENGLISH EDITION v ......Page 6
PREFACE TO THE GERMAN EDITION vi ......Page 7
Table of Contents viii ......Page 9
1. Notations, 1 ......Page 12
2. The concept of stability according to Liapunov, 5 ......Page 16
3. The idea of the direct method of Liapunov, 11 ......Page 22
4. The main theorems on stability, 14 ......Page 25
5. Theorems on instability, 18 ......Page 29
6. Fundamental remarks on the applications, 22 ......Page 33
7. Equations with definite first integrals, 24 ......Page 35
8. Construction of a Liapunov function for a linear equation with constant coefficients, 26 ......Page 37
9. Simple stability considerations for nonautonomous linear differential equations, 29 ......Page 40
10. Equations with linear principal parts, 33 ......Page 44
11. Bounds for the initial values, 35 ......Page 46
12. Estimates for the stability domain of the parameters, 39 ......Page 50
13. The problem of Aizerman and its modifications, 42 ......Page 53
14. The problem of Lur’e and its generalizations, 49 ......Page 60
15. Estimates for the solutions, 56 ......Page 67
16. Statement of the problem, 60 ......Page 71
17. Uniform stability, 61 ......Page 72
18. The inversion of the stability theorems, 68 ......Page 79
19. The inversion of the instability theorems, 73 ......Page 84
20. On the stability theory of dynamical systems, 75 ......Page 86
21. Zubov’s method of construction, 78 ......Page 89
22. Order number and exponential stability, 83 ......Page 94
23. Differential equations with homogeneous right hand sides, 88 ......Page 99
24. The stability behavior of linear differential equations, 92 ......Page 103
25. The order numbers of a linear differential equation, 96 ......Page 107
26. Stability according to the first approximation, 100 ......Page 111
27. The theorem of Liapunov on regular differential equations, 104 ......Page 115
28. Total stability, 107 ......Page 118
29. General remarks on the critical cases, 111 ......Page 122
30. The two simplest critical cases, 112 ......Page 123
31. Malkin’s comparison theorems, 117 ......Page 128
32. Special investigations of critical cases, 120 ......Page 131
33. The boundary of the stability domain in the parameter space, 123 ......Page 134
34. Stability in a finite interval, 126 ......Page 137
35. Differential equations with bounded solutions, 129 ......Page 138
36. The application of the direct method in general metric spaces, 132 ......Page 143
37. Stability in the case of partial differential equations, 137 ......Page 148
38. Application of the direct method to differential-difference equations, 139 ......Page 150
39. Application of the direct method to difference equations, 146 ......Page 157
BIBLIOGRAPHY, 151 ......Page 162
AUTHOR INDEX, 175 ......Page 186
SUBJECT INDEX, 178 ......Page 189
cover......Page 1
back cover ......Page 194
