Theory and Application of Liapunov's Direct Method

Höfundur: Wolfgang Hahn (Útgáfa: 0)
Theory and Application of Liapunov's Direct Method

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The groundbreaking work of Russian mathematician A. M. Liapunov (1857–1918) on the stability of dynamical systems was overlooked for decades because of political turmoil. During the Cold War, when it was discovered that his method was applicable to the stability of aerospace guidance systems, interest in his research was rekindled. It has remained high ever since. This monograph on both the theory and applications of Liapunov's direct method reflects the work of a period when the theory had been studied seriously for some time and reached a degree of completeness and sophistication.

Nánar um bókina

Útgefandi
Dover Publications
ISBN
9780486839868
Print ISBN
9780486833606
Format
ePub
Útgáfa
0
Höfundar
Wolfgang Hahn
Tungumál
English
Útgefið
2019-04-17
Prent takmörkun á líftíma
100

Kaflar

  • Cover
  • Title Page
  • Copyright Page
  • Contents
  • 1 Fundamental Concepts
  • 1. Notations
  • 2. The concept of stability according to Liapunov
  • 3. The idea of the direct method of Liapunov
  • 2 Sufficient Conditions for Stability or Instability of the Equilibrium
  • 4. The main theorems on stability
  • 5. Theorems on instability
  • 3 Applications of the Stability Theorems to Concrete Problems
  • 6. Fundamental remarks on the applications
  • 7. Equations with definite first integrals
  • 8. Construction of a Liapunov function for a linear equation with constant coefficients
  • 9. Simple stability considerations for nonautonomous linear differential equations
  • 10. Equations with linear principal parts
  • 11. Bounds for the initial values
  • 12. Estimates for the stability domain of the parameters
  • 13. The problem of Aizerman and its modifications
  • 14. The problem of Lur’e and its generalizations
  • 15. Estimates for the solutions
  • 4 The Converse of the Main Theorems
  • 16. Statement of the problem
  • 17. Uniform stability
  • 18. The inversion of the stability theorems
  • 19. The inversion of the instability theorems
  • 20. On the stability theory of dynamical systems
  • 21. Zubov’s method of construction
  • 5 Liapunov Functions with Certain Properties of Rate of Change
  • 22. Order number and exponential stability
  • 23. Differential equations with homogeneous right hand sides
  • 24. The stability behavior of linear differential equations
  • 25. The order numbers of a linear differential equation
  • 6 The Sensitivity of the Stability Behavior to Perturbations
  • 26. Stability according to the first approximation
  • 27. The theorem of Liapunov on regular differential equations
  • 28. Total stability
  • 7 The Critical Cases
  • 29. General remarks on the critical cases
  • 30. The two simplest critical cases
  • 31. Malkin’s comparison theorems
  • 32. Special investigations of critical cases
  • 33. The boundary of the stability domain in the parameter space
  • 8 Generalizations of the Concept of Stability
  • 34. Stability in a finite interval
  • 35. Differential equations with bounded solutions
  • 36. The application of the direct method in general metric spaces
  • 37. Stability in the case of partial differential equations
  • 38. Application of the direct method to differential-difference equations
  • 39. Application of the direct method to difference equations
  • Bibliography
  • Author Index
  • Subject Index