Stability of nonautonomous systems by Liapunov's direct method

Dirk Aeyels

Banach Center Publications (1995)

  • Volume: 32, Issue: 1, page 9-17
  • ISSN: 0137-6934

Abstract

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This paper discusses asymptotic stability for nonautonomous systems by means of the direct method of Liapunov. The existence of a positive time-invariant Liapunov function with negative semi-definite derivative is assumed. The paper focuses on the extra conditions needed in order to guarantee asymptotic stability. The proposed criterion is compared with the standard results available for autonomous systems. A specialization and extension is obtained for a class of linear nonautonomous systems.

How to cite

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Aeyels, Dirk. "Stability of nonautonomous systems by Liapunov's direct method." Banach Center Publications 32.1 (1995): 9-17. <http://eudml.org/doc/262653>.

@article{Aeyels1995,
abstract = {This paper discusses asymptotic stability for nonautonomous systems by means of the direct method of Liapunov. The existence of a positive time-invariant Liapunov function with negative semi-definite derivative is assumed. The paper focuses on the extra conditions needed in order to guarantee asymptotic stability. The proposed criterion is compared with the standard results available for autonomous systems. A specialization and extension is obtained for a class of linear nonautonomous systems.},
author = {Aeyels, Dirk},
journal = {Banach Center Publications},
keywords = {Lyapunov function; negative semi-definite derivative; asymptotic stability; nonautonomous system},
language = {eng},
number = {1},
pages = {9-17},
title = {Stability of nonautonomous systems by Liapunov's direct method},
url = {http://eudml.org/doc/262653},
volume = {32},
year = {1995},
}

TY - JOUR
AU - Aeyels, Dirk
TI - Stability of nonautonomous systems by Liapunov's direct method
JO - Banach Center Publications
PY - 1995
VL - 32
IS - 1
SP - 9
EP - 17
AB - This paper discusses asymptotic stability for nonautonomous systems by means of the direct method of Liapunov. The existence of a positive time-invariant Liapunov function with negative semi-definite derivative is assumed. The paper focuses on the extra conditions needed in order to guarantee asymptotic stability. The proposed criterion is compared with the standard results available for autonomous systems. A specialization and extension is obtained for a class of linear nonautonomous systems.
LA - eng
KW - Lyapunov function; negative semi-definite derivative; asymptotic stability; nonautonomous system
UR - http://eudml.org/doc/262653
ER -

References

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  1. [1] D. Aeyels and R. Sepulchre, On the convergence of a time-variant linear differential equation arising in identification, to appear in Kybernetika. Zbl0832.93051
  2. [2] B. D. O. Anderson, Exponential stability of linear equations arising in adaptive identification, IEEE Trans. Automat. Control 22 (1977), 84-88. Zbl0346.93014
  3. [3] E. A. Barbashin and N. N. Krasovskii, Stability of motion in the large, Dokl. Akad. Nauk SSSR 86 (1952), 453-456. 
  4. [4] H. K. Khalil, Nonlinear Systems, Macmillan Publishing Company, New York, 1992. 
  5. [5] G. Kreisselmeier, Adaptive observers with exponential rate of convergence, Trans. Automat. Control 22 (1977), 2-8. Zbl0346.93043
  6. [6] J. P. LaSalle, Stability of nonautonomous systems, Nonlinear Anal. 1 (1976), 83-91. Zbl0355.34037
  7. [7] N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov's Direct Method, Springer-Verlag, New York, 1977. Zbl0364.34022
  8. [8] J. L. Willems, Stability Theory of Dynamical Systems, Nelson, London, 1970. 

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