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Asymptotic stability for three-dimensional linear differential systems with time-varying coefficients
Author(s):
Jitsuro
Sugie;
Yuichi
Ogami
Journal:
Quart. Appl. Math.
MSC (2000):
Primary 34D05, 34D20, 34D23;
Secondary 37B25, 37B55
Posted:
May 27, 2009
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Abstract:
This paper is concerned with the asymptotic stability of the zero solution of three-dimensional linear differential systems with variable coefficients. The coefficients are not assumed to be positive. A concept innovated by László Hatvani plays a vital role in our results. Sufficient conditions are also given for the zero solution to be uniformly stable. Some suitable examples are included to illustrate our results. Finally, certain changes of variable are used to broaden the application of our results.
References:
-
- 1.
- A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York and London, 1966. MR 0216103 (35:6938)
- 2.
- J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, London and Sydney, 1969. MR 0419901 (54:918); (revised) Krieger, Malabar, 1980. MR 0587488 (82e:34001)
- 3.
- L. Hatvani, A generalization of the Barbashin-Krasovskij theorems to the partial stability in nonautonomous systems, Qualitative Theory of Differential Equations I, II (ed. M. Farkas), Colloq. Math. Soc. János Bolyai 30, North-Holland, Amsterdam and New York, 1981, pp. 381-409. MR 0680604 (84d:34048)
- 4.
- L. Hatvani, On the uniform attractivity of solutions of ordinary differential equations by two Lyapunov functions, Proc. Japan Acad. 67, 162-167 (1991). MR 1114963 (92g:34063)
- 5.
- L. Hatvani, On the asymptotic stability for a two-dimensional linear nonautonomous differential system, Nonlinear Anal. 25, 991-1002 (1995). MR 1350721 (96k:34105)
- 6.
- A. Hurwitz, Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Math. Ann. 46, 273-284 (1895). MR 1510884
- 7.
- J. P. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method with Applications, Mathematics in Science and Engineering 4, Academic Press, New York and London, 1961. MR 0132876 (24:A2712)
- 8.
- J. C. Maxwell, On governors, Proc. Roy. Soc. London 6, 270-283 (1868).
- 9.
- D. R. Merkin, Introduction to the Theory of Stability, Texts in Applied Mathematics 24, Springer-Verlag, New York, Berlin and Heidelberg, 1997. MR 1418401 (98f:34074)
- 10.
- K. P. Persidskiĭ, Über die Stabilität einer Bewegung nach der ersten Näherung, Mat. Sb. 40, 284-293 (1933).
- 11.
- N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov's Direct Method, Applied Mathematical Sciences 22, Springer-Verlag, New York, Heidelberg and Berlin, 1977. MR 0450715 (56:9008)
- 12.
- E. J. Routh, Treatise on the Stability of a Given State of Motion, Macmillan, London, 1877.
- 13.
- J. Sugie, Influence of anti-diagonals on the asymptotic stability for linear differential systems, to appear in Monatsh. Math.
- 14.
- T. Yoshizawa, Stability Theory by Liapunov's Second Method, Math. Soc. Japan, Tokyo, 1966. MR 0208086 (34:7896)
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Additional Information:
Jitsuro
Sugie
Affiliation:
Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
Email:
jsugie@riko.shimane-u.ac.jp
Yuichi
Ogami
Affiliation:
Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
PII:
S0033-569X-09-01133-X
Keywords:
Uniform stability,
asymptotic stability,
linear differential systems,
weakly integrally positive.
Received by editor(s):
May 8, 2008
Posted:
May 27, 2009
Additional Notes:
The first author was supported in part by Grant-in-Aid for Scientific Research, No. 19540182
Copyright of article:
Copyright
2009,
Brown University
The copyright for this article reverts to public domain after 28 years from publication.
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