Asymptotic stability for three-dimensional linear differential systems with time-varying coefficients
Authors:
Jitsuro Sugie and Yuichi Ogami
Journal:
Quart. Appl. Math. 67 (2009), 687-705
MSC (2000):
Primary 34D05, 34D20, 34D23; Secondary 37B25, 37B55
DOI:
https://doi.org/10.1090/S0033-569X-09-01133-X
Published electronically:
May 27, 2009
MathSciNet review:
2588230
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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
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- László Hatvani, On the uniform attractivity of solutions of ordinary differential equations by two Lyapunov functions, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), no. 5, 162–167. MR 1114963
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References
- A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York and London, 1966. MR 0216103 (35:6938)
- 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)
- 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)
- 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)
- L. Hatvani, On the asymptotic stability for a two-dimensional linear nonautonomous differential system, Nonlinear Anal. 25, 991–1002 (1995). MR 1350721 (96k:34105)
- A. Hurwitz, Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Math. Ann. 46, 273–284 (1895). MR 1510884
- 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)
- J. C. Maxwell, On governors, Proc. Roy. Soc. London 6, 270–283 (1868).
- 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)
- K. P. Persidskiĭ, Über die Stabilität einer Bewegung nach der ersten Näherung, Mat. Sb. 40, 284–293 (1933).
- 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)
- E. J. Routh, Treatise on the Stability of a Given State of Motion, Macmillan, London, 1877.
- J. Sugie, Influence of anti-diagonals on the asymptotic stability for linear differential systems, to appear in Monatsh. Math.
- 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
MR Author ID:
168705
Email:
jsugie@riko.shimane-u.ac.jp
Yuichi Ogami
Affiliation:
Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
Keywords:
Uniform stability,
asymptotic stability,
linear differential systems,
weakly integrally positive.
Received by editor(s):
May 8, 2008
Published electronically:
May 27, 2009
Additional Notes:
The first author was supported in part by Grant-in-Aid for Scientific Research, No. 19540182
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.
