Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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
Published electronically: May 27, 2009
MathSciNet review: 2588230
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. A. Halanay, Differential equations: Stability, oscillations, time lags, Academic Press, New York-London, 1966. MR 0216103
  • 2. Jack K. Hale, Ordinary differential equations, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. MR 0419901
    Jack K. Hale, Ordinary differential equations, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR 587488
  • 3. L. Hatvani, A generalization of the Barbashin-Krasovskij theorems to the partial stability in nonautonomous systems, Qualitative theory of differential equations, Vol. I, II (Szeged, 1979), Colloq. Math. Soc. János Bolyai, vol. 30, North-Holland, Amsterdam-New York, 1981, pp. 381–409. MR 680604
  • 4. 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
  • 5. L. Hatvani, On the asymptotic stability for a two-dimensional linear nonautonomous differential system, Nonlinear Anal. 25 (1995), no. 9-10, 991–1002. MR 1350721, 10.1016/0362-546X(95)00093-B
  • 6. A. Hurwitz, Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Math. Ann. 46 (1895), no. 2, 273–284 (German). MR 1510884, 10.1007/BF01446812
  • 7. Joseph LaSalle and Solomon Lefschetz, Stability by Liapunov’s direct method, with applications, Mathematics in Science and Engineering, Vol. 4, Academic Press, New York-London, 1961. MR 0132876
  • 8. J. C. Maxwell, On governors, Proc. Roy. Soc. London 6, 270-283 (1868).
  • 9. David R. Merkin, Introduction to the theory of stability, Texts in Applied Mathematics, vol. 24, Springer-Verlag, New York, 1997. Translated from the third (1987) Russian edition, edited and with an introduction by Fred F. Afagh and Andrei L. Smirnov. MR 1418401
  • 10. K. P. Persidskiĭ, Über die Stabilität einer Bewegung nach der ersten Näherung, Mat. Sb. 40, 284-293 (1933).
  • 11. Nicolas Rouche, P. Habets, and M. Laloy, Stability theory by Liapunov’s direct method, Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 22. MR 0450715
  • 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. Taro Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966. MR 0208086

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 34D05, 34D20, 34D23, 37B25, 37B55

Retrieve articles in all journals with MSC (2000): 34D05, 34D20, 34D23, 37B25, 37B55

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

DOI: http://dx.doi.org/10.1090/S0033-569X-09-01133-X
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.

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2016 Brown University
Comments: qam-query@ams.org
AMS Website