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Integral conditions on the uniform asymptotic stability for two-dimensional linear systems with time-varying coefficients


Authors: Jitsuro Sugie and Masakazu Onitsuka
Journal: Proc. Amer. Math. Soc. 138 (2010), 2493-2503
MSC (2010): Primary 34D05, 34D20; Secondary 34D23, 37C75
Published electronically: February 24, 2010
MathSciNet review: 2607879
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the uniform asymptotic stability of the zero solution of the linear system $ \mathbf{x}' = A(t)\mathbf{x}$ with $ A(t)$ being a $ 2\times2$ matrix. Our result can be used without knowledge about a fundamental matrix of the system.


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  • 1. F. Brauer and J. Nohel, The Qualitative Theory of Ordinary Differential Equations, W. A. Benjamin, New York-Amsterdam, 1969; (revised) Dover, New York, 1989.
  • 2. W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965. MR 0190463
  • 3. Jane Cronin, Differential equations, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 180, Marcel Dekker, Inc., New York, 1994. Introduction and qualitative theory. MR 1275827
  • 4. Jack K. Hale, Ordinary differential equations, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. MR 0419901
  • 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. D. W. Jordan and P. Smith, Nonlinear ordinary differential equations, 3rd ed., Oxford Texts in Applied and Engineering Mathematics, vol. 2, Oxford University Press, Oxford, 1999. An introduction to dynamical systems. MR 1743361
  • 7. Oskar Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), no. 1, 703–728 (German). MR 1545194, 10.1007/BF01194662
  • 8. Jitsuro Sugie, Convergence of solutions of time-varying linear systems with integrable forcing term, Bull. Aust. Math. Soc. 78 (2008), no. 3, 445–462. MR 2472280, 10.1017/S000497270800083X
  • 9. Ferdinand Verhulst, Nonlinear differential equations and dynamical systems, Universitext, Springer-Verlag, Berlin, 1990. Translated from the Dutch. MR 1036522

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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

Masakazu Onitsuka
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
Address at time of publication: General Education, Miyakonojo National College of Technology, Miyakonojo 885-8567, Japan
Email: onitsuka@math.shimane-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10274-3
Keywords: Uniform asymptotic stability, linear systems, Coppel's criterion, Floquet theory, integrally positive
Received by editor(s): July 16, 2009
Received by editor(s) in revised form: October 30, 2009
Published electronically: February 24, 2010
Communicated by: Yingfei Yi
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.