Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): Jitsuro Sugie; Masakazu Onitsuka
Journal: Proc. Amer. Math. Soc. 138 (2010), 2493-2503.
MSC (2010): Primary 34D05, 34D20; Secondary 34D23, 37C75
Posted: February 24, 2010
MathSciNet review: 2607879
Retrieve article in: PDF

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 being a $2\times2$ matrix. Our result can be used without knowledge about a fundamental matrix of the system.

References:

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, Heath, Boston, 1965. MR 0190463 (32:7875)
3.: J. Cronin, Differential Equations: Introduction and Qualitative Theory, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, 180, Marcel Dekker, New York-Basel-Hong Kong, 1994. MR 1275827 (95b:34001)
4.: J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York-London-Sydney, 1969; (revised) Krieger, Malabar, 1980. MR 0419901 (54:7918)
5.: L. Hatvani, On the asymptotic stability for a two-dimensional linear nonautonomous differential system, Nonlinear Anal., 25 (1995), 991-1002. MR 1350721 (96k:34105)
6.: D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, 3rd ed., Oxford Texts in Applied and Engineering Mathematics, 2, Oxford University Press, Oxford, 1999. MR 1743361 (2000j:34001)
7.: O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Zeits., 32 (1930), 703-728. MR 1545194
8.: J. Sugie, Convergence of solutions of time-varying linear systems with integrable forcing term, Bull. Austral. Math. Soc., 78 (2008), 445-462. MR 2472280 (2009k:34102)
9.: F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York-Berlin-Heidelberg, 1990. MR 1036522 (91b:34002)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 34D05, 34D20, 34D23, 37C75

Retrieve articles in all Journals with MSC (2010): 34D05, 34D20, 34D23, 37C75

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: 10.1090/S0002-9939-10-10274-3
PII: S 0002-9939(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
Posted: February 24, 2010
Communicated by: Yingfei Yi
Copyright of article: Copyright 2010, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|