Acta Mathematica Hungarica
Volume/Issue:
Volume 134: Issue 3
On the asymptotic stability for impulsive functional differential equations
Authors:
Fangfang JiangDepartment of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang, 310036, China

Search for other papers by Fangfang Jiang in
Current site
Google Scholar
PubMedClose
and
Jianhua ShenDepartment of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang, 310036, China

Search for other papers by Jianhua Shen in
Current site
Google Scholar
PubMedClose
Pages:
307–321
Online Publication Date:
09 Dec 2011
Publication Date:
01 Feb 2012
Article Category:
Research Article
DOI:
https://doi.org/10.1007/s10474-011-0180-5
Keywords:
impulsive functional differential equation; Lyapunov functional; asymptotic stability; 34K15; 34K20
Restricted access

Abstract

We consider the asymptotic stability problems by Lyapunov functionals V for a class of functional differential equations with impulses of the form x′(t)=f(t,xt), x∊Rn, t≧t0, t≠tk; △x=Ik(t,x(t−)), t=tk, k∊Z+. Some new asymptotic stability results are presented by using an idea originated by Burton and Makay [6] and developed by Zhang [1]. We generalize some known results about impulsive functional differential equations in the literature in which we only require the derivative of V to be negative definite on a sequence of intervals In=[sn,ξn] which may or may not be contained in the sequence of impulsive time intervals [tn,tn+1).

  • [1] Anokhin, A. V. Berezansky, L. Braverman, E. 1995 Exponential stability of linear delay impulsive differential equations J. Math. Anal. Appl. 193 923–941 .

    • Crossref
    • Search Google Scholar
    • Export Citation

  • [2] Bainov, D. D. Covachev, V. Stamova, I. 1994 Stability under persistent disturbances of impulsive differential-difference equations of neutral type J. Math. Anal. Appl. 187 790–808 .

    • Crossref
    • Search Google Scholar
    • Export Citation

  • [3] Ballinger, G. Liu, X. 1999 Existence and uniqueness results for impulsive delay differential equations Dynamics of Continuous, Discrete and Impulsive Systems 5 579–591.

    • Search Google Scholar
    • Export Citation

  • [4] Burton, T. A. 1978 Uniform asymptotic stability in functional differential equations Proc. Amer. Math. Soc. 68 195–199 .

  • [5] Burton, T. A. 1985 Stability and Periodic Solutions of Ordinary and Functional Differential Equations Academic Press Orlando.

  • [6] Burton, T. A. Hatvani, L. 1989 Stability theorems for nonautonomous functional differential equations by Lyapunov functional Tohoku Math. J. 41 65–104 .

    • Crossref
    • Search Google Scholar
    • Export Citation

  • [7] Burton, T. A. Hatvani, L. 1990 On nonuniform asymptotic stability for nonautonomous functional differential equations Differential and Integral Equations 2 285–293.

    • Search Google Scholar
    • Export Citation

  • [8] Burton, T. A. Makay, G. 1994 Asymptotic stability for functional differential equations Acta Math. Hungar. 65 243–251 .

  • [9] Hale, J. K. 1997 Theory of Functional Differential Equations Springer-Verlag New York.

  • [10] Hatvani, L. 1997 Annulus arguments in the stability theory for functional differential equations Differential and Integral Equations 10 975–1002.

    • Search Google Scholar
    • Export Citation

  • [11] Hatvani, L. 2002 On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals Trans. Amer. Math. Soc. 354 3555–3571 .

    • Crossref
    • Search Google Scholar
    • Export Citation

  • [12] Kato, J., A conjecture in Lyapunov method for functional differential equations, preprint.

  • [13] Krasovskii, N. N. 1959 Some Problems in the Theory of Stability of Motion Gosudarstv. Izdat. Fiz. Mat. Lit. Moscow in Russian.

  • [14] Shen, Jianhua 1996 Existence and uniqueness of solutions for impulsive functional differential equations on the PC space with applications Acta Sci. Nat. Uni. Norm. Hunan 24 285–291 in Chinese.

    • Search Google Scholar
    • Export Citation

  • [15] Shen, J. Li, J. 2005 Impulsive control for stability of Volterra functional differential equations J. for Anal. and its Appl. 24 721–734.

    • Search Google Scholar
    • Export Citation

  • [16] Shen, J. Luo, Z. 1999 Impulsive stabilization of functional differential equations via Lyapunov functionals J. Math. Anal. Appl. 240 1–15 .

    • Crossref
    • Search Google Scholar
    • Export Citation

  • [17] Yoshizawz, T. 1966 Stability by Lyapunov's Second Method Math. Soc. Japan Tokyo.

  • [18] Zhang, B. 1995 Asymptotic stability in functional differential equations by Lyapunov functionals Trans. Amer. Math. Soc. 347 1375–1382 .

  • Collapse
  • Expand
Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632

To see the editorial board, please visit the website of Springer Nature.

Manuscript Submission: HERE

For subscription options, please visit the website of Springer Nature.

Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation		 1950
Volumes
per Year		 3
Issues
per Year		 6
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Aug 2022 0 0 0
Sep 2022 0 0 0
Oct 2022 0 0 0
Nov 2022 0 0 0
Dec 2022 0 0 0
Jan 2023 1 0 0
Feb 2023 0 0 0