Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

Stability conditions for linear nonautonomous delay differential equations


Authors: Stavros N. Busenberg and Kenneth L. Cooke
Journal: Quart. Appl. Math. 42 (1984), 295-306
MSC: Primary 34K20
DOI: https://doi.org/10.1090/qam/757167
MathSciNet review: 757167
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We derive new sufficient conditions for uniform asymptotic stability of the zero solution of linear non-autonomous delay differential equations. The equations considered include scalar equations of the form

$\displaystyle x'\left( t \right) = - c\left( t \right)x\left( t \right) + \sum\limits_{i = 1}^n {{b_i}\left( t \right)x\left( {t - {T_i}} \right)} $

where $ c\left( t \right)$, $ {b_i}\left( t \right)$ are continuous for $ t \ge 0$ and $ {T_i}$ is a positive number $ (i = 1, 2,...,n)$, and also systems of the form

$\displaystyle x'(t) = B(t)x(t - T) - C(t)x(t)$

where $ B(t)$) and $ C(t)$ are $ n \times n$ matrices. The results are found by using the method of Lyapunov functionals.

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

  • [1] T. A. Burton, Stability theory for delay equations, Funkcialaj Ekvacioj 22, 67-76 (1979) MR 551253
  • [2] S. Busenberg and K. L. Cooke, Periodic solutions of a periodic nonlinear delay differential equation, SIAM J. Appl. Math. 35, 704-721 (1978) MR 512180
  • [3] L. A. V. Carvalho, E. F. Infante and J. A. Walker, On the existence of simple Lyapunov functions for linear retarded difference-differential equations, Tohoku Math. J. 32, 283-297 (1980) MR 580283
  • [4] J. Dyson and R. Villella-Bressan, Functional differential equations and non-linear evolution operators, Proc. Roy. Soc. Edinburgh 75A, 223-234 (1975/76) MR 0442402
  • [5] J. K. Hale, Theory of functional differential equations, Applied Math. Sciences, Vol. 3, Springer-Verlag, New York 1977 MR 0508721
  • [6] R. M. Lewis and B. Anderson, Insensitivity of a class of nonlinear compartmental systems to the introduction of arbitrary time delays, IEEE Trans. on Circuits and Systems, Vol. CAS27 604-612 (1980) MR 581411

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 34K20

Retrieve articles in all journals with MSC: 34K20


Additional Information

DOI: https://doi.org/10.1090/qam/757167
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society