Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

A Liapunov functional for linear Volterra integro-differential equations


Authors: D. L. Abrahamson and E. F. Infante
Journal: Quart. Appl. Math. 41 (1983), 35-44
MSC: Primary 45J05; Secondary 34K20
DOI: https://doi.org/10.1090/qam/700659
MathSciNet review: 700659
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Liapunov functionals of quadratic form have been used extensively for the study of the stability properties of linear ordinary, functional and partial differential equations. In this paper, a quadratic functional $ V$ is constructed for the linear Volterra integrodifferential equation

$\displaystyle \dot x\left( t \right) = Ax\left( t \right) + \int_0^T {B\left( {... ...\ge {t_0}, \\ x\left( t \right) = f\left( t \right), \qquad 0 \le t \le {t_0}} $

. This functional, and its derivative $ \dot V$, is more general than previously constructed ones and still retains desirable computational qualities; moreover, it represents a natural generalization of the Liapunov function for ordinary differential equations. The method of construction used suggests functionals which are useful for more general equations.

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

  • [1] R. K. Miller, Asymptotic stability properties of linear Volterra integrodifferential equations, J. Diff. Eq. 10, 485-506 (1971) MR 0290058
  • [2] S. I. Grossman and R. K. Miller, Nonlinear Volterra integrodifferential systems with L$ ^{1}$-kernels, J. Diff. Eq. 13, 551-566 (1973) MR 0348417
  • [3] G. Seifert, Liapunov-Razumikhin conditions for stability and boundedness of functional differential equations of Volterra type, J. Diff. Eq. 14, 424-430 (1973) MR 0492745
  • [4] G. Seifert, Liapunov-Razumikhin conditions for asymptotic stability in functional differential equations of Volterra type, J. Diff. Eq. 16, 289-297 (1974) MR 0460837
  • [5] R. Grimmer and G. Seifert, Stability properties of Volterra integrodifferential equations, J. Diff. Eq. 19, 142-166 (1973) MR 0388002
  • [6] T. A. Burton, Stability theory for Volterra equations, J. Diff. Eq. 32, 101-118 (1979) MR 532766
  • [7] T. A. Burton, Uniform stabilities for Volterra equations, J. Diff. Eq. 36, 40-53 (1980) MR 571126
  • [8] E. F. Infante and W. B. Castelan, A Liapunov functional for a matrix neutral difference-differential equation with one delay, J. Math. Anal. Appl. 71, 105-130 (1979) MR 545863
  • [9] E. F. Infante and W. B. Castelan, A Liapunov functional for a matrix difference-differential equation, J. Diff. Eq. 29, 439-451 (1978) MR 507489
  • [10] E. F. Infante and W. B. Castelan, On a functional equation arising in the stability theory of difference-differential equations, Quart. Appl. Math. 35, 311-319 (1977) MR 0492694
  • [11] L. A. V. Carvalho, E. F. Infante, and J. A. Walker, On the existence of simple Liapunov functions for linear retarded difference-differential equations, Tôhoku Math. J. 32, 283-297 (1980) MR 580283
  • [12] R. K. Miller, Nonlinear Volterra integral equations, W. A. Benjamin, Inc., Menlo Park, CA, 1971 MR 0511193
  • [13] J. K. Hale, Theory of functional differential equations, Springer-Verlag, New York, 1977 MR 0508721
  • [14] F. R. Gantmacher, The theory of matrices, Vol. 1, Chelsea, New York, 1977 MR 1657129

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 45J05, 34K20

Retrieve articles in all journals with MSC: 45J05, 34K20


Additional Information

DOI: https://doi.org/10.1090/qam/700659
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society