Preservation of uniform asymptotic stability under perturbations

Author:
R. K. Miller

Journal:
Proc. Amer. Math. Soc. **51** (1975), 155-158

MSC:
Primary 45M10

MathSciNet review:
0372563

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Abstract: Suppose the trivial solution of the interval initial value problem for a linear, convolution, Volterra integrodifferential equation is uniformly asymptotically stable. If the kernel in this equation is integrable, then it is shown that this stability is preserved under perturbations which are nonantisipative and of order two or more.

**[1]**J. M. Bownds and J. M. Cushing,*On preserving stability of Volterra integral equations under a general class of perturbations*, Math. Systems Theory**9**(1975), no. 2, 117–131. MR**0383025****[2]**S. I. Grossman and R. K. Miller,*Nonlinear Volterra integrodifferential systems with 𝐿¹-kernels*, J. Differential Equations**13**(1973), 551–566. MR**0348417****[3]**R. K. Miller,*Asymptotic stability properties of linear Volterra integrodifferential equations*, J. Differential Equations**10**(1971), 485–506. MR**0290058****[4]**-,*Asymptotic stability and perturbations for linear integrodifferential systems*, Delay and Functional Differential Equations and Their Applications, Academic Press, New York, 1972, pp. 257-268.**[5]**R. K. Miller,*Linear Volterra integrodifferential equations as semigroups*, Funckcial. Ekvac.**17**(1974), 39–55. MR**0350511**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1975-0372563-1

Article copyright:
© Copyright 1975
American Mathematical Society