Preservation of uniform asymptotic stability under perturbations

Miller, R. K.

doi:10.1090/S0002-9939-1975-0372563-1

Preservation of uniform asymptotic stability under perturbations
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by R. K. Miller PDF

Proc. Amer. Math. Soc. 51 (1975), 155-158 Request permission

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.

References

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 383025, DOI 10.1007/BF01704015
S. I. Grossman and R. K. Miller, Nonlinear Volterra integrodifferential systems with $L^{1}$-kernels, J. Differential Equations 13 (1973), 551–566. MR 348417, DOI 10.1016/0022-0396(73)90011-9
R. K. Miller, Asymptotic stability properties of linear Volterra integrodifferential equations, J. Differential Equations 10 (1971), 485–506. MR 290058, DOI 10.1016/0022-0396(71)90008-8

Asymptotic stability and perturbations for linear integrodifferential systems

R. K. Miller, Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac. 17 (1974), 39–55. MR 350511