Stability in systems with aftereffect when there are singularities in the integral kernels

Abstract

Systems with aftereffect are considered. The state of these systems is described by integrodifferential equations of the Volterra type, which depend on functionals in integral form and, in particular, on analytic functionals which are represented by Frechet series. The integral kernels can allow of singularities of Abel kernel singularities. The total stability (i.e. stability under persistent disturbances) is investigated, and the structure of the general solution is investigated in the neighbourhood of zero for an equation with a holomorphic non-linearity assuming asymptotic stability of the trivial solution of the linearized unperturbed equation. The conditions for instability are given in the critical case of a single zero root, which generalise results obtained previously.