Stability of a linear integro-differential equation with periodic coefficients

Abstract

Stability of a linear integro-differential equation with periodic coefficients is studied. Such an equation arises in the dynamics of thin-walled viscoelastic elements of structures under periodic compressive loading. The equation under consideration has a specific peculiarity which makes its analysis difficult: in the absence of the integral term it is only stable, but not asymptotically stable. Therefore, in order to derive stability conditions we have to introduce some specific restrictions on the behavior of the kernel of the integral operator. These restrictions are taken from the analysis of the relaxation measures for a linear viscoelastic material.