Stability of integro-differential equations with periodic operator coefficients

Stability of the zero solution is studied for a linear integro-differential equation with operator coefficients. The coefficients are assumed to be linear, selfadjoint, commuting operators explicitly depending on time. A new method for the stability analysis is derived which employs the Lyapunov approach on the one hand, and the frequency-domain technique on the other hand. New stability functionals are developed accounting for some properties of kernels of the integral operators. These properties reflect specific features of relaxation measures for a wide range of viscoelastic materials. Using these functionals, explicit restrictions are obtained for time-varying operators. These restrictions provide a fair estimation of the stability region for the eigenvalues with large numbers, but are rather far from necessary conditions of stability for the first eigenvalues. To make more precise the stability conditions for the eigenvalues with small numbers, a frequency-domain technique is used. The results obtained are applied to the stability problem for a viscoelastic bar under compressive longitudinal forces periodic in time. Explicit expressions are derived for the critical load. The effect of rheological and geometrical parameters on the critical load is studied both analytically and numerically.