STABILITY OF INTEGRODIFFERENTIAL EQUATIONS WITH PERIODIC OPERATOR COEFFICIENTS

Stability of the zero solution is studied for a linear integro-differe ntial equation with operator coefficients. The coefficients are assume d to be linear, selfadjoint, commuting operators explicitly depending on time. A new method for the stability analysis is derived which empl oys the Lyapunov approach on the one hand, and the frequency-domain te chnique on the other hand. New stability functionals are developed acc ounting for some properties of kernels of the integral operators. Thes e properties reflect specific features of relaxation measures for a wi de range of viscoelastic materials. Using these functionals, explicit restrictions are obtained for time-varying operators. These restrictio ns provide a fair estimation of the stability region for the eigenvalu es with large numbers, but are rather far from necessary conditions of stability for the first eigenvalues. To make more precise the stabili ty conditions for the eigenvalues with small numbers, a frequency-doma in technique is used. The results obtained are applied to the stabilit y problem for a viscoelastic bar under compressive longitudinal forces periodic in time. Explicit expressions are derived for the critical l oad. The effect of rheological and geometrical parameters on the criti cal load is studied both analytically and numerically.