INSTABILITY IN THE CRITICAL CASE OF A PAIR OF PURE IMAGINARY ROOTS FOR A CLASS OF SYSTEMS WITH AFTEREFFECT

The stability of motion of a system described by Volterra integrodiffe rential equations is investigated in the critical case when the charac teristic equation has a pair of pure imaginary roots. Conditions for i nstability, analogous to the well-known conditions from the theory of differential equations [1], are derived. (A similar result was establi shed previously in [2] for integro differential equations of simpler s tructure with integral kernels of exponential-polynomial type.) For th e proof, several manipulations are used to simplify the original equat ion and, in particular, to reduce the linearized equation to the form of a differential equation with constant diagonal matrix. (An analogou s approach was used to analyse instability for Volterra integrodiffere ntial equations in the critical case of one zero root in [3, 4].) As a n example, the sign of the Lyapunov constant in the problem of the rot ational motion of a rigid body with viscoelastic supports is calculate d. (C) 1998 Elsevier Science Ltd. All rights reserved.