A Liapunov functional for linear Volterra integro-differential equations

Liapunov functionals of quadratic form have been used extensively for the study of the stability properties of linear ordinary, functional and partial differential equations. In this paper, a quadratic functional $V$ is constructed for the linear Volterra integrodifferential equation \[ \dot x\left ( t \right ) = Ax\left ( t \right ) + \int _0^T {B\left ( {t - \tau } \right )x\left ( \tau \right ) dt, \qquad t \ge {t_0}, \\ x\left ( t \right ) = f\left ( t \right ), \qquad 0 \le t \le {t_0}} \]. This functional, and its derivative $\dot V$, is more general than previously constructed ones and still retains desirable computational qualities; moreover, it represents a natural generalization of the Liapunov function for ordinary differential equations. The method of construction used suggests functionals which are useful for more general equations.