An Equation of Volterra

Abstract

Consider the system

$${\rm{A\:x}}\left( {\rm{t}} \right) + {\rm{Bx}}\left( {\rm{t}} \right) = \int_0^{\rm{r}} {{\rm{F}}\left( \theta \right){\rm{x}}\left( {{\rm{t}} - \theta } \right)} {\rm{d}}\theta $$

((15.1))

where A,B,F are symmetric n × n matrices and F is continuously differentiable. Let

$${\rm{M}} = {\rm{B}} - \int_0^{\rm{r}} {{\rm{F}}\left( \theta \right){\rm{d}}\theta }$$

and write (15.1) as

$$\begin{array}{*{20}c} {{\rm{\dot x}}\left( {\rm{t}} \right)} & = & {{\rm{y}}\left( {\rm{t}} \right),} \\ {{\rm{A\dot y}}\left( {\rm{t}} \right)} & = & { - {\rm{Mx}}\left( {\rm{t}} \right) + \int_0^{\rm{r}} {{\rm{F}}\left( \theta \right)\left[ {{\rm{x}}\left( {{\rm{t - }}\theta } \right) - {\rm{x}}\left( {\rm{t}} \right)} \right]{\rm{d}}\theta} } \\ \end{array}$$

((15.2))

.