Approximation of the State of Infinite Delay and Volterra-Type Equations

Abstract

In this paper we describe a general scheme for the dynamical approximation of infinite delay equations. By this we understand that the original delay equation, which has its natural formulation as an abstract Cauchy problem in an infinite dimensional state space (compare (2.7) below), will be approximated by a sequence of differential equations in finite dimensional Euclidean spaces. There are several reasons for our interest in such approximation results. One of them is that they lead to numerically implementable schemes and another one is that these approximations are known to preserve qualitative properties of the original delay equation in the case of finite delays, as for instance location of the spectrum and bifurcation properties [1,6]. Although we have good reason to believe that the situation is similar for infinite delay equations we shall not pursue this second aspect any further in this paper. Rather we describe in some detail how the general scheme that is presented can be used for approximation of the state by employing subspaces of linear spline functions. In a forthcoming paper we shall rely on these results in order to discuss identification and control problems associated with delay equations. It is well known (see [1,2] among others) that approximation results similar to the one presented here have proven to be very efficient for such system theoretical problems in the case of finite delays.