On well-posedness of integro-differential equations in weighted $L^2$-spaces

Abstract

In this paper we consider the problem of constructing a well-posed state space model for a class of singular integro-differential equations of neutral type. The work is motivated by the need to develop a framework for the analysis of numerical methods for designing control laws for aeroelastic systems. Semigroup theory is used to establish existence and well-posedness results for initial data in weighted $L^{2}-$spaces. It is shown that these spaces lead naturally to the dissipativeness of the basic dynamic operator. The dissipativeness of the solution generator combined with the Hilbert space structure of these weighted spaces make this choice of a state space more suitable for use in the design of computational methods for control than previously used product spaces.