Discretization of solution operators for linear time invariant - Time delay systems in Hilbert spaces

In this paper a numerical scheme to discretize the solution operators of linear time invariant - time delay systems is proposed and analyzed. Following previous work of the authors on the classic state space of continuous functions, here the focus is on working in product Hilbert state spaces. The method is based on a combination of collocation and Fourier projection. Full discretization details for constructing the approximation matrices are given for the sake of implementation. Moreover, convergence results are proved and discussed, with particular attention to their pros and cons with regards to fundamental targets such as time-integration and detection of asymptotic stability of equilibria.