On discretizing the semigroup of solution operators for linear time invariant - time delay systems

in this paper we give an account of the basic facts to be considered when one attempts to discretize the semigroup of solution operators for Linear Time Invariant - Time Delay Systems (LTI-TDS). Two main approaches are presented, namely pseudospectral and spectral, based respectively on classic interpolation when the state space is C = C(–ü, 0;C) and generalized Fourier projection when the state space is X = C × L2(–ü, 0;C). Full discretization details for constructing the approximation matrices are given. Moreover, concise, yet fundamental, convergence results are discussed, with particular attention to their similarities and differences as well as pros and cons with regards to solution approximation and asymptotic stability detection.