Quasipolynomial matrices connected with stable systems of difference-differential equations of neutral type

This paper deals with stability problems for an important class of differential equations of neutral type and functional difference equations with constant delays, continuous time and matrix coefficients. Based on previous work of the second author, we give an adapted proof of a known point-wise characterization of the closure set of the real parts of the characteristic roots associated with such systems for the case where all delays vary independently of each other. We also provide explicit methods to construct quasipolynomial matrices whose characteristic roots coincide with the zeros of exponential polynomials with nonzero frequencies linearly independent over the rational numbers. As a main result, sufficient conditions for stability are derived from the spectral analysis of such classes of quasipolynomial matrices. These results are illustrated by means of several examples.