Stability and hyperbolicity of linear systems with delayed state: a matrix-pencil approach

This note focuses on the problem of asymptotic stability and hyperbolicity of a class of linear systems described by delay-differential equations including commensurable delays. An unitary approach for the considered problems is proposed via a matrix-pencil technique. Necessary and sufficient conditions, delay-independent or delay-dependent, are given in terms of the generalized eigenvalue distribution of two constant and regular matrix pencils: one associated with finite time delays and the other one associated with infinite delay. Furthermore, the proposed results are easy to check in numerical examples. An example from the neural-network field has also been considered.