Asymptotic properties of delayed matrix exponential functions via Lambert function

Abstract

In the case of first-order linear systems with single constant delay and with constant matrix, the application of the well-known by step method (when ordinary differential equations with delay are solved) has recently been formalized using a special type matrix, called delayed matrix exponential. This matrix function is defined on the intervals \begin{document} $(k-1)τ≤q t , \begin{document} $k=0,1,\dots$ \end{document} (where \begin{document} $τ>0$ \end{document} is a delay) as different matrix polynomials, and is continuous at nodes \begin{document} $t=kτ$ \end{document} . In the paper, the asymptotic properties of delayed matrix exponential are studied for \begin{document} $k\to∞$ \end{document} and it is, e.g., proved that the sequence of values of a delayed matrix exponential at nodes is approximately represented by a geometric progression. A constant matrix has been found such that its matrix exponential is the quotient factor that depends on the principal branch of the Lambert function. Applications of the results obtained are given as well.