Survey on Analysis of Time Delayed Systems via the Lambert W Function

This paper summarizes recent research on an approach for the analytical solution to systems of delay differential equations developed using the Lambert W function [26, 29], and its applications [25, 28]. Also, several outstanding research problems associated with the solution of systems of delay differential equations using the matrix Lambert W function are highlighted. The solution has the form of an infinite series of modes written in terms of the matrix Lambert W function. The essential advantage of this approach is the similarity to the concept of the state transition matrix in linear ordinary differential equations, enabling its use for general classes of linear delay differential equations. This similarity makes it possible to extend some of control methods for ordinary differential equations to delay differential equations.