Analysis of Systems of Linear Delay Differential Equations Using the Matrix Lambert Function and the Laplace Transformation

An approach for the analytical solution, free and forced, to systems of delay differential equations (DDEs) has been developed using the matrix Lambert function. To generalize the Lambert function solution for scalar DDEs to systems of DDEs, we introduce a new matrix, Q when the coefficient matrices in a system of DDEs do not commute. The solution is in the form of an infinite series of modes written in terms of the matrix Lambert function using Q. The essential advantage of this approach is the similarity with the concept of the state transition matrix in linear ordinary differential equations (ODEs), enabling its use for general classes of linear delay differential equations. Examples are presented to illustrate the new approach by comparison to numerical methods. The analytical solution to systems of DDEs in terms of the Lambert function is also presented in the Laplace domain to reinforce the analogy to ODEs.