Construction of Optimal Expressions for Transition Matrices of a Class of Double – Delay Scalar Differential Equations

Abstract

This paper constructed optimal expressions for solution matrices of a class of double – delay autonomous linear differential equations on arbitrary intervals of length equal to the delay h for non –negative time periods. The proof was achieved using skilful combinations of summation notations, multinomial distribution, greatest integer functions, change of variables techniques, multiple integrals, as well as the method of steps to obtain these matrices on successive intervals of length equal to the delay h. This theorem globally extends the time scope of applications of these matrices to the solutions of initial function problems, rank conditions for controllability and cores of targets, constructions of controllability Grammians and admissible controls for transfers of points associated with controllability problems.