Formulations and Proofs of Optimal Expressions for Control Index Matrices for a Class of Double – Delay Differential Equations.

Abstract

This paper derived the structure of the indices of control systems for a class of double – delay autonomous linear differential equations on any given interval of length equal to the delay h for non –negative time periods. The formulations and the development of the theorem relied on an earlier work by Ukwu (2013i) on the interval 11 4 [ , ] h tt  . The derivation of the associated solution matrices exploited the continuity of these matrices for positive time periods, the method of steps and backward continuation recursions to obtain these matrices on successive intervals of length equal to the delay h. The proofs were achieved using ingenious combinations of summation notations, greatest integer functions and multiple integrals. The indices were derived using the stage – wise algorithmic format, starting from the right – most interval of length h. Our results globally extend the time scope of applications of these matrices to the solutions of terminal function problems and rank conditions for controllability and cores of targets.