الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
In this dissertation, after obtaining the exact solutions of time-delay systems, the representation of solutions of these systems is used as an application to derive the Hyers-Ulam stability and controllability results. To prove the effectiveness of the proposed approach, the obtained results will be compared with the outcomes in the existing literature. The dissertation contributions are summarized as follows:For the delayed linear discrete systems with permutable or non-permutable matrices and second-order differences, we prove and derive explicit solutions of the delayed linear
discrete systems by using Z-transform. Examples are given to illustrate the main results
that we improve and extend previous ones.For the linear differential systems with pure delay and multiple delays with linear parts given by non-permutable matrices, we obtain the representation of solutions of linear differential systems with pure delay using new delayed matrix functions. Furthermore, we get the representation of solutions of nonhomogeneous linear systems of second order differential equations with multiple delays with linear parts given by non-permutable matrices via the Laplace transform. Finally, we give an example to illustrate the main results
that improve and extend previous ones.-We obtain the representation of solutions of nonhomogeneous systems of linear fractional equations with pure delay using the delayed matrix Mittag-Leffler functions. After that, we derive the representation of a solution of linear fractional systems with multiple delays with linear parts given by permutable or non-permutable matrices via the Laplace
transform. Finally, we give an example to illustrate the main results that are novel, and
we extend and improve some existing ones.We establish and prove sufficient and necessary conditions of controllability of linear differential delay systems by introducing a delay Gramian matrix. Furthermore, we form some sufficient conditions of controllability and Hyers-Ulam stability of nonlinear differential delay systems by applying Krasnoselskii’s fixed point theorem. Finally, we
give examples to illustrate the main results that we improve, extend, and complement
some previous ones by removing some restrictive conditions. The results are applicable to arbitrary matrices.