Time-delay systems occupy a place of central importance in all areas of science. Time-delays are often related to Physico-chemical processes, electric networks, hydraulic networks, heredity in population growth, the economy, and other related industries. Such time-delay systems can be represented by delay diﬀerential equations, delay discrete equations, or delay fractional diﬀerential equations. Some real-world problems can be modeled in a more accurate way by including time-delays. Such systems are often used to model phenomena in scientiﬁc and technological problems. These models are used in computer engineering, viscoelastic systems, diﬀusion processes, signal analysis, biology, forced oscillations, control theory, disease modeling, ﬁnance, and population dynamics. Recently, several attempts have been made to ﬁnd an analytical solution for time-delay systems under diﬀerent conditions, which leads to results on stability analysis, control problems, observability, and iterative learning control for linear or nonlinear continuous delay systems, discrete delay systems, impulsive delay systems or fractional order delay systems. The ﬁrst objective of this work is to obtain the exact solutions of linear or nonlinear continuous delay systems, discrete delay systems, and fractional order delay systems. After that, as an application, the representation of solutions of these systems is used to derive the ﬁnite time stability, Hyers-Ulam stability, and controllability results. To prove the eﬀectiveness of the proposed approach, the obtained results will be illustrated by applications, and compared with the outcomes in the existing literature.

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