In this thesis an efficient approximation scheme is successfully developed for solving linear optimal control problems containing delays with a quadratic performance index. Time delays are frequently encountered in various fields of applications. Many real-life phenomena and practical systems can be modeled by various types of delay differential equations. Typical examples are aerospace engineering, transmission lines, chemical processes, climate models, population dynamics, robotics, nuclear reactors, economics, communication networks, biological models, manufacturing processes and power systems. The optimal control problem for linear systems with delays is still open, depending on the delay type, specific system equation, criterion, etc. It is known that except for some special cases, it is either difficult or impossible to obtain a closed-form solution for delay di?erential equations. Indeed, time-delay systems is one of the most important subject in optimal control theory. The application of Pontryagin’s maximum principle to the optimization of control systems with time delays, results in a system of coupled two-point boundary value problem involving both delayed and advanced terms. Therefor, the main object of all computational aspects of optimal control of time-delay systems has been to devise a methodology to avoid the solution of the mentioned two-point boundary value problem. It is worth noting that the corresponding solution to this ltr" A wide variety of optimal control problems with delays are investigated to verify the efficiency, accuracy and applicability of the proposed numerical scheme. The numerical results obtained by the developed approach are compared with the existing results in the literature . The proposed method is simple, easy to implement and provides very accurate results.