In this thesis an efficient approximation scheme is successfully developed for solving linear optimal control problems with a piecewise constant delay function. Time delays are frequently encountered in various fields of science and engineering such as chemical processes, transmission lines, power systems, biological methods, communication networks, economics and finance. It is known that except for some simple cases. It is extremely difficult or impossible to derive. A closed-form solution for such a class of systems. Indeed, time-delay systems is one of the most important subject in optimal control theory. The situation becomes much more complicated when time-delay is a piecewise constant function. Accordingly, an efficient and effective numerical method has to be adopted. Up to now, many research works have been dedicated to the numerical treatment and theoretical aspects of optimal control problems with a constant delay. To our knowledge, linear optimal control problems involving a piecewise constant function has not yet been considered in the literature. The main purpose of this thesis is to propose a hybrid approximation method for solving the mentioned problems. The method is based on a hybrid of block-pulse functions and Legendre polynomials. It is worth noting that the exact solution of a delayed optimal control problems is a piecewise smooth function. Accordingly, none of the smooth basis functions is able to provide a satisfactory approximation for such a class of systems. It should be mentioned that the approximation of a piecewise smooth function by a finite number of smooth functions often fails to converge. This is due to the existence of the well-known Gi phenomenon. The exact response of these systems cannot be obtained solely either by block-pulse functions or by Legendre polynomials. Combining block-pulse functions and Legendre polynomials allows one to simultaneously make use of the best characteristics of the two mentioned bases