Time delays are frequently encountered in various fields of applications. Many real-life phenomena and practical systems can be described by delay differential equations. Time delays arise in diverse areas of science and engineering such as nuclear reactions, rolling mills, hydraulic systems, chemical processes, transmission lines, robotics and communication networks. It is noted that the exact solution of a time-delay system is a piecewise smooth function. For this reason, piecewise constant basis functions such as block-pulse functions or continuous basis functions including Lagrange interpolating polynomials cannot provide a satisfactory approximation for the mentioned systems. In this thesis, a hybrid approximation scheme is successfully employed for solving linear optimal control problems with inequality constraints on the state and control variables. This thesis includes two parts. The first part is devoted to the numerical solution of constrained linear optimal control problems with a quadratic performance index. The second part is concerned with the numerical investigation of linear delayed optimal control problems with inequality constraints on the state and control variables. Numerical methods for solving optimal control problems are classified into two categories: direct methods and indirect methods. The foundation of the indirect methods is based on the necessary conditions of optimality resulting from Pontryagin’s maximum principle. These methods are collectively called indirect methods. There are many successful implementation of indirect methods in the literature. Although indirect methods enjoy some nice properties, they suffer from many drawbacks. For instance, the boundary value problem resulting from the necessary optimality conditions is extremely sensitive to initial guesses. Furthermore, these necessary conditions must be explicitly derived. Over the past two decades, an alternative approach based on discrete approximations has gained wide popularity. The main idea of this method is to discretize the optimal control problem and solve the resulting finite-dimensional optimization problem. The simplicity of direct methods belies a wide range of deeply theoretical issues that lie at the intersection of approximation theory, control theory and optimization. The method implemented in this thesis is based on direct approach using a hybrid of block-pulse functions and the well-known Lagrange interpolating polynomials. A key advantage of the hybrid functions is the good representation of piecewise smooth functions. Because hybrid functions consists of block-pulse functions and Lagrange interpolating polynomials which are both complete, the set of hybrid functions is a complete set in the Hilbert space $L^{2} [0, t_f)$. The excellent properties of the hybrid functions together with the associated operational matrices of integration and delay are used to transform the optimal control problem under consideration into a parameter optimization problem whose solution is much easier than original one. The mentioned operational matrices are sparse. The sparsity of these matrices makes the proposed approach computationally attractive. In addition, a penalty function method is applied to convert the inequality constraints on the state and control variables into equality constraints. The resulting optimization problem is then solved by the well-known Lagrange multipliers method. Various types of optimal control problems are included to demonstrate the efficiency, accuracy and applicability of the proposed numerical method.