Analysis , identification and optimal control of time-delay systems have been of considerable concern . Delays occur frequently in many practical systems and different branches of engineering and sciences such as chemical processes , transmission lines , robotics , communication networks , manufacturing and power systems . Therefore time-delay systems are very important as their analysis , identification , stability and optimization to many investigators . The presence of delay makes analysis and control design much more complicated . 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 whose solution except in some special cases is very difficult . Therefore , 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 . Orthogonal functions have been widely used to solve various problems of dynamic systems . The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations , thus greatly simplifying the problem . Up to now many research works have been devoted to the numerical treatment and theoretical analysis of various types of linear delay systems . The operational matrices of delay , product and integration are sparse matrices , hence making the method computationally attractive without sacrificing the accuracy of the solution . In recent years , different types of hybrid functions have been successfully applied for solving various types of problems arising in diverse areas of engineering and science . An essential property of hybrid function is the good representation of smooth and especially piecewise smooth functions by finite hybrid expansion . Typical examples are Walsh functions , block-pulse , Legendre polynomials , Chebyshev polynomials and Fourier series . In this thesis , we are concerned with the analysis , identification and optimal control of linear multi-delay systems with a quadratic performance index . The presented approach is based on direct method using a hybrid of block-pulse functions and Legendre polynomials . The first part of the thesis is devoted to the numerical treatment of linear multi-delay systems . The second part of the current thesis is relevant to parameter identification of time-invariant multi-delay systems . Finally , the optimal control of multi-delay systems is discussed . To solve optimal control of multi-delay systems; an effective numerical method is successfully implemented . The excellent properties of hybrid functions together with the associated operational matrices of integration , product and delay are then used to transform the mentioned optimal control problem into a mathematical optimization problem whose solution is more easier than the original one . Various types of multi-delay systems are investigated to demonstrate the effectiveness and computational efficiency of the proposed method . The method has a simple structure , easy to implement and provides very accurate solutions . Due to the inherent behavior of time-delay systems , the analytical solution of this class of systems is piecewise smooth . Accordingly , the exact solution of multi-delay systems cannot be obtained solely either by block-pulse functions or by Legendre polynomials .