Bang-Bang type controls arise in some well-known application areas , such as industrial robots , aerospace engineering , cranes , applied physics , game theory , and biological systems . Because of difficulty in obtaining switching times and optimal solution , the simulation an d numerical approximation of bang-bang optimal control problems have received considerable attention . Over the last two decades , pseudospectral methods have been successfully used to solve a wide variety of optimal control problems arising in diverse areas of engineering and science . These methods can effectively solve optimal control problems whose solutions are smooth . Although pseudospectral methods enjoy some nice properties ,, they also suffer from many drawbacks . For instance , they do not provide a satisfactory approximation for non-smooth problems such as bang-bang optimal control problems . Due to the fact that obtaining the analytical solution for bang-bang optimal control problems is difficult , therefore , it is important to provide a numerical solution for solving such problems. In this thesis , a modified Legendre pseudospectral method is used to obtain accurate and efficient solution of bang - bang optimal control problems . In this method , control and state variables are considered as piecewise constant and piecewise continuous functions , respectively , and the switching points are also taken as decision variables . This method has two major differences with the traditional pseudospectral methods . First , instead of approximating the states and controls by a polynomial in the whole computational domain , as suggested in the dir=ltr Thus , the problem converts to a non-linear mathematical programming whose solution is much more easier than the original one . The main advantages of this approach are : -1 It obtains good results even by using a small number of collocation points and the rate of convergence is high . -2 The switching times can be captured accurately . -3 If the number of switching points is not selected correctly , it is possible to correct this mistake with simulation and numerical results . Various types of bang-bang optimal control problems are included to show the efficiency and the accuracy of the proposed discretization scheme .