Abstarct: Aabstract In this thesis pseudospectral method is presented for solving a continuous-time optimal control problem using collocation at Legendre–Gauss–Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived baxsed on the difference between the Lagrange polynomial approximation of the state and a Legendre– Gauss–Radau quadrature integration of the dynamics within a mesh interval. The derived relative error estimate is then used to decide if the degree of the approximating polynomial within a mesh should be increased or if the mesh interval should be divided into subintervals. The degree of the approximating polynomial within a mesh interval is increased if the polynomial degree estimated by the method remains below a maximum allowable degree. Otherwise, the mesh interval is divided into subintervals. The process of refining the mesh is repeated until a specified relative error tolerance is met. the approach is more computationally efficient and produces significantly smaller mesh sizes for a given accuracy tolerance when compared with fixed-order methods. also in this thesis mesh refinement method is developed for solving optimal control problems with discontinuous control profiles. The method assumes that the optimal control Hamiltonian is linear in the control. The method employs Legendre-Gauss-Radau collocation to accurately approximate the solution in segments where the solution is smooth, and the mesh refinement method determines the locations of discontinuities in the control by examining the sign in the function that multiplies the control in the Hamiltonian. by presenting several examples, we examine the efficiency of extended pseudospectral methods.