Abstarct: In recent years, the application of fractional calculus in modeling dynamical systems with long-term memory has attracted considerable attention. Fractional optimal control problems (FOCPs), as a generalization of classical optimal control problems, have found wide appli-cations in various fields including structural control, biological modeling, and energy system optimization. However, analytical solutions of these problems, especially those with time delays and higher dimensions, are often impossible, necessitating the development of efficient. and accurate numerical methods. This thesis presents two novel computational frxameworks for solving fractional opti-mal control problems. First, a method baxsed on the combination of block-pulse functions and Lerch polynomials is developed for solving delay fractional optimal control problems (DFOCPS). In this method, the Riemann-Liouville fractional integral operational matrix for hybrid Lerch functions is derived, and the problem is transformed into a system of algebraic equations using the Galerkin method. The convergence of the method is theoretically proven, with an estimated convergence rate of O(N 2) for problems with delay. Second, a method baxsed on generalized Lerch polynomials is proposed for solving nonlin-ear two-dimensional fractional optimal control problems (N2DFOCPs). In this method, the fractional-fractal derivative operational matrix in the Atangana-Riemann-Liouville sense is derived, and the problem is converted into a parameter optimization problem using Lagrange multipliers and the Ritz method. The convergence of this method in the two-dimensional L2 space is proved. The efficiency and accuracy of both methods are investigated by solving numerous nu-merical examples. Results show that the first method provides higher accuracy in solving delay problems compared to existing methods such as those baxsed on Bernstein and Leg-endre polynomials. The second method also demonstrates the capability to solve complex two-dimensional problems with relative errors less than 10-10. Error analysis indicates rapid convergence and satisfactory numerical stability for both methods. This research not only provides efficient numerical methods for solving two important classes of fractional optimal control problems but also establishes a frxamework for developing similar methods with other function baxses.