Hybrid Functions to Solve Fractional Optimal Control Problems Using the Collocation Method

This article aims to introduce a modern numerical method based on the hybrid functions, consisting of the Bernoulli polynomials and Block-Pulse functions. An indirect approach is proposed for solving the fractional optimal control problems (FOCPs). Firstly, the two-point boundary value problem (TPBVP) is calculated for a class of FOCPs, including integer- fractional derivatives, leading to a system of fractional differential equations (FDEs), which have the left and right-sided Caputo fractional derivatives (CFD). Therefore, a new approach is proposing to achieve the left Riemann-Liouville fractional integral (LRLFI) and right Riemann-Liouville fractional integral (RRLFI) operators for Bernoulli hybrid functions. Then, hybrid functions approximation, LRLFI and RRLFI operators, and the collocation method are used to solve the TPBVP. The error bounds for the hybrid function and LRLFI and RRLFI operators are also presented. Moreover, the convergence of the proposed method is proved. Finally, the simplicity and accuracy of the method are illustrated using some numerical examples.