A Numerical Solution of Fractional Optimal Control Problems Using Spectral Method and Hybrid Functions

In this paper‎, ‎a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly‎. ‎First‎, ‎the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs)‎. ‎Then‎, ‎the unknown functions are approximated by the hybrid functions‎, ‎including Bernoulli polynomials and Block-pulse functions based on the spectral Ritz method‎. ‎Also‎, ‎two new methods are proposed for calculating the left Caputo fractional derivative and right Riemann-Liouville fractional derivative operators of the hybrid functions that are proportional to the Ritz method‎. ‎The FOCP is converted into a system of the algebraic equations by applying the fractional derivative operators and collocation method‎, ‎which determines the solution of the problem‎. ‎Error estimates for the hybrid function approximation‎, ‎fractional operators and‎, ‎the proposed method are provided‎. ‎Finally‎, ‎the efficiency of the proposed method and its accuracy in obtaining optimal solutions are shown by some test problems.