Ritz approximate method for solving delay fractional optimal control problems

In this paper, a numerical method based on the Muntz-Legendre polynomials for solving a class of time-delay fractional optimal control problems (TDFOCPs) is presented. The concept of fractional derivatives is considered in the Caputo sense. The key point of the applied method is that the initial and boundary conditions are imposed upon the approximations of state and control functions. First, the unknown state or control and their delayed functions are approximated by the Muntz-Legendre polynomials basis using the Ritz method then the next one is calculated through the given fractional differential equation. By substituting the estimated values in the cost function, the TDFOCP is converted to an unconstrained linear or nonlinear optimization problem. The convergence of the suggested method is extensively argued and several illustrative examples are presented to show the applicability and effectiveness of the method. It is shown that only a few terms of Muntz-Legendre polynomials are needed for obtaining the high accuracy and satisfactory results. The obtained results are compared with the available results in the literature to demonstrate the superiority of the applied approach. Furthermore, the approximated solutions agree with exact solutions and as alpha approaches an integer value, the method provides solution for the integer-order optimal control problem. (C) 2022 Elsevier B.V. All rights reserved.