An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix

This paper deals with the Ritz spectral method to solve a class of fractional optimal control problems (FOCPs). The developed numerical procedure is based on the function approximation by the Bernstein polynomials along with fractional operational matrix usage. The approximation method is computationally consistent and moreover, has a good flexibility in the sense of satisfying the initial and boundary conditions of the optimal control problems. We construct a new fractional operational matrix applicable in the Ritz method to estimate the fractional and integer order derivatives of the basis. As a result, we achieve an unconstrained optimization problem. Next, by applying the necessary conditions of optimality, a system of algebraic equations is obtained. The resultant problem is solved via Newton's iterative method. Finally, the convergence of the proposed method is investigated and several illustrative examples are added to demonstrate the effectiveness of the new methodology.