A numerical scheme for solving two-dimensional fractional optimal control problems by the Ritz method combined with fractional operational matrix

In this article, we present a new method to solve a class of two-dimensional fractional optimal control problems based upon the numerical polynomial approximation. In the proposed method, the fractional derivatives are expressed in the Caputo sense. The approach used here is to approximate the state and control functions by the Legendre orthonormal basis by using the Ritz method. To approximate derivative of the basis, the operational matrix of Caputo derivative is taken into account. Then we apply two-dimensional Legendre-Gauss quadrature rule to approximate double integral of the performance index functional. Next, the problem is converted into an equivalent non-linear unconstrained optimization problem. This problem is solved via the Newton's iterative method. At last, the convergence of the proposed method is extensively investigated and an example is included to illustrate the effectiveness and applicability of the new procedure.