Generalized fractional-order Bernoulli-Legendre functions: an effective tool for solving two-dimensional fractional optimal control problems

In this article, we present a new method to solve a class of two-dimensional fractional optimal control problems. The fractional derivative is defined in the Caputo sense. Our approach is based upon to approximate the state and control functions by the elements of generalized fractional-order Bernoulli functions in space and generalized fractional-order Legendre functions in time with unknown coefficients. First the properties of these basis functions are presented. Second, the Riemann-Liouville fractional integral operators of generalized fractional-order Bernoulli-Legendre functions are proposed. 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. Finally, convergence of the proposed method is extensively investigated and two examples are included to demonstrate the validity and applicability of the presented new technique.