We define a new operational matrix of fractional derivative in the Caputo type and apply a spectral method to solve a two-dimensional fractional optimal control problem (2D-FOCP). To acquire this aim, first we expand the state and control variables based on the fractional order of Bernstein functions. Then we reduce the constraints of 2D-FOCP to a system of algebraic equations through the operational matrix. Now, one can solve straightforward the problem and drive the approximate solution of state and control variables. The convergence of the method in approximating the 2D-FOCP is proved. We demonstrate the efficiency and superiority of the method by comparing the results obtained by the presented method with the results of previous methods in some examples.

The aim of this paper is solving optimal control problems governed by non-local diffusion equations via a mesh-less method. The diffusion equation and in particular, the heat conduction equation is essential in sciences. This equation appears in many fields, such as engineering, electrostatic, and mathematics. For solving the mentioned optimal control problems, the method is established upon expanding of variables by the basis of Bezier functions. We apply, for the first time, the Bernstein approximation in solving an optimal control problem governed by the diffusion equation. A direct algorithm is given for solving this problem. Bernstein polynomials expand the trajectories and control functions with unknown control points. Then the optimal control problem is converted to a mathematical programming problem. By solving the mathematical programming problem, the approximated solution of trajectories and control are driven. The convergence of the method in approximating of the optimal control problem is proved. Some numerical examples for demonstrating the effectiveness of the method are included.