Discrete Chebyshev polynomials for two-dimensional fractional optimal control problems involving Caputo-Hadamard derivative

In this paper, a new class of two-dimensional nonlinear optimal control problems governed by a fractional partial differential equation including the Caputo-Hadamard derivative is introduced. A numerical method based on the orthogonal shifted discrete Chebyshev polynomials (DCP) is developed to solve these problems. To do this, a new operational matrix for the Hadamard fractional integral of the shifted DCPs is derived. In the proposed method, by approximating the state and control variables through tensor products of the shifted DCPs, using the classical and fractional operational matrices of the aforementioned basis functions, together with the Lagrange multipliers technique, the solution of the main problem is transformed into the solution of a system of nonlinear algebraic equations. The robustness, efficiency, and high accuracy of the proposed method are investigated by solving two numerical examples.