Solving a class of two dimensional optimal control problem for fractional order differential systems involving fractal-fractional derivatives

In this paper, an operational method based on Chelyshkov polynomials is used for solving a class of two dimensional optimal control problem for fractional order differential system involving fractal-fractional derivatives. The operational matrix of the corresponding fractional integration operator is calculated. First, the control signal and the differential of the state signals are approximated with unknown coefficients by orthogonal basis. Next, by replacing the approximate signals in objective functions, using two dimensional Gauss–Legendre quadrature rule and necessary optimal conditions the main problem is converted to a system of algebraic equations, which can be solved easily. Theoretically, the convergence analysis of the proposed method is stated. Moreover, to demonstrate the efficiency of the method, three test problems solved.