Numerical solution of nonlinear 2D optimal control problems generated by Atangana-Riemann-Liouville fractal-fractional derivative

Abstract

This paper is dedicated to introduce a new category of nonlinear 2D optimal control problems (OCPs), generated by dynamical systems involved with fractal-fractional derivatives and to develop an operational matrix (OM) scheme for their numerical solutions. The fractal-fractional derivatives are defined in the Atangana-Riemann-Liouville sense with Mittag-Leffler non-singular kernel. The proposed approach is based on the Chelyshkov polynomials (CPs) and their OM of fractal-fractional derivative (which is derived in the present study). More precisely, the proposed method transforms solving such problems into solving systems of nonlinear algebraic equations. To this end, at first the state and control variables are approximated by the CPs with unknown coefficients, and are substituted in the objective function, dynamical system and the initial conditions. Then, the 2D Gauss-Legendre quadrature rule with the OM of fractal-fractional derivative of the CPs are utilized to construct a constrained problem, which is solved by the Lagrange multipliers method. The accuracy of the presented method is investigated on two test problems. The obtained results confirm that the presented scheme is highly accurate for the numerical solution of such problems.