Chebyshev cardinal functions for a new class of nonlinear optimal control problems generated by Atangana–Baleanu–Caputo variable-order fractional derivative

Abstract

This paper introduces a novel class of nonlinear optimal control problems generated by dynamical systems involved with variable-order fractional derivatives in the Atangana–Baleanu–Caputo sense. A computational method based on the Chebyshev cardinal functions and their operational matrix of variable-order fractional derivative (which is generated for the first time in the present study) is proposed for the numerical solution of this class of problems. The presented method is based on transformation of the main problem to solving system of nonlinear algebraic equations. To do this, the state and control variables are expanded in terms of the Chebyshev cardinal functions with unknown coefficients, then the cardinal property of these basis functions together with their operational matrix are employed to generate a constrained extremum problem, which is solved by the Lagrange multipliers method. The applicability and accuracy of the established method are investigated through some numerical examples. The reported results confirm that the established scheme is highly accurate in providing acceptable results.