A Novel Approach to Nonlinear 2D Fractional Optimal Control Problems Via Generalized Lerch Polynomials

Abstract

In this research paper, we introduce an optimal approach based on a category of basis functions known as generalized Lerch polynomials (GLPs) for solving nonlinear two-dimensional fractional optimal control problems (N2DFOCPs) arising from nonlinear fractional dynamical systems (NFDSs) that involve fractal–fractional derivatives in the sense of Atangana–Riemann–Liouville (ARL) under the Goursat–Darboux condition (GDC). We first develop the GLPs and then use them as a new set of basis functions to approximate the unknown parameters of control and state. The Lagrange multipliers method is employed, utilizing the quadrature rule of 2D Gauss–Legendre and operational matrices of fractal–fractional derivatives to solve a system of algebraic equations rather than the original problem. It is also demonstrated that the obtained solution converges to the optimal solution of the problem under investigation. Finally, through a series of numerical experiments, we evaluate the performance of our proposed method and compare it to existing approaches. Our findings reveal significant improvements in solution precision and computational efficiency. Ultimately, this research aims to contribute to the broader field of fractional calculus by establishing new methodologies and identifying potential applications of GLPs in various scientific domains.