Development of Chelyshkov Wavelets Neural Network Method for Solving Nonlinear Fractal–Fractional Optimal Control Problems

Abstract

This paper employs Chelyshkov wavelets neural network method for solving nonlinear fractal– fractional optimal control problems (FFOCPs) in the Atangana–Riemann–Liouville sense. The described neural network is comprised of three layers: the input layer, the hidden layer, and the output layer. This three-layer structure allows the neural network to learn complex relationships between input and output by applying Chelyshkov wavelets in the hidden layer and nonlinear scaling in the output layer. Initially, the problem under investigation is transformed into an equivalent variational problem by applying the definition of the fractal– fractional integral. Subsequently, by utilizing the Chelyshkov wavelets, the neural network method, and the Gauss–Legendre integration, the problem is reformulated as a system of algebraic equations. Finally, this system is solved using Newton’s iterative method. Additionally, we show the convergence of the proposed approach within the Hilbert space framework. To assess the applicability and efficacy of the proposed scheme, four examples are given.