Fuzzy wavelet neural networks for a class of time delay fractional optimal control problems under Atangana–Baleanu–Caputo variable-order fractional derivative

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This study presents an innovative approach that employs fuzzy wavelet neural networks (FWNNs) to solve time delay fractional optimal control problems (TDFOCPs). This methodology incorporates variable-order fractional derivatives in the Atangana–Baleanu–Caputo (ABC) sense, characterised by its non-singular and non-local kernel properties. By utilising the calculus of variations and the fractional integration by parts formula, the optimality conditions related to the mentioned problem are derived. To address the optimality system, the process is initiated by approximating the system's solution with FWNN solutions, each equipped with adjustable parameters. An unsupervised approach based on a loss function is then used to learn and adjust the parameters of FWNN models. The validity and performance of our proposed approach are validated through a series of illustrative examples pertaining to TDFOCPs. These examples not only demonstrate the efficiency of our methodology but also highlight its accuracy in solving complex problems.