A Fast Iterative Method for Variable‐Order Time‐Fractional Optimal Control Problem

ABSTRACT Variable‐order fractional optimal control models are typically formulated as a coupled system of forward and backward variable‐order fractional differential equations, alongside a variational inequality. In this work, we propose a fast iterative method to efficiently solve this coupled system with the spatial computational domain as an interval, where two matrix splitting formats are utilized and an alternating iteration scheme is adopted to obtain a corresponding linear system. By integrating these strategies, we develop a two‐parameter alternating iterative method that requires only the inversion of tridiagonal matrices, achieved using the Thomas method and fast sine transform, which greatly reduces the computational cost. Furthermore, we provide a rigorous convergence analysis of the proposed iterative method. The effectiveness and efficiency of our method are validated by numerical examples.