Fixed-Time Synchronization of Fractional-Order Hopfield Neural Networks with Unbounded Proportional Delay and Bounded Parameter Uncertainties

This paper investigates the fixed-time synchronization of fractional-order proportional delay Hopfield neural networks (PDHNNs) with bounded parameter uncertainties. Unlike constant delay and bounded variable delay, proportional delay has time-varying and unbounded characteristics, which pose challenges for the synchronization control of primary–secondary fractional neural networks. To achieve fixed-time synchronization, we propose a new nonlinear multi-module feedback controller. It consists of three key functional modules: eliminating the impact of proportional delay on system stability; ensuring convergence within a fixed time frame without being limited by initial conditions; and expanding the selectable range of parameters. Combining the stability lemma and inequality techniques, synchronization criteria of PDHNNs are derived based on the construction of a Lyapunov function with a negative fractional derivative. The settling time can be effectively estimated, which depends on the control parameters and is independent of initial values. Two numerical experiments verify the effectiveness of the theorem and corollary in this study.