Sliding mode control for finite-time projective synchronisation in distinct Caputo fractional-order delayed neural networks with inconsistent orders

This manuscript delves into the realm of projective synchronisation in delayed neural networks governed by Caputo fractional-order dynamics, addressing the intricate challenges of attaining finite-time synchronisation in networks with distinct structures, functions, and orders. To start, the integration of a compensation controller within the control architecture is examined, aiming to formulate the projection error dynamics. By leveraging projection error system analysis alongside output observation and sliding mode control, a meticulously crafted derivative-based sliding mode surface along with a sliding mode controller are devised, ensuring both convergence and seamless sliding motion. Afterwards, the accessibility of the surface is assessed through relevant lemmas, Caputo derivative characteristics and the Lyapunov direct approach. The subsequent stability of sliding path is corroborated, and indispensable prerequisites for achieving finite-time projective synchronisation are outlined. Furthermore, the establishment of an upper bound for synchronisation time of distinct Caputo fractional-order delayed neural networks (CFDNNs) with inconsistent orders is carried out. Ultimately, simulations on distinct CFDNNs with consistent and inconsistent orders are conducted to exemplify our approach, employing the continuous frequency distribution model.