A Generic Framework for Fixed-Time Synchronization of Large Delayed Impulsive Neural Networks

This article investigates fixed-time synchronization (FxTS) for neural networks with delayed impulses. There are two main challenges associated with the delay in this area. One is that it causes the networks to oscillate close to the equilibrium. The other is that it causes the synchronization criterion to be very conservative, particularly when the impulses contain large (exceeding impulsive intervals) delays. To overcome these challenges, we propose the concept of equivalent impulsive sequence and the method of delayed impulsive sequence decomposition, respectively. Meanwhile, we establish a framework that equivalently transforms the networks with large delays into a collection of networks with small (shorter than impulsive intervals) delays. Particularly, this transformation is a sufficient and necessary condition, and thus does not induce any conservatism. Following this, we delineate the FxTS criteria for the networks containing synchronizing and desynchronizing impulses. Interestingly, it is shown that under some conditions, the large delay has no effect on the FxTS criterion of networks with synchronizing impulses, but destroys those of networks with desynchronizing impulses. Furthermore, we prove the necessity and rationality of the FxTS criterion. Finally, the conclusions are substantiated through numerical demonstrations.