Prescribed-Time Synchronization of Multiweighted and Directed Complex Networks

In this note, we study the prescribed-time (PT) synchronization of the multiweighted and directed complex networks (MWDCNs). Unlike finite-time and fixed-time synchronization, the time for synchronization can be prescribed as needed, which is independent of initial values and parameters, such as coupling matrix. At first, we define a class of regulatory functions whose improper integrals diverge, and it is proved to play a key role in PT stability for dynamical systems. Many special functions established previously for PT stability can be included in this class. Then, we apply this new result on MWDCNs as an application. The synchronization error at the PT is discussed carefully, so PT synchronization can be reached. The network topology can be directed and disconnected, which means that the outer coupling matrices (OCMs) can be asymmetric and not connected. The relationships between nodes are allowed to be cooperative or competitive, so elements in OCMs and inner coupling matrices (ICMs) can be positive or negative. We use the rearranging variables' order technique to combine ICMs and OCMs together to get the sum matrices, which can make a bridge between multiweighted and single-weighted networks. Finally, simulations are presented to illustrate its effectiveness.