Delay Synchronization of Drive-Response Boolean Networks and Boolean Control Networks

This article investigates the delay synchronization of drive Boolean network (DBN) and response BN (RBN) [response Boolean control network (RBCN)]. Synchronization and time delay are two common phenomena in real life. In aircraft formation, the latter's trajectory is consistent with the former's, but there exists a time gap when they arrive at the same position, which illustrates that time delay can happen in the process of synchronization. According to the delay step depending on initial states or not, two types of delay synchronization are proposed: 1) weak and 2) strong delay synchronization. Compared with synchronization, delay synchronization is only the theoretical generalization and has different application fields. Using the results of limit sets, several criteria for delay synchronization of the DBN and RBN are derived. This article provides an important result that delay synchronization of the DBN and RBCN can be equivalent to set stabilization of the augmented BCN, then the necessary and sufficient conditions are obtained. Moreover, two algorithms are established to design all time-optimal state feedback controllers. Finally, two examples are shown to demonstrate the effectiveness of the results.