In this article, the set stabilization problem of Boolean control networks (BCNs) using output-feedback controllers is investigated comprehensively. First, a novel method for stabilizing BCNs, namely, the state dynamics matrix-based method, is proposed. Then, based on the above method, the time-invariant output-feedback (TIOF) and the time-variant output-feedback (TVOF) laws are derived. The necessary and sufficient conditions of both TIOF and TVOF stabilization of BCNs are also given. In addition, corresponding algorithms are provided such that all possible TIOF stabilizers and TVOF stabilizers for given BCNs can be designed. Finally, some illustrative examples are demonstrated, where the results are discussed and the effectiveness of the presented method is validated.

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.