This paper first investigates the synchronization design problem for both Boolean networks (BNs) and singular BNs. According to the complete family of reachable sets, all possible synchronized response BNs and all possible weakly synchronized response singular BNs can be constructed. Subsequently, it is noted that there is similarity between synchronized response (singular) BNs and state observers. The synchronized response (singular) BN is constructed based on the state of a given drive (singular) BN, making its state gradually tend to be consistent with the state of the given drive (singular) BN. While the state observer is designed based on the output of a given real (singular) BN, making its state (state estimation) gradually tend to be consistent with the state of the given real (singular) BN. Therefore, consider applying the results for constructing synchronized response (singular) BNs to designing state observers for (singular) BNs. Specifically, the real (singular) BN to be observed can be regarded as a drive (singular) BN. Consequently, the problem for designing state observers can be transformed into the problem for constructing response (singular) BNs that achieve synchronization with a given real (singular) BN, and then the gain matrix of the designed state observer can be obtained by solving a matrix equation. The method for designing state observers is superior to other methods in that the accurate estimated time for designed state observers can be determined and adjusted to some certain extent, and all possible state observers in the considered form can be designed.

This article aims to stabilize probabilistic Boolean networks (PBNs) via a novel pinning control strategy. In a PBN, the state evolution of each gene switches among a collection of candidate Boolean functions with preassigned probability distributions, which govern the activation frequency of each Boolean function. Due to the existence of stochasticity, the mode-independent pinning controller might be disabled. Thus, both mode-independent and mode-dependent pinning controller are required here. Moreover, a criterion is derived to determine whether mode-independent controllers are applicable while the pinned nodes are given. It is worth pointing out that this pinning control is based on the$n\times n$network structure rather than$2^{n} \times 2^{n}$state transition matrix. Therefore, compared with the existing results, this pinning control strategy is more practicable and has the ability to handle large-scale networks, especially sparsely connected networks. To demonstrate the effectiveness of the designed control scheme, a PBN that describes the mammalian cell-cycle encountering a mutated phenotype is discussed as a simulation.

This paper studies the set stabilization of Boolean control networks (BCNs) under sampled-data state feedback control (SDSFC). The main research content is divided into two parts. First, the topological structure of BCNs under given SDSFC is investigated. The fixed point and sampled cycle are defined, respectively. It is found that sampled cycles allow elements to be repeated and not every element can be regarded as an initial state, and this is quite different from conventional cycles of BCNs. A theorem is presented to calculate the number of fixed points and an algorithm is given to find all fixed points and sampled cycles. Second, the set stabilization problem of BCNs by SDSFC is investigated based on the sampled point set and the sampled point control invariant set (SPCIS). A necessary and sufficient condition is derived for the global set stabilization of BCNs by SDSFC, and further sampled-data state feedback controllers are also designed. The interesting thing is that if a state enters the SPCIS as an unsampled point, then it may run out of the given set again, which is in sharp contrast to conventional BCNs. Finally, an example is given to illustrate the efficiency of the obtained results.