Stabilization of probabilistic Boolean networks via pinning control strategy

The stabilization of probabilistic Boolean networks with pinning control is investigated. Only a part of nodes are chosen to be controlled for the aim of high efficiency. Stabilization with probability one and stabilization in probability are respectively discussed. Since the probability of stabilization is not required to be strict one, stabilization in probability is a more practical extension of the former, which is also proven in this work. Stabilization with probability one needs the target state to be transferred to itself with 100% certainty, while stabilization in probability cannot even guarantee the existence of such a possibility. Thus, stabilization in probability is a different and challenging problem. Some necessary and sufficient conditions are proposed for both types of stabilization via the semi-tensor product of matrices. Based on them, approaches to controller design are also developed. Finally, illustrative examples are provided to demonstrate the effectiveness of the derived results. (C) 2019 Elsevier Inc. All rights reserved.