Distributed Pinning Control Design for Probabilistic Boolean Networks

This paper investigates the stabilization of probabilistic Boolean networks (PBNs) via a novel pinning control strategy based on network structure. In a PBN, each node needs to choose a Boolean function from candidate Boolean function set at each time instance with certain probability. Owing to the stochasticity, the uniform state feedback controllers, which is independent of switching signal, might be out of work. Thereby, a criterion is derived to determine that under what condition uniform controllers can be applied, otherwise non-uniform controllers need to be utilized. Accordingly, an algorithm is designed to find a series of state feedback pinning controllers, under which such a PBN is stabilized to a prespecified steady state. It is worth pointing out that the pinning control used in this paper only requires local in-neighbors' information, rather than global information. Hence, it is also termed as distributed pinning control and reduces the computational complexity to a large extent. Profiting from this, it provides a potential to deal with some large-scale networks. Finally, the mammalian cell-cycle encountering a mutated phenotype is described as a PBN, and presented to demonstrate the obtained results.