This study is concerned with probabilistic Boolean control networks (PBCNs) with state feedback control. A novel definition of bisimilar PBCNs is proposed to lower computational complexity. To understand more on bisimulation relations between PBCNs, we resort to a powerful matrix manipulation called semi-tensor product (STP). Because stabilization of networks is of critical importance, the propagation of stabilization with probability one between bisimilar PBCNs is then considered and proved to be attainable. Additionally, the transient periods (the maximum number of steps to implement stabilization) of two PBCNs are certified to be identical if these two networks are paired with a bisimulation relation. The results are then extended to the probabilistic Boolean networks.

这项研究涉及具有状态反馈控制的概率布尔控制网络（PBCN）。提出了一种新的双相似PBCN定义，以降低计算复杂度。要了解有关PBCN之间的双仿真关系的更多信息，我们诉诸一种称为半张量积（STP）的强大矩阵操作。由于网络的稳定至关重要，因此考虑并证明了在双相似PBCN之间以概率1进行稳定的传播。此外，如果两个PBCN的瞬态周期（实现稳定的最大步骤数）与双仿真关系配对，则被证明是相同的。然后将结果扩展到概率布尔网络。