Pinning Stabilization of Stochastic Networks With Finite States via Controlling Minimal Nodes

In this article, pinning control has been applied to stabilize a kind of stochastic network with finite states called probabilistic logical networks (PLNs). First, the solvability of pinning controllers, including the selection of pinning nodes and the corresponding control design, is addressed to guarantee the stabilization of PLNs. Then, based on the complete matrices set, one necessary and sufficient condition is obtained for PLNs to be stabilized by exact $p$ nodes. In addition, an algorithm is presented for obtaining the minimal number of pinning nodes and how exact they are. As an application, our algorithm is used to calculate the minimal number of pinning nodes for the stabilization of logical networks (LNs). Examples are given to illustrate the process of choosing pinning nodes and show the efficiency of the obtained results.