Pinning Asymptotic Stabilization of Probabilistic Boolean Networks: A Digraph Approach

This article investigates the pinning asymptotic stabilization of probabilistic Boolean networks (PBNs) by a digraph approach. In order to stabilize the PBN asymptotically, a mode-independent pinning control (MIPC) is designed to make the target state a fixed point, and transform the state transition digraph into one that has a spanning branching rooted at the target vertex. It is shown that if there is a mode-dependent pinning control that can asymptotically stabilize the PBN, then there must exist an MIPC that can do the same with fewer pinned nodes and control inputs. A necessary and sufficient condition is given to verify the feasibility of a set of pinned nodes based on an auxiliary digraph. Three algorithms are proposed to find a feasible set of pinned nodes with the minimum cardinality. The main results are extended to the case where the target is a limit cycle. Compared with the existing results, the total computational complexity of these algorithms is strongly reduced. The obtained results are applied to a numerical example and a cell apoptosis network.