Iterative Algorithms for Set Stabilization of Probabilistic Boolean Control Networks and Applications in State-Based Games

A probabilistic Boolean control network (PBCN) is a binary discrete-time system that utilizes probability values. An important application of PBCN is in gene regulatory network (GRN), providing a more flexible model, and lay a foundation for the study of gene interactions in complex environments. In this research, the set stabilization with probability 1 of PBCNs is investigated. Utilizing semi-tensor product (STP) of matrices, an iterative algorithm for calculating the largest control invariant subset of PBCNs is established and a necessary and sufficient condition for set stabilization with probability 1 is provided. Moreover, we design an algorithm for the stabilization controller to achieve stabilization of the system in the shortest time. As an application, the results of PBCNs are used to solve the stabilization problems at recurrent state equilibriums (RSEs) of state-based games. Examples are given to verify the feasibility of the proposed method and results.