State feedback design for set stabilization of probabilistic Boolean control networks

Abstract

Abstract Set stabilization of probabilistic Boolean control networks (PBCNs) is investigated in this paper and some interesting results are derived. The main results consist of three parts. (1) A definition of set stabilizability with probability one by closed-loop control is proposed for PBCNs, which is not a natural extension from deterministic Boolean control networks to PBCNs due to the random feature of PBCNs. (2) A necessary and sufficient set stabilizability condition is provided for PBCNs. (3) An algorithm for designing a state feedback controller is developed. It is guaranteed that all designed controllers can stabilize a PBCN to a given subset with probability one. The design method is constructive, so it is convenient to use this method in practical application. The results derived above are fundamental and important, since based on them many problems about PBCNs can be solved, for example partial stabilization, synchronization, and so on. Finally, a practical example is employed to show the effectiveness of our results.