State-flipped control design for the stabilization of probabilistic Boolean control networks

Stabilization is a fundamental issue in modern control theory. In the past decades, significant efforts have been invested in deriving necessary and sufficient conditions for verifying the global stabilization of probabilistic Boolean control networks (PBCNs). However, systematic methods and general criteria for exploring the local stabilization and determining the domain of attraction of PBCNs are still lacking in the existing literature. Motivated by this research gap, this paper investigates the local state feedback stabilization of PBCNs, including local finite-time state feedback stabilization with probability one (FTSFS) and local state feedback stabilization in distribution (SFSD). Firstly, a sequence of reachable sets with probability one is constructed, based on which, the largest domain of attraction is derived for the FTSFS of PBCNs by designing the state feedback controllers. Secondly, by constructing a sequence of reachable sets with positive probability, the largest domain of attraction is determined for the SFSD of PBCNs. Finally, when the largest domain of attraction is not the whole state space, the state-flipped control is designed to achieve the global FTSFS or SFSD of PBCNs via the largest domain of attraction.