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

Area of Science:

• Control Theory
• Network Science
• Systems Engineering

Background:

• Stabilization is a core problem in modern control theory, with extensive research on global stabilization of probabilistic Boolean control networks (PBCNs).
• Existing literature lacks systematic methods for analyzing local stabilization and determining the domain of attraction for PBCNs.

Purpose of the Study:

• To address the research gap by investigating local state feedback stabilization of PBCNs.
• To explore local finite-time state feedback stabilization with probability one (FTSFS) and local state feedback stabilization in distribution (SFSD).
• To design state feedback controllers and state-flipped controllers for achieving stabilization and determining domains of attraction.

Main Methods:

• Construction of a sequence of reachable sets with probability one to derive the largest domain of attraction for FTSFS.
• Construction of a sequence of reachable sets with positive probability to determine the largest domain of attraction for SFSD.
• Design of state feedback controllers for local stabilization and state-flipped controllers for global stabilization.

Main Results:

• The largest domain of attraction for FTSFS of PBCNs was derived using constructed reachable sets and designed controllers.
• The largest domain of attraction for SFSD of PBCNs was determined using constructed reachable sets and designed controllers.
• State-flipped control was designed to achieve global FTSFS or SFSD when the largest domain of attraction is not the entire state space.

Conclusions:

• This paper successfully develops systematic methods for local stabilization and domain of attraction determination in PBCNs.
• The proposed methods provide criteria for both finite-time stabilization and stabilization in distribution.
• The study offers a pathway to achieve global stabilization by leveraging the derived domains of attraction.