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    Area of Science:

    • Systems Biology
    • Control Theory
    • Network Science

    Background:

    • Probabilistic Boolean control networks (PBCNs) are complex systems requiring robust control strategies.
    • Event-triggered control (ETC) offers an efficient, intermittent approach to managing such networks.
    • Ensuring control invariance is crucial for system stability and predictable behavior.

    Purpose of the Study:

    • To investigate the robust control invariance problem in PBCNs using event-triggered control (ETC).
    • To develop theoretical conditions for the existence and design of event-triggered controllers for PBCNs.
    • To ensure a given set remains a robust ETC invariant set (RETCIS) under control.

    Main Methods:

    • Utilizing the semi-tensor product (STP) technique to transform PBCNs with ETC into an algebraic linear system.
    • Deriving a matrix testing condition to verify if a set qualifies as a robust ETC invariant set (RETCIS).
    • Developing necessary and sufficient conditions for the existence of suitable event-triggered controllers.

    Main Results:

    • An equivalent algebraic linear system representation for PBCNs with ETC was established.
    • A matrix testing condition was derived to identify robust ETC invariant sets (RETCIS).
    • Conditions for the existence of event-triggered controllers were determined, and feasible controllers were designed.

    Conclusions:

    • The study successfully addresses the robust control invariance problem for PBCNs using ETC.
    • Theoretical results provide a framework for designing stabilizing controllers for these networks.
    • A biological example validated the effectiveness of the proposed theoretical results.