A Graphical Approach for Robust Set Stabilization of Boolean Control Networks With Lowest Time Complexity

This paper investigates the robust set stabilization of Boolean control networks (BCNs) under the influence of random disturbances. In contrast to prevailing algebraic methods, we propose an efficient approach based on a novel graphical perspective. First, we introduce the expanded state transition graph (ESTG), an extension of the traditional state transition graph (STG) to handle disturbances within a graphical framework. Subsequently, we devise two graphical algorithms based on the ESTG. These algorithms aim to compute the largest robust control invariant subset (LRCIS) and achieve time-optimal robust set stabilization via state feedback. We present the necessary and sufficient conditions for constructing time-optimal state feedback gain matrices using the outputs of these algorithms. Remarkably, we prove that this approach theoretically achieves the lowest time complexity for robust set stabilization. When compared to an algebraic method, the proposed approach demonstrates significant speedup across networks of varying sizes, with simulations conducted on a 30-node genetic network, highlighting its scalability and efficiency. More importantly, it even outperforms existing algebraic methods designed exclusively for deterministic set stabilization without disturbances in terms of time efficiency. Therefore, in addition to its theoretical innovation, the proposed approach can effectively handle relatively large BCNs, far beyond the capacity of current methods. Potential extensions include its application to disturbance-free BCNs, various types of disturbances, constraints, and other logical control networks and problems, thus opening new avenues for future research and practical applications.