Set stability and set stabilization of Boolean control networks based on invariant subsets

This study addresses the set stability of Boolean networks (BNs) and set stabilization of Boolean control networks (BCNs). Set stability determines whether a BN converges to a given subset, whereas set stabilizability addresses the issue of whether a BCN can be stabilized to a given subset. Many problems can be viewed as special cases of set stability and set stabilization, including synchronization, partial stability, and partial stabilization problems. The concepts of invariant subset and control invariant subset are introduced. Then, algorithms for the largest invariant subset and the largest control invariant subset contained in a given subset are proposed. Based on the invariant subsets obtained, the necessary and sufficient conditions for set stability and set stabilizability are established, and formulae are provided to calculate the shortest transient periods for respective initial states. A design procedure is proposed for finding all the time-optimal set stabilizers. Finally, an example is used to show the application of the proposed results to the synchronization problem of BNs.