Time-optimal open-loop set stabilization of Boolean control networks

We show that for stabilization of Boolean control networks (BCNs) with unobservable initial states, open-loop control and close-loop control are not equivalent. An example is given to illustrate the nonequivalence. Enlightened by the nonequivalence, we explore open-loop set stabilization of BCNs with unobservable initial states. More specifically, this issue is to investigate that for a given BCN, whether there exists a unified free control sequence that is effective for all initial states of the system to stabilize the system states to a given set. The criteria for open-loop set stabilization is derived and for any open-loop set stabilizable BCN, every time-optimal open-loop set stabilizer is proposed. Besides, we obtain the least upper bounds of two integers, which are respectively related to the global stabilization and partial stabilization of BCNs in the results of two literature articles. Using the methods in the two literature articles, the least upper bounds of the two integers cannot be obtained.