State-controlling Sets for Conjunctive Boolean Networks

A Boolean network is a discrete-time finite state dynamical system, whose variables take values from the binary set {0,1}, and the value update rules are Boolean functions. A conjunctive Boolean network is a special type of Boolean network, whose value update rule for each variable is comprised only of “AND” operations. Recently there have been extensive investigations on conjunctive Boolean networks. Questions about asymptotic behaviors, stabilities of periodic orbits, and reachability and observability have all been addressed to some extent. We focus in this paper on controllability of a conjunctive Boolean network. Specifically, assuming that there is a selected subset of variables whose values are determined by external control inputs, we pose and answer the question of whether (and how) one can steer the system from any initial state to any final state. We establish a necessary and sufficient condition, via a graphical approach, for a conjunctive Boolean network to be controllable. An explicit control law is also presented along the analysis.