Orbit-controlling Sets for Conjunctive Boolean Networks

A Boolean network is a finite dynamical system, whose variables take values from a binary set. The value update rule for each variable is a local function, i.e., it depends only on a selected subset of variables. Boolean networks have been widely used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, termed as conjunctive Boolean networks. A Boolean network is called conjunctive if the associated value update rule is comprised of only AND operations. It is known that any trajectory of a finite dynamical system will enter into a periodic orbit. Periodic orbits of a conjunctive Boolean network have been completely understood. In this paper, we investigate the orbit-controllability of a conjunctive Boolean network: We ask the question of how one can steer the network from any initial state to a state in any periodic orbit. Specifically, we assume that there is a selected subset of variables in the network whose values are under control. We then establish a necessary and sufficient condition for the conjunctive Boolean network to be orbit-controllable. Furthermore, we present an explicit control law for steering the network from any initial condition to any periodic orbit.