Given a conjunctive Boolean network (CBN) with n state-variables, we consider the problem of finding a minimal set of state-variables to directly affect with an input so that the resulting conjunctive Boolean control network (CBCN) is controllable. We give a necessary and sufficient condition for controllability of a CBCN; an O(n(2))-time algorithm for testing controllability; and prove that nonetheless the minimal controllability problem for CBNs is NP-hard. (C) 2018 Elsevier Ltd. All rights reserved.