Controllability of Boolean control networks with time delays both in states and inputs

This paper investigates the controllability of Boolean control networks (BCNs) with time-invariant delays both in states and inputs. The necessary and sufficient conditions on the controllability via two kinds of controls are presented by providing the corresponding reachable sets. The proposed results generalize the controllability analysis on Boolean control networks without time delay. And we also consider the optimal control problem of BCNs from a given initial state to a desired state in minimal time. Three examples include a biological example are given to show the effectiveness of the proposed results.