Controllability of Boolean Control Networks With Multiple Time Delays

In this paper, the controllability of Boolean control networks (BCNs) with multiple time delays is investigated. The concept of controllability matrices for BCNs with multiple time-varying delays is proposed, and an iterative algorithm is proposed to calculate all controllability matrices. Based on this, a concise criterion for controllability is proposed, which states that a BCN with delays is controllable if and only if all elements of its controllability matrix are nonzero. The proposed method is applied to BCNs with periodic and constant delays, respectively. Sufficient and necessary conditions for controllability are presented. Finally, two examples are given to illustrate the main results.