Controllability of delayed generalized asynchronous Boolean control networks under disturbances

For delayed generalized asynchronous Boolean networks with random disturbances, it is still uncertain about their robust controllability. This paper uses the semi-tensor product to transform these Boolean networks into linear representations to derive the dependence matrix of the network transition on the control mode. After that, three different control modes are utilized to define controllability. Each control mode provides both reachable trajectories set under a specific initial trajectory and an initial trajectory set under a specific target trajectory. It is also calculated for the likelihood of a transition between two states. Lastly, two concrete examples validate the theoretical findings under various control modes.