Controllability of probabilistic Boolean control networks with time-variant delays in states

This paper investigates the controllability of probabilistic Boolean control networks with time-variant delays (not necessarily bounded) in states. After a brief introduction to converting the network to an equivalent discrete-time dynamical system via the semi-tensor product of matrices, the system is split into at most a countably infinite number of subsystems with no time delays that are pairwise independent by using the concept of constructed forest proposed by us, where subsystems are essentially a special type of paths of the forest. Then the controllability of the system is investigated by verifying that of any subsystem, especially one of the longest paths, called the controllability constructed path. For the deterministic case, the controllability is equivalent to that of any controllability constructed path, which is not true for the probabilistic case. When the controllability constructed path is of infinite length, an equivalent condition for the controllability is given. When it is of finite length, a necessary condition and a sufficient condition for the controllability are given, respectively.