Controllability of probabilistic Boolean control networks based on transition probability matrices

In this paper, we propose a new approach to investigate the controllability and reachability of probabilistic Boolean control networks (PBCNs) with forbidden states. We first give a simple algebraic formula for the transition probability between two states in a given number of time-step, while avoiding a set of forbidden states. Then we construct the controllability matrix based on a new operator, and some necessary and sufficient conditions are obtained for the controllability and reachability of PBCNs. A numerical example is given to illustrate the efficiency of the obtained results.